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PLO from scratch Part 1-12

This is mainly a PLO beginner series, but we’ll also discuss topics that will be useful for more experienced players. Our goal is to learn a solid fundament for winning PLO play, both preflop and postflop.
1. Introduction
This is part 1 of the article series “PLO From Scratch”. The target audience is micro and low limit players with some experience from limit or no-limit Hold’em, but little or no PLO experience. My goal with this series is to teach basic PLO strategy in a systematic and structured way.In part 1 I will first discuss the background for this series and how it will be structured. Then I’ll give an overview of the (in my opinion) best PLO learning material on the market today, and we’ll end part 1 with a study plan for learning basic PLO theory from literature and videos. We will then start discussing PLO strategy in part 2.2. The background for this article series

When I started playing poker in the spring of 2005, limit and no-limit Hold’em were the dominating games, and the skills of the average player were low in both games. All you needed in order to climb up from the FL or NL Hold’em low limits to the middle and higher limits was normal intelligence and some dedicated effort.

Armed with this you could climb from the low to the middle limits in a few months and start to make good money. Many winning players learned the necessary skills and strategies strictly “on the job”, and did nothing in particular to continue to improve systematically.

These days are mostly over. Limit and no-limit Hold’em have become much tougher games since the golden age of online poker (the years 2003-2006 or thereabouts). There are several reasons for this, but it’s beyond doubt that a lot of the average player’s improvement stems from the fact that good strategy has become common knowledge through books, forums and coaching videos.

There are many smart people in the online poker player pool, and in the 6 years that have passed since online poker exploded (in 2003), these people have played, analyzed, and discussed optimal strategy. This has lead to a rapid development of FL and NL Hold’em strategy. Today you can easily find low limit tables that play just as tough as the middle limit games did a few years ago. If you want to start at the bottom in Hold’em and work your way up to the middle and high limits, you have to be prepared to work very hard.

So what are the consequences for ambitious players in today’s online environment? For starters, you have to be willing to work hard to improve your skills continually and systematically. If you don’t, your edge will slowly be reduced as your average opponent continues to improve. Another consequence is that you have to put more effort into game selection, both with regards to the games you play today, and with regards to learning new games to give yourself more good games to play in.

And this brings us to pot-limit Omaha (PLO). For me, PLO sailed under the radar for a long time. I heard a lot of talk about how fun and profitable it was, but I didn’t give it a try until 2008, and I played it mostly for variation (I grinded Hold’em at the time). I splashed around without much knowledge about how the game was supposed to be played, but I gradually started to get a feel for the game. I also observed that the average player in this game often made horrible mistakes, and that the skill level of the player pool reminded me of the Hold’em games of old.

This gave me the motivation to learn the game properly. In the autumn of 2009 I therefore decided to start a systematic learning process and teach myself solid PLO strategy from scratch. And since I like writing about poker theory, I decided to simultaneously write an article series for Donkr’s micro and low limit players.

In this series I will write about PLO strategies and concepts I have worked with in my own learning process, and my goal is to lay out a theoretical framework for PLO learning, aimed at beginning players. I hope the series will help the readers getting started with PLO, and that they can use it as a starting point both for learning PLO strategy and for learning how to think about PLO (which can be very different from the way we think about Hold’em).

3. The plan for the article series

I have previously written an article series (“Poker From Scratch”) for limit Hold’em where I discussed basic limit Hold’em strategy and ran a bankroll building project on the side (grinding up a 1000 BB limit Hold’em bankroll from $0.25-0.50 to $5-10). I plan to use the same form for this series. We will start with preflop strategy and principles of starting hand strength. Then we will move on to postflop play.

Also, the general principles for “big bet poker” (pot-limit and no-limit) will be a common thread throughout the series. Many of the strategic principles of PLO are consequences of the game’sbetting structure(pot-limit) and not of the game type (a flop game where we use starting hands with 4 cards, and we have to use 2 cards from the hand and 3 from the board). Thinking about any poker game as a combination of betting structure and game type makes it easier to understand why proper strategy is the way it is.

We will also include a micro/low limit bankroll building project in this article series, and there are several good reasons for this. The series is aimed at beginners, which means most of the target audience will be playing at the lowest limits. I have never grinded microlimit PLO, so I should ensure that the strategies I discuss are appropriate for the limits the readers are playing. This means I have to gather experience from these limits myself.

A grinding project will also be a source of situations and hands that can be used in the article series. Finally, a grinding project will hopefully give us an indication of the win rates a solid and disciplined player can achieve at the micro and low limits, and how fast he can move up the limits using a sensible bankroll management scheme. This could serve as inspiration for small stakes players new to the game.

So where to begin the grind? I decided to start with an article series bankroll of $250, since my impression is that most micro limit players start with similar bankrolls. The next step is to pick a bankroll management scheme, and I have chosen a scheme I call “50+10”. This means playing with a 50 BI minimum bankroll (so we start out at $5PLO), and we can start taking shots at the next limit whenever we have 50 BI for the current limit plus 10 BI for the next limit.

If we lose the shotting capital, we move back down to rebuild and try again (grind in 10 new BI for the next limit and take another shot). So we take shots with 10 BI at a time, and we always move down when the bankroll drops to 50 BI for the previous limit.

The next question is where to end the project. I like a challenge, so I plan to make this article bankroll ready for taking a shot at $200PLO. This means we end the project when we have 50 BI ($5000) for $100PLO plus 10 BI ($2000) for $200PLO. In other words, we will turn our $250 into $7000.

How much time (e.g. how many hands) will we realistically have to use for this project? First we find out how many buy-ins we have to win (minimum) for the different limits:

  • $5PLO to $10PLO:Grind in 20 BI ($100) at $5PLO and build the roll to 50+10 BI ($350) for a shot at $10PLO.
  • $10PLO to $25PLO:Grind in 40 BI ($400) at $10PLO and build the roll to 50+10 BI ($750) for a shot at $25PLO.
  • $25PLO to $50PLO:Grind in 40 BI ($1000) at $25PLO and build the roll to 50+10 BI ($1750) for a shot at $50PLO.
  • $50PLO to $100PLO:Grind in 35 BI ($1750) at $50PLO and build the roll to 50+10 BI ($3500) for a shot at $100PLO.
  • $100PLO to $200PLO:Grind in 35 BI ($3500) at $100PLO and build the roll to 50+10 BI ($7000) for a shot at $200PLO.

If all shots succeed at the first try, we have to grind in 20 + 40 + 40 + 35 + 35 =170 BI. If we (somewhat arbitrarily) assume an average win rate of 7.5 ptBB/100 (ptBB =2 x big blind), we will make 1.5 BI per 1000 hands on average. So we have to play a minimum of 170/(1.5 per 1000 hands) =113,000 hands.

Piece of cake for a grinder with a minimum of professional pride. We have made some assumptions here, so take this estimate with a grain of salt. But we are probably close to the realities.

(And by the way .. if I haven’t already said so we are playing 6-max in this house. Not, and I repeat not, full ring)

4. Learning material and poker tools for PLO

Until recently there was not much to be found for PLO on the book and coaching video market. But in the last couple of years several good books have been published, and most coaching sites have started to produce plenty of high quality PLO videos.

In this section I will give an overview of the best (in my opinion) books, videos and tools for PLO. I will also design a brief study plan for those who want to take up a systematic study of PLO theory and concepts.

4.1 PLO books
Below are short reviews of the best (again, in my opinion) PLO literature on the market today:

Pot-Limit Omaha Poker – The Big Play Strategy (Hwang 2008)
As far as I’m concerned, the publish date of this book marks year zero with regards to good PLO literature. The book discusses full ring strategy, and it’s main theme is to set up profitable situations where we play for deep stacks as a favorite. In order to achieve this, we need to understand starting hand structure, and this is where the book really shines in my opinion.

Regardless of whether we’re playing full ring or shorthanded PLO, we need to know what makes a good starting hand. We also need to know which hands are suitable for winning big pots, and which hands are more suitable for winning small pots.

Hwang’s discussion of PLO starting hands is the most thorough in print as of today. He classifies starting hands both according to type and according to strength. He also thoroughly explains structural defects, and the consequences of getting involved with hands that have poor structure.

Hwang’s main game plan for deep-stacked full ring play is to get involved as a favorite in big pots, and that’s why he devotes so much of the book to understanding starting hand strength and structure, and which type of postflop scenarios the different starting hand types prefer.

We will be playing 6-max, but Hwang’s discussion of starting hands will be very valuable to us, since we will frequently find ourselves in “big play” situations where our good hand clashes with another good hand in a big pot.

Hwang then moves on to postflop play and discusses the principles of postflop ABC poker in pot-limit Omaha. In addition to playing for stacks with quality hands we also need to be skilled in small pot play, and Hwang discusses both big pot and small pot postflop scenarios.

Advanced Pot-Limit Omaha – Volume 1: Small Ball and Short-Handed Play (Hwang 2009)
The is the follow-up toPot-Limit Omaha Poker – The Big Play Strategy, and it’s the first book in a planned series of (probably) 3 books on advanced pot-limit Omaha. Hwang assumes that the reader is familiar with the principles laid out in his first book, and he now takes a big leap forward. The book’s main theme is utilizing position, and Hwang demonstrates through discussion and hand examples how good use of position gives us new opportunities for profit. It also allows us to loosen up our starting hand requirements, sometimes dramatically.

“The Big Play Strategy” from Hwang’s first book is still our core strategy, but by learning to utilize position we will get more opportunities to win small pots in situations where we suspect nobody has much of a hand (this is frequently the case in heads-up and shorthanded pots). Hwang calls this strategy “small ball”, and it’s his preferred strategy in shorthanded play.

Secrets of Professional Pot-Limit Omaha (Slotboom 2006)
A book mainly targeted at full ring players, and it isthebook for learning the principles of shortstacking (our filosophy is that shortstacking is nothing but an annoyance, but that doesn’t mean it isn’t profitable). Slotboom explains his (sometimes unconventional) full ring PLO strategies in great detail, both his shortstacking strategies and his strategies for deep stack play. He does not give an integrated game plan like Hwang does, but he explains how he thinks about PLO, and this should give the reader lots of things to think about (at least it did for me).

Secrets of Short-Handed Pot-Limit Omaha (Slotboom/Hollink 2009)
Like Hwang, Slotboom followed up his full ring book with a book on shorthanded PLO. He uses a structure similar to the first book, which means he discusses his own strategies, and explains how and why they work for him. His process of moving from full ring to shorthanded games (which became necessary partly because the full ring games got flooded with shortstackers who had read his first book) is described in detail, and he discusses the strategic adjustments he had to make.

The last 1/3 of the book is written by coauthor Rob Hollink (a well known high stakes player). Hollink analyzes 33 PLO hands played by himself at limits ranging from $25-50 to $200-400. Many of the hands involve well known online nicks like durrrr, Urindanger, OMGClayAiken, etc.

How Good is Your Pot-Limit Omaha? (Reuben 2003)
This little gem of a book contains 57 hand quizzes taken from live play. Stewart Reuben is a very loose-aggressive player with a relaxed attitude towards starting hand requirements and such. This works well for him, since he is skilled in live deep stack play. But trying to emulate his play in today’s 100 BB buy-in online games will probably lead to bankroll suicide.

But this is not a book you read in order to copy strategies, you read it to train your PLO though processes. I recommend that you take the quizzes seriously and solve them as best you can before you check the answers. You get a score for each hand, and Reuben does a good job of explaining his recommended strategies.

You can learn a lot from comparing your own though processes with those of a strong player. You will sometimes discover logical inconsistencies in your own play, and you learn to think about things you previously didn’t consider.

4.2 PLO videos
Here are some of my favorites among the coaching videos currently on the market. Note that how much you learn from a particular coach can be a matter of personal preference. Different coaches have different playing styles and teaching styles, and a coach that I learn a lot from does not necessarily have to be the best one for you. That said, here are some good videos from some of the different coaching sites:

Deucescracked.com
– The video series2 X 6(Vanessa Selbst & Whitelime)

An introductory series i 8 parts where PLO specialist Vanessa Selbst (who also has a WSOP bracelet in PLO) helps NLHE specialist Whitelime making the transition to PLO. Whitelime is good at asking relevant questions, and many interesting topics emerge from the discussions.

– The video seriesPLO(Whitelime & Phil Galfond)

Whitelime continues his PLO education in another 8 part series, this times with the one and only Phil Galfond (OMGClayAiken/Jman28). When you listen to Phil Galfond explaining PLO concepts, your brain will be filled with light.

Cardrunners.com
– Everything by Stinger (19 videos).
– Everything by lefty2506 (11 videos)

Stinger is a PLO god, that’s it and that’s that. He is also very good at explaining his thought processes. Stinger’s approach to the game is not the most mathematical, and this makes his explanations easy to follow. He mostly uses sound poker logic and reads, and these are things all players can understand.

Note that Stinger uses a pretty loose preflop style. This is fine for a player of his caliber, but probably not something a beginner should start out with. So don’t try to copy everything Stinger does, but pay close attention to his decision making processes.

lefty2506 is a solid TAG player who also explains things very well. Watching a good TAG play makes poker seem simple (and when you play solid poker, thingsarein fact simple most of the time).

Pokersavvy.com
– Everything by LearnedFromTV (16 videos)

LearnedFromTV has a very analytical approach to the game, and he is good at explaining theory. I recommend that you start with the two videosLearnedFromTV #16: PLO Fundamentals – Part 1andLearnedFromTV #18: PLO Fundamentals – Part 2(note that these are not his first videos).

These are theory videos where he explains the most important PLO principles. His live videos are also of high quality with very good explanations of his play.

Continue reading PLO from scratch Part 1-12

C-beting in NLHE 6-max- Part 2

1. Introduction
This is Part 2 of the series “C-Betting in NLHE 6-max” where we take a closer look at flop c-betting in NLHE 6max. In Part 1 we looked at c-betting heads-up and out of position as the preflop raiser. We studied c-betting with “air” (worthless hands) on two example flops:

Coordinated flop

Dry flop

We assumed that the raiser had opened our standard 25% CO range:

22+
A2s+ A9o+
K9s+ KQo
Q9s+ QTo+
J8s+ JTo
T8s+
97s+
87s
76s
65s

326 combos
25%

While the flatter used our standard ~10% “IP flat list”, defined in the article series “Optimal 3/4/5-betting in NLHE 6-max”, and given in the summary document below:

Download link (right-click and choose “Save as …”): IP_3-bet_summary.doc

We wanted to find out whether or not c-betting any two cards was profitable on these two flop textures, against this preflop flatting range. First we let the flatter defend optimally against the c-bet on both flop textures. When he does, the preflop raiser can (per definition) not profit from c-betting any two cards as a bluff. The flatter defends just enough to prevent it (1/(1 + 0.75) =57% defense if the c-bet is 0.75 x pot).

Next, we let the flatter deviate from optimal flop play. We let him play closer to the way a typical weak-tight opponent plays, namely folding too much on certain flop textures and not defending aggressively enough. More specifically, we gave him the following restrictions on the flop:

  • 1. He is unwilling to bluff raise
  • 2. He is unwilling to call c-bets with pairs lower than two of the board cards (e.g. he will fold 77 and lower pairs on a A 8 2 flop).
  • 3. He is unwilling to float naked overcards or naked gutshots without additional draws

In other words, we assumed that the flatter would play straightforward against c-bets, and that he would see each hand as an isolated case. He does not think about defending his total range sufficiently against c-bets, but thinks only about whether or not the hand he has right now can be played profitably on the flop in a vacuum.

Folding a lot on the flop can be better for him than calling c-bets with lots of weak hands, if he does a poor job of stealing on later streets (you need to be willing to sometimes steal on the turn and river if you are floating a lot of weak hands on the flop). But note that if you’re not willing to defend correctly on the flop, you might lose money by flatting preflop. For example, if you’re not willing to sometimes raise J9 as a bluff on a T72 flop, or float and bluff turns when checked to, you might not have a profitable flat preflop with this hand.

Based on the assumptions above we reached the following conclusions:

  • It was unprofitable for the raiser to c-bet any two cards on the coordinated example flop, even with restrictions on the flatter’s flop defense strategy
  • It was clearly profitable for the raiser to c-bet any two cards on the dry flop texture, when we imposed restrictions on the flatters flop defense strategy

We concluded that the preflop raiser should check and give up with his total “air” hands (like 22, 22, A3, and 76) on the very coordinated example flop. Also when the flatter defends in a weak-tight manner on the flop. Simply put, such very coordinated flops are very easy to defend correctly, and there is nothing the preflop raiser can do about it.

However, on the very dry flops we can c-bet all our “air” hands against an opponent who plays weak-tight on the flop. If he is not willing to defend with all his pairs and some naked overcards and weak draws on dry flops, we can fire away. The reason is that very dry flops mostly miss a typical preflop flatting range. So in order to defend optimally on these flops, it becomes necessary to defend with some very weak hands. Most players are uncomfortable doing that.

In Part 2 we’ll build on the modeling we did in Part 1. There we let the preflop flatter use our standard ~10% “IP flat list” that we introduced in “Optimal 3/4/5-betting in NLHE 6-max – Part 2”. This is a flatting range we defined as our standard range in position outside of the blinds, regardless of the raiser’s position.

Now we’ll give the flatter the option to vary his flatting range. We’ll give him two more choices:

– A tight ~5% flatting range
– A loose ~15% flatting range

We’ll repeat the modeling process from Part 1 using these two ranges, and we’ll see if our conclusions change. We’ll find answers to the following questions:

  • Which range is easier to defend on a coordinated flop?
  • Which range is easier to defend on a dry flop?
  • Will the weak-tight restrictions we impose on the flatter’s flop defense strategies be more limiting for him with a tight range or with a loose range?

When this work is done on the very dry and very coordinated example flops. we’ll look at some more intermediate flop textures in Part 3. This will give us more insight into how various preflop flatting ranges interact with various flop textures, and the consequences this has for the profitability of c-bet bluffing with any two cards.

2. Assumptions about ranges
Assume the following model:

  • Alice (100 bb) raises to 3.5 bb preflop with her standard 25% CO open range. She gets flatted by Bob (100 bb) in position
  • Alice c-bets 0.75 x pot on the flop, and we want to know if this is automatically profitable for her with any two cards

We let Bob use 3 different preflop flatting ranges:

– A tight 5% range
– A medium 10% range (our standard “IP flat list”)
– A loose 15% range

Bob’s 10% “IP flat list” range was given earlier in the article. His other two options are defined as:

Tight 5% flatting range

JJ-55
AQs-AJs AQo
KQs

66 combos
5.0%

Bob here chooses to 3-bet or fold his lowest pocket pairs 44-22, and then he flats his remaining pairs and the best high card hands that he doesn’t 3-bet for value ({QQ+,AK} are value hands for Bob against Alice’s 25% CO range). This is a very tight flatting range, and Bob is giving up some profit by folding hands like 44-22, ATs and QJs. On the other hand, this range should be easy to defend on many flops, since it’s so strong.

Loose 15% flatting range

JJ-22
AQs-A6s AQo-ATo
K9s+ KQo
Q9s+
J9s+
T8s+
97s+
76s
65s

200 combos
15.1%

Bob now flats all pairs plus a wide range of high/medium unpaired hands. The unpaired hands are weighted towards suited and coordinated hands that will often flop draws (while hands like ATo depends more on flopping a decent pair).

We expect this flatting range to be harder to defend correctly postflop, since it often flops medium/weak hands and draws. When we start out with a wide and weak range, we will often have to defend with weak hands against a flop c-bet. If we’re not willing to do that, we risk folding so much that the preflop raiser can exploit us by c-betting any two cards profitably.

It follows that in order to flat preflop with a wide and weak range, we have to be comfortable bluffing and floating with weak hands postflop. If we’re not, many of the hands we flat preflop might be unprofitable for us. This is something we want to look at in our model study.

3. C-betting on coordinated flop

We’ll now build Bob’s defense strategies on the coordinated example flop from Part 1 with the 3 preflop flatting ranges he has at his disposal (and the work for the 10% range was done in Part 1). For each range we first estimate his optimal flop strategy. On coordinated flops, Bob’s defense consists of:

– Raising his best hands
– Flatting his next best hands
– Bluff raise with some weak hands in a 1 : 1 value/bluff ratio

Then we build a strategy that the non-optimal version of Bob can use under the following weak-tight restrictions:

  • 1. He is unwilling to bluff raise
  • 2. He is unwilling to call c-bets with pairs lower than two of the board cards (e.g. he will fold 77 and lower pairs on a A 8 2 flop).
  • 3. He is unwilling to float naked overcards or naked gutshots without additional draws

When Bob defends optimally on the flop, Alice can’t c-bet any two cards profitably per definition. When Bob deviates from optimal play, she might be able to. She c-bets 0.75 x pot, so she can c-bet any two cards with a profit if Bob folds more than 1/(1 + 0.75) =57%.

If we conclude from our analysis that the non-optimal version of Bob will defend less than 57%, Alice has an automatically profitable c-bet bluff, regardless of her cards. We can then estimate the EV of her bluff with an EV calculation.

3.1 Defense against c-bets with a tight 5% flatting range
On this flop, 55 combos remain in Bob’s 5% flatting range, as shown below:

Optimal defense against a 0.75 x pot c-bet means Bob has to defend 57% of his total range, which is 0.57 x 55 =31 combos. Here is one way to do it:

  • Value raise:
    {TT,55} =6 combos
  • Flat:
    {AQ,KQs,AJ,JJ} =22 combos
  • Bluff raise:
    {AJ,AJ,AJ,99,99,99} =6 combos
  • Total: 34 combos (optimal: 31)

Bob can easily get to the optimal defense and then some. Note that a queen high flop texture “smashes” his flatting range, since almost all of his unpaired hands contain a Q. A king high flop would have given him fewer pairs to use, but on the other hand a K high and coordinated flop would have given him various draws he could use.

Now we restrict Bob’s flop defense strategy and see what we get. A possible strategy for Bob to use under these conditions is:

  • Value raise:
    {TT,55} =6 combos
  • Flat:
    {AQ,KQs,AJs,JJ} =25 combos
  • Bluff raise:
    None
  • Total: 31 combos (optimal: 31)

Bob has to stretch a bit by floating AJ,AJ, and AJ that only give him overcard + gutshot combos. He is unwilling to float naked overcards or naked gutshots, but he can float hands that give him a combination of such weak draws. AJs makes the cut.

We see that the non-optimal version of Bob manages to (barely) get to optimal defense with his tight 5% flatting range on our coordinated example flop. Alice can not c-bet any two cards profitably in this scenario. But note that she might have been able to, if the flop had been king high instead of queen high (we can always to a separate analysis if we want to look further into this).

3.2 Defense against c-bets with a medium 10% flatting range
This scenario was discussed in Part 1, and we only include the results here:

The remaining number of combos in Bob’s range is 120:

Optimal 57% defense with 0.57 x 120 =68 combos:

  • Value raise:
    {TT,55,QTs,AQ,AJ,KJ} =23 combos
  • Flat:
    {KQ,QJs,JJ,ATs} =24 combos
  • Bluff raise:
    {KTs,JTs,T9s,KJ,KJ,KJ,98,AJ,AJ,AJ,AJ,AJ,AJ,98,98,98} =22 combos
  • Total: 69 combos (optimal: 68)

Non-optimal defense under weak-tight restrictions:

  • Value raise:
    {TT,55,QTs,AQ,AJ,KJ} =23 combos
  • Flat:
    {KQ,QJs,JJ,ATs,KTs,JTs,T9s,98,KJs,AJ,AJ,AJ,AJ,AJ,AJ} =43 combos
  • Bluff raise:
    None
  • Total: 66 combos (optimal: 68)

Bob can easily get to optimal defense with his 10% flatting range on our coordinated example flop. Alice can’t c-bet any two cards profitably in this scenario either.

3.3 Defense against c-bets with a loose 15% flatting range
The number of remaining combos in Bob’s 15% flatting range is 174:

Optimal 57% defense means Bob has to defend 0.57 x 120 =99 combos. Here is one way to do it:

  • Value raise:
    {TT,55,QTs,AQ,AJ,KJ,J9} =24 combos
  • Flat:
    {KQ,QJs,Q9s,JJ,AT,KTs,A9,A8,A7,A6,98,97,87,76,65} =48 combos
  • Bluff raise:
    {JTs,T9s,KJ,KJ,KJ,J9,J9,J9,AJ (not AJ)} =27 combos
  • Total: 99 combos (optimal: 99)

It’s still easy for Bob to defend optimally on the coordinated flop, even with a loose preflop flatting range. His range is dominated by suited and coordinated high card hands, and it hits this type of flop very hard. He has more than enough strong/medium hands and draws to use.

When Bob is given weak-tight restrictions, defending enough will be harder. Mainly because he now loses the option to bluff raise, which is an important component of the defense on coordinated flops. Now he has to call more, but it might be difficult for him to come up with enough flatting hands, since he can’t use naked overcard/gutshot draws or his lowest pairs.

Here is one way to defend under weak-tight restrictions:

  • Value raise:
    {TT,55,QTs,AQ,AJ,KJ,J9} =24 combos
  • Flat:
    {KQ,QJs,Q9s,JJ,AT,KTs,JTs,T9s,T8s,A9,A8,A7,A6,98,97,87,76,65,KJ,KJ,KJ,JS9,J9,J9,AJ (not AJ)} =72 combos
  • Bluff raise:
    None
  • Total: 96 combos (optimal: 99)

Bob can get to optimal defense is he is willing to call the c-bet with all pairs 2nd pair or better, as well as AJ for a overcard + gutshot draw. Alice still can’t c-bet any two cards profitably on our coordinated example flop.

4. C-betting on dry flop

Now we build Bob’s defense strategies on the dry example flop from Part 1. For each range we first build his optimal strategy. On dry flops, Bob’s defense consists of

– Flatting with all his defense hands

The reason for using a flatting-only strategy on dry flop textures has been thoroughly discussed in the article series “Optimal Postflop Play in NLHE 6-max”. When the optimal strategies have been found, we impose the weak tight restrictions:

  • 1. He is unwilling to bluff raise
  • 2. He is unwilling to call c-bets with pairs lower than two of the board cards (e.g. he will fold 77 and lower pairs on a A 8 2 flop).
  • 3. He is unwilling to float naked overcards or naked gutshots without additional draws

Raising is not an option on dry flops regardless, so the restrictions only concern the hands Bob is willing to flat with on the flop.

4.1 Defense against c-bets with a tight 5% flatting range
Bob has 62 remaining combos in his 5% flatting range after accounting for card removal effects_

Optimal defense means defending 57% of these, which is 0.57 x 62 =35 combos. Here is one way to do it:

  • Value raise:
    None
  • Flat:
    {99,KQs,JJ-TT,88-66} =36 combos
  • Bluff raise:
    None
  • Total: 36 combos (optimal: 35)

Bob can easily get to optimal defense with his tight 5% range, without having to float with naked overcards. Then we impose the weak-tight restrictions and see how that changes things. Now Bob can’t flat naked overcards, naked gutshots or pairs lower than the 9 on the board. This makes it impossible for Bob to defend enough. If he goes as far as he possibly can, he ends up with:

  • Value raise:
    None
  • Flat:
    {99,KQs,JJ-TT} =18 combos
  • Bluff raise:
    None
  • Total: 18 combos (optimal: 35)

Bob’s problem in this scenario is that he is not willing to flat his lowest pairs and best overcards (AQ). When he folds these hands, he can only get to about 1/2 of the necessary defense. He defends only 18/62 =29% of his range (as opposed to the optimal 57%), and folds 100 – 29 =71%. Alice can now exploit him by c-betting any two cards.

Alice’s EV for a pure c-bet bluff that can never win unless Bob folds on the flop is:

EV (c-bet)
=0.71 (P) + 0.29 (-0.75P)
=+0.49P

Where P is the pot size on the flop. If the preflop raise was 3.5 bb, the pot is P =2(3.5) + 0.5 + 1 =8.5 bb. The EV of Alice’s c-bet bluff is then 0.49 x 8.5 bb =4.2 bb.

Note that when Bob’s preflop flatting range is tight, our conclusions are very dependent on the exact cards that come on the flop, as well as the exact hands Bob’s range is made up of. For example, if Bob had elected to flat the 12 KQo combos instead of the 12 66/55 combos, he would have been able to defend about optimally on this king high flop texture, also with the restricted strategy.

When Bob’s range is very tight, we can gain a lot from paying close attention. Some players flat all pairs, others fold or 3-bet-bluff the lowest pairs and flat more Broadway hands instead. Observe hands that go to showdown, and take notes. If your PokerTracker/HEM database has many hands on a player, you can use it to extract information and take notes between sessions (this is a smart thing to do for opponents you meet regularly).

4.2 Defense against c-betting with a medium 10% flatting range
This work was done in Part 1, and below is a summary of the results:

The number of combos after card removal is 126:

Bob defends 0.57 x 126 =72 combos when playing optimally. Here is one way to do it:

  • Value raise:
    None
  • Flat:
    {99,22,KQ,KJs,KTs,JJ-TT,T9s,98s,88-66,AQ} =76 combos
  • Bluff-raise:
    None
  • Total: 76 combos (optimal: 72)

And here is one way Bob can defend under the weak-tight restrictions:

  • Value raise:
    None
  • Flat:
    {99,22,KQ,KJs,KTs,JJ-TT,T9s,98s} =42 combos
  • Bluff-raise:
    None
  • Total: 42 combos (optimal: 72)

Bob now defends only 42/126 =33% of his range and folds 100 – 33 =67%. Alice can exploit this by c-bet bluffing any two cards. Her EV for a c-bet bluff with a worthless hand is:

EV (c-bet)
=0.67 (P) + 0.33 (-0.75P)
=+0.42P

Where P is the pot size on the flop. With a pot of 8.5 bb, the EV is 0.42 x 8.5 bb =3.6 bb.

4.3 Defense against c-betting with a loose 15% flatting range
We’ll see that this is a difficult job for Bob when we impose weak-tight restrictions. The number of combos that remain in his range after accounting for card removal effects is 180:

Optimal 57% defense means Bob has to use 0.57 x 180 =103 combos. Here is one way to do it:

  • Value raise:
    None
  • Flat:
    {99,22,K9s,KQ,KJs-KTs,JJ,TT,A9s,Q9s,J9s,T9s,98s-97s,88-55,AQ,QJs,JTs} =104 combos
  • Bluff raise:
    None
  • Total: 104 combos (optimal: 103)

Bob has to flat almost all of his pairs, plus some overcard hands (AQ) and gutshots (QJs, JTs). It’s hard enough to defend optimally when Bob can use all hands, and when we impose weak-tight restrictions, it becomes impossible. Here is what Bob comes up with when he goes as far as he can:

  • Value raise:
    None
  • Flat:
    {99,22,K9s,KQ,KJs-KTs,JJ,TT,A9s,Q9s,J9s,T9s,98s-97s} =56 combos
  • Bluff-raise:
    None
  • Total: 56 combos (optimal: 103)

The defense is more or less identical to the optimal defense, except that we have dropped all pairs lower than 9, all naked overcard hands (AQ) and all naked gutshots (QJs, JTs). Bob now defends about 1/2 of the optimal amount: 56/180 =31% of his range. So he folds 100 – 31 =69% on the flop, and the EV for Alice’s’ c-bet bluffs becomes:

EV (c-bet)
=0.69 (P) + 0.31 (-0.75P)
=+0.46P

Where P is the pot size on the flop. With P =8.5 bb, the EV becomes 0.46 x 8.5 bb =3.9 bb.

So a c-bet bluff will be automatically profitable on the flop, but note something else as well: Bob is forced to defend on the flop with many low pairs and weak draws, also under weak-tight restrictions. So Alice should have many opportunities to 2-barrel profitably on the turn. Bob can protect himself somewhat against 2-barrel bluffs by slowplaying his strongest hands on the flop, but life will still be tough for him on the turn if Alice decides to bluff a lot.

So a good player with knowledge about Bob’s preflop flatting range and his postflop tendencies should be able to make even more money from c-bet bluffing by sometimes continuing to bluff on the turn and the river. But note that we don’t have to continue out bluffs in order to have a nicely profitable c-bet bluff in isolation on the flop.

5. Summary
We used the two example flop textures (very coordinated and very dry) from Part 1 and continued our modeling of c-bet bluffing. This time we let Bob use 3 preflop flatting ranges:

– A tight 5% range
– A medium 10% range (our standard “IP flat list”)
– A loose 15% range

Based on our modeling, we conclude the following:

  • We can’t c-bet bluff profitably with any two cards on a very coordinated flop against any reasonable flatting range, even if our opponent defends weak-tight
  • On very dry flops we can c-bet bluff profitably with any two cards, if our opponent defends weak-tight

We noted that the profitability of a c-bet bluff against the tight 5% range on a dry flop was very sensitive to the exact flop texture and the exact composition of the flatting range. At the other end of the spectrum, this became relatively unimportant against the loose 15% range.

A wide and weak preflop flatting range is impossible to defend correctly against c-bets on a very dry flop, unless the player is willing to flat just about any pair plus lots of overcard and gutshot combos. Exactly what the flop is, and exactly which hands we flat is now less important, since we have to defend lots of weak hands/draws regardless.

We summarize:

On very coordinated flops we can’t get away with any two cards c-bet bluffing regardless of our opponents preflop flatting range. If he defends weak-tight, this does not help you a lot, since very coordinated flop textures are so easy to defend.

On very dry flops you can probably get away with any two cards c-bet bluffing regardless of your opponent’s flatting range, as long as he isn’t willing to always defend optimally. A wide flatting range gives you the best opportunities, since wide ranges are very hard to defend optimally on very dry flops. Of course, against an opponent that always defends optimally, we can’t buff any two cards profitably, per definition. But most players are unable or unwilling to defend enough on dry flops. So our starting assumption can be that any-two-cards c-bet bluffing is profitable on very dry flops. If we are wrong against a particular opponent, we can adjust later, and start checking more hands.

In Part 3 we’ll look at some other flop textures in the region between very coordinated and very dry flops. We’ll also introduce a software tool (“Flopzilla“) that lets us quickly analyze the profitability of a c-bet bluff, without having to write out complete strategies like we have done up to this point.

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/c-beting-in-nlhe-6-max–part-2-274#sthash.IbkJIeKk.dpuf

C-beting in NLHE 6-max- Part 1

1. Introduction
This is the first part of an article series about flop c-betting in NLHE 6-max. In the previous series “Optimal Postflop Play in NLHE 6-max” we looked at postflop play in the scenario where one player has raised preflop and gotten called by another player in position. We discussed how the player in position can defend optimally against c-bets on the flop, and against 2- and 3-barrels on the turn and river. Then we discussed how the preflop raiser can play the turn and river optimally after c-betting the flop and getting called, to prevent his opponent from exploiting him by floating.

For both the preflop raiser out of position and the flatter in position we built postflop strategies that prevents their opponent from exploiting them by betting or floating with any two cards on any street. The flatter in position has to defend enough against c-bets to prevent the preflop raiser from c-bet bluffing any two cards on the flop. And those times the preflop raiser has c-bet the flop and gotten called, she has to play the turn in such a way that she prevents the flatter from floating on the flop with any two cards (planning to steal the pot on the turn).

In our discussion of turn and river play for this scenario, we simply assumed that the preflop raiser had started postflop play by c-betting her entire range on the flop. When we looked at turn and river barreling we limited our study to dry flop textures, so this assumption was reasonable.

In this article we’ll look more closely at c-betting with “air” on the flop, heads-up as the preflop raiser. We’ll use a model where one player (Alice) openraises out of position and gets flatted by another player (Bob) in position. The flop comes, and Alice has a c-bet decision to make. We want to train our ability to recognize flop textures where Bob’s preflop flatting range has connected poorly, so that Alice can c-bet any two cards profitably on the flop.

We are then assuming that Bob is not willing to defend optimally. Because if he does, we can’t profit from c-betting any two cards per definition of optimal play. So we are assuming that Bob will fold more than the optimal amount on flop textures that are bad for his preflop flatting range (for example, a dry flop like J 6 2). In “Optimal Postflop Play in NLHE 6-max” we built optimal postflop strategies for Bob to use in this scenario, but now we’ll assume he behaves more like the players we meet in practice. And they will typically fold too much on flop textures that mostly misses their range.

In Part 1 of this series we’ll study how well different flop textures hit a typical preflop flatting range (we’ll use our standard 10% “IP flat list” range). Based on this, we can estimate the EV of a c-bet bluff with a worthless hand on different textures. In Part 2 we’ll vary the preflop flatting range and see how the EV of the c-bet changes when we’re up against a tight (~5%) and a loose (~15%) flatting range. This analysis will train our ability to identify profitable c-bet bluffing spots based on the flop texture and our knowledge about the preflop flatter’s range.

The modeling we do in these articles is inspired by the video Alans Common C-betting Spots by Bluefirepoker coach Alan Jackson.

Our approach in this article series about c-betting is exploitive. We make assumptions about various opponent mistakes, and then we move away from optimal play in order to exploit these mistakes. Our previous work on optimal play gives us a starting point, and tells us in which direction we should move our strategy. The main mistake we focus on in this article series is the mistake of folding too much to a c-bet. We want to find spots where our opponent is making this mistake, thus giving us an opening for c-bet bluffing any two cards profitably.

2. Our model
Here are the assumptions we’ll use in this article:

2.1 Assumptions about preflop ranges

 

  • Alice (100 bb) openraises in CO
  • Bob (100 bb) is on the button and follows the previously defined optimal strategies for 3/4/5-betting. Other than that he flats his standard range in position (“IP flat list”).
  • Alice knows Bob’s flatting range based on observations and HUD stats

Alice uses our standard 25% opening range from CO:

22+
A2s+ A9o+
K9s+ KQo
Q9s+ QTo+
J8s+ JTo
T8s+
97s+
87s
76s
65s

326 combos
25%

We assume Bob uses the optimal 3/4/5bet strategy against a 25% opening range, given in the table of optimal strategy pairs built in “Optimal 3/4/5-betting in NLHE 6-max – Part 2”:

Download link (right-click and choose “Save as”): IP_3-bet_summary.doc

So Bob will use the following preflop strategy against Alice’s 25% CO raise:

  • 3-bet {QQ+,AK, 12 air} for value, planning to 5-bet all-in against a 4-bet
  • 3-bet bluff 70% of “IP 3-bet air list”, planning to fold against a 4-bet
  • Flats the entire “IP flat list”: {JJ-22,AQs-ATs,AQo-AJo,KQs-KTs,KQo,QJs-QTs,JTs,T9s,98s} =140 combos when {QQ+,AK} is 3-bet for value

Bob’s standard preflop flatting range then has 140 combos, which is 140/1326 =10.6% of all hands. This range is representative for what many players will flat on the button in this scenario, and it’s a reasonable assumption to use against unknowns.

When the flop comes, Alice tries to determine whether or not she has a profitable c-bet for 0.75 x pot with any two cards. She bases her analysis on her knowledge about Bob’s preflop flatting range, the flop texture, and assumptions about which hands Bob is willing to defend with.

Since Alice c-bets 0.75 x pot, she is giving herself pot odds 1 : 0.75 on a bluff with any two cards. A c-bet bluff will be automatically profitable if Bob folds more than 0.75/(1 + 0.75) =57%. If Alice’s analysis concludes that Bob in practice will fold more than 57%, she can c-bet her entire range profitably on the flop. If not, she has to check and give up with some of her weakest hands. How much hand strength we need to c-bet proftiably in this case will be discussed in future articles.

The purpose of the work we do in Part 1 is to train our ability to come up with a qualitative yes/no answer to the question about whether or not we can c-bet any two cards profitably. We look at the flop, we think about our opponents preflop flatting range, and we analyze how the flop and the range interact. We then introduce some assumptions about which hands opponent will defend in practice against our c-bet, and we have our answer.

2.2 Assumptions about Bob’s flop strategy
We’ll look at two example flop textures in this article:

– Very coordinated
– Very dry

For both textures we’ll first build Bob’s optimal defense against Alice’s c-bet. The optimal defense strategy is designed to prevent her from c-betting any two cards profitably. If Bob uses this strategy, there is nothing Alice can do to exploit him by bluffing a lot on the flop.

Then we’ll make some assumptions about the strategy Bob will use in practice. We’ll assume that Bob will fold some weak hands (for example overcards and weak pairs) that he should not fold on flops where his range is weak and difficult to defend correctly. Then we’ll analyze whether or not Bob’s deviation from optimal play will make it possible for Alice to exploit him by c-bet bluffing any two cards.

Exactly how Bob deviates from optimal play will be a function of the flop texture. Here are three general assumptions we’ll use for the non-optimal version of Bob:

  • 1. He is not willing to bluffraise against Alice’s c-bet
  • 2. He is not willing to call the c-bet with pairs lower than two of the cards on the board (for example, he will fold 77 and all lover pairs on a A 8 2 flop)
  • 3. He is not willing to call the c-bet with naked overcards and gutshots, with no additional draws

In addition we can make specific assumptions about how Bob will play on specific flop textures. If we do make extra assumptions, we’ll use good poker sense and let Bob play the way a typical opponent in our games will play.

In general, we’ll assume that the non-optimal Bob plays like a typical decent-but-not-great low limit player. He plays mostly straightforward, he bluffs little when others have the initiative, and he has limited knowledge about the interaction between flop texture and hand ranges. Also, he does a poor job changing his postflop strategies and ranges based on the pot odds he’s getting.

The non-optimal version of Bob mostly sees each hand as an isolated case, and he does not think about the hand as a part of an overall range. This is typical for how the majority of poker players think. They think things like “I have top pair, which is a good hand” or “I have bottom pair, which is a very weak hand“, and they don’t think about all the other possible hands they could hold in this particular scenario.

3. C-betting on a coordinated flop
In “Optimal Postflop Strategies in NLHE 6-max” we concluded that our standard positional flatting range “IP flat list” is easy for Bob to defend on coordinated flops like J 9 3, since these flops hit his preflop range hard.

Now we’ll show through analysis why c-bet bluffing any two cards on these flops is a bad idea heads-up and out of position against a preflop flatter, even if our opponent is tight and straightforward. This is something most players intuitively understand, but not we’ll “prove” it using theory, and we’ll get a much clearer picture of exactly why this is so. Then we’ll repeat the process on a dry flop, and we’ll see that dry textures give us opportunities for profitable any-two-cards c-bet bluffing if our opponent is somewhat tight.

3.1 Optimal defense against c-betting on a coordinated flop
Alice (100 bb) raises her standard 25% range from CO, and Bob flats his standard 10.6% flatting range “IP flat list” ={JJ-22,AQs-ATs,AQo-AJo,KQs-KTs,KQo,QJs-QTs,JTs,T9s,98s} =140 combos.

Our coordinated flop is:

When Alice c-bets, she bets 0.75 x pot, and Bob needs to defend at least 1/(1 + 0.75) =57% to prevent her from c-betting any two cards with automatic profit.

Bob has 120 remaining combos in his range after adjusting for card removal effects, as shown below:

Bob’s optimal defends is then to defend 0.57 x 120 =68 combos. We remember from “Optimal Postflop Play in NLHE 6-max” that Bob’s defense on coordinated flops has three components:

– Raise the best hands for value
– Flat the next best hands
– Bluff raise some weak hands using a 1 : 1 value/bluff ratio

Below is a suggestion for a near-optimal flop strategy for Bob. At this point in the analysis our only concern is to defend with 68 combos (or thereabouts) overall. If this leads us to folding or bluffing with hands that could have been played more profitably by calling, this is not a problem for us.

  • Value raise:
    {TT,55,QTs,AQ,AJ,KJ} =23 combos
  • Flat:
    {KQ,QJs,JJ,ATs} =24 combos
  • Bluffraise:
    {KTs,JTs,T9s,KJ,KJ,KJ,98,AJ,AJ,AJ,AJ,AJ,AJ,98,98,98} =22 combos
  • Total: 69 combos (optimal: 68)

As we have seen in previous articles, the optimal flop defense ranges are strong on very coordinated flops after we have flatted our default “IP flat list” preflop. We have so many strong hands to use that we can get away with only flatting top pair + best underpair (JJ) + 2nd pair/top kicker (ATs). All lower pairs can be folded or used as bluff raises.

3.2 Non-optimal defense against c-betting on coordinated flop
Now we’ll limit the strategies Bob is willing to use when he defends against Alice’s c-bet:

  • 1. He is not willing to bluffraise against Alice’s c-bet
  • 2. He is not willing to call the c-bet with pairs lower than two of the cards on the board (for example, he will fold 77 and all lover pairs on a A 8 2 flop)
  • 3. He is not willing to call the c-bet with naked overcards and gutshots, with no additional draws

We remember that Bob has to defend less than 57% to give Alice a profitable any-two-cards bluffing opportunity when she c-bets 0.75 x pot. So the question we want to answer is this:

Will the restrictions above make it impossible for Bob to defend at least 57% on the flop?

If this is the case, Alice can c-bet her entire range profitably. We now try to build a defense strategy for Bob where he defends 57% (68 combos) given the limitations above:

  • Value raise:
    {TT,55,QTs,AQ,AJ,KJ} =23 combos
  • Flat:
    {KQ,QJs,JJ,ATs,KTs,JTs,T9s,98,KJs,AJ,AJ,AJ,AJ,AJ,AJ} =43 combos
  • Bluff raise:
    None
  • Total: 66 combos (optimal: 68)

We can easily get to around the optimal defense, even if we’re not willing to bluffraise, call with 3rd pair or lower, or float with naked overcards and gutshots. The weakest draw Bob has to call with is AdJx/AxJd (overcard + gutshot + backdoor flush draw).

3.3 Conclusion for defense against c-betting on coordinated flop
Both the optimal and the non-optimal versions of Bob could easily defend the optimal 57% on this flop texture. These flops hit Bob’s preflop flatting range so hard that the can get away with folding lots of marginal hands, and still defend enough.

A range analysis with Pokerazor illustrates this with numbers:

On this flop we have 2nd pair or better 45% of the time (see the list “Cumulative frequency” to the right). So we can cover most of the optimal 57% defense with good one pair hands. And the rest is easily covered by our draws. Even if we never bluff raise, flat pairs below 2nd pair, or flat naked overcards/gutshots, we can get to 57% defense.

We therefore conclude:

Alice can’t c-bet any two cards profitably on our very coordinated flop texture, even if Bob plays tightly and isn’t necessarily willing to defend an optimal amount. He can easily build defense strategies that defend the optimal amount, even with strong limitations on the hands he is willing to defend.

In future articles we’ll talk more about how much hand strength we need to have a profitable c-bet on these flops. We obviously have to be willing to semibluff a bit, and c-bet some weak draws. But we should check-fold our pure air, like 76, 22 and A4 on this flop. Bob simply doesn’t fold often enough to make it profitable, even if he plays somewhat tight postflop.

4. C-betting on a dry flop
Next we’ll show why c-bet bluffing with any two cards on very dry flops generally is a good idea Even players who defend loosely on the flop will find it difficult to defend the optimal amount, since this requires them to float with lots of air.

4.1 Optimal defense against c-betting on dry flops
Alice (100 bb) raises her standard 25% range from CO, and Bob (100 bb) flats his standard 10.6% flatting range “IP flat list” ={JJ-22,AQs-ATs,AQo-AJo,KQs-KTs,KQo,QJs-QTs,JTs,T9s,98s} =140 combos.

Our dry flop is:

This is the classic super-dry flop with one Broadway card, two medium/low cards, and no flush or open-ended straight draws possible. Again, Alice c-bets 0.75 x pot, and Bob needs to defend 57% to prevent her from bluffing with any two cards, as in the previous case.

After adjusting for card removal effects, Bob has 126 remaining combos in his range:

Bob’s optimal defense requires him to defend 0.57 x 126 =72 combos. We remember from “Optimal Postflop Play in NLHE 6-max” that Bob defends only by flatting on very dry flops. So he slowplays all his monsters (only sets are monsters on this flop), together with hos good hands, and some weak hands. He is often forced to flat with all his one pair hands, and perhaps also float some naked overcards and gutshots in order to defend optimally.

Below is a suggestion for an optimal flop defense strategy for Bob:

  • Value raise:
    None
  • Flat:
    {99,22,KQ,KJs,KTs,JJ-TT,T9s,98s,88-44} =72 combos
  • Bluff raise:
    None
  • Total: 72 combos (optimal: 72)

Bob has to flat all his pocket pairs, except 33. As an alternative, he can fold some low pocket pairs and float his best overcards instead (AQ):

  • Value raise:
    None
  • Flat:
    {99,22,KQ,KJs,KTs,JJ-TT,T9s,98s,88-66,AQ} =76 combos
  • Bluff raise:
    None
  • Total: 76 combos (optimal: 72)

But regardless of how he chooses to do it, Bob has to flat lots of weak hands on this flop texture in order to defend the optimal 57%.

4.2 Non-optimal defense against c-betting on a dry flop:
Again, we introduce limitations for Bob:

  • 1. He is not willing to bluffraise against Alice’s c-bet
  • 2. He is not willing to call the c-bet with pairs lower than two of the cards on the board (for example, he will fold 77 and all lover pairs on a A 8 2 flop)
  • 3. He is not willing to call the c-bet with naked overcards and gutshots, with no additional draws

Then we see how far he can go:

  • Value raise:
    None
  • Flat:
    {99,22,KQ,KJs,KTs,JJ-TT,T9s,98s} =42 combos
  • Bluff raise:
    None
  • Total: 42 combos (optimal: 72)

It turns our that if Bob is unwilling to flat with naked gutshots, naked overcards, and pairs lower than two of the cards on the board, it is impossible for him to defend the optimal amount. He gets to 42/126 =33% defense, and folds 100 – 33 =67%.

Let’s say Alice c-bets with a worthless hand that will never win the pot when Bob doesn’t fold on the flop. Her EV for the bet is:

EV (c-bet)
=0.67 (P) + 0.33 (-0.75P)
=+0.42P

Where P is the pot size on the flop. If the preflop raise was 3.5 bb, the pot is P =2(3.5) + 0.5 + 1 =8.5 bb on the flop. Alice’s c-bet bluff is then worth 0.42 x 8.5 bb =3.6 bb. This is a very nice profit for an any-two-cards bluff with a hand that can only win when Bob folds.

4.3 Conclusion for defense against c-betting on dry flops
Bob could defend our dry flop texture optimally without floating with extremely weak hands, but he had to drop down to the “cellar” and use his lowest one pair hands. Alternatively, he could fold some low pairs and float with some of his best overcard hands instead.

When Bob’s strategies were limited, it was impossible for him to defend enough. If he is unwilling to call with his lowest one pair hand, good ace high hands, or gutshots, he can’t defend our dry example flop optimally. This opens him up for getting exploited by Alice’s any-two-cards c-bet bluffs.

A range analysis with Pokerazor illustrates this with numbers:

On the coordinated example flop we had 2nd pair or better 45% of the time, in addition to many draws. On the dry example flop we have 2nd pair or better only 33% of the time, and we have no strong draws, only naked overcards and gutshots.

Most NLHE players know (or intuitively see) that our dry example flop is an excellent flop to bluff at. So you can expect the preflop raiser to c-bet a lot when you are the preflop flatter on such a flop. Therefore, if you believe the preflop raiser will try to exploit you by c-bet bluffing with any two cards, don’t be afraid to float!.

Remember that you will also call with some good hands like sets, top pair, and good 2nd pair/underpair hands. So if he 2-barrels a lot with air, he will get punished by your strong flatting hands. Think about what his range looks like on this type of flop. If he has raised from CO, his range is full of garbage like A8o, 76s, etc. Force him to play turns with these hands if he is aggressive enough to c-bet any two cards on the flop.

If he keeps betting on the turn, you have to fold low pairs like 55 and floats like AQ, but you will still plenty of hands to continue with (remember, you have slowplayed sets and top pair hands in your range). So your turn range will be decently strong, even if you floated the flop with a weak range.

5. Summary
In this article we have begun studying c-betting on the flop in heads-up pots as the preflop raiser.

We saw that coordinated flops are easy to defend optimally for the preflop flatter, even if he isn’t necessarily willing to defend optimally. When we did the same model study on a dry flop, we saw that it was impossible for the flatter to defend optimally if he was unwilling to float his weakest one pair hands, and/or some floats (overcards/gutshots type hands. When you have identified such players at the table (and they are common at the low limits), you can c-bet dry flops with your entire range against them, and “print money”.

The gist of it is that all flops can be defended optimally, in principle (it’s only a matter of including more and more weak hands, as the flop texture gets drier), but many players are unwilling to do so if it requires them to defend with very weak hands. These players can be exploited by c-bet bluffing a lot on dry flop textures. On the driest of flops, you can c-bet your entire range profitably,.

In Part 2 we’ll continue our modeling. Now we’ll let Bob use two other preflop flatting ranges (tight =5% and loose =15%) in addition to his standard 10% “IP flat list”. This gives us an opportunity to learn about how various preflop flatting ranges hit various types of flop textures, and the consequences this has for our c-betting strategy.

Being able to classify flop textures as coordinated or dry gives us possibilities to c-bet profitable with any two cards, and this was what we learned in this article. If we also train our ability to distinguish between different opponent ranges, we add one extra dimension to our analysis. This will enable us to find even more profitable c-bet bluffing spots. A particular flop texture can give us a profitable c-bet bluff against one opponent range, but not against another. This is the topic for the next article.

Note that the work done in this article defines a method for training our ability to recognize profitable c-bet bluffing opportunities. You can generate random flops using Flopgenerator.com and perform this type of analysis, using assumptions about your opponent’s flatting range and postflop tendencies. This will tell you whether or not you have a profitable any-two-cards c-betting opportunity on the given flop.

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/c-beting-in-nlhe-6-max–part-1-263#sthash.rntipxOO.dpuf

Optimal Postflop Play in NLHE 6-max – Part 7

1. Introduction
This is Part 7 of the article series “Optimal Postflop Play in NLHE 6-max” where we’ll study optimal strategies for heads-up postflop play in NLHE 6-max.

In this article we’ll continue the work started in Part 5 and Part 6, where we studied postflop strategy for a preflop raiser out of position in a heads-up scenario. In Part 5 we designed an optimal barreling strategy for the raiser that protected her against random floating done by her opponent in position. In Part 6 we verified mathematically that this strategy made her opponents any-two-cards-floats break even, which means he can not float random weak hands profitably against her flop c-bet. We also studied the effect of changing the raiser’s preflop opening range. We found that a looser preflop range forced her to play looser ranges postflop, if she began postflop play by c-betting her entire range on the flop.

The topic for this article is to look more closely at:

– The effect of the preflop raiser’s postflop bet sizing
– The effect of her opponent slowplaying his strong hands postflop

In previous postflop articles we have assumed our players are using a standard postflop bet sizing scheme of 0.75 x pot on the flop, 0.75 x pot on the turn, and 0.60 x pot on the river. But if we always stick with standard bet sizing, we risk giving up +EV in some spots. What makes NLHE one of the most profitable games for a strong player is the freedom she has to vary her bet sizing. This enables her to exploit weaker players’ mistakes maximally.

Here we’ll look at a specific example where the raiser c-bets the flop, 2-barrels the turn, and 3-barrels the river with an overpair on a dry flop texture. Conventional wisdom is we can/should use small bet sizing on dry flop textures, since our opponent will have fewer draws on such flops. So there’s less risk of getting drawn out on, and we can bet smaller to protect our hand against draws. But this does not necessarily mean we maximize our EV for the hand by betting small on these board textures.

If we find ourselves heads-up against a player who we know has a range full of medium/weak hands (so that it’s easy for us to know when we’re ahead and when we’re behind(, we’ll see that we maximize our EV by using big value bets on all streets. But of course with a balanced mix of value hands and bluffs, since we’re trying to play close to optimally.

The scenario we’ll study in this article is valuebetting/barreling an overpair on a dry flop texture heads-up and out of position against a weak opponent range. We’ll study the effect of varying bet sizing for the raiser out of position, and the effect of slowplaying for the player in position.

We begin by defining the model scenario we’ll work with throughout the article. Then we define the two bet sizing schemes (“standard” and “alternate”) that the preflop raiser (Alice) will be using postflop. Next, we define the postflop strategies for the player in position (Bob), and we use Pokerazor to compute the EV for Alice’s c-bet/2-barrel/3-barrel postflop line with overpairs against Bob’s range/strategy.

We end up with EV calculations for Alice’s postflop play under 4 combinations of circumstances:

  • Alice’s standard bet sizing against Bob who doesn’t slowplay
  • Alice’s standard bet sizing against Bob who slowplays
  • Alice’s alternate bet sizing against Bob who doesn’t slowplay
  • Alice’s alternate bet sizing against Bob who slowplays

Based on this we can draw conclusions about how Bob should defend in position on dry flops. We’ll verify that slowplaying on dry flops is a good strategy for him, which is something we have simply assumed in previous articles. We’ll also draw conclusions about how Alice can vary her bet sizing to increase her EV against an opponent that she knows has a weak postflop range.

What Alice wants is to use the information she has about Bob’s postflop range after he flats preflop (with a medium strong preflop range) and the flop comes dry and uncoordinated (which means it mostly misses Bob’s preflop range). On these flops Alice’s good one pair hands (e.g. her overpairs) can extract lots of value from Bob’s weaker pairs. One way to achieve this is to use big turn and river bets so that her final bet is all-in on the river (as opposed to the standard bet sizing scheme where ~1/2 the stack has been put into the pot after the river bet).

We’ll test this alternate bet sizing scheme for Alice by computing the EV for her barreling the three overpairs AA-QQ on a dry flop against Bob who defends according to the strategies designed for him in Part 1, Part 2, Part 3 and Part 4 of this article series. Alice will of course also barrel other hands on the turn and river, including an optimal number of bluffs, but here we simply want to find the EV for her best overpair hands in a vacuum. They are a part of an overall optimal barreling strategy for her, but we don’t have to know her total strategy in order to find the EV for these hands in isolation. However, we will need Bob’s complete defense strategy in position, in order to find Alice’s EV with AA-QQ against his strategy.

2. Definition of our model scenario
Alice (100 bb) raises her ~15% UTG-range:

Alice’s Default 15% UTG-range

22+
A9s+ AJo+
KTs+ KQo
QTs+
J9s+
T9s
98s
87s
76s
65s

194 combos
15%

Bob (100 bb) flats his standard “IP flat list” on the button:

IP flat list after ~15% UTG openraise

QQ-22
AKs-ATs AKo-AJo
KTs+ KQo
QTs+
JTs
T9s
98s

162 combos
12%

The flop comes:

Alice then begins postflop play by c-betting 0.75 x pot with her entire preflop range on the flop. Bob now calls. We give Bob the option of choosing between always slowplaying and never slowplaying his strong hands on the flop:

Bob’s postflop strategy 1: Bob never slowplays
In this case we can assume that Bob’s flat on the flop eliminates the few possible monster hands (66 or 22) from his range, since he would have raised them for value. We will also assume that Bob would have raised for value with his 3 best overpairs (QQ-TT) as well. Beyond his choice of slowplaying/nor slowplaying his monster hands, Bob’s strategies follow the principles outlined in previous articles. So when he flats a dry flop in a situation where he would have raised all his strong hands, he must have a range of mostly weak one pair hands and overcards. His plan for the turn and river (barring improvement) is to call down optimally, in order to prevent Alice to profitably barrel any two cards as a bluff.

If Alice has a read on Bob as a player who never slowplays the flop, she now knows that his postflop range is weak after the flop call. He can never have anything better than a medium one pair hand, and Alice can use this knowledge to make big turn and river value bets with her good one pair hands, mixed with an optimal number of bluffs. Note that this is something she can do because the flop is dry and because she knows Bob’s range is weak (and likely to stay weak all the way to the river). On a coordinated flop, where Bob’s flatting range would have been stronger (and more likely to improve on many turn and river cards) value betting hard with her good one pair hands would have been much more dangerous for Alice.

Bob’s postflop strategy 2: Bob slowplays until the river
If Bob slowplays the flop, he will also slowplay the turn with his few monster hands to give Alice a chance to lose more money by bluffing the river with her weakest barreling hands. This is a reasonable strategy for Bob, and by slowplaying his strongest hands he also protects the weakest hands in his call-down range. His range is weak overall, and Alice can put pressure on him, but she can’t automatically fire big turn and river bets with her optimal value/bluff range without sometimes getting punished.

If Alice knows that Bob slowplays, there isn’t really a lot she can do with this information, since Bob’s range is still pretty weak. So she should still bet for value with her good one pair hands. But we expect that Bob’s slowplaying will counter the positive effect of Alice alternate bet sizing, where she bets big on the turn and river to get all-in for value with her good (and probably winning) one pair hands (as well as her monster hands, and some bluffs for balance. Whether or not Alice should revert to standard bet sizing against a slowplaying Bob remains to be seen.

Regardless of her turn/river betting scheme, Alice starts out with a 0.75 x pot c-bet on the flop. We’ll then estimate the EV for Alice’s turn/river barreling with her 3 best overpairs AA-QQ, using the Pokerazor analysis software.

The turn is:

Alice will now 2-barrel the turn with AA-QQ after Bob’s flop flat, and Bob calls again, regardless of whether he’s using a slowplay strategy or not (since he will always slowplay to the river, when he slowplays). Bob then uses the theory for optimal postflop play in position, defined in Parts 1-4 in this article series, and he calls with a range designed to make Alice’s weakest 2-barreling hands (i.e. her bluffs) break even). Note that we have chosen a turn card that doesn’t improve Bob, so that we won’t have to think about how the few cases where one of his medium/weak flop flatting hands improves to a value hand on the turn.

Here Alice can use two different bet sizing schemes, and we’ll study her EV for both using Pokerazor.

The river is:

Alice will now 3-barrel the river for value with AA-QQ after Bob’s turn flat. We have let the river card be a card that could have improved Bob. But if Bob doesn’t slowplay, it can’t have improved him to anything better than one pair, since he would have raised TT for value on the flop. So if Bob isn’t slowplaying postflop, he will now have a range of bluffcatchers on the river after flatting the flop and turn. He will defend against Alice’s riverbet by calling down an optimal amount that prevents her from profitably 3-barreling any two cards against him.

In the case that Bob slowplays, he will now raise all-in with all his slowplayed monster hands. If Alice has used the alternate bet sizing scheme, her 3-barrel will put bob all-in, and he will of course call with his monsters. And he will also call with enough bluffcatchers to prevent a profitable any-two-cards bluff from Alice. Bob’s monsters are 66 (1 combo), 22 (1 combo) and TT (3 combos). In the case where he raises all-in for value, he also raises some bluffs for balance.

3. Defining Alice’s two bet sizing schemes

Standard bet sizing

– 0.75 x pot on the flop
– 0.75 x pot on the turn
– 0.60 x pot on the river

Alice and Bob then get to the river with 74 bb left in their stacks, and the pot is 53.5 bb (100 – 74 =26 bb from each of them, plus 1.5 bb from the blinds). Alice then bets 0.60 x 53.5 =32 bb on the river, and Bob calls or shoves all-in to 74 bb. When Bob shoves, Alice gets pot-odds 159.5 : 42 =3.8 : 1 on a call. Since Bob is shoving a balanced range, she is indifferent towards calling or folding with her overpairs (they are now bluffcatchers). Since her EV is the same (0) for calling or folding against Bob’s optimal river raising strategy, we simply choose to let her bet-call the river.

Alternate bet sizing
Alice and Bob have 96.5 bb in their stacks after preflop play, and the pot is 8.5 bb before postflop betting begins. Alice c-bets 0.75 x pot (6.5 bb rounded to the nearest half big blind), and the pot grows to 21.5 bb with 90 bb behind.

Alice now chooses her turn and river bet sizing so that she bets the same fraction of the pot on both streets, and her river bet is all-in. To accomplish this, she bets the same fraction (r) of the pot on both the turn and river so that the final pot becomes 201.5 bb when Bob calls the river.

She begins by betting r times the pot on the turn, and the pot grows to:

flop-pot + 2r x flop-pot =flop-pot x (1 + 2r)

Then she bets r times the pot on the river, and the pot grows to:

turn-pot + 2r x turn-pot
=turn-pot x (1 + 2r)
=flop-pot x (1 + 2r) x (1 + 2r)
=flop-pot x (1 + 2r)^2

The flop pot is 21.5 bb, and we know that the final river pot should be 201.5 bb, so we can write:

21.5(1 + 2r)^2 =201.5
(1 + 2r)^2 =201.5/21.5
(1 + 2r)^2 =9.37

We take the square root on both sides and get:

1 + 2r =3.06
r =(3.06 - 1)/2
r =1.03

We find that Alice should bet 1.03 x pot on both the turn and the river. This puts her all-in on the river, using two bets slightly bigger than pot. Let’s check that this is correct:

Alice bets 1.03 x 21.5 =22 bb on the turn, the pot grows to 21.5 + 2 x 22 =65.5 bb, and both players have 90 – 22 =68 bb behind. Then she bets the remaining 68 bb on the river into the 65.5 bb pot (ratio: 68 : 65.5 =1.04) and gets all-in. So we get very close to the desired bet sizing of 1.03 x pot on both streets.

Before we move on, lets ask: Why does Alice want to use a bet sizing scheme where she bets the same fraction of the pot on the turn and the river, planning to get all-in?

We will not delve into the theory here, but simply accept that this is a reasonable thing to do. Matthew Janda has discussed this in his game theory video series at Cardrunners, and you can also find a more in-depth discussion in the book The Mathematics of Poker (Chen/Ankenman)). If Alice has a range of nuts/air hands (i.e. hands that either always win or always lose at a showdown), and Bob has a range of bluffcatchers (i.e. hands that lose to all of Alice’s value hand and beat all her bluffs), Alice maximizes her EV by betting in such a way that she:

– Gets all-in on the river
– Bets the same fraction of the pot on each street

Alice then bets a balanced ratio of nuts air, so that Bob becomes indifferent towards calling down or folding with his bluffcatchers. If Bob folds too much, Alice’s bluffs become more profitable, and if he calls too much, her value hands become more profitable. When Alice’s value/bluff ratio is optimally balanced, she is guaranteed a minimum profit regardless of what Bob does.

We choose this alternate bet sizing scheme for Alice, since the situation after Bob calls the flop is similar to the nuts/air scenario described above. For example, Alice knows that when Bob calls the flop, and he never slowplays, her overpairs AA-TT have to be ahead on our example flop:

This is because Bob would have:

– Raised AA-KK preflop
– Raised QQ-TT and house/quads on the flop (we assumed this earlier in the article)

Therefore Alice can bomb away with big turn and river bets against Bob’s very weak range, after he has revealed is as such by calling the flop (assuming Alice knows that Bob doesn’t slowplay). It’s easy for her to know which of her hands are value hands (all monsters and her highest overpairs), which hands are bluffcatchers (medium one pair hands), and which hands are air (everything else). She balances her value/bluff ratio according to the postflop strategies we designed for her in Part 5 and Part 6, and we’ll use Pokerazor to show that this alternate bet sizing scheme (0.75x/1.03x/1.03x) yields a higher EV than the standard scheme (0.75x/0.75x/0.60x) when Bob never slowplays

The next step is to build Bob’s postflop strategies on the flop, turn and river. Then we’ll use these strategies as Pokerazor input, and estimate the EV for Alice’s turn/river betting with AA-QQ. If you need to brush up on these strategies, read Parts 1-4.

4. Bob’s postflop strategies as a function of Alice’s bet sizing
Alice’s choice of bet sizing scheme (“standard” or “alternate”) determines the pot-odds Bob is getting on the flop and turn, so his defense strategies will vary with the bet sizing. This means we have to build two sets of postflop strategies for him, one for standard bet sizing and one for alternate bet sizing.

We remember that regardless of Alice’s bet sizing scheme, and regardless of whether or not Bob slowplays, the postflop play goes like this:

– Alice c-bets the flop, Bob calls
– Alice 2-barrels the turn, Bob calls
– Alice 3-barrels the river, Bob calls or shoves

And this is because:

When Bob slowplays, he always slowplays to the river, so he will always call the flop and the turn when he defends. Those times he doesn’t slowplay, the turn and river cards will not improve him to a monster hand, so he will be stuck with a calling range on all streets.

4.1 Bob’s postflop play against standard bet sizing
We begin with Bob’s defense on the flop:

Standard 0.75 x pot c-bet sizing means that Alice is getting pot-odds 1 : 0.75, and she will automatically profit if Bob folds more than 1/(1 + 0.75) =43%. Bob prevents this by defending 100 – 43 =57% of his range on the flop. His preflop flatting range is reduced from 162 to 154 combos on this particular flop:

So Bob needs to defend 0.57 x 154 =88 combos on the flop. In the case where he doesn’t slowplay, we’ll assume he raises 66, 22, QQ-TT =22 combos for value. He balances this with 2 bluff combos, and raises a total of 22 + 22 0 44 combos. Then he needs to flat 88 – 44 =44 combos in order to defend 88 combos in total:

Flop defense against standard bet sizing, without slowplay:

  • Value raise:
    {66,22,QQ-TT} =22 combos
  • Flat:
    {99-77,55-44,AK} =46 combos
  • Bluff raise:
    {KQ,KJs,K JK J} =22 combos
  • Total: 90 combos (Optimal: 88)

If Bob slowplays, he will not use a raising range, and he flats with his ~88 best combos:

Flop defense against standard bet sizing, with slowplay:

  • Value raise:
    None
  • Flat:
    {66,22,QQ-77,55-33,AK,AQ} =90 combos
  • Bluff raise:
    None
  • Total: 90 combos (Optimal: 88)

So when Bob flats the flop, he has a range of marginal one pair hands and overcards ({99,88,77,55,44,AK} =46 combos) when he doesn’t slowplay, and a somewhat stronger range of monsters, marginal one pair hands and overcards ({66,22,QQ-77,55-33,AK,AQ}) =90 combos) when he slowplays. He brings these two ranges with him to the turn:

Alice now bets 0.75 x pot on the turn, and Bob defends 57% like he did on the flop. When he doesn’t slowplay, he has the flop range {99,88,77,55,44,AK} =46 combos, which doesn’t change with this turn card (no card removal effects). When he slowplays, he has the range {66,22,QQ-77,55-33,AK,AQ} =90 combos, which is reduced to 88 combos given this turn card:

When Bob doesn’t slowplay, he has no value raising hands on the turn, and he defends the optimal 57% by flatting 0.57 x 46 =26 combos:

Turn defense against standard bet sizing, without slowplay:

  • Value raise:
    None
    None
  • Flat:
    {99-77,55,4 4,4 4,} =26 combos
  • Bluff raise:
    None
  • Total: 26 combos (Optimal: 26)

When he slowplays, he has some value hands on the turn, but he keeps slowplaying them to the river and he defends the optimal 57% by flatting 0.57 x 88 =50 combos:

Turn defense against standard bet sizing, with slowplay:

  • Value raise:
    None
  • Flat:
    {66,22,QQ-77,55-44} =50 combos
  • Bluff raise:
    None
  • Total: 50 combos (Optimal: 50)

Bob brings the ranges {99-77,55,4 4,4 4,} =26 combos and {66,22,QQ-77,55-44} =50 combos to the river

In the case where Bob isn’t slowplaying, he gets to the river with the range {99-77,55,4 4,4 4} =26 combos which doesn’t change with this river card. In the case where he’s slowplaying, he gets to the river with the range {66,22,QQ-77,55-44} =50 combos, which is reduced to 47 combos:

Alice now bets 0.6 x pot, which gives her pot-odds 1 : 0.6 on a bluff. She has an automatic profit with any two cards if Bob folds more than 0.6/(1 + 0.6) =38%. Bob prevents this by defending 100 – 38 =62% of his river ranges. So Bob defends 0.62 x 26 =16 combos when he hasn’t slowplayed, and 0.62 x 47 =29 combos when he has slowplayed.

In the case where he has slowplayed, Bob gets to the river with the range {99-77,55,4 4,4 4} =26 combos, all of them bluffcatchers. He calls Alice’s river bet with the 16 best combos:

River defense against standard bet sizing, without slowplay:

  • Value raise:
    None
  • Flat:
    {99-88, 7 77 7,7 7,7 7} =16 combos
  • Bluff raise:
    None
  • Total: 16 combos (Optimal: 16)

Bob’s slowplayed range now has value hands he can raise, namely {66,22,TT} =5 combos. The stacks are 74 bb on the river after Alice’s standard 0.75x/0.75x/0.60x betting scheme, and her river bet is 32 bb into a 53 bb pot. Bob then raises his value hands all-in, and the pot grows to 159 bb with 42 bb for Alice to call. Her pot-odds are 159 : 42 =3.8 : 1, and Bob bluffs just enough to make her indifferent towards calling or folding with her bluffcatchers (and all her overpairs are now bluffcatchers).

Bob accomplishes this by raising 1 bluff combo for every 3.8 value combos, which is 1/3.8 =0.26 bluff combos per value combo. So he needs 5 x 0.26 =1.3 bluff combos, which we round to 1. Since it’s the same for Alice whether she calls or folds against an optimally balanced raising range, we’ll simply assume she is bet-calling with all her value hands on the river. When Bob has built his raising range, he does the rest of the defense by adding calls with bluffcatchers until he’s defending 29 combos in total:

River defense against standard bet sizing, with slowplay:

  • Value raise:
    {66,22,TT} =5
  • Flat:
    {QQ-JJ,99,all 88 except 8 8} =23 combos
  • Bluff raise:
    {8 8} =1 combo
  • Total: 29 combos (Optimal: 29)

The next step is to find Bob’s flop, turn and river strategies for the alternate 0.75x/1.03x/1.03x betting scheme.

4.2 Bob’s postflop play against alternate bet sizing
Since the flop c-bet is the same in both the 0.75x/1.03x/1.03x scheme and the 0.75x/0.75x/0.60x scheme, Bob’s flop play is the same in both. So we begin by finding is new turn strategies:

When Bob isn’t slowplaying, he has the range {99,88,77,55,44,AK} =46 combos, which doesn’t change with this turn card. When he is slowplaying, his range is {66,22,QQ-77,55-33,AK,AQ} =90 combos, which is reduced to 88:

Alice now bets 1.03 x pot, and gives herself pot-odds 1 : 1.03. She will have an automatic profit if Bob folds more than 1.03/(1 + 1.03) =51%. Bob prevents this by defending 100 – 51 =49% of his range. So he defends 0.49 x 46 =23 combos when he isn’t slowplaying, and 0.49 x 88 =43 combos when he is slowplaying.

In both cases he defends the turn entirely by flatting, and we get the turn strategies.

Turn defense against alternate bet sizing, without slowplay

  • Value raise:
    None
  • Flat:
    {99-77,5 5,5 5,5 5,5 5,5 5} =23 combos
  • Bluff-raise:
    None
  • Total: 23 combos (Optimal: 23)

Turn defense against alternate bet sizing, with slowplay:

  • Value raise:
    None
  • Flat:
    {66,22,QQ-77,5 5,5 5,5 5,5 5,5 5} =43 combos
  • Bluff raise:
    None
  • Total: 43 combos (Optimal: 43)

Bob brings the ranges {99-77,5 5,5 5,5 5,5 5,5 5} =23 combos and {66,22,QQ-77,5 5,5 5,5 5,5 5,5 5} =43 combos with him to the river:

His two ranges are reduced to 23 and 40 combos, given the river card:

Alice now bets the rest of her stack all-in with a 1.03 x pot river bet, and the pot-odds are identical to the situation on the flop. Bob defends 49% of his ranges, and he has to do this by calling all-in. He calls 0.49 x 23 =12 combos when he isn’t slowplaying, and 0.49 x 40 =20 combos when he is slowplaying:

River defense against alternate bet sizing, without slowplay

  • Value raise:
    None
  • Flat:
    {99-88} =12 combos
  • Bluff raise:
    None
  • Total: 12 combos (Optimal: 12)

River defense against alternate bet sizing, with slowplay:

  • Value raise:
    None
  • Flat:
    {66,22,TT,QQ-JJ,9 9,9 9,9 9} =20 combos
  • Bluff raise:
    None
  • Total: 20 combos (Optimal: 20)

Now we have built Bob’s postflop strategies against Alice’s barreling, and we can plug them into Pokerazor and estimate the EV for Alice’s barreling with the overpairs AA-QQ:

5. EV simulations for Alice’s 3-barreling with the overpairs AA-QQ
In the standard 0.75x/0.75x/0.60x betting scheme, Alice bets AA-QQ for value on the flop, turn and river on this dry board, and then she calls those times raises the river (but since Bob’s river raising range is optimally balanced, it doesn’t matter whether she calls or folds). Bob follows the strategies outlined above. In the alternate 0.75x/1.03x/1.03x betting scheme, Alice bets for value on the flop, turn and river, and gets all-in with the river bet. So Bob’s river defense is done by calling all-in.

Note that we have built Bob’s postflop strategies without taking our knowledge about Alice’s hands into consideration (since Bob can’t know that we’re only looking at AA-QQ in isolation in our model study). For example, we haven’t reduced the number of AK combos in Bob’s ranges to adjust for the fact that many of the aces and kings are in Alice’s range (card removal effects). We accept this as a simplifying approximation.

We now compute the EV for Alice’s turn/river bet-bet line with her overpairs AA-QQ:

5.1 Results from the Pokerazor simulations:
Standard betting scheme, without slowplay
EV (AA) =+44.9 bb
EV (KK) =+44.9 bb
EV (QQ) =+40.7 bb

In the case where Bob raises all his strong hands on the flop, he defends the turn and river with a weak calling range of one pair hands and overcards. Alice’s overpair are basically “the nuts” against Bob’s weak range, and we extract a lot of value by betting the turn and river. Checking the turn or river for pot control is NOT recommended in this scenario, and we’ll see in a minute that we profit even more from “bombing” the turn and river with big value bets, putting ourselves all-in with the final bet.

Note that AA and KK are basically the same hand against Bob’s weak range. The same goes for QQ, but the for QQ differs from the EV for AA/KK because of the card removal effects discussed previously. For example, AA/KK makes it less likely that Bob has AK. We’ll ignore these effects for simplicity.

Bob can reduce Alice’s EV significantly by slowplaying his monsters, as shown by the next set of simulations:

Standard betting scheme, with slowplay
EV (AA) =+34.7 bb
EV (KK) =+33.2 bb
EV (QQ) =+30.0 bb

Bob’s slowplay strategy reduces the EV for Alice’s overpair by 23-26%. This confirms that slowplaying is a much better strategy on this type of dry flop than raising our few monsters on the flop and being stuck with a very weak calling range on later streets. As we’ll see in a minute, Alice’s alternate betting scheme can really punish Bob when he only flats the flop with weak hands. If Alice knows this, she can punish him by overbetting the turn and the river:

Alternate betting scheme, without slowplay
EV (AA) =+55.2 bb (+44.9 bb)
EV (KK) =+55.2 bb (+44.9 bb)
EV (QQ) =+49.1 bb (+40.7 bb)

The EVs for the standard betting scheme is given in parentheses for comparison. The effect is what we expected. When Bob never slowplays, Alice can increase her EV for the turn/river betting by 21-23% relative to the standard betting scheme. She does this by making sure she gets her entire stack in with her overpairs against Bob’s weak range of bluffcatchers. Interestingly, this increase is of the same order of magnitude as the effect of Bob slowplaying in the standard betting scheme (23-26%).

Alternate betting scheme, with slowplay:
EV (AA) =+38.2 bb (+34.7 bb)
EV (KK) =+36.4 bb (+33.2 bb)
EV (QQ) =+30.0 bb (+30.0 bb)

The EVs for the standard betting scheme is given in parentheses for comparison. When Alice uses pot-sized betting on the turn and river, the effect of Bob’s slowplaying is increased. He can now reduce Alice’s EV by 29-39%, relative to not slowplaying. Note that even if Bob slowplays against Alice alternate scheme of big turn and river bets, she still makes more money than from the standard betting scheme. Bob’s slowplaying keeps her profit down, but Bob can’t stop Alice from overbetting profitably.

5.1 Conclusions from our Pokerazor simulations
The best strategy for Bob is to always slowplay dry flops. Below are Alice’s EVs for the standard betting scheme, with and without slowplay:

Standard betting scheme with/without slowplay
EV (AA) =+34.7 bb / +44.9 bb
EV (KK) =+33.2 bb / +44.9 bb
EV (QQ) =+30.0 bb / +40.7 bb

The difference between slowplaying/not slowplaying is 11-12 bb in favor of Bob, when Alice uses the standard bet sizing.

When Alice uses big turn and river bets, it’s even more important for Bob to slowplay:

Alternate betting scheme with/without slowplay
EV (AA) =+38.2 bb / +55.2 bb
EV (KK) =+36.4 bb / +55.2 bb
EV (QQ) =+30.0 bb / +49.1 bb

The difference between slowplaying/not slowplaying is now 16-18 bb in favor of Bob, when Alice maximizes her EV with big turn and river bets.

We conclude:

Bob should always slowplay his monster hands on dry flops, regardless of Alice’s betting scheme. If he chooses to not slowplay, he can get lucky and lose less than maximum, if Alice chooses to use small turn and river bets. But if Alice bets big on the turn and river, Bob will loose significantly by not slowplaying. Since Bob can use slowplaying to keep Alice’s EV down, regardless of her bet sizing, he should always do so. 

Note that our conclusion isn’t necessarily valid on coordinated flops where both players have many draws in their ranges. But on dry and uncoordinated flops, Bob should slowplay.

6. Summary:
We have studied the scenario where the preflop raiser 3-barrels overpairs in a dry flop against a flatter in position. We studied the effects of bet sizing for the preflop raiser, and slowplaying for the flatter.

We concluded that:

  • On dry flop textures where the flatters preflop range has flopped mostly marginal one pair hands and overcards, the raiser can maximize her EV by using big turn and river bets that puts her all-in on the river
  • The flatter should always slowplay in these flops to keep the raiser’s EV down
  • But even if the flatter slowplays, the raiser can profitably overbet the pot on the turn and river, so she should do so

These very dry flop textures give the preflop raiser an opportunity to extract additional EV by putting pressure on the flatter’s weak postflop range with big bets. The flatter can limit the damage by slowplaying, but he can’t eliminate all of the raiser’s advantage from using big bet sizing.

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/optimal-postflop-play-in-nlhe-6-max—part-7-834#sthash.GgJlIV1g.dpuf

Optimal Postflop Play in NLHE 6-max – Part 3

1. Introduction
This is Part 3 of the article series “Optimal Postflop Play in NLHE 6-max” where we’ll study optimal strategies for heads-up postflop play in NLHE 6-max.

In Part 1 and Part 2 we introduced fundamental theory for heads-up flop play in position after flatting preflop. Alice raises from some position, Bob flats in position, and all other players fold. Alice c-bets most flops, and Bob has to defend enough to prevent Alice from c-betting any two cards profitably.

Bob’s response to Alice’s c-bet is to choose:

– A range for value-raising
– A range for flatting
– A range for bluff-raising

And the he folds the rest of his hands. We found that Bob had to defend minimum 57% against a 0.75 x pot c-bet. We also estimated that Bob should use a 1 : 1 ratio of value hands to bluffs when he raises. Our method for estimating Bob’s flop ranges are:

  • 1. Choose a value range (for example, top pair/top kicker or better, plus monster draws). Then we also know how many bluff combos we need (number of bluffs =number of value hands)
  • 2. When the number of value hands/bluffs is counted, we pick enough flatting hands to give us a total defense of 57%. Our flatting hands are chosen from the best hands not strong enough to raise for value (for example, top pair hands weaker than top pair/top kicker, some lower pairs, and non-monster draws).
  • 3. Lastly, we choose our bluff combos from the best hands not strong enough to raise for value or flat (typically the weakest one pair hands, the best overcard hands, and gutshot draws)

In this article we’ll put these principles to work on two different flops:

– A coordinated flop with many draws
– A dry flop without draws

And we’ll place Bob in two different preflop flat scenarios:

– On the button after a CO openraise from Alice
– In the big blind after an SB openraise from Alice

So Bob will defend against Alice’s c-bet on two different flop types, and with two different preflop flatting ranges. This gives us 4 scenarios:

– Bob on the button with a coordinated flop
– Bob on the button with a dry flop
– Bob in the big blind with a coordinated flop
– Bob in the big blind with a dry flop

We’ll work through these scenarios systematically for practice. After reading this article you should be able to do the same type of analysis on your own, so that you can practice optimal heads-up flop play away from the table, using your own standard preflop flatting ranges.

2. Our two practice flops
We go to FlopgGenerator.Com and generate a coordinated (wet) flop and an uncoordinated (dry) flop:

2.1. Coordinated flop

A coordinated flop with two possible straights, and also high cards that will connect with many hands in Bob’s preflop flatting ranges. So we expect this flop to be an easy one to defend.

2.2 Uncoordinated flop flop

A low, rainbow flop that mostly misses Bob’s preflop flatting ranges. There are some possible straight draws, but few of Bob’flatting hands connects with these draws (none, when he has flatted on the button). So we expect this flop to be a tough one to defend enough.

This is also a flop where we have to consider slowplaying the few monster hands in our flop range (basically, our sets) in order to make it harder for Alice to play the turn and river after we flat the flop (since our flop flatting range will be weak on this type of low, dry flop texture). More about this later.

2. Alice’s and Bob’s preflop ranges
We’ll work with two scenarios:

Scenario 1: Alice in CO and Bob on the button
Alice opens her default 25% CO range:

22+
A2s+ A9o+
K9s+ KQo
Q9s+ QTo+
J8s+ JTo
T8s+
97s+
87s
76s
65s

326 combos
25%

Bob flats with his default flatting range outside of the blinds (“IP flat list”) given in the overview below:

Here is a download link for this document (right click and choose “Save as”):
IP_3-bet_summary.doc

With Alice in CO, Bob 3-bets {QQ+,AK} for value, so his flatting range contains 140 combos:

“IP flat list” after a 25% CO openraise:

JJ-22
AQs-ATs AQo-AJo
KQs-KTs KQo
QTs+
JTs
T9s
98s

140 combos

This flatting range is weighted towards high/medium suited and coordinated hands. So it will connect well with high/medium coordinated flop textures and be easy to defend on these flops.

But on low, uncoordinated flop textures it might be difficult for us to defend enough, since we simply don’t have enough strong combos in our range. So we might have to accept that we won’t be able to defend the required minimum 57% on very dry flops. But this is not necessarily a problem for us, since we should be able to defend a bit more than minimum on the coordinated flops. So in the long run, these two factors should even out.

Scenario 2: Alice in the small blind and Bob in the big blind
Alice now opens her 35% button range as default from the small blind:

35% button openrange:

22+
A2s+ A7o+
K2s+ K9o+
Q6s+ Q9o+
J7s+ J9o+
T7s+ T9o+
96s+
86s+
76s
65s

458 combos
35%

We discussed this flatting scenario in detail in Part 7 of the preflop series. Since Bob is the only player left to defend the blinds, he has all of the responsibility of defending the blinds enough to prevent Alice from stealing with any two cards. We found that Bob needs to defend with 37.5% of his hands preflop, and he will use a combination of optimal 3/4/5-bet strategies and flatting.

We assumed Bob would 3-bet {JJ+,AK} for value, together with an optimal amount of 3-bet bluffs, and the rest of the defense was done by flatting. We ended up with the following suggestion for a default flatting range (“Blind vs blind flat list”) for Bob to use in the big blind after an openraise from the small blind:

Blind vs Blind flat list

TT-22
ATs-A6s AJo-A7o
K8s+ K9o+
Q8s+ Q9o+
J7s+ J9o+
T7s+ T8o+
96s+
86s+
75s+
65s

362 combos

This flatting range contains many more low combos that the flatting range we use on the button (“IP flat list”), so it will hit more of the low/dry flops. Since we can hit any flop hard, we have the possibility to credibly represent strength on any flop, and thereby create postflop difficulties for the small blind.

But on the other hand we will now have lots of low hand combos in our range that can’t be used to defend high/medium coordinated flops. Whether or not this will create problems for us on the coordinated flops remains to be seen.

In all scenarios we’ll use the strength principle when designing ranges:

– Raise the best hands for value
– Flat with the next best hands
– Bluff with the best of the weakest hands, and fold the rest

4. Bob’s flop strategies after flatting on the button
We now go through Bob’s flop play systematically. First for the coordinated flop, then for the dry flop:

4.1 Play on the coordinated flop after flatting on the button

“IP flat list” after 25% CO openraise:

JJ-22
AQs-ATs AQo-AJo
KQs-KTs KQo
QTs+
JTs
T9s
98s

140 combos

First we count all remaining combos in Bob’s preflop flatting range, given the cards on the board. ProPokerTool’s count function gives us:

So Bob has 117 combos in his range on the flop. In order to defend a total of 57%, he needs to defend 0.57 x 117 =67 combos in total. We choose his value combos first.

Assume Bob will value-raise all his made hands two pair and better on this coordinated flop (so we let all top pair hand go in the flatting range). Bob with then raise a range made up of two pair (QTs, T9s), sets (TT, 99) and straights (KJs). In addition we let him raise the monster draw combo QJs (top pair + open-ended straight draw). This gives us 17 value combos as shown below:

Bob balances this with 17 bluff-raise combos, but before we choose these we pick his flatting hands. Bob needs 67 – 2 x 17 =33 flatting combos to get to 57% total defense. We pick his flatting hands from the next tier of hands on the equity ladder:

– One pair hands
– Open-ended straight draws

It seems obvious to choose from top pair/top kicker (AQs, AQo), top pair/2nd kicker (KQs, KQo), underpair + open-ender (JJ), middle pair + open-ender (JTs). This gives us the 33 combos we need:

Note how strong the ranges for value-raising and flatting are on this flop. We only raise two pair or better + monster draws for value, and our weakest flatting hand is middle pair + open-ended straight draw.

So we have somewhat of a “luxury problem” on these flops after flatting our tight and solid “IP flat” list on the button. We can pick and choose from some very good hands, and we can easily defend the required 57% by only continuing past the flop with quality hands that have good equity.

The last step of the process is to choose Bob’s 17 bluff combos. We step down to the last rung on the equity ladder and choose hands from the low pairs and weak draws (weak one pair hands, overcard hands, gutshots). Note that some open-ended straight draws are counted as weak draws on this flop, since we have so many better made hand and draws to use.

For example, we can pick KTs (2nd pair + gutshot + overcard), ATs (middle pair + overcard), 98s (3rd pair + gutshot) and AJ (open-ender + overcard). This gives us a few too many bluff combos, so we can drop some of the AJ combos. We end up with the following bluffraising range:

Summary
Bob’s total flop strategy on the coordinated flop Q T 9 after flatting on the button is:

  • Raise for value
    {QTs,T9s,TT,99,KJs,QJs} =17 combos
  • Flat
    {AQs,AQo,KQs,KQo,JJ,JTs} =33 combos
  • Bluffraise
    {KTs,ATs,AJs,A J , A J , A J , A J , A J , A J , A J } =17 combos

Bob then defends 17 + 33 + 17 =67 combos in total, which is exactly 67/117 =57% of his total range on the flop. This is the optimal defense percentage we found in Part 1, and Bob’s flop strategy now makes Alice’s random c-bet bluffs break even. We could have designed Bob’s flop strategy in slightly different ways, but our strategy is very reasonable.

But note that we haven’t bothered to defend more than the optimal 57% here, even if we could have. For example, we let Bob fold some draw combos like A J , and the weak pair + draw combos 987s. We have also used potential flatting hands (the weakest middle pair hands) as bluffs, since we had so many better hands to use for value raising and flatting.

We won’t be able to defend the very dry flops as easily, and we should consider overdefending a bit on the coordinated flops to make up for this. For example, we could have moved ATs up to the flatting range and moved the AJ/98s combos we folded up from the folding range to the bluffing range. We have lots of flexibility on this type of coordinated flop, and if we can easily defend more than 57%, we should consider doing so.

We now move on to Bob’s defense with “IP flat list” on the button when the flop comes low and uncoordinated. We’ll see that this flop texture is much harder to defend sufficiently:

4.2 Play on dry flop after flatting on the button

“IP flat list” after 25% CO openraise:

JJ-22
AQs-ATs AQo-AJo
KQs-KTs KQo
QTs+
JTs
T9s
98s

140 combos

As before we begin by counting the remaining combos in Bob’s preflop flatting range:

The poor match-up between this flop texture and Bob’s preflop flatting range is reflected in the number of remaining combos (130 of the original 140). On the coordinated flop we lost a much bigger chunk of our preflop range (117 of the original 140 remained), since our range connected much harder with that flop.

Our standard procedure is to begin by choosing Bob’s value range, but before we do this we should ask: Should Bob have a value range at all on this extremely dry flop?

There are no draws on this flop, and our only monster hands are 9 set combos (3 of each of 88, 55 and 33). If we decide to raise these for value, together with our best overpairs (e.g. JJ and TT), we’ll have an extremely strong value range, but also an extremely weak and easily readable flop flatting range. The reason is that our flatting range will then be made up of two types of hands: Mediocre one pair hands, and some strong overcards (e.g. AQ).

This makes it easy for Alice to play the turn with her value hands. For example, when she has QQ she can bet confidently for value on basically all turn cards, knowing that the best hand we could have on the flop was a pair lower than her. Remember that we would have raised AA/KK preflop, we would have raised all sets for value on the flop, and there are no two pair hands in our range on this flop.

To avoid this problem we can drop all value/bluff raising on the flop and defend entirely by flatting. Then we put all hands worth playing (sets, one pair, good overcards) into our flatting range. Our flop defense range will still be a bit weak, but now Alice can’t bet safely for value with all of her good one pair hands without risking running into a concealed monster hand. If she does, she will every so often get punished by a slowplayed set.

So let’s design a flop flatting for Bob. We want to defend 57% of our range, so we need to find 0.57 x 130 =74 playable combos. It might be impossible to do so without having to flat some unreasonably weak hands, but we’ll see.

We begin with all sets and one pair hands: {88,55,33,JJ,TT,99,77,66,44,22}. This gives us 51 combos, so sets and pairs do most of the work for us. Then we add the best overcard hands: {AQ,AJ} =32 combos.

This gives us 83 combos, and a bit more than we need. We can now use a bit of good poker sense and drop the 6 22 combos. Note that if we are behind a better pair on the flop, it’s better to have AQ/AJ than 22, since the overcard hands have more outs. So we land on the following defense strategy for Bob on the 8 5 3 flop after flatting on the button preflop:

We defend 3 combos more than we need, but that’s fine.

Summary:
Bob’s total flop strategy on the dry flop 8 5 3 after flatting on the button:

  • Raise for value
    None
  • Flat
    {88,55,33,JJ,TT,99,77,66,44,AQ,AJ} =77 combos
  • Bluffraise
    None

So we managed to defend the minimum 57%, but we had to use overcard hands to get there. Of course we technically don’t have the pot-odds to draw to overcard outs, but keep in mind that our overcards are sometimes ahead of Alice on the flop (she has lots of low card hands in her c-betting range). We should also have a bit of implied odds, since Alice might barrel a lot of turn cards that hit our overcards, assuming they are scare cards for us. So she might choose to bluff the turn if a Q falls to barrel us off our weakest one pair hands. Then she donates implied odds to our top pair with AQ, and sometimes she will bet into our slowplayed sets.

We’ll now go through the two example flops one more time, but now with Bob in the big blind after flatting a preflop steal raise from Alice in the small blind. Bob’s preflop flatting range is now wider, and therefore more difficult to defend.

5. Bob’s flop strategies after flatting in the big blind
We’ll now go through Bob’s flop strategies on the coordinated flop and then on the dry flop after flatting in the big blind after a steal raise from the small blind.

5.1 Play on coordinated flop after flatting in the big blind

Blind vs Blind flat list

TT-22
ATs-A6s AJo-A7o
K8s+ K9o+
Q8s+ Q9o+
J7s+ J9o+
T7s+ T8o+
96s+
86s+
75s+
65s

362 combos

Bob has 294 remaining combos in his range, given this flop:

To defend this preflop flatting range optimally, Bob needs to defend 57% of 294 combos on the flop, which is 0.57 x 294 =168 combos. So compared to playing the button preflop range, we will now have to climb further down on the equity ladder and “promote” some button folding hands to flatting and bluffraising hands in the big blind. Note that this is consistent with the fact that we’re up against a weaker raising range (Alice opens her 35% button range in the small blind, but her 25% CO range in CO). So it makes sense that we can value raise and flat with weaker hands than we could on the button.

We now have much more worthless trash in our range, but on the other hands we also have more two pair combos (wider ranges make more “raggedy” two pair combos postflop), and this helps our defense. Which of these two effects is more significant remains to be seen.

We do as we did on the button and put top pair in the flop flatting range. So we value raise two pair(T9s, T9o, Q9s, Q9o, QTs, QTo), sets (TT,99), and straight straighter (J8s, KJs,KJo). This gives us 53 value combos of strong made hands. Then we can add the best pair + draw combos QJs/QJo (top pair + open-ender), and we end up with a value range of 65 combos:

Now we need 65 bluff combos and and 168 – 2 x 65 =38 flatting combos. We pick the flatting hands first from the next rung on the equity ladder (one pair hands and non-monster draws):

For example:

– The remaining top pair hands: AQs,AQo,KQs,KQo,Q8s
– The best middle pair + gutshot hands: KTs,KTo

This gives us 39 combos as shown below:

So we end up with a situation similar to the one we had on the button. We use a tight value range of only two pair and better plus monster draws, and we have plenty of good hands to use as flatting hands. We also have a wide range of mediocre hands to use as bluffs (weak one pair hands and weak draws).

Again, note that we’re not particularly concerned with how to best play a hand like AT on this flop. We simply use the strength principle together with the requirement of 57% total defense, and then we let the hands fall into reasonable categories. In this example AT ended up in the bluffraising range, but this is not very important for us. What counts the most is that we end up with a solid total defense strategy, and that we have a reasonable system for labeling hands as value hands, flatting hands, bluffraising hands and folding hands.

At any rate, what remains is to choose the 65 bluff combos. We pick hands from the remaining one pair hands and draws. For example.:

– The remaining middle pair hands: ATs,ATo,T8s,T8o,T7s
– Bottom pair + open-ender/gutshot: J9s,J9o,98s
– Underpair + gutshot: 88
– Remaining open-enders: AJ,J7s

This gives us 64 combos (close enough) as shown below:

Summary:
Bob’s total flop strategy on coordinated flop Q T 9 after flatting in the big blind is:

  • Raise for value
    {T9s,T9o,Q9s,Q9o,QTs,QTo,TT,99,J8s,KJs,KJo,QJs,QJo} =65 combos
  • Flat
    {AQs,AQo,KQs,KQo,Q8s,KTs,KTo} =39 combos
  • Bluffraise
    {ATs,ATo,T8s,T8o,T7s,J9s,J9o,98s,88,AJ,J7s} =64 combos

Bob then defends 65 + 39 + 64 =168 combos in total, which is 168/294 =57% of his total flop range. Again we see that it’s easy to design a strategy that defends the minimum requirement 57% when the flop comes medium/high and coordinated. We have more weak hands in our preflop range after flatting in the big blind, but we also flop more value hands (more two pair combos).

Like we did in the button scenario we ended up putting some potential flatting hands in the bluffraising range. We used the strength principle as our starting point, chose a solid value range, and let the rest more or less follow from mathematics.

Our last scenario is the most difficult one, namely defending on a dry flop with a wide and weak preflop flatting range:

5.2 Play on dry flop after flatting in the big blind

Blind vs Blind flat list

TT-22
ATs-A6s AJo-A7o
K8s+ K9o+
Q8s+ Q9o+
J7s+ J9o+
T7s+ T8o+
96s+
86s+
75s+
65s

362 combos

Bob has 337 remaining combos in his range, given this flop:

Again we see that most of Bob’s preflop range is intact on a low and dry flop, since the flop connects poorly with our range. We have a flop range of 337 combos and we have to defend with 57%, which corresponds to 0.57 x 337 =192 combos. We use the same philosophy as before, and choose to defend this low and dry flop with only a flatting range.

As we’ll see in a minute, it’s impossible to get to 57% defense without flatting a very wide range of overcard hands. But we start by counting all our combos of one pair or better, and see what we get:

– Sets: 88,55,33
– One pair: TT,99,77,66,44,22,A8,K8s,Q8s,J8s,T8,98s,87s,86s,75s,65s

We have 93 combos of one pair or better:

So with a theoretical 57% total defense, we have to flat 192 – 93 =99 no pair combos. This means we have to reach far down the overcard hierarchy, and we conclude that:

Defending an extremely low/dry flop optimally with a very wide preflop flatting range might me impossible in practice

So we have to accept lots of folding in this scenario, unless we want to defend with lots of ace high and king high hands. We remember that with a tight/solid “IP flat list” on the button (with only 140 preflop combos) we managed to defend this flop 57% by only flatting sets, one pair, and the best overcard hands AQ/AJ. But with a big blind flatting range we have to play many more overcard hands.

Let’s build an optimal 57% defense range, so that we can see what it looks like. We begin by adding our only decent draw (an open-ender with 76s) and then we add overcard hands. If we flat all ace high/king high combos with minimum a T kicker, we get 193 combos (1 more then the 12 we need):

Here we could also have chosen the gutshot + overcard combos 97s/96s, but this will not make a big difference. The gist of it is that we have to defend a very wide and weak range on the flop, and that more than half our flats are no-pair hands.

Summary
Bob’s total flop strategy on the dry flop 8 5 3 after flatting in the big blind is:

  • Raise for value
    None
  • Flat
    {88,55,33,TT,99,77,66,44,22,A8,K8s,Q8s,J8s,T8,98s,87s,86s,75s,65s,AQ-AT,KQ-KT} =193 combos
  • Bluff raise
    None

6. Summary
We have worked our way through 4 flop scenarios where we tried to defend optimally against a c-bet after flatting preflop. We looked at the following scenarios:

– Coordinated flop with a tight preflop flatting range
– Dry flop with a tight preflop flatting range
– Coordinated flop with a loose preflop flatting range
– Dry flop with a loose preflop flatting range

We saw that defending a coordinated flop is an easy task with both preflop flatting ranges. On dry flops we run into the problem of not having enough one-pair-or-better hand or good draws, so we have to resort to overcard hands to reach 57% total defense. On the driest flops we might have to give up more than optimally, but we might be able to make up for this by defending a bit more than optimally on the coordinated flops.

On the extremely dry flops we chose to defend with only a flatting range to avoid polarizing our flop defense ranges into a very strong raising range and a very weak flatting range. If we choose to defend this way, we slowplay all strong hands by flatting them on the flop, planning to raise for value on later streets.

In the next article in this series we’ll go one step further and discuss play on the turn and river after executing our defense strategies in position on the flop.

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/optimal-postflop-play-in-nlhe-6-max—part-3-808#sthash.m4RmAhOM.dpuf

Optimal Postflop Play in NLHE 6-max – Part 2

In this series we apply game theory principles to heads-up postflop play in singly raised pots. Our goal is to design strategies that are hard to exploit. We focus mainly on flop play, but turn and river play will also be mentioned.

1. Introduction
This is Part 2 of the article series “Optimal Postflop Play in NLHE 6-max” where we’ll study optimal strategies for heads-up postflop play in NLHE 6-max.

In Part 1 we introduced a model and some basic theory for playing the flop heads-up in position after flatting a raise preflop. Alice raises from some position, Bob flats her raise in position, and all other players fold. Alice now c-bets the flop, and Bob has to defend enough to prevent Alice from profitably c-betting any two cards as a bluff.

Bob defends against the c-bet by raising his best hands for value, flatting the c-bet with his next best hands, bluff raising some of the best hands too weak to raise for value or flat, and folding the rest. In Part 1 we found that Bob needs to defend 57% against a 0.75 x pot c-bet to prevent a profitable any-two-card c-bet. We then estimated the optimal value/bluff ratio for Bob’s flop raising range to be 1 : 1. So Bob should raise one bluff combo for every value combo.

Bob’s method for defending against Alice’s c-bets is then:

  • Bob first chooses his value hands. These are the hands Bob raises on the flop with the intent of getting all-in if Alice 3-bets (and keep betting on most turn cards if Alice calls the raise and checks the turn). He now automatically knows how many bluff combos he needs, and the total number of combos he will be raising
  • Then he picks enough flatting hands to bring his total defense percentage (the sum of raising hands and flatting hands) to 57% of his total range on the flop
  • Then he picks his bluffing hands from the best of the remaining hands (the hands not good enough to raise for value or flat). These are the hands Bob raises on the flop, planning to fold to a 3-bet. If Alice calls the raise and checks the turn, he will sometimes keep bluffing on the turn, sometimes bet for value (when he improves), and sometimes check and give up

In Part 1 we worked through a detailed example, where we let Alice raise with our default ~15% UTG range:

Alice’s default ~15% UTG range:

22+
A9s+ AJo+
KTs+ KQo
QTs+
J9s+
T9s
98s
87s
76s
65s

194 combos
15%

Then Bob flatted on the button with the default “IP flat list” defined in Part 2 of the NLHE preflop article series:

Download link for this table (right click and choose “save as”):
IP_3-bet_summary.doc

The flop came:

We estimated the following optimal value, flat and bluff ranges for Bob:

Value raise:

  • Sets: {JJ,99,44} =9 combos
  • Overpairs: {QQ} =6 combos
  • Top pair/top kicker: {AJ} =12 combos
  • Monster draws: {A K , A Q , K Q , Q T } =4 combos
  • Total: 31 combos

Flat:

  • Top pair without top kicker {K J , K J , K J , Q J , Q J , Q J , J T ,J T ,J T } =9 combos
  • 2nd pair: {T 9 , T 9 , T 9 } =3 combos
  • Underpair higher than 2nd pair: {TT} =6 combos
  • Draws: {A T , K T } =2 combos
  • Total: 20 combos

Bluff raise:

{9 8 , 9 8 , 9 8 , Q TQ T , Q T , A K , A K , A K , A K , A K , A K , A K , A Q , A Q , A Q , A Q , A Q , A Q , A Q , A T , K Q , K Q , K Q , K Q , K Q , K Q , K Q , K T } =29 combos

By going through this process many times on many different flop textures, we can train our ability to estimate these ranges quickly at the table (we won’t have the time to find the ranges as precisely as in the example). In Part 1 we focused on the method for estimating Bob’s flop strategy, and how to train this method by repeating it over and over on many flops.

In Part 2 we’ll study this method in more detail, and we’ll discuss some points we only touched on in Part 1. Among other things, we’ll talk about:

  • How Bob’s value range changes as a function of the raiser’s preflop range and the flop texture
  • Play on coordinated flops versus play on dry flops (when should we slowplay?)
  • What should Bob do when Alice checks instead of c-betting?

We’ll illustrate the principles with some simulations done with the analysis softwarePokerazor (which lets us estimate EV for postflop play). So Part 2 will be mostly about theory and modeling. When this work is done, we’ll work through some more examples in Part 2, where we’ll estimate Bob’s flop strategies on two different flop textures (one very coordinated flop, and one very dry flop). We’ll then let Bob use his default “IP flat list” (defined in Part 2 of the preflop series) on the button after a raise from early position, and the “Blind vs blind flat list” (defined in Part 7 of the preflop series) in the big blind after a small blind openraise.

2. Principles for choosing a value range on the flop
When we build a value range for Bob to raise on the flop, we want this to be hands that profit from raising and getting all-in when Alice 3-bets us. For simplicity, assume Alice either 3-bets our raise or folds, and that she never 3-bet bluffs. In this model, our equity has two components:

– What we make when Alice c-bets and folds to our raise
– The all-in equity we have when she c-bets, we raise, she 3-bets, we shove and she calls

It’s obvious that our value hands and our bluff hands make the same when Alice bet-folds (ignoring card removal effects), and the difference between them is the equity from getting all-in against Alice’s value hands.

Choosing value hands that are the favorite against Alice’s presumed all-in range is a good starting point (then both value components are positive), but it’s not an absolute must if Alice bet-folds a lot. It could be that we have a +EV raise with a hands that is a small underdog when getting all-in, if Alice bet-folds so much that the chips we win when she folds outweigh the chips we lose when she 3-bets us and we get all-in. However, for this type of hand we might be better off flatting the c-bet and playing the turn against her total c-betting range. This is analogous to flatting QQ preflop in position against a tight openraiser instead of 3-betting for value, even if 3-betting is +EV in isolation (but flatting is more +EV).

Let’s begin the process of building a value range by making some assumptions:

  • All hands two pair or better are automatic value hands on all flops (unless we elect to slowplay some monster hands)
  • Overpairs and top pair/top kicker can be value hands, but not necessarily. We shall see that this depends on the flop texture (coordinated or dry), the raisers range (tight or loose), and how high our pair is
  • Monster draws can be played for value. The most common monster draws are strong flush draws with extra (flush draws with a straight draw, a pair, or overcards)
  • Made hands top pair without top kicker, and all lower pairs, are candidates for the flatting range

We’ll now do some simple modeling on 4 scenarios to illustrate the effect of Alice’s open range (tight or loose) on two different flop types (coordinated or dry). We will then generalize and draw some conclusions about how to select our value range under different circumstances

2.1 Modeling of a value range
We’ll let Alice openraise two different ranges:

Alice’s default ~15% UTG range:

22+
A9s+ AJo+
KTs+ KQo
QTs+
J9s+
T9s
98s
87s
76s
65s

194 combos
15%

Alice’s default ~25% UCO range:

22+
A2s+ A9o+
K9s+ KQo
Q9s+ QTo+
J8s+ JTo
T8s+
97s+
87s
76s
65s

326 combos
25%

And we’ll use two different flops:

Coordinated flop:

This is the flop we worked with in Part 1. There we used top pair/top kicker as a value hand, and now we’ll check whether or not this was a good choice:

Dry flop:

This is a very dry rainbow flop without strong draws. This limits Bob’s possibilities for defending against the c-bet, since he now doesn’t have any draws to use. On the driest flops it might be impossible for him to defend sufficiently (57%) without flatting very weak hands. We’ll also see that we run into trouble on these flops if we automatically raise the few monster hands we have (mostly sets) on these flops, since this makes our flatting range weak and transparent. The turn can then be difficult to play when Alice knows we flatted the flop with a weak range of mostly one pair hands and overcards. .

Having a medium/weak flatting range is not a problem on a coordinated flop, since we’re flatting many draws with decent potential for improvement. So Alice can’t simply barrel her whole range again on every turn card, just because she suspects our flop flatting range was weak. If she does, she will often bet into our improved hands. But when we flat on a dry flop, the weak hands in our flatting range are unlikely to improve on the turn (since one pair hands have few outs).

If we never slowplay on dry flops, a flat tells Alice the following:

  • Our flatting range on a dry flop consists mostly of marginal one pair hands and overcards
  • These hands rarely improve on the turn

So Alice can safely bet for value with both her monster hands and her good one pair hands, and she can balance her value bets with bluffs and put a lot of pressure on our marginal hands. We’ll talk more about this problem later, but it’s obvious that we can fix this problem (at least partly) by slowplaying on dry flops, thus making our flatting range stronger.

We now define a model for Alice’s and Bob’s postflop play. We define top pair/top kicker as a hand in between the obvious value hands (where two pair or better are always value hands) and the obvious flatting hands (where one pair hands lower than top pair/top kicker are always flatting hands). Note that we have made these choices to get a simple model that is easy to work with. The assumptions will not always be the best for all situations, but they are reasonable.

Now we’ll model how Bob’s top pair/top kicker perform against:

– Alice’s UTG range on the coordinated flop J 9 4
– Alice’s CO range on the coordinated flop J 9 4
– Alice’s UTG range on the dry flop J 3 2
– Alice’s CO range on the dry flop J 3 2

In all scenarios Bob uses his default “IP flat list” for flatting on the button preflop. We’ll use Pokerazor to compute the EV for Bob’s value raise with top pair/top kicker in these 4 scenarios. Before we can do this, we also need some assumptions about how Alice plays on the flop.

  • Alice raises pot (3.5 bb) preflop, Bob flats his “IP flat list” on the button, and all other players fold. Both players start with 100 bb stacks, so the pot is 8.5 bb on the flop with 96.5 bb left in the stacks.
  • Alice c-bets 6.5 bb with all her hands, Bob raises to 17 bb (~1/2 pot) with top pair/top kicker, Alice folds, calls, or 3-bets to 34 bb ((~1/2 pot) and calls a shove
  • Bob shoves all-in if Alice 3-bets his raise. If she calls his raise, both players check the hand down

Alice’s postflop strategy after Bob’s raise is:

  • 3-bet (and call a shove) with top pair/top kicker or better, together with flushdraw + pair, flushdraw + straight draw, and flushdraw + 2 overcards
  • Call the raise with top pair without top kicker, and let the hand be checked to showdown
  • Fold everything else

Note that we’re not trying to design a perfect flop strategy for Alice. We’re simply making some reasonable assumptions that we can model over. What we are interested in is finding out how the EV for Bob’s value raise with top pair/top kicker changes with the flop texture and with Alice’s openrange, given the assumptions we have chosen for the model.

So it’s the EV differences between the various scenarios that are of most interest to us, not the absolute EVs for each separate scenario. The trends we find for Bob’s EV when he raises to pair/top kicker for value will tell us something about how he should select his value range, given Alice’s openrange and the flop texture. Top pair/top kicker is a hand that can be used both as a value hand and as a flatting hand, so a model study for this hand will tell us a lot about where to draw the line between value range and flatting range.

2.2 Model scenario 1: 15% UTG openrange on coordinated flop J 9 4
Bob’s “IP flat-list” after Alice’s 15% UTG raise is {QQ-22,AKs-ATs,AK-AJ,KTs+,KQo,QTs+,JTs,T9s,98s}, where QQ and AK are in the flatting range preflop. The top pair/top kicker hands on this flop are the 12 AJ combos, and we use all of them to get an average EV (they are not 100% equivalent, since some of them have backdoor flush draws).

Alice’s presumed value and flatting hands from her UTG range on this flop are:

  • Top pair/top kicker or better:
    {JJ,99,44,J9s,AA,KK,QQ,AJ}
  • Monster draws:
    {A K , A Q , K Q , K T , Q T , 8 7 }
  • Flatting hands (top pair without top kicker):
    {KJs,QJs,JTs,J9s}

The EV for Bob’s flop raise with his AJ hands on the J 9 4 flop against Alice’s 15% UTG range is:

EV (raise TPTK) =-3.55 bb

Raising top pair/top kicker in this scenario does not quite work for Bob against Alice’s tight openrange, even if the loss is not catastrophic. Bob makes money when Alice folds to his raise, and when she calls with some worse top pair hands, but he loses too much when he gets all-in against her tight value range.

Let’s see what happens against Alice’s CO range:

2.3 Model scenario 2: 25% CO openrange on coordinated flop J 9 4
Bob’s “IP flat-list” after Alice’s 25% CO raise is {JJ-22,AQs-ATs,AQ-AJ,KTs+,KQo,QTs+,JTs,T9s,98s}, where QQ and AK now gets 3-bet for value preflop. Everything else is as in Scenario 1.

Alice’s presumed value and flatting hands for her CO range on this flop are:

  • Top pair/top kicker or better:
    {JJ,99,44,J9s,AA,KK,QQ,AJ}
  • Monster draws:
    {A 4 , A K , A Q , K Q , K T , Q T , T 8 , 8 7 }
  • Flatting hands:
    {KJs,QJs,JTs,J9s,J8s}

Alice has two monster draws more than in Scenario 1 (since she now also opens A4s and T8s), and she now also flats the raise with the top pair hand J8s. These rest of her postflop strategy is identical to Scenario 1.

The EV for Bob’s flop raise with AJ on the J 9 4 flop against Alice’s 25% CO range is:

EV (raise TPTK) =+1.33 bb

Bob’s EV goes from negative to positive. Since Alice uses almost the same 3-betting and flatting ranges as in Scenario 1, most of the difference must come from Alice’s folding. She c-bets a much wider range now (25% vs 15%), but the hands she defends with against the range has not changed much. Bob’s value raise then picks up a lot of pots uncontested, and his EV increases.

Note that this problem for Alice is something we have worked with in the preflop series as well. It’s much easier to defend a tight range correctly against aggression than a wide and weak range. This is true both preflop and postflop.

Now we do the same simulations on the dry flop:

2.4 Model scenario 3: 15% UTG openrange on dry flop J 3 2
Bob’s “IP flat-list” after Alice’s 15% UTG raise is {QQ-22,AKs-ATs,AK-AJ,KTs+,KQo,QTs+,JTs,T9s,98s}, where QQ and AK are in the flatting range. His top pair/top kicker hands on this flop are the 12 AJ combos, and we use all of them to get an average, as before.

Alice’s presumed value and flatting hands from her UTG range on this flop are:

  • Top pair/top kicker or better:
    {AA,KK,QQ,JJ,33,22,AJ}
  • Monster draws:
    None
  • Flatting hands:
    {KJs,QJs,JTs,J9s}

The EV for Bob’s value raise with AJ on the J 3 2 flop against Alice’s 15% UTG range is:

EV (raise TPTK) =-0.96 bb

The EV for value raising top pair/top kicker against Alice’s 15% UTG range is a bit better on the dry flop texture than on the coordinated one, but still negative. Alice must fold a lot, but a tight c-betting range is still easy to defend.

Then we let Alice openraise from CO:

2.4 Model scenario 4: 25% CO openrange on dry flop J 3 2
Bob’s “IP flat-list” after Alice’s 25% CO raise is {JJ-22,AQs-ATs,AQ-AJ,KTs+,KQo,QTs+,JTs,T9s,98s}, where QQ and AK now bet 3-bet for value preflop. Everything else is as before.

Alice’s presumed value/calling hands from her CO range on this flop are identical to the hands she played on the coordinated flop, except one additional flatting hand (since she now also openraises J8s):

  • Top pair/top kicker or better:
    {AA,KK,QQ,JJ,33,22,AJ}
  • Monster draws:
    None
  • Flatting hands:
    {KJs,QJs,JTs,J9s,J8s}

The EV for Bob’s flop raise with his AJ hands on the J 3 2 flop against Alice’s 25% CO range becomes

EV (raise TPTK) =+3.02 bb

Raising TPTK on this dry flop becomes even better when Alice starts with a wide range. This is obvious, since Alice now has to bet-fold a lot in our model (she has no draws to defend with, and only a few value hands). We are now in poor shape when we get 3-bet, but all the fold equity we have makes this a nicely +EV raise.

2.5 Summary of the modeling of EV for value raising flops with top pair/top kicker

– Against Alice’s UTG range on coordinated flop: -3.55 bb
– Against Alice’s CO- range on coordinated flop: +1.33 bb
– Against Alice’s UTG range on dry flop: -0.96 bb
– Against Alice’s CO range on dry flop: +3.02 bb

We can draw some conclusions from this:

1. Raising top pair/top kicker for value can be a slightly losing play
In all model scenarios we were either a small loser or a small winner. This is not totally unexpected, since top pair/top kicker is a good-but-not-great hand somewhere in the region between obvious value hands and obvious flatting hands. We have good equity against all other one pair hands and against draws, but we struggle against better one pair hands, and all hands two pair or better. But overall we did not lose much when raising was -EV.

2. Raising top pair/top kicker goes down in value against a tight openraising range
When Alice starts out with a tight UTG range, she has an easy job defending it against Bob’s flop raises, even if she bet-folds a lot. On both the dry and the coordinated flops her UTG range contained enough value and flatting hands to make Bob’s value raise with top pair/top kicker a slightly losing play, given the assumptions in our model.

But Bob’s raise was +EV on both flops when Alice started with the much looser CO range. Now she had to bet-fold much more, so Bob cashed in on fold equity.

3. Raising top pair/top kicker for value is more profitable on dry flops than on coordinated flops against a raiser that c-bets her entire preflop range on the flop
Note the assumptions used in our model. We have assumed that Alice c-bets her entire preflop raising range on both flops, which isn’t entirely realistic. A good NLHE player will check more of her weak hands on coordinated flops, because she knows that these pots will be hard to win uncontested (she is out of position with a weak hand, and coordinated flops hits her opponent’s preflop flatting range hard). So she will check-fold more weak hands, and her c-betting range will become stronger.

But against a “primitive” player who c-bets too much, and who does not distinguish between flop textures, our model and our conclusions are more valid. We then have much more fold equity on dry flops than on coordinated flops, so the EV for any raising hand increases (and against such a player we can also consider bluff raising a lot more than against a good player).

So we have learned that top pair/top kicker isn’t always a value hand. We have to take the preflop raiser’s c-betting range into consideration, as well as flop texture, before we raise for value, planning to shove all-in after a 3-bet. Furthermore, tight c-betting ranges contain much less “air” than loose ranges, regardless of flop texture. So we have less fold equity when we raise against a tight range, and this reduces the EV for all raising hands.

As a rule of thumb for later postflop modeling, we can assume that it’s fine to draw the line for value raising at top pair/top kicker, regardless of the flop. As a default we will never raise worse hands for value, and we will always raise better hands for value (assuming we don’t want to slowplay). With top pair/top kicker we will sometimes raise for value and sometimes flat. But as we shall see, we won’t necessarily have a value raising range on all flop textures. On the driest flops we might want to slowplay all our strong hands, planning to raise the turn instead.

3. Slowplaying on dry flops
As mentioned previously in this article, we have to take care not to give away too much information about our range when we flat on very dry flops like J 3 2 . For example, let’s say we elect to defend against Alice’s c-bet on this flop by raising all hands top pair/top kicker or better, flatting all lower one pair hands and the best overcard hands (KJ/99/AK/AQ, etc.), and bluffraising some of the better weak hands (KQ, QT, etc).

Some consequences of this flop strategy are:

  • Our value raising range is now easier for Alice to read than when we raise on a coordinated flop (where we can have many draws). This is not a big problem, since we balance our value raises with bluffs, but it will be easy for Alice to see which hands we are representing
  • Our flatting range becomes very transparent. This is our biggest problem when we raise all our best hands on a dry flop and flat with our marginal hands. Now Alice knows that our flatting range contains only marginal one pair hands and some overcard hands, and that we have few outs to improve on the turn.

Let’s study this in more detail on the example flop J 3 2 where we have flatted out default “IP flat list” after a CO raise from Alice. So we begin with the preflop range {JJ-22,AQs-ATs,AQ-AJ,KTs+,KQo,QTs+,JTs,T9s,98s}, and we decide to never slowplay on the flop. Alice c-bets the flop, and we call. We would have raised all hands top pair/top kicker or better on the flop, and Alice knows this based on the reads she has picked up on us. She also knows our preflop flatting range.

Alice can now draw some strong conclusions:

  • We don’t have any overpairs in our flop flatting range (we would have raised AA-QQ preflop).
  • We don’t have top pair/top kicker or sets in our flop flatting range (we would have raised these on the flop)
  • So the best hand we can have after flatting the flop is KJs, and most of our flatting hands are weaker than this

Alice now has easy pickings on the turn. She can continue to bet safely for value with all her made hands top pair/top kicker and better, regardless of the turn card. Of course some turn cards will improve some of our flatting hands, but usually they won’t, and the percentage play for Alice is to keep value betting all her best hands. And on a flop without draws, Alice don’t have to worry about clashing with a flush or a straight on the turn.

Furthermore, Alice can easily take advantage of both scare cards and blanks on the turn to force us to fold our weak one pair hands and overcard hands. Let’s say we’re at the turn with a mix of marginal hands like AK, 99, 88, etc after flatting the flop c-bet. It will be difficult for us to call another big bet on the turn, even if it’s a blank. An in addition, scare cards can come (typically overcards) that will make it extremely difficult for us if Alice bets again.

Problem:
Playing straightforwardly with strong hands on dry flops makes it difficult to play our flop flatting range well on later streets.

Solution:
On dry flops we have to consider slowplaying some (or all) of our best hands, and flat them together with our marginal hands. This serves two purposes:

  • We make our flatting range stronger
  • We make more money from Alice’s bluffs. Sometimes she improves and keeps betting (but now for value). Other times she keeps bluffing, assuming our range is weak after we flatted the flop

Do we need to slowplay on coordinated flops like J 9 4 ? We can if we want to, but our flop flatting range will be strong on many turn cards, since we flat with a lot of draws. So Alice can’t simply barrel away on the turn, assuming our flop flatting range is still weak on the turn.

The thing about coordinated flops is that many of our flatting hands can improve to the nuts or near-nuts on the turn. And since our flop flatting range in practice will cover all turn scare cards, Alice can’t simply bluff at any turn scare card and get away with it.

Also, slowplaying on coordinated flops is more risky with regards to giving Alice a cheap shot at drawing out. Slowplaying top set on a J 9 4 flop is much more risky than on a J 3 2 flop, since Alice will have many flush and straight draws on the first flop.

We’ll talk more about slowplaying on dry flops in Part 3. There we will look at Bob’s flop play on two different flop textures (coordinated and dry) with two different preflop flatting ranges (IP flat list” and “Blind vs blind flat list”). On very dry flops we might not want to raise at all. If this is the case, we do all the flop defense with a flatting range, planning to raise our monster hands for value on a later street.

4. What do we do when Alice checks the flop?
It should now be clear from examples and discussion that Bob’s flop strategy of value raising, bluff raising, flatting and sometimes slowplaying will prevent Alice from profitably c-betting blindly on all flop textures, particularly on the coordinated flops that hit Bob’s preflop flatting range hard. So Alice will have to check some hands, planning to give up. What should Bob do when Alice checks?

It’s obvious that we want to bet our value hands and balance this by buffing with some weak hands, particularly if Alice rarely checkraises in these scenarios. There are always lots of hands without a pair, a draw or showdown value in our range. We should bluff a lot of these hands when Alice has checked, since she has told us that she is probably weak, and it will be difficult for us to win the pot unless we bluff. We can also consider turning our weakest one pair hands into bluffs when checked to, if we believe this will be more profitable than trying to sneak cheaply to showdown.

For example, if Bob has 22 on the flop J 9 4 , it’s fine to bet the hand when checked to, mostly to win it right there, even if his bet isn’t for value (he can’t continue after a checkraise). Betting can be a bad idea when no worse hands call and no better hands fold, but there can be merit to bet to collect dead money, even if we never get action from hands we beat. If Alice is usually weak and will rarely checkraise, betting to protect a hand with weak showdown value can be fine.

But if our marginal hand is strong enough to call a turn bet if we check the flop, and/or if it’s difficult for Alice’s worse hands to draw out on us, we have more reason to check behind. For example with QQ on a A 9 3 rainbow flop. Now we can check the flop, planning to call at least one bet, should Alice come out betting later. And giving Alice a free card will rarely cost us the pot, since she her worse hands can never have many outs against us.

If Alice checks the flop, we can also semibluff with the draws we would have flatted a c-bet with, and we can bet for thin value/protection with some of the better one pair hands that we would have flatted with (typically the top pair hands). In general, we don’t want to give Alice a free card when we have a vulnerable hand, but we have to be cautious about betting marginal hands behind players that also slowplay good hands. Against players that checkraise a lot and give us tough decisions, we should check more of our marginal hands on the flop, and then we play the turn (planning to call a lot of turn bets if we get bet into by an aggressive player).

On the other hand, against a straightforward player who rarely check good hands, we can bet more turns. Either for value (our flop value raising range), as a semibluff (draws not in our value range), or as a thin bet for value/protection (some marginal one pair hands that we would have flatted against a c-bet). And of course we will bet lots of pure bluffs (the bluff raising hands, and some more).

Getting these betting opportunities is a bonus effect of defending strongly against Alice’s c-bets, using our optimal value raise/bluff/raise/flat strategy. Alice sees that we defend more than half the time she c-bets (57% to be exact), so she has to give up any-two-cards bluffing, and she is forced to check many weak hands to us. Note that she can in principle balance her weak checks with some strong checks (slowplaying) but this is difficult to do well. Her strong hands also need to balance her c-betting range, and she only has so many strong hands to use. Balancing both a c-betting range and a checking range is very hard, and against most players you can get away with a lot of betting after they check to you on the flop. And many of your bets should be pure bluffs.

5. Summary
In this article we have talked about choosing our value raising range against a c-bet on the flop, and we have modeled value raising with top pair/top kicker in 4 scenarios:

– Against a tight UTG range on a coordinated flop
– Against a loose CO range on a coordinated flop
– Against a tight UTG range on a dry flop
– Against a loose CCO range on a dry flop

We saw that top pair/top kicker is a hand that can sometimes be used as a value hand (particularly against a loose c-betting range), even if it’s an underdog against the range the c-bettor continues with after we raise. But top pair/top kicker was a marginal value hand at best in our model, and sometimes flatting the c-bet will be a better way to play it, for example against an UTG raiser with a tight range.

Then we discussed slowplaying on dry flops, and the consequences of always value raising our best hands for value on these flops. Our flatting range then becomes transparent and easy to play against on later streets, so we have to consider slowplaying some (or all) of our best hands as well. On extremely dry flops we might not want to raise for value at all, and only defend with a flatting range (more about this in Part 3).

Lastly, we talked about what to do when Alice checks to us instead of c-betting. Our defense strategy against her c-bets forces her to check some hands, and her checking range will usually be weak and easy to play against. This gives us the opportunity to bluff a lot when checked to, particularly against a player that rarely checks strong hands or checkraise bluffs in this situation.

In Part 3 we’ll see how these principles are applied in practice. We’ll pick a coordinated flop and a dry flop, and see how Bob designs his defense against Alice’s c-bets on these flops, taking the positive effects of slowplaying into account on the dry flop. To practice using our default preflop ranges postflop, we’ll let Bob use two different preflop flatting ranges: First the standard “IP flat list” on the button, and then the “Blind vs blind flat list” in a blind vs blind scenario where he flats an openraise made by the small blind.

God luck!
Bugs

– See more at: http://en.donkr.com/Articles/optimal-postflop-play-in-nlhe-6-max—part-2-761#sthash.dzqhKRUQ.dpuf

Optimal Postflop Play in NLHE 6-max – Part 1

1. Introduction
This is Part 1 of the article series “Optimal Postflop Play in NLHE 6-max” where we’ll study optimal strategies for heads-up postflop play in NLHE 6-max.

We’ll base this article series on:

– Principles from game theory
– The preflop article series “Optimal 3-bet/4-bet/5-bet strategies in NLHE 6-max” parts 1 to 7

This postflop series is a follow-up to the preflop series. We use our default preflop ranges and optimal strategies for 3/4/5-betting and flatting, and move on to postflop play.

In the preflop series we mostly looked at preflop play in heads-up scenarios. Alice raises, and it’s folded to Bob, who 3-bets or flats. All other players fold, and we have a heads-up pot between Alice and Bob. Those times Bob 3-bets in position there is no postflop play, since Alice’s defense strategy against 3-bets is to 4-bet or fold (and when she 4-bets, Bob 5-bets or folds). Those times Bob 3-bets out of position, Alice will flat some 3-bets in position, and there is postflop play. And when Bob flats Alice’s raise in position, there will be postflop play as well.

In this article we’ll discuss optimal postflop strategies (only play on the flop) in the following scenario:

– Alice raises
– Bob flats in position
– All other players fold

This results in a heads-up scenario with Alice out of position in a singly raised pot. In most pots the postflop play will begin with Alice c-betting. Bob now needs a flop strategy that prevents Alice from profitably c-bet bluffing with any two cards on the flop. If Alice automatically makes money by c-betting any two cards as a bluff on any flop, Bob is obviously doing something wrong on the flop.

Bob’s response to Alice’s c-bet follows the now well-known strength principle that we have used throughout the article series:

  • Raising the best hands for value
  • Flatting the c-bet with the best hands not strong enough to raise for value
  • Bluff-raising with some of the best hands that aren’t strong enough to raise for value or flat
  • Fold all other hands

And Bob has to make sure he value raises, flats and bluff raises enough to prevent Alice from profitably c-betting any two cards as a bluff. In an optimal postflop strategy, Bob wants to defend just enough to make Alice’s weakest c-bet bluffs break even.

As we shall see in future articles, Bob might have to let Alice get away with profitable any-two-cards bluffing on some flop textures (extremely dry flops that are hard to hit for Bob’s preflop flatting range, and therefore hard for Bob to defend postflop). But he can make up for this on coordinated flops that hits his flatting range hard. The important thing for Bob is that Alice on average should not be allowed to c-bet any two cards profitably on any flop.

In order to design an optimal flop strategy for Bob against Alice’s c-bets, we need to know a few things:

  • How often does Bob have to defend against Alice’s c-bets?
  • Which hands should he raise for value?
  • Which hands should he flat?
  • Which hands should he bluff raise?
  • What should the value/bluff ratio be for Bob’s raising range?
  • Should Bob slowplay some of his strongest hands instead of raising all of them for value on the flop?

We’ll find the answers to these questions using simple mathematics (the pot odds Alice and Bob are getting on their bets and raises) and common poker sense.

The theory for optimal postflop play heads-up after flatting preflop was given a thorough discussion by Cardrunners instructor Matthew Janda in his Stoxpoker video series“Optimal Positional Flop Play” in 3 parts (this series unfortunately became unavailable when Stoxpoker merged with Cardrunners in 2010). This article is based on the principles laid out in this video series, and I’ll show how I have incorporated the theory into my own NLHE game.

We’ll base all postflop situations on the preflop ranges and preflop theory discussed in the article series “Optimal 3-bet/4-bet/5-bet strategies in NLHE 6-max” parts 1 to 7. We will mainly talk about “core strategy” where we use tight preflop ranges that are easy to work with theoretically. This will give us a general knowledge about optimal postflop play, and with a little work we can easily adapt and apply this knowledge to the ranges we’re using in practice.

The work done in this postflop article is less suited for memorization and direct application at the felt. The reason is the vast amount of possible outcomes we have when a preflop range meets a random flop. Therefore, our goal is not to memorize everything, but to design a training method for postflop play.

First we learn all the theoretical principles we need. Then we sit down with pen and paper and train postflop strategies by studying how we should play our preflop range on various flops. For each particular flop, we write out our complete flop strategy. By repeating this process over and over on many different flop textures, patterns will begin to emerge, and the thought processes will become more and more automatic. Through repetition we will slowly build knowledge and feel for how to play on different flop types.

So the purpose of our work is to define the necessary theory, plus design a training method that you can work with on your own. The more you practice postflop play away from the table, the faster you’ll learn, and the better your understanding of optimal postflop play will become. As a bonus, you will get a much better understanding of your own default preflop ranges, and how these interact with flops.

In Part 1 we’ll go through the necessary theory for defending optimally against a c-bet after flatting a raise in position preflop. We choose the scenario where Alice raises her default ~15% UTG range and Bob flats on the button with the default flatting range “IP flat list” used throughout the preflop article series. Then we pick an example flop and discuss how Bob should play his preflop flatting range postflop on that particular flop texture.

We’ll continue this work in Part 2 where we’ll talk about play on coordinated flops versus uncoordinated flops, and how Bob’s postflop strategy is a function of how draw-heavy the flop is (which determines how many hands Bob should slowplay). We will also talk about how Alice’s open-range influences Bob’s postflop strategy (since Alice’s various open-ranges hits different flops in different ways).

2. The necessary theory and mathematics
The scenario we’ll use to illustrate postflop theory throughout this article is defined by the following preflop and postflop models:

2.1 Preflop model
Alice (100 bb) openraises pot (3.5 bb) with her 15% UTG range, defined in the preflop series’ Part 2:

Alice’s ~15% UTG range:

22+
A9s+ AJo+
KTs+ KQo
QTs+
J9s+
T9s
98s
87s
76s
65s

194 combos
15%

Bob (100 bb) flats her raise on the button with his default “IP flat list”, found in the overview over optimal 3/4/5-bet strategy pairs with the raiser out of position, defined in Part 2 of the preflop series:

Download link for this table in document form (right click and choose “save as”):
IP_3-bet_summary.doc

When Alice raises from UTG, Bob flats QQ/AK on the button, so his “IP flat list” for this scenario is:

Bob’s “IP flat list versus ~15% UTG range:

QQ-22
ATs+ AJo+
KTs+ KQo
QTs+
JTs
T9s
98s

162 combos

Both blinds fold, and Alice and Bob sees a flop in a 3.5 bb (Alice’s raise) + 3.5 bb (Bob’s call) + 1.5 bb (the blinds) =8.5 bb heads-up pot. Both players have 96.5 bb remaining stack.

2.2 Postflop model
Next we’ll outline the theory for our estimate of en optimal postflop strategy for Bob. Alice c-bets the flop, and Bob’s response is to raise some hands for value, raise some hands as a bluff, flat with some hands, and fold the rest of his hands.

We want to know:

– How often does Bob have to defend on the flop?
– Which value/bluff ratio should he use for his flop raising range?

We’ll not discuss how Bob should play the turn and river, and we’ll look at flop strategy only. Alice c-bets, and Bob’s response is to raise, flat or fold. We want to know how often Bob has to defend on the flop to prevent Alice from c-bet bluffing any two cards profitably, which hands he should defend with, and how he should play them (raise or flat) on the flop.

How often does Bob have to defend against Alice’s c-bet?
We’ll assume that Alice’s c-bet is to ~3/4 of the pot, which is 6.5 bb into a 8.5 bb pot. Alice risks 6.5 bb to win 8.5 bb, and her pot odds on a c-bet bluff are 8.5 : 6.5. If she wins more than 6.5/(8.5 + 6.5) =43%, her bluffs become automatically profitable.

So we conclude:

Bob needs to defend 100 – 43 =57% of the time against Alice’s c-bets to prevent her from having an automatically profitable bluff with any two cards

Bob defends with a combination of value raising (his best hands), bluff raising (some of the best of his weakest hands), and flatting (good hands that are not good enough to raise for value). Note that when Bob flats, he lets Alice freeroll flops with her bluffs. So in practice, Bob should defend a bit more than 57% overall to compensate for this.

As in the preflop series, “value raising” means raising with the intent of getting all-in when Alice 3-bets or calls. Of course we can change our mind, for example when bad cards come on the turn or river. But as a starting point, Bob’s plan with a value hand is to commit his stack when Alice does not fold to his raise. For example, he can have a made hand strong enough that he will happily get all-in on the flop when 3-bet, or keep betting for value on the turn when called (for example, a set). Or he could have a monster draw (for example, a nut flush draw with two overcards and a gutshot) that can get profitably all-in on the flop when 3-bet, while having profitable semibluffing opportunities on many turn cards when called.

The next question is:

What is the optimal value/bluff ratio for Bob’s flop raising range?
Alice c-bets 6.5 bb into a 8.5 bb pot. We now assume Bob’s flop raise is about 1/2 pot, or 17 bb. And we assume that when Alice 3-bets, she also reraises to 1/2 pot, or 34 bb.

Alice now risks 27.5 bb more (34 bb minus her 6.5 bb c-bet) to win a 32 bb pot (8.5 bb flop pot + Alice’s 6.5 bb c-bet + Bob’s 17 bb raise). Her pot-odds on a 3-bet bluff becomes 32 : 34, and she needs to win 34/(32 + 34) =52% to have an automatically profitable 3-bet bluff. Bob can’t allow this, so he needs to defend 48% against her 3-bets. So 48% of his flop raising range should be for value. We round this to 50% to keep things simple.

We conclude:

50% of Bob’s flop raises should be for value, and 50% should be bluffs

Note that Bob can get away with a bit more bluffing in practice, since Alice sometimes calls his raise and lets him freeroll turn cards. When Alice calls his flop raise, Bob’s bluffing hands get a chance to either bluff profitably on some turn cards (if Alice signals more weakness by checking to him on the turn), or improve to the best hand on the turn. For example, if Bob elects to raise a gutshot straight draw as a bluff on the flop (a typical bluff raising hand, since it’s too weak to raise for value or call for implied odds with 100 bb stacks), he will hit his draw about 10% of the time on the turn those times Alice calls and lets him see a turn card with his bluff.

Having an extra 10% chance to win the pot on the turn does not sound like much, but it will increase the EV of Bob’s bluff raise significantly. So when we raise the flop as a bluff, we never use completely worthless hands, but the best hands among those that are too weak to raise for value or flat. Hands with two overcards, gutshots, and backdoor draws are fine for this purpose. Picking our bluff raising hands from the hands with such bits and pieces of equity also randomizes our bluff raises, in addition to giving us an escape hatch those times our bluff raises get called.

We found the value/bluff ratio to be 50% by calculating how often Bob needs to defend against Alice’s 3-bet. But Alice can also flat his raise. One can use mathematics and assumptions to show (and Matthew Janda did this in his video series) that Bob can increase his bluffing percentage to 60% on flops where he expects Alice to mostly call his raises. But we will keep things simple and use a 50/50 value/bluff ratio on all types of flops. This is easy to remember and relatively easy to apply in practice after a bit of training away from the felt.

3. A training method for learning optimal flop play after flatting in position preflop
We now have a simple theory for Bob to use when defending optimally against Alice’s c-bets after flatting heads-up in position preflop:

  • Bob needs to defend at least 57% against Alice’s c-bet, using a combination of value raising, bluff raising and flatting
  • Bob uses a 50/50 ratio of value hands to bluffs for his flop raising range
  • Bob uses the strength principle to determine which hands goes into which range:
    • Raise the best hands for value
    • Flat the best hands not strong enough to raise for value
    • Bluff raise some of the best hands not strong enough to flat
    • Fold his weakest hands

3.1 A method for training Bob’s flop strategy against Alice’s c-bets after flatting in position preflop
Unlike the preflop strategies defined in our preflop series, we can’t write out our postflop strategies once and for all, since we play our range differently on different flops. In theory we could write out a set of rules for how to play on all possible flops, but in practice we have to limit ourselves to a set of qualitative guidelines. For example, we have already defined one such guide line in the strength principle. So our approach to learning optimal flop play against Alice’s c-bets is:

We’ll train the flop strategy by defining a step-by-step process, and then repeat this process over and over on various randomly generated flops:

Below is our method:

  • Bob begins by counting the number of hand combinations (combos) in his range, given the cards he can see on the board
  • Then he calculates of many combos he needs to defend (57% of all his hands)
  • He picks a value range from his best combos (string made hands and monster draws). Then he also knows how many bluff combos he needs (number of bluff combos =number of value combos), and how many hands he raises in total
  • Then he adds flatting combos from the hands a bit too weak to raise for value (for example, medium strong one pair hands, flush draws, straight draws, god overcard hands) so that he ends up with 57% total defense
  • Finally, he picks his bluff combos from the best hands not good enough to use as flats. Bob then picks hands with various weak equity components like overcards, gutshots, and backdoor draws.

When Bob has gone through this process on a particular flop, he has learned how to play his preflop flatting range on precisely this flop (and only this flop). But this knowledge can be generalized.

For example, if you have worked your way through your flop strategy on the flop Q J 6 , you should have a pretty good idea about how to play any flop with two coordinated high cards, one uncoordinated low card, and a flush draw (for example, K J 4 , J 9 3 , etc).

And similarly, the work on a 8 8 2 flop can be generalized to other very dry flops with one low pair and another low card (for example 5 5 3 , 2 2 7 . etc.).

Therefore, by working through many different flop types, we’ll slowly but surely build knowledge about classes of flop textures and how our preflop flatting range should be played against Alice’s c-bet on these flop texture classes. Then, for each flop texture type, we divide our own range into classes (value raising hands, bluff raising hands, flatting hands). For example, we’ll typically classify hand category “underpair” as calling hands on dry flops (since they will be among our better hands on these flops), but demote underpairs to bluff raising hands or even folding hands on coordinated flops (since we will generally have better hands to use as flatting hands on these flops).

In this article we’ll go through one detailed example of using the training method outlined above. Then we’ll do more work in future articles, and also talk about some important differences between coordinated and uncoordinated flops. But for now we’ll let a flop be a flop, and the only thing we’re interested in right now is to train our understanding of bob’s flop strategy by working through various example flops. So for now we’re following the training method to the letter.

Note that the mystical poker concept “feel”, probably is just this type of understanding built through sheer repetition. You see a situation, and based on similar situations you have found yourself in in the past, you have a pretty good idea about how to play this one.

The flop training we do with the method outlined in this article builds experience, and you will probably notice that your “feel” for heads-up flop play in position improves noticeably after a while, even if you aren’t always able to articulate your thoughts. For example, you could “feel” that raising your gutshot + backdoor flush draw is the best play, and then you execute your strategy. And if you want to, you can always sit down and analyze your play away from the table later, using the principles for optimal flop play you have learned in this article.

3.2 An example of designing an optimal flop strategy for Bob
We go to Flopgenerator.com and generate a random flop:L

Then we work our way through the training method, step by step:

How many combos are there in Bob’s range on the flop
Previously in this article we found that Bob’s “IP flat list” has 162 combos after flatting an UTG raise by Alice:

QQ-22
ATs+ AJo+
KTs+ KQo
QTs+
JTs
T9s
98s

162 combos

This changes a bit when the flop comes, since we now have to take card removal effects into consideration. For example, Bob no longer has four combos of JTs in his range (J T , J T , J T , and J T ) since the J is on the board. Bob now loses one JTs combo (J T ).

It’s not too complicated to do card removal adjustments manually, but the simplest way is to plug Bob’s “IP flat list” into ProPokerTools together with the flop, and then use the “count” function:

Bob’s preflop flatting range is reduced from 162 to 144 combos on this particular flop. Note that the flop cards are listed as “dead cards” in the output, and that the original number of combos in the preflop range (162) is given as “Base count”.

How many combos does Bob need to defend on the flop against Alice’s c-bet?
Bob needs to defend 57% of his range (we’re ignoring the effect of Alice freerolling some flops), which is 0.57 x 144 =82 combos. He does not yet know which combos to play and how to play them, but he knows that he should use one bluff combo for every value combo.

The next step of the process is:

Which hands should Bob raise for value?
Here it’s important to think about which hands we’re raising for value against. In general, we can say that made hands two pair and better are always value hands, together with true monster draws (flush draw + straight draw, flush draw + top pair, nutflush draw + any pair, nutflush draw + two overcards, etc), but it’s not certain that all top pair/overpair hands are strong enough to raise for value

Which top pair/overpair hands that can be played profitably for value (planning to get all-in if Alice plays back at us) depends on:

– Alice’s openrange
– The flop texture

Against a tight openrange, our marginal top pair/overpair hands go down in value, since the raiser will now have a larger percentage of better overpairs in his range than if he had openraised a wide range (for example, there is a bigger probability Alice has AA in her range when she openraises a 15% UTG range than when she openraises a 25% CO range). But against a wide openraising range, our good-but-not-great one pair hands go up in value.

We will not continue this line of thinking in this article, since our goal here is to train a simple method for learning flop play. But in the next article we’ll add some “polish” to our method. We will then begin thinking about Alice’s range, and how this range connects with the flop, when we choose our own value range.

At any rate, on this flop we might elect to choose our value hands like this (and note that we don’t have any two pair hands in our range on this particular flop, since J9s isn’t in our “IP flat list”):

  • Made hands: Sets, overpairs, and top pair/top kicker (TPTK)
  • Monster draws: Nut flush draw + 2 overcards, flush draw + 2 overcards + gutshot, flush draw + open-ended straight draw

Which gives:

  • Sets: {JJ,99,44} =9 combos
  • Overpair: {QQ} =6 combos
  • TPTK: {AJ} =12 combos
  • Monster draws: {A K , A Q , K Q , Q T } =4 combos
  • Total: 31 combos

Bob now knows that he also needs 31 bluff combos to get a 50/50 value/bluff ratio for his raising range. But before he picks his bluffs, he designs a flatting range. These are the hands that have good equity against Alice’s range, but they are not strong enough to raise for value.

Bob’s raising range will contain 31 + 31 =62 combos when he has picked his bluffs. Since he needs to defend 82 combos in total, he only needs to pick 20 flatting combos to defend exactly 57%. In practice we might want to defend a bit more, but let’s start by picking the 20 best flatting candidates and see where that takes us:

Which hands should Bob use for flatting?
Now we use the strength principle and pick our flatting hands from the tier below the value raising hands. Which hands we pick is of course a matter of equity, but our choice is also influenced by how many combos we need to defend optimally. Remember that the purpose of this work is to train a flop strategy for Bob that defends enough (57%) against Alice’s c-bets. This is not the same as squeezing every bit of value out of our range on the flop.

So we focus more on building sound ranges, and less on how we play particular hands. Some hands are clear value hands (sets on any flop), flatting hands (2nd pair with top kicker on a dry flop), or bluffing hands (a gutshot with an overcard on a draw-heavy flop), while other hands are less clear cut (for example 2nd pair with top kicker on a draw-heavy flop). For the in-between hands we don’t worry much about how we play them, as long as our total strategy meets the requirement of minimum 57% total defense. For example, whether we use 2nd pair/top kicker on a draw-heavy flop as a flatting hand or a bluff raising hand does not matter much to us.

Note that various categories of hands (for example, underpairs) can be assigned different “jobs” on different flops. If we have a large value range on a coordinated flop that hits our preflop flatting range hard (like our example flop here), there might not be room for any underpairs in our flatting range, since we have sufficiently many better hands to use. But on a dry flop that misses our preflop flatting range, we might have to call a c-bet with all our underpairs to get to 57% total defense.

This is logical, also from an equity point of view, since underpairs/low pairs have rather poor equity on average on coordinated flops (like 77 on a Q T 6 flop), and we have many hands with better equity to put in our value and flatting ranges. But on the dry flops where we don’t have draws (and neither does the preflop raiser), we will usually operate with a flop flatting range that contains many underpairs (like 77 on a T 5 2 rainbow flop). On dry flop textures pairs lower than top pair has decent equity against the preflop raiser’s range, and we call more c-bets with them.

Let’s start with the following list of flatting candidates:

  • Made hands: Top pair without top kicker, 2nd pair, underpairs higher than 2nd pair
  • Draws: Any flush draw, nut open-ended straight draw

We list the hands we have in these categories, and then we can remove some of the weakest candidates later. Of course we now have to remember which hands we have already used as value hands (for example, the best flush draws):

  • Top pair without top kicker: {K J , K J , K J , Q J , Q J , Q J , J T ,J T ,J T } =9 combos
  • 2nd pair: {T 9 , T 9 , T 99 8 , 9 8 , 9 8 } =6 combos
  • Underpairs higher than 2nd pair: {TT} =6 combos
  • Draws: {A T , K T , Q TQ T , Q T } =5 combos
  • Total: 26 combos

We find a few more combos than we need (20). we can choose to keep this range and over-defend a bit to compensate for Alice freerolling flops when we flat. In that case we raise 31 + 31 =62 combos and flat 26 combos for 88 combos total. The results in a 88/144 =61% total defense against Alice’s c-bet.

Alternatively, we can choose to use exactly 57% defense, and remove the 6 weakest flatting hands to end up with 20. For example, we can remove {9 8 , 9 8 , 9 8 , Q TQ T , Q T } =6 combos. Note that if we remove these potential flatting candidates, it’s obvious to use them as bluff raising candidates (since these per definition should be hands not quite good enough to flat).

We’ll assume that this is our choice, so we end up with the following flop flatting range:

  • Top pair without top kicker: {K J , K J , K J , Q J , Q J , Q J , J T ,J T ,J T } =9 combos
  • 2nd pair: {T 9 , T 9 , T 9 } =3 combos
  • Underpaid higher than 2nd pair: {TT} =6 combos
  • Draws: {A T , K T } =2 combos
  • Total: 20 combos

The last item on our to-do list is to find the 31 bluff raising combos Bob needs to complete his flop strategy:

Which hands should Bob bluff raise?
We need 31 combos, and they should be picked from the hands a little bit weaker than the flatting hands. We already have 6 candidates here, namely the 6 combos of 2nd pair not quite strong enough to flat:

{9 8 , 9 8 , 9 8 , Q TQ T , Q T } =6 combos

So we need 25 more combos. We don’t have more flush or open-ended straight draws to use, so we can turn to the hand category overcards + weak draws. We have some of these, so we’ll find what we need there. If this hand category isn’t large enough, we can always move down to underpairs (88-22) and pick what we need there, but note that we prefer hands with as many outs as possible when we’re bluffing.

For example, a naked underpair like 5 5 has only 2 outs, while the overcard hand A T could have as much as 8 outs(6 pair-outs, some of them clean, backdoor nut flush draw, backdoor straight draw). So A T is a better bluff raising candidate than 5 5 .

Here are some potential candidates of the type overcards + weak draw:

– All remaining AK with backdoor flush draw
– All remaining AQ with backdoor flush draw
– ATs with backdoor nut flush draw
– All remaining KQ with backdoor flush draw
– All remaining KTs with backdoor flush draw

In other words, we bluff raise:

{A K , A K , A K , A K , A K , A K , A K , A Q , A Q , A Q , A Q , A Q , A Q , A Q , A T , K Q , K Q , K Q , K Q , K Q , K Q , K Q , K T } =23 combos

2 less than the 25 overcard bluffs we need. If we want to be very precise we need to pick two more, for example two KQ combos without backdoor flush draws. But in practice we will not be able to design such precise ranges at the table, so we will not bother doing it here. In general, we want to avoid playing hands of the same type (like KQ combos without backdoor flush draws) in different ways like flatting 2 of them and folding the rest). At the table we will think like this: “I raise all AK with backdoor flush draw, all AQ with backdoor flush draw,” etc) and we should be satisfied if we get the value/bluff ratio approximately correct in the heat of battle.

Summary of Bob’s flop strategy
We have now designed a complete defense strategy for Bob against Alice’s c-bets on the flop J 9 4 . This was a lot of work for only one flop, and when the work is done we should generalize our results.

We can start by classifying the flop according to its type, and use it as a reference/template for playing similar flops. We might choose to call it “MMx (two-tone)” to describe it as a flop with two coordinated medium cards (J and 9) plus a blank (4) with a 2-flush (two clubs).

Our strategy on this MMx (two-tone) flop was:

  • Raise overpairs and TPTK for value, together with monster draws
  • Flatting top pair without top kicker, underpairs above 2nd pair, and the best 2nd pair hands, together with the remaining flush draws
  • Bluff raising with the weakest 2nd pair hands, the open-ended straight draws without a flush draw, and the best overcard hands with backdoor flush draws

This generalization, formulated with words and not lists of ranges, is useful to remember. When you are playing, you will only have time to formulate your flop strategy in this manner, but that is good enough for our purposes. By systematically working your way through various flop types like we did here, you will see patterns emerging. The general shapes of your value, bluff, and flatting ranges on different flop textures will become clearer and clearer with practice.

For example, based on our example you now know that on flops similar to J 9 4 two overcards with a backdoor flush draw and/or a gutshot will typically be bluffing hands. The same goes for weak open-ended straight draws (those without flush draws) and some of the marginal one pair hands.

Work through lots of flops, and this type of generalized knowledge will slowly become feel/intuition, or whatever you want to call it. You can generate flops with Flopgenerator.com, or you can pick flops from hands you have played.

When I trained this method, I marked a couple of flatting-heads-up-in-position spots in HoldemManager for every session I played. After the session I worked through the flop strategy to see if I had played my particular hand correctly according to the optimal overall strategy. Acquiring a good feel for positional flop play was surprisingly easy. It’s only a matter of repetition, repetition, repetition.

4. Summary
We have now embarked on a series of articles about optima postflop play. We begun with a study of the scenario where Bob flats Alice’s raise in position, sees a flop heads-up, and has to defend on the flop against Alice’s c-bet

We started by determining how often Bob needs to defend against Alice’s c-bets to prevent her from c-betting profitably with any two cards. Then we defined a method for finding Bob’s ranges for value raising, bluff raising and flatting on the flop. This method is easy to learn, even if the resulting strategies are too complicated to memorize afterward. So I recommend that you memorize the method, use it to train a lot on various flop textures, and then generalize your strategies for different type of flops.

The method we have defined so far is simple, and we have ignored some things. One important factor we have ignored is Alice’s openrange, and how her various openranges hits various flop types. For example, defining all top pair/top kicker hands as value hands on all flops is a decent starting point, but we might have to demote some of them to flatting hands when Alice starts out with a tight openraising range (a tight openrange increases the likelihood of her having a better pair than us).

We will discuss these things in more detail in “Optimal Postflop Play in NLHE 6-max – Part 2”.

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/optimal-postflop-play-in-nlhe-6-max—part-1-750#sthash.vQZoUCMJ.dpuf

Optimal 3-bet/4-bet/5-bet strategies in NLHE 6-max – Part 7

1. Introduction

This is Part 7 in the series Optimal 3-bet/4-bet/5-bet-strategies i NLHE 6-max, and the last theoretical part of the article series (a practical part might come later, and we’ll talk about this at the end of this article). In Part 6 we began testing the strategies laid out in Part 1 to Part 5, and we’ll continue this work in Part 7.

As in Part 6 we’ll use modeling with Pokerazor to estimate EV for our heads-up 3/4/5-bet strategies (including flatting) to confirm that they are fundamentally sound. The work done in Part 6 showed that both the raiser’s and the 3-bettor’s strategies were solid, and that they defended well against an opponent trying to bluff with any two cards. This was what the strategies were designed to do, and we can now be sure that they work the way we want them to.

Part 7 will be about:

  • Flatting in position and comparing the EV for flatting vs 3-betting for value with hands in between the regions of clear value hands and clear flatting hands
  • Adjusting the heads-up 3/4/5-bet theory to blind vs blind scenarios

 

1.1 Introduction to Pokerazor simulations of flatting in position

In Part 7 we’ll discuss the part of the 3-bettors strategy that comes in addition to 3-betting. When Alice has openraised and Bob has position, he will 3-bet a range of hands according to his part of an optimal strategy pair, and in addition he will flat some range of hands he thinks can be played profitably. In Part 2 we defined the following default flatting range for Bob in position:

IP flat list

ATs+ AJo+
KTs+ KQo
QTs+
JTs
T9s
98s

Without {KK+}: 162 combos
Without {QQ+}: 156 combos
Without {QQ+,AK}: 140 combos
Without {JJ+,AK}: 134 combos

And we remember that the number of combos in the flatting range depends on how wide of a range Bob 3-bets for value. Against a ~15% openraise from Alice, Bob’s value range is only {KK+} (plus 7 combos of Axs that he 3-bets as a bluff, planning to 5-bet bluff if Alice 4-bets). So QQ/AK are put in the flatting range, and Bob now has 162 combos that he flats. Against Alice’s ~25% CO raising range, Bob 3-bets {QQ+,AK} for value (plus 12 combos of Axs 5-bet bluffs), so he has 140 flatting combos in this scenario.

All of this is summarized in the overview over optimal strategy pairs that we made in Part 2:

Below is a link for downloading this document (right click and choose “save as”):
IP_3-bet_summary.doc

In a similar way, when Alice has position after she has openraised and Bob has 3-bet her from the blinds, she will respond to the 3-bet with a mix of optimal 4-betting and flatting. In Part 3 we defined the following standard defense range for Alice after she has opened her default 35% openrange and Bob has 3-bet from the blinds:

– 4-bet: {QQ+,AK} for value + {ATo,A9s-A7s} as a bluff
– Flat JJ-88,AQ-AJ,ATs,KQ-KJ,KTs,QJ,QTs,JTs[/pre]

Bob’s total defense strategy from the blinds heads-up against Alice’s button openraise is summarized here:

Download link (right click and choose “save as”):blind_defense_vs_button_summary.doc

In this article we’ll focus mostly on Bob’s flatting in position after an openraise by Alice. We’ll use the standard positional flatting range as our starting point, and then study how the best flatting candidates (for example QQ and AK) move between the value range and the flatting range when Alice’s openrange varies. To illustrate this we’ll use Pokerazor to estimate the EV of using QQ as a value 3-betting hand and as a flatting hand. This will give us insight into the best way of playing QQ preflop, as a function of Alice’s openrange.

We shall see that some hands can be played profitably both ways. For example, QQ is +EV against Alice’s ~15% EP openrange both when we 3-bet it for value and when we flat. So when we decide to flat QQ as our default play in this scenario, it’s because we assume that flatting is more profitable than value 3-betting. we’ll use modeling and Pokerazor simulations to show this.

We’ll also discuss adjusting our 3-betting/flatting ranges when we play against weak players that flat lots of medium strong hands out of position after a 3-bet. Bob’s optimal 3-betting strategy in position is based on the assumption that Alice either 4-bets or folds out of position, so he polarizes his 3-betting range into premium hands (for example, {QQ+,AK} and bluffs (for example K9s), and then he flats some hands in the region between his value 3-betting hands and his 3-bet bluffs (for example, AQ). But when the raiser flats a lot of 3-bets out of position, it might be better for Bob to move his best flatting hands up to the value range. For example, Bob might elect to 3-bet all pairs AA-JJ for value, together with AK-AQ. The reason for this adjustment is that JJ and AQ should do well as value hands against a player that flats JJ-99, AQ-AT, KQ-KT, QJ and similar hands out of position (so we’re raising for value mainly against the range that calls us and not necessarily against the range that 4-bets us).

The last scenario we’ll model is flatting versus value 4betting with JJ after Alice has openraised JJ on the button and gotten 3-bet by Bob in the blinds. This choice was discussed in Part 3, and we talked about the consequences of putting JJ in the value range versus flatting the 3-bet with it. Pokerazor will give us an estimate of the best way to play JJ against a 3-bet in this scenario.

1.2 Introduction to the scenario “blind vs blind”

The last topic of Part 7 is a look at the heads-up scenario blind vs blind. Small blind openraises and big blind defends by 3-betting or flatting. We then get two possible scenarios:

– A heads-up 3/4/5-war preflop
– Postflop play in a raised pot with the big blind in position

First we’ll look at how mathematics (the pot-odds small blind is getting on a steal raise) dictates how often big blind needs to defend preflop. Then we’ll use the theory for heads-up 3/4/5-betting from previous articles as a starting point, and then adjust it to the blind vs blind scenario. We know this theory well by now, and this work will be straightforward application of familiar concepts.

2. Pokerazor simulations of flatting in position

We’ll estimate EV for 3-scenarios using Pokerazor and simple modeling:

  • Flatting versus 3-betting with QQ heads-up with position on a ~15% UTG raiser who defends against 3-bets by 4-betting or folding
  • Flatting versus 3-betting with QQ heads-up with position on a ~15% UTG-raiser who defends against 3-bets by 4-betting, calling or folding
  • Flatting of 3-bet vs 4-betting with JJ heads-up on the button against a 3-bet from the blinds (where the 3-bettor is using our default blind defense strategy outlined in Part 3)

In the first two model studies we’ll justify our choice of flatting QQ (and similarly, AK) in position against a tight ~15% openrange (a typical tight-aggressive UTG range in 6-max play) where the raiser defends against 3-bets by 4-betting or folding. In previous articles we noted that QQ is not strong enough to be a favorite against the ~15% raiser’s optimal value range {QQ+,AK}, so we’d rather flat QQ and play it postflop with position on his total openraising range.

But if the raiser defends against 3-bets by also flatting some medium strong hands (e.g. JJ-99, AQ, AJ, KQ) out of position, we’ll see that the EV for value 3-betting QQ increases. The reason is obviously that we can extract more value from all of these medium strong flatting hands by playing against them postflop instead of winning the pot against them preflop (since the raiser will mostly fold them to our 3-bet if he follows our optimal strategy out of position).

The last model study we’ll do is comparing the EV for flatting vs value 4-betting when we have openraised JJ on the button and have gotten 3-bet from the blinds. In Part 3 we designed a default defense strategy for this scenario, and we used JJ as a flatting hand. But we commented that JJ would also work as a value 4-betting hand, and we defined an alternative defense strategy with JJ in our 4-bet value range. Here we’ll use modeling to determine what works best (and we’re guessing right now that it’s a close decision).

2.1 A simple model for estimating preflop + postflop EV

When we flat hands preflop we are setting ourselves up for seeing a flop and then playing postflop. Postflop strategies are impossible to write out in full detail, since we have to take into consideration all possible combinations of flops, turn cards and river cards. We’ll simplify things by using the following simple model:

  • Both players start with 100 bb stacks
  • The raise and the 3-bet are pot-sized
  • A 4-bet is to 27 bb (a little less than pot-sized)
  • A 5-bet is all-in
  • We specify preflop ranges and preflop strategies exactly, based on our default openranges and optimal 3/4/5-bet strategy pairs from previous articles
  • We assume that all other players fold preflop
  • Those times we have to play postflop, we assume both players check to showdown

This is a very simplified model that won’t give us precise estimates for specific EV-values. But what we want is to compare EVs for various scenarios. If we can assume that the model gives about the same error for all scenarios, we can assume that EV differences can give us useful information. For example, of the model tells us that flatting has higher EV than 3-betting for value, we shall assume that this is the case, even if we can’t determine the individual EVs for flatting or 3-betting accurately.

In addition to the numbers we crunch out using this model, we will use logic and sound poker sense where we can. For example, we can assume that if flatting sets you up for playing postflop with a hand that is the favorite against the raiser’s total range (e.g. QQ against a ~15% UTG range), you should be able to extract some EV postflop (e.g. when you flop an overpair or a set with QQ).

2.2 Flatting versus 3-betting with QQ heads-up in position against a ~15% UTG-range

We let Alice openraise from UTG with out default ~15% UTG range:

22+
A9s+ AJo+
KTs+ KQo
QTs+
J9s+
T9s
98s
87s
76s
65s

194 combos
15%

Scenario 1: Alice 4-bets or folds against a 3-bet
First we let Alice use the optimal 3/4/5-bet strategy corresponding to a 15% UTG range out of position. From the overview over optimal 3/4/5-bet strategy pairs with the raiser out of position (see document presented earlier in this article), we see that Alice then 4-bets {QQ+,AK} for value, 4-bets {AQ,AJs-ATs} as bluffs, and folds everything else.

We now use Pokerazor to calculate the EV for flatting and the EV for 3-betting QQ for value. We get:

EV (flat) =+2.49 bb
EV (3-bet) =+3.11 bb

We remember that there is 1.5 bb dead money in the pot from the blinds (we assume they always fold). So by flatting, we on average pocket the blinds plus 2.49 – 1.50 =0.99 bb from UTG’s stack. This is intuitively obvious, since we play postflop with a hand that is 70% favorite against Alice’s total range:

When we 3-bet QQ, we make 3.11 bb in total. 1.5 bb from the blinds and 3.11 – 1.50 =+1.61 bb from UTG’s stack. This is +0.62 bb relative to flatting, so 3-betting is more profitable than flatting when we ignore postflop betting (remember, both players are checking to showdown when they see a flop).

But here we should add a manual adjustment based on poker sense. There are two arguments for flatting being better than 3-betting in practice when the model EV difference is as small as here:

  • We should be able to make more than +0.62 bb postflop with QQ after flatting versus UTG’s total 15% range
  • The blinds will sometimes get involved with ranges we have very good equity against

So the +0.62 bb head start that 3-betting has over flatting in our model should be easy to overcome in practice, since we can extract value from postflop betting. For example, we will often flop an overpair (or a set) on an uncoordinated flop where Alice elects to continuation-bet her whole preflop raising range into us on the flop. And we’re a big favorite against this range.

We therefore conclude:

Flatting QQ in position against a tight ~15% openraising range should be more profitable than 3-betting and planning to 5-bet all-in after a 4-bet

Of course, this is against Alice’s optimal defense strategy against 3-bets where she 4-bets a value range + some 4-bet bluffs, and folds everything else (including most hands we beat). But what if we let Alice deviate from optimal play, and tell her to flat the 3-bet with various medium strong hands that are forbidden to play out of position in the optimal strategy?_

Scenario 2: Alice 4-bets, calls or folds against a 3-bet
This defense strategy is common among weak players, and you will see lots of flatting with weak hands out of position at loose-passive tables. There you will often see the raiser flatting 3-bets heads-up and out of position with decent aces, medium/low pocket pairs, and various suited/coordinated high/medium cards. Let’s give Alice permission to flat 3-bets with the following range out of position: {JJ-99,KQ,KJs,QJs,JTs}.

Note that the only change we make relative to the optimal strategy is to allow Alice to flat some hands in addition to 4-betting optimally. So she will still have 4-bet bluffs in her 4-betting range. This is not totally “in character” for a loose-passive player, but we keep the 4-bet bluffs in our strategy to make it simple to study the effect of flatting (since adding a flatting range is the only change we make).

So Alice’s new defense strategy against 3-bets heads-up and out of position becomes:

– 4-bet {QQ+,AK} for value and {AQ,AJs-ATs} as bluffs
– Flat {JJ-99,KQ,KJs,QJs,JTs}

Pokerazor gives us the EV for playing QQ in position behind Alice’s new loose-passive strategy:

EV (flat) =+2.49 bb
EV (3-bet) =+3.67 bb

EV for flatting QQ against Alice’s range is of course the same as before. But when Alice introduces some medium strength flatting hands to her defense strategy, the EV for 3-betting increases (from +3.11 bb to +3.67 bb).

The difference between flatting and 3-betting for value was +0.62 bb in favor of 3-betting when Alice used the optimal strategy. When she adds a flatting range, the difference increases to +1.18 bb in favor of 3-betting.

In the previous simulation we used some qualitative arguments to conclude that flatting should be able to “catch up” to 3-betting in practice because of postflop betting. We can probably conclude the same thing here, since the difference still is only ~1 bb.

But the modeling we have done with a flatting range for Alice is of course not a realistic model for a loose-passive player. So we will not draw strong conclusions about the profitability of flatting versus 3-betting. Instead, we conclude that:

Against a raiser who defends against 3-betting partly by flatting a range of medium strange hands out of position, the EV of 3-betting QQ for value increases relative to the same raiser not flatting.

Then we’ll have to use judgment to decide whether the raiser is flatting sufficiently many/sufficiently weak hands to make 3-betting better than flatting in practice when we have a hand that we flat as a default. As a final simulation, let’s give Alice an extremely loose-passive strategy with lots of flatting and no 4-bet bluffing:

– 4-bet {QQ+,AK} for value
– Flat {JJ-22,AQ-AJ,ATs,KQ,KJs-KTs,QJs,JTs,T9s,98s}

We get:

EV (flat) =+2.49 bb
EV (3-bet) =+2.66 bb

3-betting still makes more than flatting, but the difference is less than when Alice defended optimally. So giving Alice an extremely wide flatting range is seemingly not an argument for value 3-betting QQ. Or?

This is an interesting result, and we’ll look into it more closely. One thing that has happened here is that Alice has stopped 4-bet bluffing. And a significant chunk of our EV for value 3-betting comes from 5-betting and forcing Alice to fold her 4-bet bluffs after putting 27 bb into the pot. When Alice stops 4-bet bluffing, our value 3-bet with QQ sets us up for getting all-in against Alice’s value range {QQ+,AK}. Against this range our QQ is a 40% underdog, so we’re losing chips as of the moment when Alice 4-bets her value range and forces us to get the rest of the stack in as an underdog (without getting any compensation from picking up the pot against her 4-bet bluffs):

This means we have to be cautious when we 3-bet loose-passive players aggressively with “thin” value hands. We expect to make a lot of money from their folding or flatting against our 4-bet, but when they 4-bet, it might be best for us to fold our hand, even if it started out as a value 3-betting hand. Always 3-betting {QQ+,AK} for value and getting all-in when 4-bet is a fine standard line to take, but what if we have elected to 3-bet JJ for value against a loose player who flats a lot of 3-bets with a weak range out of position?

We can test this by repeating the last simulation, but this time we have JJ. Alice 4-bets {QQ+,AK} as before (no 4-bet bluffs) and flats this wide range {JJ-22,AQ-AJ,ATs,KQ,KJs-KTs,QJs,JTs,T9s,98s} against our 3-bet. Let’s first play JJ as a value hand and 5-bet it all-in against a 4-bet:

EV (flat) =+2.04 bb
EV (3-bet) =-4.39 bb

Ouch! Playing JJ as a value hand against a loose-passive player that never 4-bet bluffs makes our 3-bet a losing play, even if he flats out 3-bet with a wide and weak range preflop. We now get all-in against a range {QQ+,AK} that does not have a single hand we’re a big favorite against. So if the choice is between 3-betting JJ for thin value (planning to get all-in if 4-bet) and flatting, flatting is clearly best.

But there is a third option we can choose against a loose-passive player who flats a lot of 3-bets out of position, 4-bets a tight value range, and never 4-bet bluffs. We can simply 3-bet for thin value against his flatting range, but fold those (few) times he 4-bets us with his strong value range!.

We now 3-bet JJ for value (against the hands that call us), but fold to a 4-bet. We get:

EV (flat) =+2.04 bb
EV (3-bet and fold to 4-bet) =+2.52 bb

Bingo! 3-betting is now +EV, and more profitable than flatting (in our model). Next we can use judgment to determine whether flatting or value 3-betting/folding to a 4-bet is best in practice. 3-betting is probably our best option, since a hand like JJ is easier to play postflop heads-up than in a multiway pot (a 3-bet will probably isolate the raiser while flatting will often pull inn more players). The same argument can of course be used for QQ, but there are more good flops for QQ than for JJ.

Note that in the optimal strategies we have used throughout this article series, the term “3-betting for value” has been equivalent to 3-betting with the plan of 5-betting all-in against a 4-bet. But against a player who flats extremely loose against a 3-bet, but only 4-bets his strongest hands (and no bluffs), we might be better off 3-betting for value against his calling range, but folding those few times he has a strong value hand and 4-bets us. This is obviously an exploitative line that we should only use with reads. This type of player is fairly common in soft low limit games, so pay attention!

We saw in a previous simulation that QQ is too strong to fold against a tight 4-betting range, but with hands like JJ, TT, and perhaps also AQ (that blocks AA, QQ and AK in Villain’s value range) a 3-bet-for-value-but-fold-to-4-bet line could be the best line. We then deviate from optimal 3/4/5-betting to exploit the tendencies of a known loose-passive player.

Note that 3-betting for thin value in this way also as a bonus effect: We will usually isolate the raiser and get to play him heads-up postflop (since he calls far more often than he 4-bets). Isolating is good for us with hands like JJ, TTT and AQ, since they are more difficult to play in multiway pots (which we will often get when we flat) than QQ is.

Summary of modeling of flatting QQ in position
Below are some of the things we have learned from this series if simulations of playing QQ in position versus a ~15% UTG openraise:

  • Flatting QQ is probably more profitable than 3-betting for value when the raiser has a tight ~15% openrange that she defends optimally against 3-bets
  • 3-betting QQ increases in value relative to flatting when the raiser adds a flatting range of medium strong hands to her defense against 3-bets
  • But 3-betting for value with QQ against a ~15% UTG range is not necessarily better against a loose-passive player who flats a lot against 3-bets, but never 4-bet bluffs
  • Against that type of player we can 3-bet for thin value (against the range that flats us) with QQ and probably also some weaker hands like JJ, TT and AQ, but we might have to do exploitative folding against 4-bets to make this profitable (at least with JJ/TT/AQ)

Remember that all simulations done here are done with a tight openraising range. This is a range that is easy to defend correctly against 3-bets, so 3-betting with QQ (and probably also AK) is not a big earner for us. Therefore, let’s do one last simulation where we have QQ against Alice with a ~25% CO openraising range that she defends optimally:

Flatting vs 3-betting with QQ against optimally defended ~25% CO openrange

EV (flat) =+2.65 bb
EV (3-bet) =+9.28 bb

EV for flatting increases a little bit from +2.49 bb to +2.65 bb compared to flatting against a 15% UTG range. The EV for 3-betting makes a big jump from +3.11 bb to +9.28 bb. The reason is obviously that Alice’s value range now includes some hands (JJ, TT and AQ) that are crushed by our QQ. We conclude that against a loose openraising range it’s obligatory to 3-bet QQ for value.

Then we are done with our discussion of flatting versus 3-betting for value with QQ against an openraiser. Next we’ll do a model study of flatting vs 4-betting for value with JJ after openraising on the button and getting 3-bet by a player in the blinds.

2.3 Flatting versus 4-betting with JJ heads-up against a 3-bet from the blinds

The scenario is:

– We openraise JJ on the button
– Bob 3-bets us from the blinds
– We flat the 3-bet, or we 4-bet for value (calling a 5-bet)

We remember from Part 3 that Bob’s 3-betting range from the blinds against a button steal raise is:

– {TT+,AQ+} for value
– {66-22,A9s-A6s,K9s-K8s,QTs-Q9s,J9s-J8s,T9s-T8s,98s-97s,87s,76s,65s} as bluffs

In this range the weakest value hands TT/AQ effectively work as 5-bet bluffs, and Bob does not use dedicated 5-bet bluffing hands (like the Axs 5-bet bluff hands he uses in position).

We now use the same preflop and postflop models as previously. We either flat JJ and let the hand get checked down, or we 4-bet for value and call a shove.

Pokerazor gives us the EVs for flatting and value 4-betting:

EV (flat) =+4.32
EV (4-bet) =+5.91 bb

As we guessed in Part 3 both alternatives are nicely profitable for us, and the difference between them isn’t large. So we can play JJ both ways. Note that even if 4-betting has a head start of 5.91 – 4.32 =+1.59 bb relative to flatting, we might be able to catch up because of postflop betting.

In this spot we can use reads to help us decide. If you think Bob will make big postflop mistakes if you let him see a flop with all of his 3-bet bluffs, flatting could be better for you than 4-betting and making the rest of the hand automatic (Bob will shove his value hands and fold his bluffs, so he has no decisions to make, and neither have you). But if you think Bob will be able to outplay you postflop (or at least give you some tough decisions), just make it simple for yourself and 4-bet to end the decision making process right there.

Since we have begun looking at flatting of 3-bets on the button, let’s do the same simulation for AQ. This is a hand we immediately can see is a profitable flatting hand against Bob’s range, this his total 3-betting range is full of hands (the 3-bet bluffs) that we are a favorite against. But using AQ as a value 4-betting hand is probably too thin:

EV for flatting vs value 4-betting with AQ:

EV (flat) =+0.63
EV (4-bet) =-0.84 bb

We have an easy conclusion:

Value 4-betting AQ against the optimal blind defense strategy is not profitable. Flatting the 3-bet is marginally profitable in our model. In practice, flatting should be more profitable than in the model, since we’re playing postflop with position against a range we’re a small favorite against:

So we should have more opportunities to outplay Bob postflop than he has to to outplay us, and our postflop EV should be positive if we play well. But note that this requires more than a simple fit-or-fold strategy postflop. Calling the 3-bet with AQ and then folding to Bob’s c-bet on all flops where we don’t have a pair or a good draw will not be a good strategy for us. We have to be prepared to do things like floating without a pair or draw, or raising all-in on the flop as a semibluff. Not every time of course, but on some flops (and we’ll use flop texture and our knowledge about Bob’s range to determine which flops).

3. Blind versus blind

Our last topic in this article is the blind vs blind scenario:

– It’s folded to the small blind, who openraises
– The big blind 3-bets, flats, or folds

This is a heads-up scenario with the raiser (Alice) heads-up and out of position against an opponent (Bob) who defends by optimal 3/4/5-betting or flatting, so we can use the theory from previous articles. The only difference is that when Alice and Bob are in the blinds, the bet sizing changes a little. For example. if Alice openraises pot from the small blind, she raises to 3 bb and not 3.5 bb. Similarly, Bob’s pot-sized 3-bets become 9 bb and not 12 bb. This changes the value/bluff ratios in the 3-bet and 4-bet ranges somewhat.

We can make things simple by assuming that the optimal ranges we designed with both players outside of the blinds will work in the blind vs blind scenario as well. But this is a good opportunity to repeat the mathematics and the method for constructing optimal strategy pairs, so we’ll build them from the ground up.

Those of you that don’t want to memorize more ranges can use the previous strategy pairs. You then use the overview document and use Alice’s openraise percentage to pick a corresponding optimal strategy pair:

3.1 Bob’s optimal defense percentage

We begin with the fundamental principle of defense:

Bob has to defend enough to prevent Alice from stealing profitably with any two cards

Then we use some additional assumptions:

– Alice openraises her default 35% button range from the small blind
– Bob 3-bets his best hands for value
– He flats with he best hands not good enough to 3-bet for value
– Han 3-bet bluffs with the best hands not god enough to flat
– And then he 3-bets some Axs hands, planning to 5-bet bluff against a 4-bet
– Alice 4-bets or folds against a 3-bet

Our assumptions about stack sizes and bet sizes are:

– Both players start with 10 bb stacks
– Alice openraises pot (3 bb)
– Bob 3-bets pot (9 bb)
– Alice 4-bets to about 3/4 pot (20 bb)
– Bob 5-bets all-in

The assumption about Alice’s openrange is simply a choice we make. But opening 25-40% is typical for a good, aggressive player in the big blind, depending on how well the big blind defends. So assuming a 35% opening range for Alice in the small blind should give us a strategy pair that will work well for most small blind players.

From the overview over optimal strategy pairs we see that Bob’s 3-bet range varies little when Alice’s openrange goes from 30% to 40% (for example, Bob’s value range is {JJ+,AK} against the 30%, 35% and 40% opening ranges). So it makes sense to use a 35% opening range and then assume that the strategy pair we end up with is a good starting point for most small blind openraising scenarios.
We remember that when Bob had position on Alice outside the blinds, there was no minimum defense requirement for his total defense. We constructed his optimal 3/4/5-bet strategy, and then we said that Bob also would flat the hands in “IP flat list”. But when Bob is the only player between Alice and the pot, he has all of the blind defense responsibility. So let’s find out how often Bob needs to defend to prevent Alice from stealing profitably with any two cards:

Alice raises to 3 bb and risks 2.5 bb (remember, she has already posted a 0.5 bb small blind) to win the 1.5 bb pot. Her effective pot odds on a steal raise are 1.5 : 2.5. She needs to win more than 2.5/(1.5 + 2.5) =62.5% to have an automatic profit with any two cards.

Conclusion: Bob needs to defend at least 100 – 62.5 =37.5% by 3-betting and flatting

3.2 The relations between opening range, 3-bet range, 4-bet range and 5-bet range

Now we go through all the steps we went through when we outlined the 3/4/5-bet theory in in Part 1. The only difference is that we’re using different bet sizes, pot sizes and pot odds in the blind vs blind scenario:

What is Alice’s optimal 4-bet%?
The process begins when Alice openraises some range (we’re assuming the default 35% button range) that is known to both her and Bob. When Bob 3-bets, Alice has to 4-bet enough to prevent him from profitably 3-bet buffing with any two cards.

Bob’s 3-bet risks 8 bb (9 bb minus the big blind he posted) to win a 3 + 1 =4 bb pot, so his effective pot odds for a 3-bet bluff are 4 : 8 =1 : 2. He will automatically make a profit if Alice folds more than 1/(2+1) =1/3 =33%. So Alice’s optimal defense against Bob’s 3-bets means she defends 33% of her opening range with a value/bluff ratio we’ll find in a moment.

What is Bob’s optimal value/bluff ratio for the 3-bet range?
When Alice 4-bets to 20 bb, she risks 17 bb more (20 bb minus her 3 bb raise) to win a 3 + 9 =12 bb pot. The effective pot odds for her 4-bet bluffs are 12 : 17. So she profits from a 4-bet bluff with any two cards if Bob folds more than 17/(17 + 12) =59%.

Bob can’t allow this, and he defends optimally against Alice’s 4-bets by defending 100 – 59 =41%. He should then 5-bet all-in with 41% of his 3-betting range and fold the remaining 59%. We round this to 40/60, and end up with the same 40/60 value/bluff ratio we have used in previous articles.

What should Bob’s 5-bet range look like?
We know from Part 1 that the Axs hands (A5s-A2s) work well as 5-bet bluffs. They block Alice’s AA/AK/AQ hands, and they have about 30% equity when called, even against a strong calling range of good aces and big pairs. For example against {QQ+,AK}:

When Bob 5-bet bluffs an Axs hand all-in and gets called, he has ~30% equity in a 200 bb pot where he has invested 91 bb with his 5-bet (a 100 bb stack minus his 3-bet to 9 bb). From this pot he gets back ~0.30 x 200 =60 bb on average, so his net loss when his 5-bet bluff gets called is 91 – 60 =31 bb.

The pot before the 5-bet is 20 bb (Alice’s raise + 4-bet) + 9 bb (Bob’s 3-bet) =29 bb. So Bob risks 31 bb to win 29 bb. His effective pot odds on the 5-bet bluff are 29 : 31. Bob thus needs to win at least 31/(29+31) =52% of the time.

This means that Alice needs to call Bob’s 5-bet at least 48% of the time to prevent him from 5-bet bluffing with automatic profit. We round this to 50%. Alice’s value/bluff ratio for the 4- bet range is then 50/50 (and not 60/40 as in previous articles).

As discussed in Part 1, Bob adds enough Axs 5-bet bluffs to make Alice indifferent towards calling or folding the 5-bet with her weakest value hands.

3.2 Summary of Alice’s optimal 3/4/5-bet strategy from the small blind in a blind vs blind scenario

Alice opens her default 35% button range:

22+
A2s+ A7o+
K2s+ K9o+
Q6s+ Q9o+
J7s+ J9o+
T7s+ T9o+
96s+
86s+
76s
65s

458 combos
35%

When Bob 3-bets, Alice defends 33% of the time with a 4-bet. She then plays 0.33 x 458 =151 combos. She uses a 50/50 value/bluff ratio, so she 4-bets 75 combos for value and 75 combos as a bluff.

75 value combos is approximately the value range {88+,AJs+,AQo+} =78 combos. She now picks an equivalent amount of bluffs, for example {AJo-A7o,ATs-A7s} =76 combos. Here we have used Ax hands for blocker value (reduces the probability that Bob has one of the value hands AA/AK when when he 3-bets.

– Alice 4-bets {88+,AJs,AQo} for value
– Alice 4-bets {AJo-A7o,ATs-A7s} as bluffs

And this is Alice’s total defense strategy against Bob’s 3-bets after she has openraised from the small blind. Over to Bob:

3.3 Summary of Bob’s optimal 3/4/5-bet strategy from the big blind in a blind vs blind scenario

Bob first finds his value range. He uses hands that are at least 50% against Alice’s value range {88+,AJs+,AQo+}, and this gives him the value range {JJ+,AK} =40 combos.

Then we find the optimal number of 5-bet bluffs for Bob. When he 5-bets all-in, Alice has to call 80 bb more to win a 100 bb (Bob’s stack) + 20 bb (Alice’s raise + 4-bet) =120 bb. So her effective pot odds are 120 : 80 =1.5 : 1. To profit from calling with the weakest hands in her value range, she needs at least 1/(1.5 + 1) =40% equity.

Bob then picks Axs hands from the top and works his way from A 5 down to A 2 . From the equity calculation below we see that Bob has to use all the 16 Axs hands A5s-A2s to make Alice’s weakest value hands break even:

So Bob 3-bets the following total value range, including 5-bet bluffs, planning to 5-bet all-in after a 4-bet: {JJ+,AK,A5s-A2s} =56 combos.

Bob should use a 40/60 value/bluff ratio, so he needs 60/40 =1.5 times as many bluffs as value combos. Bob then picks 1.5 x 56 =84 bluff combos. We’ll make a list of these, but first we define his flatting range. Regardless, Bob’s total 3-bet range contains 56 + 84 =140 combos. This is 140/1326 =10.6% of all hands.

His flatting range needs to be wide to get to 37.5% total defense, so he will flat some of the hands we used as 3-bet bluffs outside the blinds. So we design a separate list of flatting hands to use in the blind vs blind scenario. We can call this list “Blind vs blind flat list”.

To defend 37.5% Bob needs to flat 37.5 – 10.6 =26.9% of all hands. This is 0.269 x 1326 =357 combos, and we can put together this range in various ways. Below is one way to do it:

Blind vs Blind flat list:

  • Pairs: TT-22 =54 combos
  • Suited aces: ATs-A6s =20 combos
  • Offsuit aces: AJo-A7o =60 combos
  • Suited Broadways: KQs-K8s,QJs-Q8s,JTs-J7s =52 combos
  • Off-suit Broadways: KQo-K9o,QJo-Q9o,JTo-J9o =108 combos
  • Suited connectors: T9s-T7s,98s-96s,87s-86s,76s-75s,65s =44 combos
  • Offsuit connectors: T9o-T8o =24 combos
  • Total: 362 combos

Note that we here have only used mathematics to tell us how many hands we need to flat to prevent Alice from opening any two cards with automatic profit. We have not given though to which hands we are able to play profitably after flatting them, and how we should play this range postflop (but we will have more to say about that in a later article series about optimal postflop play).

Using this flat list and the previously defined value range, now only have to pick our 3-bet bluffs (we need 84 combos) from the remaining (and rather trashy) hands. For example, we can use {A6o-A2o,K8o,Q8o} =84 combos.

Now we finally have one possible defense strategy for Bob, designed with optimal 3/4/5-betting against Alice’s 35% openrange from the small blind, and designed to prevent her from having an automatic profit from stealing:

  • 3-bet value range (including 5-bet bluffs): {JJ+,AK,A5s-A2s} =56 combos
  • 3-bet bluff range: {A6o-A2o,K8o,Q8o} =84 combos
  • Flatting range: {TT-22,ATs-A6s,AJo-A7o,K8s+,K9o+,Q8s+,Q9o+,J7s+,J9o+,T7s+,T8o+96s+,86s+,75s+,65s} =362 combos

This total strategy is a handful, particularly the flatting range. But in the next article series about optimal postflop play (“Optimal Postflop Play in NLHE 6-max”) we’ll see than postflop play with a wide range after flatting preflop becomes easier when we use a systematic approach based on principles from game theory.

Note that if we need to, we can construct a 3/4/5-bet strategy pair for any openraise percentage Alice uses from the small blind. we only picked a 35% here, and this strategy pair will be a good default to use in a blind vs blind battle with an unknown small blind. But we could also have made a list of optimal strategy pairs for various small blind openraise ranges like we did in Part 2 for the scenario where both players were outside the blinds.

4. Summary:

We have done a series of numerical simulations to estimate the EV of flatting and value 3-betting for hands in the region between obvious value hands and obvious flatting hands. We did this by calculation the EVs of flatting and 3-betting with QQ against a tight ~15% UTG opening range.

The simulations showed that flatting a very strong hand like QQ can be correct when the raiser has a tight range. We also found that 3-betting becomes mandatory against wider ranges, for example with QQ against a ~25% CO opening range.

We also did some simulations to study the effect of the raiser flatting out of position. Even if this increases the EV for value 3-betting, we also have to take into consideration how often the raiser 4-bet bluffs. Against a loose-passive raiser who flats a lot of 3-bets out of position but never 4-bet bluffs, we can use a “hybrid strategy”. we then 3-bet “thin” value hands (for example JJ), planning to fold to the raisers squeaky tight value 4-bet range. The rationale behind this is that we profit from the raiser’s folding and calling, but on the rare occasions he 4-bets us, we are crushed and can fold. Note that we are exploiting his lack of 4-bet bluffing by making safe folds.

Then we we studied the scenario blind vs blind where the small blind openraises and the big blind defends in position by optimal 3/4/5-betting and flatting. we saw that the big blind has to defend a very wide range (37.5%) to prevent the small blind from profitably stealing with any two cards. Of course, the big blind then has to play his wide flatting range well postflop, and we’ll discuss this further in the coming postflop article series.

I am planning to publish a Part 8 in this preflop series some time in the future. This will be a practical part where we look at how our strategies perform in practice. I’m thinking about grinding a decently large sample of low limit NLHE hands where I focus on playing close to the core strategy we have defined in this preflop article series. This should give us an idea about how our core strategy performs at the limits most of the readers play.

I have already tested the optimal 3/4/5-bet strategies at the middle limits ($400NL to $1000NL), and they work very well as solid defaults, and as a starting point for exploitative adjustments against players I have reads on. But at the middle limits I of course mix up my play a lot, depending on my opponents, so for testing purposes it would be better for the readers to see how the strategies perform against unknowns at the limits they play.

I will not give a date for Part 8, but it will be some time after we have finished the theoretical series about optimal postflop play. The series “Optimal Postflop Play in NLHE 6-max” comes next. There we’ll use strategies and ranges from this preflop series and see how principles for optimal play can be used postflop, using our default preflop strategies and preflop ranges to set up postflop scenarios to study.

Good luck!
Bugs – See more at: http://en.donkr.com/forum/optimal-3-bet-4-bet-5-bet-strategies-in-nlhe-6-max—part-7-533567#sthash.xn4CohAV.dpuf

Optimal 3-bet/4-bet/5-bet Strategies in NL Hold’em 6 Max – Part 6

1. Introduction
This is Part 6 in the series Optimal 3-bet/4-bet/5-bet-strategies i NLHE 6-max, and the next to last theoretical part of the series (there will possibly be a practical part later this year, and we’ll talk about that in Part 7). In Part 1 to Part 5 we built a foundation for default NLHE preflop play based on mathematical principles from game theory, plus some common poker sense. In this and the next article we’ll test these strategies numerically.

The article series started with a simple scenario in Part 1 where we studied 3/4/5-betting heads-up with the raiser out of position. Then we generalized the strategies we found to other heads-up 3/4/5-bet scenarios, and also to a few select multiway scenarios. Along the way we also defined default ranges for open-raising from all positions.

Below is a summary of the content in Part 1 to Part 5:

  • Part 1: Introduction to the mathematics behind game theory optimal 3/4/5-betting heads-up, studying the scenario where the raiser is out of position
  • Part 2: We discussed in greater detail how to implement the theory from Part 1, and we defined default openranges for all positions. Then we defined the heads-up 3/4/5-bet theory for a wide range of openranges with the raiser out of position.
  • Part 3: We let the raiser and the 3-bettor switch positions, and we studied the scenario where the raiser opens on the button and gets 3-bet by a player in the blinds.
  • Part 4: We generalized the theory from Part 4 and looked at 3/4/5-betting heads-up with the raiser opening from any position outside of the blinds, and the 3-bettor 3betting from out of position in the blinds
  • Part 5: We discussed 3/4/5-betting for two multiway scenarios (squeezing in a 3-way pot and cold 4-betting in a 3-way pot).

Throughout Part 1 to Part 5 we have gone through most of the possible preflop scenarios and discussed good default strategies for them. In all 3/4/5-bet scenarios we have used the theory from Part 1 as our starting point, and then adjusted it for similar scenarios. We have used a mix of mathematical reasoning and good poker sense.

The plan for Part 6 is to test the strategies for heads-up 3/4/5-betting using the poker analysis software “Pokerazor“. The final test for a strategy is of course to try it out at the tables and see how it performs. But we can also study our strategies numerically using analysis software. Today there are two programs available that let us study complete pre- and postflop strategies for any number of players:

– Pokerazor
– StoxEV

Pokerazor is for the time being no longer commercially available, but a new version is expected some time in the future. StoxEV is available and being actively developed. I have elected to use Pokerazor for this article, since this is the program I am most familiar with. But StoxEV will work just as well if you are interested in doing this type of analysis work on your own.

What we’ll do first in this article is to study the typical ABC poker new players are advised to use when they get started with NLHE at the lowest limits (“play tight”, “bluff little”, “fold a lot when you get 3-bet”, etc.) Then we’ll show how this ABC poker makes us vulnerable for attacks from aggressive opponents (particularly when they have position on us). We will here only look at preflop play, but the same principles apply postflop as well.

Then we’ll go one step further and show how we can improve on ABC preflop strategy by adding strategy components that fully or partly neutralize the attacks aggressive players subject us to (for example, we add 4-bet bluffing to our preflop strategy to defend against 3-bet bluffing). Then we go back to our opponents’ strategies and discuss how they can adjust to our adjustments, and so on.

In this manner we’ll show how the optimal 3/4/5-bet strategies we have designed can be viewed as the final product of an evolutionary process based on our desire to defend against profitable bluffing with any two cards from aggressive opponents. The main point is that we don’t want to put ourselves in a situation where our opponent(s) can exploit us by bluffing profitably with any two cards, be it open-raising, bluff 3-betting, bluff 4-betting, or bluff 5-betting. An optimal strategy “plugs” all such openings for our opponents, but of course this defense does not come entirely without cost.

Through this discussion we’ll also shed light on the difference between optimal play and exploitative play, and when we should use one or the other. Optimal strategies put a lot of weight on defense, and they are not necessarily the most profitable strategies against players with big leaks. One reason is that optimal strategies include defensive components (for example, 4-bet bluffing as a defense against light 3-betting) that are often unnecessary against weak players (for example, we don’t need to 4-bet bluff against an opponent who only 3-bets premium hands like {JJ+,AK}).

Against players with big and easily exploitable leaks, we’d rather deviate from optimal play and play exploitatively to take full advantage of these leaks. But we need to be aware that by doing so we are creating openings in our strategies that can be exploited by observant opponents. So we have to find a balance between optimal and exploitative play, and we should use different strategies against different opponents. We will do our best to exploit weak players’ big mistakes, but we can always fall back on optimal play against good opponents without big leaks. We can also return to optimal play if the player we’re trying to exploit with exploitative play suddenly changes his strategies to take advantage of the openings created by our exploitative strategies.

For example, let’s say we choose to never 4-bet bluff against a passive player who never 3-bet bluffs. He might now notice this, and adjust to our tight play by starting to 3-bet bluff us. Our exploitative adjustment against this particular opponent then runs the risk of getting counter-exploited if he starts 3-bet-bluffing us often with random weak hands. If this happens, we should return to our optimal optimal 3/4/5-bet strategy. Alternatively, we can make another exploitative adjustment to his adjustment by 4-bet bluffing him a lot (since we know he often is weak and have to fold). But the optimal strategy is always an alternative if we aren’t sure whether or not we can exploit his aggressive 3-betting.

In my opinion, this mindset is at the core of the thought processes of a strong NLHE player. He doesn’t have to use mathematics like we have done, but he will have a good feel for what an optimal (or near optimal) strategy is in the situation he is in. So he has a strong default strategy to fall back on against unknown players or known strong players, so that he can’t be easily exploited. But at the same time he knows how to deviate from optimal strategies to exploit his opponents’ systematic leaks. So he can adjust his play in a controlled manner against each individual opponent instead of being locked into a static strategy that he uses against everyone.

Rules of thumb such as “never 4-bet bluff against fish” or “don’t 3-bet hands that perform poorly when called” are then replaced by a dynamical mindset that gives is strong control over our choice of strategies. Using optimal play as a starting point (and as a strategy we can always fall back on regardless of who we’re playing against), we can move around freely in “strategy land” and exploit opponent leaks as we pick up information about how they play.

Optimal play is never bad play, but exploitative play is always better. But we need information about our opponents’ strategies before we can exploit them. If we don’t have this information, we can always fall back on optimal play as a good default.

2. Testing preflop strategies using the analysis software “Pokerazor”
In this part of the article we’ll use Pokerazor to study 2 things:

  • 1. How a tight openraise strategy without defense against light 3-betting is vulnerable to 3-bet bluffing with any two cards
  • 2. What the raiser can do to plug this leak, and how this leads to an optimal strategy pair for the raiser and the 3-bettor

2.1 ABC preflop strategies and how these can be exploited
Those of you who have played for a while probably remember the good old days (up to around 2007 or thereabouts) when micro and low limit NLHE was easily beatable by sticking close to the following rules of thumb for preflop play:
Those

  • Open tight from all positions (say, 10-12% from UTG/MP, ~20% from MP and ~30% from the button
  • 3-bet only for value with {QQ+,AK}, and possibly {JJ+,AQ+} against a loose raiser
  • When you get 3-bet and you are out of position, fold everything but {QQ+,AK} regardless of your position and where the 3-bet comes from
  • Defend the blinds very tightly (typically 10%)

Believe it or not, but this was more or less the standard getting-started preflop strategy recommended to beginning players at the micro and low limits up to $100NL or so. And it worked well, since the games were so loose and passive that it was correct both to openraise tight, and to fold a lot against 3-bets.

Those of you who have been members of Cardrunners for a while might remember Brystmar’s beginner video series “Small Stakes NL” in 6 parts (published during the spring of 2007). This series began with tight-aggressive preflop recommendations based on tight opening ranges and 3-betting only for value:

Brystmar’s preflop strategy for micro/low limit NLHE
Let’s take a trip down memory land and study Brystmar’s preflop recommendations given 3.5 years ago. Those who want to read discussion about his video series or download his preflop scheme can look at this Cardrunners forum thread.

Below is a summary of the default openraising ranges (note that “KTs” and “KTo” denote suited and offsuit hands, while “KT+” means both suited and offsuit:

  • UTG openraise:
    {22+,AJ+,KQs} =9.8%
  • MP openraise:
    {22+,AJ+,KQ} =11%
  • CO openraise:
    {22+,A7s+,A9o+,KT+,QTs+,QJo,J9s+,JTo,T9s} =19%
  • Button openraise:
    {22+,A4s+,A7o+,KT+,QTs+,QJo,J9s+,JTo,T8s+,
    T9o,98s,98o,87s,87o,76s} =26%
  • Raising from the small blind:
    Openraise the button range if it gets folded to you. In a limped pot, raise {JJ+,AK} for value and overlimp all other hands from your button range, plus all Axs and Kxs.
  • Raising from the big blind:
    If the small blind openlimps, raise the button range and otherwise check. Out of position in a limped pot, raise {JJ+,AK} for value and otherwise check

Tight opening ranges all around. This is of course not a leak our opponents can exploit, but we might perhaps say that we are exploiting ourselves by folding some profitable hands, particularly on the button.

But the strategies become easy to exploit when we get to playing against a raise:

  • In MP with position on a raiser:
    Reraise {JJ+,AK} for value and call with {TT-22,AJs+,AQo}
  • In CO with position on a raiser:
    Reraise {JJ+,AK} for value and call with {TT-22,AJ+,KQ,QJs,JTs}
  • On the button with position on a raiser:
    Reraise {JJ+,AK} for value (and AQo if the raise came from CO) and call with {TT-22,AJ+,KQ,QJs,JTs}. With callers between you and the raiser, also call with {JTo,T9s,98s,87s}
  • In SB after a raise:
    Reraise {QQ+,AK} for value (and also JJ/AQ if the raise came from CO or the button) and call with {TT-22,AJ+,KQ}
  • In BB after a raise:
    Reraise {QQ+,AK} for value (and also JJ/AQ if the raise came from CO or the button) and call with {TT-22,AJ+,KQ}. With callers between you and the raiser, also call with {QJs,JTs}

We note two systematic errors in these strategies:

– We’re 3-betting more or less the same range regardless of the raiser’s position
– We’re never 3-bet bluffing

We remember from Part 1 and Part 2 that an optimal 3-betting range on the button varied from 3.6% against a ~15% EP openraise to 8.7% against a ~25% CO openraise. And in all scenarios we used an optimal bluffing frequency of 60%. In other words, more than half of our 3-bets were bluffs- In Brystmar’s strategies the 3-betting range is a tight and static value range {JJ+,AK} =3.0%, which is sometimes widened to {JJ+,AQ+} =4.2% against a wide openraising range. We also note that Brystmar chooses to include AQ when loosening up. This is a hand we never 3-bet for value in position when we’re playing optimally (since AQ works better as a flatting hand in position).

Brystmar’s strategies don’t mention defense against 3-betting, but we can assume that default defense is to 4-bet a tight range {QQ+,AK} from all positions. Another significant leak is the squeaky tight blind defense. For example, of button openraises, the preflop scheme tells us to 3-bet {JJ+,AQ} =4.2% from the small blind and flat {TT-22, AJ,KQ} =7.4%. This results in a total defense of 11.6%, which is way lower than the optimal defense threshold of 16% that we estimated in Part 3.

So there are huge openings in Brystmar’s preflop recommendations, and these openings can be easily exploited by an aggressive and observant opponent. We’re also leaving money at the table because we’re openraising to tight, and the main reason for this is that Brystmar does not take full advantage of position. We can openraise a ton of hands on the button when it’s folded to us, and we can make life hell for a raiser by 3-bet bluffing him in position, but Brystmar chooses not to do so.

NB! Before we move on I want to point out that I am not trying to put Brystmar’s low limit preflop defaults from 2007 in a negative light. His preflop recommendations for beginning NLHE players were very useful back in the day, and gave many new players an easy start. His strategies are best viewed as “training wheels” for staying out of trouble (and he no doubt saw them as such himself) and they were tailored towards the micro/low limit conditions that existed at the time. They will probably still work okay at the lowest micro limits, but I would not recommend anyone to play $25NL and higher with such tight and easily exploitable preflop strategies.

It’s clear for everyone who plays $25NL and higher these days that common NLHE strategy has developed in leaps and bounds since Brystmar’s 2007 recommendations. Light 3-betting was rare in the “old days”, even at $100NL and $200NL. Today it’s common, even if you begin as low as $5NL.

The next step of the development of the average low limit NLGE regular back in the day was to add some light 3-bets in position (and Green Plastic’s 2006/2007 NLHE videos at Cardrunners inspired many to do so), call more raises and 3-bets in position, and in general get better at using positional advantage. A common mistake many aggressive players did was to 3-bet bluff with hands that were too strong to use as bluffs (for example, JTs). However, this did not cause man problems since most players defended poorly against 3-bets, particularly from out of position.

So the standard recipe in the good old days for an advanced low limit player who wanted to ramp up the aggression was to LAG it up in position. But not necessarily with balance in mind, and not necessarily with a good understanding of how to chose his value range, bluffing range and flatting range in a consistent manner. But this was not a big deal. He played tight out of position, opened a very wide range in position, and 3-bet something fierce in position against weak opponents. The 3-betting was very effective, since the raisers often did one of the following two mistakes:

– Folded a lot out of position and never 3-bet bluffed
– Called a lot with non-premium hands out of position

The first mistake lets the 3-bettor print money by giving him an opening to (in principle) 3-bet bluff any two cards. The second mistake occurs when the raiser tries to correct the first mistake, but he goes about it the wrong way. Defending against 3-bets by flatting weak hands out of position is ineffective, since the raiser now has to play postflop out of position in a scenario where it’s difficult for him to win without hitting the flop well. Playing weak starting hands well out of position against a good LAG player is hard, and often results in you losing more money postflop than if you had just folded to the 3-bet preflop.

The cure against light, positional 3-betting is of course to respond by 4-betting a correct value range (which follows from the size of our opening range), balanced with a correct amount of 4-bet bluffing. We have studied this in previous articles, and we have defined optimal strategies for the raiser from all positions. The 3-bettor uses similar thinking to design his 3-bet strategy so that the raiser can not 4bet bluff any two cards profitably. This way an equilibrium gets established.

This equilibrium is given by the optimal strategy pairs for the raiser and the 3-bettor defined in Part 1 and Part 2. We used mathematics to define these strategy pairs, but we can also think about them as a product of an evolutionary process.

The 3-better starts out by exploitative ant-two-cards 3-bet bluffing against a raiser that defends way too tight and folds too much. Then the raiser adjust by choosing a correct value range and introducing 4-bet bluffing. The 3-bettor responds by adjusting his value range and introducing 5-bet bluffing. To prevent the opponent from bluffing with any two cards anywhere, both players fine-tune their ranges until both are using an optimal set of ranges for 3/4/5-betting. “Optimal” here means that neither player can improve his EV by adjusting further. If one of them tries to do so, he is giving the other player an opportunity to increase his EV by making and exploitative adjustment.

We will now illustrate such an evolutionary process using Pokerazor simulations:

2.2 Numerical testing of optimal heads-up 3/4/5-bet strategies
We start with the following model:

  • Alice (100bb) openraises to 3.5bb with her standard 25% range from CO
  • Bob (100bb) is on the button and 3-bets to 12 bb or folds
  • Alice defends against 3-betting by 4-betting to 25 bb or folding
  • Bob defends against 4-betting by 5-betting all-in or folding
  • Alice defends against 5-betting by calling all-in or folding
  • The blinds always fold, no matter what Bob does

So we are studying a scenario where Alice openraises, Bob 3-bets or folds, and the blinds never get involved. Alice then makes 1.5 bb (the blinds) per raise when Bob folds, which is a win rate of 150 bb/100. This is her baseline EV for the simulation.

Before we begin the simulations, let’s repeat the ranges and optimal strategy pairs we defined in Part 2:

Alice’s 25% open-range from CO

22+
A2s+ A9o+
K9s+ KTo+
Q9s+ QTo+
J8s+ JTo
T8s+
97s+
87s
76s
65s

326 combos

25%

The corresponding optimal strategy pair that’s being used when Bob 3-bets in position can be found from the summary of optimal strategy pairs in Part 2:

Here is a download link for this document (right-click and choose “Save as”):
IP_3-bet_summary.doc

The optimal strategy pair is then:

  • Bob:
    3-bets {QQ+,AK,12 air} ={QQ+,AK,A5s-A3s} for value (including 5-bet-bluffing with Axs hands) and 70% av “IP 3-bet air list” as a bluff using a randomizer
  • Alice:
    4-bets {TT+,AQ+} for value and {AJ,AT,A9s-A7s} as a bluff

Pokerazor simulation 1 (Bob folds)
The baseline simulation is to let Bob fold 100%. Alice then picks up the blinds, and makes 1.5 bb each time (=150 bb/10):

– Alice openraises 25% from CO
– Bob folds

EV (baseline for Alice) =150 bb/100

Bob now begins 3-betting so that Alice ends up with EV < 150 bb/100. It’s obvious that Alice’s EV now becomes lower than the 150 bb/100 baseline, since she can not prevent Bob from making money by 3-betting only his best hands, for example his {QQ+,AK} default value range against her 25% CO openrange. Furthermore, it’s correct for Alice to never 4-bet bluff when Bob never 3-bet bluffs, so she has to fold the hands not strong enough to 4-bet for value against Bob’s tight value range and let Bob pick up the blinds.

We start by assuming that Bob never 3-bet bluffs and that Alice defends against the 3-bet by 4-betting the value range {KK+} after observing that Bob’s 3-betting range is {QQ+,AK}. This is correct since Alice does not want to get all-in preflop with QQ or AK against Bob’s {QQ+,AK} range (both are ~40% underdog) with only 3.5 bb invested.

Pokerazor simulation 2 (Bob 3-bets only for value)
Bob 3-bets his value hands. We let him use the pure value hands {QQ+,AK} that he would have used in an optimal strategy against Alice’s 25% CO openrange, and the drops his 5-bet bluffs for now:

– Alice openraises 25% from CO
– Bob 3-bets {QQ+,AK} for value and 5-bets all-in against a 4-bet
– Alice 4-better {KK+} for value

And the rest follows automatically. We get:

EV (Alice; KK+) =141 bb/100

If Alice also had 4-bet QQ/AK for value, the EV would have become:

EV (Alice; QQ+,AK) =139 bb/100

So Alice’s choice of value range against Bob’s extremely tight 3-betting range is correct. Note that Alice is now exploiting Bob by only 4-betting an extremely tight {KK+} value range! What happens here is that Bob mostly leaves Alice alone so that she can pick up the blinds almost every time. Bob only pops up with a value 3-bet the 2.56% of the time he has {QQ+,AK}, and Alice then responds by folding everything but {KK+}.

So Alice exploits Bob’s squeaky tight 3-betting by not paying off his value hands with weaker hands. And since Bob never 3-bet bluffs, there is never any doubt about his range. Alice can then play perfectly against a 3-bet and drop all her (now unnecessary) 4-bet bluffs as well as her weakest value hands (and we remember that Alice’s optimal value range in CO is {TT+,AQ+}). Exploiting someone by folding a lot to his aggression is not the first thing that comes to mind when poker players think about exploitative play. But avoiding paying off a strong range unnecessarily is just as profitable as playing aggressively against players that fold too much.

If Alice had responded with her optimal 4-bet strategy which is {TT+,AQ} for value and {AJ-AT,A9s-A7s} as bluffs, we get:

EV (Alice; optimal strategy) =126 bb/100

And we see that Alice’s optimal 3/4/5-bet strategy out of position loses relative to the best exploitative strategy (which gave her 141 bb/100). This is an example of a general rule: If we have an opportunity to exploit someone, we will make more money from this than from continuing to use an optimal strategy. The reason is that Alice now sacrifices EV by defending against a non-existing threat. She has 4-bet bluffs in her 4-betting range to defend against Bob’s 3-bet bluffs, but Bob never 3-bet bluffs.

So even if Alice’s optimal strategy can not be exploited by any-two-cards 3-bet bluffing from Bob, this defense is costing her EV relative to a strategy that exploits Bob’s very tight never-bluff 3-betting strategy. A good analogy would be a nations defense budget in peace time. There has been no warfare on Norwegian soil since 1945, but Norway has still had a defense budget in all the years since then. We are simply paying for a military defense so that other nations don’t get an opportunity to invade us without risk.

Now, let’s assume that Bob has been sitting there and 3-betting only {QQ+,AK} for value and folding everything else, while Alice has responded by 4-betting only {KK+} for value and folding everything else. Bob has observed that Alice almost always folds. It’s now easy for him to reach the conclusion that he could make a lot more money by throwing in some 3-bet bluffs, planning to fold then when Alice 4-bets. Since Bob does not have to worry about the blind players waking up with a hand, let’s allow him to 3-bet any two cards to maximally exploit Alice’s super tight defense against 3-bets:

Pokerazor simulation (Bob 3-bets any two cards)
Bob uses his read on Alice, and changes his strategy to exploit her. He keeps 3-betting {QQ+,AK} for value, and then he 3-bets all other hands as a 3-bet bluff. Alice, not knowing that Bob suddenly has changed his strategy, keeps 4-betting {KK+} for value and folding everything else:

– Alice openraises 25% from CO
– Bob 3-bets {QQ+,AK} for value + any two cards as a 3-bet bluff
– Alice 4-bets {KK+} for value

EV (Alice) =-286 bb/100

Alice now gets slaughtered by Bob’s any-two-cards 3-bet-bluffing and she actually loses money from trying to steal the blinds from CO. Her first adjustment is to return to the optimal value range {TT+,AQ+} associated with her 25% CO openrange. This gives us:

EV (Alice; optimal 4-bet value range) =-39 bb/100

It helps, but not enough. She will still lose money for every hand she openraises, unless she also starts 4-bet bluffing. Alice now returns to her complete optimal strategy in CO, designed to defend against any-two-cards 3-bet bluffing:

EV (Alice; optimal total 4-bet-range) =+179 bb/100

Bingo! Alice’s optimal 4-bet-strategy not only prevents Bob from exploiting her by 3-bet bluffing with any two cards, it also punishes him for it. Alice now makes 179 – 150 =+29 bb/100 more than if Bob had simply folded every time and let her pick up the blinds.

But is Alice’s optimal strategy the most profitable strategy against Bob’s any-two-cards 3-bet bluff strategy? No, and to find Alice’s most profitable strategy we use common sense. When Bob is 3-betting any two cards, but only continues with {QQ+,AK} after a 4-bet, he is extremely vulnerable for 4-bet bluffing. He only continues with 2.56% of his hands and folds the remaining 97.44% against Alice’s 4-bets

Alice can then maximally exploit Bob’s attempt to exploit her by 4-bet bluffing him with any two cards (or rather, any two cards in her original opening 25% opening range). We let Alice continue with her optimal {TT+,AQ+} value range. Then she deviates from the optimal strategy by widening her 4-bet bluffing range to the rest of her 25% CO opening range. Bob, unaware that Alice has suddenly changed her strategy to exploit his strategy, keeps 3-bet bluffing any two cards:

EV (Alice; 4-bet bluff any two cards) =+1261 bb/100

A huge increase in EV for Alice, and Alice now makes about 8 times more than if Bob had folded every hand. This is an extremely clear illustration of the difference between optimal and exploitative play. Alice’s optimal defense strategy against Bob’s 3-bets guarantees that he can not exploit her by 3-bet bluffing any two cards, but he optimal defense did nothing to counter-exploit Bob extreme strategy. Alice made a little bit more than if Bob had folded every hand, but not a lot ((179 bb/100 vs 150 bb/100) .

But when Alice responds by choosing the strategy that exploits Bob’s any-two-cards-bluffing strategy maximally, her EV explodes. Still, this does not come without risk for her, since Bob can take his exploitative strategy to the next level and begin 5-bet bluffing any two cards to exploit Alice’s any-two-cards 4-bet bluffing. Alice must then make a new exploitative adjustment (for example, widening her value range dramatically and calling Bob’s 5-bets with a very wide range of value hands) to stay one step ahead of Bob.

We can view this exploit/counter-exploit process as “strategic ping-pong” where both players zig and zag, using extreme strategy changes in different directions to maximally exploit their opponent. When one of them has made a big strategy change to exploit her opponent, she also creates an opening that the opponent can exploit by making an adjustment of his own. Then the first player has to make another big strategy adjustment, and the process repeats itself ad infinitum.

When we are playing optimally, we are using a different mindset. Instead of trying to stay one step ahead of our opponents by sudden “gear changes”, we can fall back on a strategy that performs more or less equally well no matter what our opponent does. If he uses an extreme strategy, our optimal strategy will win a bit from him, but maximizing our profit from his mistake(s) is not our main goal. Instead, we simply want to prevent him from exploiting us. In the simulations above we saw that this worked well for Alice, but not as well as the maximally exploitative strategy she can use against Bob’s any-two-cards 3-bet bluffing.

Okay Bob, now what? Bob can of course respond to Alice’s any-two-cards 4-bet bluffing by going to the next level and start 5-bet bluffing with any two cards. Alice then gets exploited for a while, until she realizes this and adjusts. But we will not take any-two-cards bluffing beyond what we did in the previous simulations, and we assume that Bob now adjusts by falling back on his optimal 3-betting strategy. Alice in turn falls back on her optimal 4-betting strategy:

Pokerazor simulation 4 (both players use optimal strategies)
Both Alice and Bob now uses the optimal strategy. As we have seen in previous articles, this optimal strategy pair follows from Alice’s openrange. Both players are now protecting themselves against any-two-cards bluffing from their opponent in all phases of the 3/4/5-bet war.

Alice’s EV now becomes:

EV (Alice; both players 3/4/5-bet optimally) =+129 bb/100

In the last round of simulations we will let one player stick to the optimal strategy while the other player is let “off the leash”, free to try anything to increase her or his EV against the other player’s optimal strategy. Then we compare the resulting EV with EV when both players play optimally (129 bb/100 for Alice). We start out with Bob playing optimally, while Alice is testing out some deviations from her optimal strategy:

Pokerazor simulation 5 (Bob plays optimally and Alice can do what she wants)
We remember that Alice makes 150 bb/100 when Bob doesn’t interfere, and Bob’s optimal 3-betting strategy reduces this to 129 bb/100 when Alice plays optimally too. We shall now see that Alice can’t do anything to increase her EV significantly when Bob sticks with his optimal strategy.

For example, let’s assume that Alice drops the weakest hands TT/AQ from her value range when she sees that Bob uses the strong value range {QQ+,AK} (and remember that he also 3-bets/5-bets with his 5-bet bluffs A5s-A3s). Alice then chooses to 4-bet TT/AQ as before, but she folds them when Bob 5-bets all-in:

EV (Alice; folds TT/AQ to 5-bet) =+128 bb/100

Alice’s EV drops a little, and she can’t increase her EV by playing a tighter value range against Bob’s strong value range. The reason is that Bob has put exactly so many 5-bet bluffs in his 5-betting range that Alice becomes indifferent to folding or calling with her weakest value hands. When Alice tries to avoid paying off Bob’s better value hands, his 5-bet bluffs makes more money.

Can Alice increase her EV by 4-bet-bluffing more than optimally? We test this by letting her 4-bet bluff any two cards against Bob’s optimal strategy, keeping everything else optimal:

EV (Alice; 4-bet-bluffs any two cards) =+139 bb/100

A small EV increase, but nothing comparable to the results of the extreme exploitative any-two-cards adjustments in previous simulations. Note that a perfectly optimal strategy for Bob should make it impossible for Alice to increase her EV, but our optimal strategy implementations probably contain some “numerical noise”, since we have made some approximations and rounding along the way.

At any rate, Alice’s attempt to exploit Bob’s optimal strategy with any-two-cards 4-bet bluffing only results in a small EV change. Bob’s optimal strategy therefore protects him well against exploitative 4-bet bluffing from Alice. And if Bob wanted to, he could probably fine-tune his strategy (for example, remove a couple of 3-bet bluff combos) to eliminate Alice’s small EV increase completely.

So we have seen that Bob’s side of the optimal 3/4/5-bet strategy pair seems robust against Alice’s extreme adjustments. Let’s now turn to Alice, and let her play the optimal strategy while Bob is allowed to do whatever he wants:

Pokerazor simulation 6 (Alice plays optimally, while Bob can do what he wants)
We saw previously that Bob’s attempts to exploit Alice’s optimal strategy by 3-bet bluffing any two cards didn’t work for him. He lost against her optimal strategy (her EV increased from the baseline 150 bb/100 to 179 bb/100), and he lost a lot when Alice counter-exploited him by 4-bet bluffing any two cards (her EV increased to 1261 bb/100). We shall now repeat this simulation by using Bob’s optimal strategy as a starting point, and then we make the adjustment that he 3-bet bluffs any two cards on top of that. All other ranges for value and 5-bet bluffing are as in the optimal strategy:

EV (Alice; Bob 3-bet-bluffs any two cards) =+179 bb/100

Alice’s EV increases with +50 bb/100 (from 129 bb/100) relative to Bob’s optimal strategy, and +29 bb/100 relative to the baseline EV when Bob always folds (150 bb/100). And we remember that if Alice wants to, she can exploit Bob hard by 4-bet bluffing any two cards and pocket more than 1200 bb/100 until Bob adjusts back. So Bob can’t increase his EV by aggressive 3-bet bluffing, which is what we expected.

Then we let Bob 3-bet bluff any two cards and also 5-bet bluff any two cards. In other words, we let him play like a complete maniac, where he 3-bets any two cards and then 5-bets any two cards if he gets 4-bet.

EV (Alice; Bob 3-bet-bluffs/5-bet-bluffs any two cards)
=+300 bb/100

This causes Alice’s EV to more than double relative to the 129 bb/100 she has against Bob’s optimal strategy. We conclude that Alice’s optimal strategy is waterproof against any-two-cards 3-bet bluffing, and that Bob only hurts himself if he tries.

The results from the last two simulations are worth noticing, since they can be uncomfortable scenarios to play when you don’t know whether or not you are defending correctly. But as we have seen, even against a total maniac who 3-bets you from position at every opportunity, you don’t have to do anything else than respond with your memorized optimal 4-bet strategy. If you do this, he will loose money relative to playing an optimal strategy himself, and he will probably end up losing money overall (so that 3-betting is worse for him than folding).

Note that an aggressive 3-bettor can still reduce our EV relative to the baseline EV (i.e. we will make less when he sometimes 3-bets than when he always folds), which is of course intuitively obvious since he has a lot of strong hands in his 3-betting range as well. For example, he could choose to 3-bet only AA, and we could not do anything to prevent him from making money in this situation and reduce our EV. ). So if we always respond with our optimal strategy, we can’t deny the 3-bettor some +EV.

For example, if Bob deviates from his optimal strategy by increasing his 3-bet bluffing from 70% of “IP 3-bet air list” to 100% of the list, we get:

EV (Alice; Bob 3-bet-bluffs all of "IP 3-bet air list")
=+130 bb/100

We still make a little bit more (129 bb/100 –> 130 bb/100) when Bob increases his 3-bet bluff percentage beyond the optimal percentage, but he still reduces our EV relative to our baseline EV when he always folds (150 bb/100 –> 130 bb/100). We can’t prevent Bob from making some money in this situation, and we just have to accept that a player in position has the right to make money by 3-betting us. Of course, our openraise will still be nicely profitable overall, just less profitable than if he had always folded behind us.

As we saw previously, we can exploit a complete maniac by deviating from optimal play to take advantage of the gaping holes in his strategy, particularly if he folds too much to 4-bets. If he lets us exploit him, we can make more money from an exploitative strategy than from our optimal strategy. But then we have to play guessing games with him, and we also run the risk of offering big openings to the other players at the table (they can deviate from optimal play to exploit our non-optimal play). Since an optimal strategy will protect us (and then some) from getting exploited by a wild 3-bettor, this trade-off might not be worth it

A couple of obvious adjustments we can use to exploit a very aggressive with position on us 3-bettor are:

  • 4-bet bluff more, if he folds easily to 4-bets (in other words, he defends his loose 3-betting range to tightly)
  • Drop 4-bet bluffing, but 4-bet more hands like AJ, AT, 99, 88, etc. for value, if he folds too little to 4-bets and calls and 5-bets a lot with weak hands

But we don’t have to make these adjustments to defend out of position against overly aggressive 3-betting. Our optimal strategy is more than enough. It might feel like we’re getting exploited, and some of the reason for that is that a strategy where we fold a lot (70% in the optimal strategy, as explained in Part 1) feels “weak”. But the reality is that a maniac 3-bettor in position ends up costing himself if he starts 3-betting any two cards against our optimal strategy. Keep this in mind every time you feel exploited by a 3-bettor in position.

3. Summary
We have tested optimal strategy pairs for heads-up 3/4/5-betting using the analysis software Pokerazor. We started with a discussion of ABC preflop strategies without 3-bet bluffing or 4-bet bluffing. We then used simulations to show how ineffective and vulnerable these strategies are against players who are capable of reraising as a bluff with any two cards. As a part of this simulation we looked at exploitative adjustments we can make against players with big leaks in their 3/4/5-bet strategies.

Then we tested the robustness of the optimal 3/4/5-bet strategies we defined in previous articles, with the raiser out of position. We concluded that both the raiser’s and the 3-bettor’s optimal strategies were robust, and that they did not give the opponent openings he could exploit by bluffing with any two cards.

In Part 7 we’ll do numerical simulations for flatting heads-up in position. Among other things we’ll compare EV for flatting versus 3-betting for value with hands that are in between clear value hands and clear flatting hands (for example QQ against a tight UTG raiser). In the last half of Part 7 we’ll adjust our heads-up 3/4/5-bet strategies for blind vs blind scenarios.

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/optimal-3-bet-4-bet-5-bet-strategies-in-nl-holdem-6-max—part-6-728#sthash.iNJOVogt.dpuf

Optimal 3-bet/4-bet/5-bet strategies in NLHE 6-max – Part 5

1. Introduction

This is Part 5 in the series Optimal 3-bet/4-bet/5-bet-strategies i NLHE 6-max. In Part 1, Part 2, Part 3 and Part 4 we studied 3/4/5-betting heads-up, with the raiser either in position or out of position. In Part 5 we’ll look at two cases of 3/4/5-betting in multiway pots, namely squeezing (3-betting after the raise has been called) and cold 4-betting (4-betting when the pot has been raised and 3-bet before it’s our turn to act). In this work we’ll use the poker simulation software Pokerazor to estimate the EV for cold 4-betting.

Multiway scenarios are far more complex to model than heads-up scenarios, so the work done for squeezing and cold 4-betting will be less exact than what we have done in the previous articles. But we can use our understanding of heads-up scenarios plus simple modeling to find qualitative guidelines for multiway scenarios.

The structure for Part 5 is:

– Squeezing
– Cold 4-betting

2. Squeezing

The definition of “squeezing” is to 3-bet a raiser after the raise has already been called. The raiser now has to respond to the 3-bet with another player left to act, and we say that he is in a “squeeze” between the 3-bettor and the caller, thus the name.

Below are three examples of squeezing with the raiser in position and out of position:

Example 2.1: Squeezing with the raiser out of position

$100NL

MP ($100) raises pot to $3.5, CO ($100) calls, button ($100) 3-bets pot to $15.50, the blinds fold, and it’s MP’s turn to act.

Example 2.2: Squeezing with the raiser out of position

$100NL

CO ($100) raises pot to $3.5, button ($100) calls, SB folds, BB ($100) 3-bets pot to $14.50, and it’s CO’s turn to act.

Example 2.3: Squeezing with the raiser in position

$100NL

Button ($100) raises pot to $3.5, SB ($100) calls, BB ($100) 3-bets pot to $14, and it’s button’s turn to act.

If the raiser in these examples should choose to defend against the 3-bet by calling, he is setting himself up for difficult postflop scenarios. He will then often have a weak or marginal hand postflop, and he will often have to respond to the 3-bettor’s c-bet without closing the betting (when the preflop coldcaller is left to act). All who have played a bit of NL understand intuitively that this is a difficult situation to play well, and many therefore fold a lot to squeezes when they aren’t strong enough to 4-bet for value.

As we shall see soon, the mathematics of the situation dictates that the raiser and the cold-caller have to defend a lot against the 3-bet to prevent the 3-bettor from having a profitable bluff with any two cards. Since many players can’t (or won’t) defend as actively as they should in an optimal strategy, squeezing is generally a very profitable strategy against weak opposition.

We shall approach the theory behind squeezing using the theory for heads-up 3/4/5-betting as a starting point. We let Alice open-raise pot from some position outside of the blinds, and then she is called by a player between her and Bob. Bob now 3-bets (squeezes) pot with a polarized range made up of value hands and 3-bet bluffs with an optimal value/bluff ratio, Note that this optimal ratio will be slightly different from the corresponding heads-up scenario since the presence of the caller changes the pot size and therefore the pot odds for 3/4/5-betting.

Alice defends against the squeeze by 4-betting/folding out of position and 4-betting/flatting/folding in position. When she 4-bets, she will make her 4-bet a bit less than pot-sized (e.g. to 32 bb in example 2.1 instead of 4-betting pot to 46 bb), and she uses an optimal value/bluff ratio. Bob’s response to a 4-bet is to 5-bet his value hands all-in, and fold his 3-bet bluffs. Alice’s response to an all-in 5-bet is to call with her value hands and fold her 4-bet bluffs.

We’ll now construct a model for a squeeze scenario with the raiser out of position, and then estimate optimal strategy pairs for the raiser and the 3-bettor like we did for heads-up 3/4/5-betting in Part 1 and Part 2

We use the following model:

  • All players start with 100bb stacks
  • Alice open-raises pot (3.5bb) from EP (UTG or MP)
  • A player in CO cold-calls
  • Bob squeezes with an approximately pot-sized 3-bet (14 bb) on the button with an optimal mix of value hands and 3-bet bluffs
  • Alice defends against the squeeze by 4-betting to 32 bb (a bit less than pot) with an optimal mix of value hands and 4-bet bluffs, and otherwise folding
  • We’ll assume that CO always folds to Alice’s 4-bet
  • Bob defends against Alice’s 4-bets by 5-betting his 3-bet value hands all-in, and otherwise folding
  • Alice defends against Bob’s 5-bet by calling all-in with her 4-bet value hands and otherwise folding

This model is similar to the one we used for heads-up 3/4/5-betting with Bob in position. An important difference is that the pot is bigger because of CO’s call when it’s Bob’s turn to act. The optimal strategy pair for Alice and Bob will therefore change relative to the strategy pairs we found for the corresponding heads-up scenario. We’ll assume that CO never continues after a 4-bet from Alice, so that his chips are “dead money” when a 3/4/5-bet war arises between Alice and Bob. We can then estimate the optimal strategy pair using the same method we used heads-up.

We use 14 bb for Bob’s 3-bet size as an average of his bet sizing from various positions. From the examples above we see that Bob risks 15.5 bb when he squeezes with a pot-sized 3-bet on the button, but only 13 bb (beyond the big blind he has already posted) when he squeezes from the big blind. So we use 14 bb as a representative 3-bet size for all positions.

We also assume that Bob uses the heads-up ranges for 3-bet bluffing (“IP 3-bet air list”), 5-bet bluffing (“IP 5-bet air list”) and flatting (“IP flat list”) when he chooses his bluffing and flatting hands:

IP 3-bet air list

A9s-A6s
K9s-K6s
Q9s-Q6s
J9s-J6s
T8s-T7s
97s-96s
87s-86s
76s-75s
65s

100 combos

IP 5-bet air list

A5s-A2s

16 combos

IP flat list

22+
ATs+ AJo+
KTs+ KQo
QTs+
JTs
T9s
98s

Without {KK+}: 162 combos
Without {QQ+}: 156 combos
Without {QQ+,AK}: 140 combos
Without {JJ+,AK}: 134 combos

So Bob’s candidate hands for 3-bet bluffing are the same as when 3-betting heads-up. But since the pot now is bigger, Bob’s optimal distribution of value hands and bluff hands will change relative to the heads-up scenario. Except for this, we’re using a model identical to the heads-up scenario.

We start by asking 3 important questions:

  • How often do Alice and the coldcaller have to defend against the 3-bet squeeze to prevent Bob from profitably 3-bet buffing any two cards?
  • How is the defense against the squeeze shared between Alice and the coldcaller?
  • What is the optimal strategy pair for the heads-up 3/4/5-bet war that occurs between Alice and Bob after Alice 4-bets and the coldcaller folds?

Next we’ll find the answers to these questions:

2.1 Optimal defense frequency against a 3-bet squeeze

When Alice and Bob were heads-up, Bob 3-bet to 12 bb to win a 3.5 + 0.5 + 1 =5 bb pot. He got effective pot odds 5 : 12, and had to win at least 12/(5 + 12) =70% to have an automatic profit with any two cards. Heads-up Alice had the whole responsibility for defending sufficiently often to prevent this. So Alice had to defend 30% of the time in an optimal strategy (and a bit more in position where she sometimes defends by calling and lets Bob freeroll flops with his 3-bet bluffs).

But when Alice’s raise has gotten called by CO, the pot is 3.5 + 3.5 + 0.5 + 1 =8.5 bb when it’s Bob’s turn to act. His 14 bb 3-bet squeeze then risks 14 bb to win 8.5 bb and the effective pot odds becomes 8.5 : 14. Bob needs to win at least 14/(8.5 + 14) =62% to have a profitable 3-bet squeeze with any two cards, and Alice and the coldcaller need to defend at least 100 – 62 =38% to prevent this.

The next question is how this 38% defense job should be shared between Alice and the coldcaller. This question can not be answered exactly, but we can state some qualitative guidelines:

  • The coldcaller has signaled a range with few premium hands when he chooses not to 3-bet Alice
  • Alice must therefore expect that the coldcaller will often fold to the squeeze
  • So most of the job of defending will fall on Alice

To get further, let’s assume that Alice uses her corresponding heads-up defense strategy as a starting point for the squeeze scenario, and then she makes adjustments in the value/bluff ratio to adapt to the new pot size. In other words, she starts with a defense strategy where she defends 30% (only 4-betting and never calling, since she is out of position), and that the cold caller takes care of the rest by defending some percentage x% . The probability of both Alice and the coldcaller folding is then (1-0.30)(1-x), and the probability of at least one of them defending is 1 – (1-0.30)(1-x). This should be 38% in an optimal strategy, and we get:

1 - (1-0.30)(1-x) =0.38
1 - 0.70(1-x) =0.38
0.70(1-x) =0.62
1-x =0.62/0.70
x =1 - 0.62/0.70 =0.11 =11%

So to make the total defense percentage 38%, the coldcaller needs to defend 11% of his range if Alice defends 30% of her range by 4-betting or folding. Furthermore, if the coldcaller defends partly by flatting, he should defend a bit more than 11%, since flatting lets Bob freeroll flops with his 3-bet bluffs instead of having to fold them to a 4-bet. But here we’ll focus on Alice’s strategy, and simply assume that the coldcaller defends enough.

We’ll see later that Alice ends up defending a bit less than 30% after adjusting her strategy to the new pot size, so CO has to defend a bit more than 11%. But we’ll assume that the distribution of the defense responsibility is 30% and 11% before Alice begins adjusting her strategies.

After choosing this starting point for her defense strategy, Alice needs to find the value/bluff ratio for 4-betting that corresponds to the actual pot size. We make a new simplifying assumption and let Alice use the same value range she would have used in the heads-up scenario. Then we only have to adjust the number of 4-bet bluffs to the new optimal ratio, which follows from the new pot size.

We remember that Alice’s ~15% EP opening range is:

22+
A9s+ AJo+
KTs+ KQo
QTs+
J9s+
T9s
98s
87s
76s
65s

194 combos
15%

And when working with the corresponding heads-up scenario we found that Alice used the value range {QQ+,AK} when defending her EP opening range optimally out of position against Bob’s heads-up 3-bets. So we have simplified our way down to this:

  • Alice uses the corresponding heads-up strategy as a starting point for her defense against the squeeze, and then she adjusts it to match the new pot size
  • Alice uses the same value hands she would have used in a heads-up scenario, so that her only adjustment is to change the number of 4-bet bluffs to get to the new optimal ratio (which follows from the new pot size)
  • Alice assumes the coldcaller will take care of the remaining defense, so that the total defense adds up to 38%

What remains is to estimate how many 4-bet bluffs Alice needs to get to the new optimal value/bluff ratio for her 4-betting range. Heads-up this ratio was 60/40, and next we’ll recalculate this ratio as a function of the new pot size.

2.2 Bob’s value/bluff-ratio for 3-bet squeezing

Bob knows that Alice and the coldcaller will defend a total of 38% against his squeeze 3-bet (a bit more when the coldcaller defends partly by flatting). When Alice re-squeezes with a 4-bet to 32 bb, she risks 28.5bb more (32 bb minus the original raise to 3.5 bb) to win a 3.5 + 3.5 + 14 + 0.5 + 1 =22.5 bb pot.

Alice then gets effective pot odds 22.5 : 28.5, and she needs to succeed 28.5/(22.5 + 28.5) =56%. So Bob needs to defend against a 4-bet by 5-betting 100- 56 =44% of his 3-betting range to prevent Alice from having a profitable 4-bet with any two cards. Therefore, 44% of Bob’s hands need to be value hands. We can round this to the nearest 5% to keep things simple, and we find that the optimal value/bluff ratio for Bob’s 3-bet squeezing range is 45/55 (compared to 40/60 for the heads-up scenario).

2.3 Alice’s value/bluff ratio for 4-betting

When Alice re-squeezes Bob’s 14 bb squeeze by 4-betting to 32 bb, and the coldcaller between them folds, the pot grows to 32 + 3.5 + 14 + 0.5 + 1 =51 bb. When Bob shoves his remaining 86 bb, he’s getting effective pot odds 51: 86.

Bob always has some equity when his 5-bet bluffs get called, and we’ll make the same assumption we made in Part 1. There we showed that Bob’s weakest 5-betting hands (the Axs hands he used as 5-bet bluffs) had about 30% equity when they got called by Alice’s value 4-betting hands. So Bob’s 5-bet bluffs win back about 30% of a 100 + 3.5 + 100 + 0.5 + 1 =205 bb pot, or 0.30 x 205 =61.5bb. So Bob effectively risks 86 – 61.5 =24.5 with his 5-bet bluffs and not 86 bb.

The effective pot odds for Bob’s 5-bet bluffs is then 51 : 24.5, and he needs to win 24.5/(51 + 24.5) =32%. To make Bob’s 5-bet bluffs break-even, Alice needs to defend 100 – 32 =68% against Bob’s 3-bets, which we round to 70%.

It follows that Alice’s 4-betting range needs to contain 70% value hands (compared to 60% in the heads-up scenario). Alice’s optimal value/bluff ratio for 4-betting is then 70/30.

2.4 Adjusting to squeeze scenarios in practice

We have now established that Bob should change his value/bluff ratio for 3-betting from 40/60 to 45/55, which means his 3-betting range should be more weighted towards value hands. Alice’s value/bluff ratio for 4-betting should change from 60/40 to 70/30, so range also becomes more weighted towards value.

Do these changes make sense intuitively? Yes, since both players should be less willing to fold when the cold caller’s dead money has made the pot bigger, giving them a better risk/reward ratio when continuing in the hand. So bluffing becomes less effective, and both players adjust by reducing their bluffing frequency.

We have already done a systematic discussion of Alice’s and Bob’s 3/4/5-bet strategy pairs in previous articles. In Part 2 and Part 3 we estimated specific ranges for both of them when Alice raises out of position and Bob 3-bets her in position. In Part 4 we did the same for the scenario where Alice has position on Bob after he has 3-bet from the blinds.

So instead of going through these scenarios one more time with the new value/bluff ratios, we’ll instead look at an example that illustrates how we can adjust in practice. We’ll then use the previously established heads-up optimal strategy pairs as our starting point.

When we’re in a potential squeeze situation, there are two different ways to approach it:

  • We can used precisely defined ranges based on a value range + “IP 3-bet air list” and “OOP 3-bet air list” together with a randomizer. In other words, we’re trying to squeeze 3-bet optimally (the topic for this article)
  • We can realize that we’re in a squeeze and squeeze with whatever cards we have, if we think the situation is good for it (but we’re rarely squeezing with pure trash hands). We’re now playing exploitatively, probably with an unbalanced range (weighted towards an excess of 3-bet bluffs) in selected spots

For example, let’s say button open-raises and SB flats. Button folds often against 3-bets, and SB is loose-passive with a wide flatting range, and he also folds often to 3-bets. You have K7s in the big blind. K7s is to weak for flatting, and it’s not a member of the 3-bet bluff candidate list (“OOP 3-bet air list”) that we use out of position in the blinds.

So if you’re using a strictly defined optimal strategy based on lists + a randomizer, you fold. You know that in the long run you’ll squeeze an optimal amount (which is pretty aggressive) by sticking to your strategy, and you don’t have to add more bluffing hands to get there (and if you do add more bluffing hands, your strategy will become unbalanced, which isn’t necessarily what you want).

Another approach is to exploit whatever good squeezing opportunities that come your way, without worrying about moving away from an optimally balanced 3-betting range. If you want to play this way (deviating from optimal strategy whenever you see an opportunity to exploit a profitable scenario), you’ll 3-bet K7s and similar hands in the scenario described above. You do this because you expect to make a good profit from picking up the pot preflop against two players who fold too much to 3-bets (and when the loosest player calls, you will have position on him postflop). This is obviously a fine way to play these scenarios.

But if you choose the exploitative approach, be aware that you might have to tighten up your 3-betting if your opponents realize you are 3-betting too loosely and decide to fight back (for example by 4-bet bluffing you more). On the other hand, if you choose an optimal strategy, your opponents’ strategy adjustments will have less impact. If you use an optimal value/bluff ratio for 3-betting, they can’t exploit you with any change they make. So you don’ have to make any changes in your strategy, unless you want to deviate from optimal play in order to exploit your opponents new tendencies.

Below are adjustments (based on optimal heads-up strategy pairs) for Alice and Bob in a squeeze scenario where Alice open-raises 35% on the button, small blind calls, and Bob sits in the big blind. This is a common squeeze spot, and you will profit from training solid default strategies for it (both as the raiser and as the squeezer) so that you both can squeeze and defend against squeezes with strong control preflop.

Example 2.4: Squeezing from the blinds against a button steal-raise

$100NL

Alice ($100) raises pot to $3.5 from the button, small blind ($100) calls, Bob ($100) is in the big blind.

Alice uses her default 35% button range defined in Part 2:

22+

A2s+ A7o+

K2s+ K9o+

Q6s+ Q9o+

J7s+ J9o+

T7s+ T9o+

96s+

86s+

75s+

65s

458 combos

35%

Bob’s strategy
Let’s start with Bob’s 3-betting range against Alice. We have assumed that Alice uses the same value range {QQ+,AK} that she would use heads-up on the button against a 3-bet from the blinds. So Bob’s response is to use the same value range for squeezing that he would have used heads-up. Then he adds 3-bet bluffs until he has a 45/55 value/bluff ratio for his 3-betting range.

Using the optimal heads-up strategy pair from Part 3 as our starting point, we get:

  • Bob 3-bets {TT+,AQ+} =62 combos for value from the blinds against a 35% button open-raise, planning to 5-bet all-in against a 4-bet
  • Bob then needs (55/45) x 62 =76 3-bet bluff combos to get a 45/55 value/bluff ratio. So he 3-bets 76% of “OOP 3-bet air list” as a bluff. We can round this to 75%.

We remember that “OOP 3-bet Air list” is:

OOP 3-bet air list
66-22

A9s-A6s

K9s-K8s

QTs-Q9s

J9s-J8s

97s+

87s

76s

65s

98 combos

Since the list has about 100 combos, we can convert directly between number of combos and percentages to use with a randomizer. So Bob 3-bets {TT+, AQ+} for value, and when he has a hand from “OOP 3-bet air list” he uses the randomizer. He 3-bet bluffs if the randomizer returns a number between 0 and 75, and otherwise he folds. This gives him the optimal 45/55 value/bluff ratio for squeeze 3-bets in a 3-way pot.

Alice’s strategy
From Part 3 we remember that Alice’s value range after opening her 35% button range and getting 3-bet heads-up was {QQ+,AK} =34 combos. Then she added the 4-bet bluffs {ATo,A9s-A7s} =24 combos to get a heads-up optimal 60/40 ratio between 4-bet value hands and 4-bet bluffs.

We have chosen a model where Alice uses the same value range in squeeze scenarios, but now with a 70/30 value bluff ratio instead of 60/40. So Alice needs 30/70 bluff combos for every value combo, She therefore 4-bet bluffs with (30/70) x 34 =15 combos. For example, we can drop A8s/A7s from the heads-up 4-bet bluffing range and use {ATo,A9s} =16 combos. The value/bluff ratio then becomes 34/16 =68/32 which is close to the 70/30 that we want.

In addition, Alice defends by flatting a wide range in position, also when there is a cold-caller between her and Bob. Heads-up in position we gave Alice the flatting range {JJ-88,AQ-AJ,ATs,KQ-KJ,KTs,QJ,QTs,JTs} =120 combos, and we can use this as a starting point also in a squeeze scenario. We can adjust as needed, for example by calling tighter if the cold-caller is tight and plays well postflop.

2.5 Summary of the theory for squeezing

We used a model to estimate the new optimal value/bluff ratios that arose in a 3-way squeeze scenario. We found that these were 45/55 for Bob’s 3-betting and 70/30 for Alice’s 4-betting.

Then we looked at an example with Alice on the button, the coldcaller in the small blind and Bob in the big blind to illustrate how we can adjust to these new optimal ratios. We made some simplifications along the way. For example by assuming that Alice uses the same defense frequency (30%) as in a heads-up 3-bet scenario. We also assumed she uses the same value range. Adjusting the ranges to the new value/bluff ratios then simply becomes an adjustment of the number of 3-bet/4-bet bluffs, while the value ranges are the same as in the heads-up scenario.

This method is of course only an approximation, but it captures the essence of the difference between heads-up pots and multiway pots, namely that both the raiser and the 3-bettor should bluff less and 3/4/5-bet more for value. We can make more accurate adjustments, but I recommend you keep things simple and stick with the simple model we have used here when you find yourselves in a squeeze scenario. Use the corresponding heads-up strategy pair for Alice and Bob as a starting point, and tighten up the bluffing ranges somewhat.

The most important points to take with us from this discussion are:

  • The raiser and the cold-caller have to defend a lot (38%) against the squeeze to prevent the squeezer to have a profitable 3-bet with any two cards
  • A bigger pot before it’s the 3-bettors turn to act means a higher value/bluff ratio for all players involved. A bigger pot means better risk/reward ratios and therefore less folding. The players adjust by bluffing less.

If you understand these things and use the model presented above to design (or at least think about) new value/bluff-ranges for 3/4/5-betting adjusted to the new pot-size, you should feel comfortable playing squeeze scenarios.

We discussed one specific example here to show how these adjustments can be done. Those of you who have trained the 3/4/5-bet strategy pairs for the heads-up scenarios can now work through any squeeze scenarios on your own and implement the necessary adjustments, based on the model used in this article.

3. Cold 4-betting

“Cold 4-betting” is 4-betting in a multiway pot after a raise and a 3-bet has occurred before it’s your turn to act. When the pot has been raised and 3-bet, we expect to often clash with a value hand, and the range we 4-bet for value should therefore be very tight. Having a default 4-betting range of only {KK+} in this scenario is reasonable and fairly standard.

Below is an example of a cold 4-bet for value:

Example 3.1: Cold 4-betting

$100NL

UTG ($100) raises to $3.5, button ($100) 3-bets to $12, you have K K in the small blind and 4-bet to $25. UTG and button folds.

The example above illustrates what often happens in these situations. A typical UTG-raiser will open a tight range. Therefore, the 3-bettor will also have a tight value range (of course mixed with a lot of 3-bet bluffs if he plays optimally). So when you 4-bet, you are telling both opponents that you have an extremely tight value-range. Two thinking opponents will put you on mostly {KK+}, fold all medium and weak hands, and only 5-bet all-in with their absolutely best hands.

Since cold 4-betting with an ultra tight value-range forces your opponents to fold a lot, it’s obvious that you should balance your value hands with some 4-bet bluffs. This does two things for you:

  • Your bluffs make an immediate profit if your opponents fold too much
  • Even if they don’t fold too much, thus making your bluffs break-even or close to it, you have now guaranteed that your opponents can’t “escape” your extremely tight value range by folding all weaker hands. If they do, your bluffs will make more money

The last points needs a bit more explaining. Let’s say that you choose to only cold 4-bet {KK+} for value in this situation. Two observant opponents can now save money by folding everything but AA! They will fold KK without hesitation if they are certain you’re only 4-betting {KK+}, since the AA/KK ratio in your range is 6 : 1 when they have KK (there are now 6 possible AA combos, but only 1 possible KK combos in your range). So the probability of their KK hands running into your AA hands is 6/(6 + 1) =86%, so they have an automatic fold with KK against your {KK} value range if you never cold 4-bet bluff.

So you will pick up lots of pots with your {KK+} value range, but you will never get action from worse hands, assuming your opponents are observant and play well. Of course, your opponents will not play perfectly in reality, but we should remember that playing optimally means removing opportunities for our opponents to exploit us. And if we never cold 4-bet bluff, they can exploit us by folding all hands we beat.

Therefore, by adding some cold 4-bet bluffs to our {KK+} value range, we’re forcing our opponents to choose between:

– Give our {KK+} value range action with more hands than AA
– Or lose to our 4-bet bluffs

An optimal mix of {KK+} value hands and some cold 4-bet bluffs guarantees a better average profit than only 4-betting for value, regardless of what our opponents do to defend themselves. We will not estimate what the optimal ratio is, but instead talk about qualitative guidelines for cold 4-betting, so that you can recognize the good cold 4-betting spots when they occur.

We’ll use an exploitative mindset where we’re trying to cold 4-bet bluff in situations where we expect our bluffs to make money. As a bonus we’ll balance our {KK+} value range, but we’re not necessarily trying to use an optimal value/bluff ratio for all situations. This is fine if we save our cold 4-bet bluffs for situations where expect our opponents to fold too much, so that we can exploit them by 4-bet-bluffing more than optimally.

We’ll look at how the following 4 factors influence the EV of cold 4-bet bluffing:

  • The effect of opponent ranges for openraising and 3-betting (which are functions of the openraiser’s position)
  • The ranges they choose to go all-in with against our cold 4-bet
  • Our choice of cold 4-bet bluffing hands (where we use the blocker effect to our advantage)
  • Our choice of value range. {KK+} is a sensible default, but when our opponents start out with wide ranges, we might want to also include QQ and AK

We start by defining the model we’ll use to study the situation:

3.1 Model for cold 4-betting

We’ll use the following model:

  • All players start with 100 bb stacks
  • The raiser (Alice) and the 3-bettor (Bob) use our default ranges for openraising and our estimates of optimal heads-up 3/4/5-bet strategies
  • Alice raises pot (3.5bb) from some position outside the blinds, and Bob 3-bets pot from some position between Alice and us
  • We cold 4-bet to 25 bb (a little less than pot) from the big blind with a mix of value hands and 4-bet bluffs
  • Alice and Bob defend against our 4-bet by 5-betting all-in with some value range and otherwise folding
  • If we get 5-bet all-in, we fold our bluffs and call with our value range (where {KK+} is a good default to use against unknown or tight opponents)

We will in the following only study the EV of our cold 4-bet bluffs, and not the EV of our total cold 4-betting range (remember: We’re trying to exploit our opponents in this situation, so we’re looking for the spots where cold 4-bet bluffing is most profitable)

We start by investigating the effect of opponent ranges, which is a function of the position Alice openraises from. For example, it’s reasonable to assume that the profitability of a cold 4-bet bluff will increase as Alice moves from UTG (tight openrange) to the button (loose openrange). When Alice’s openrange widens, Bob will respond by 3-betting a wider range, and both of them will have to fold more hands to a cold 4-bet.

Of course, if Alice and Bob are trying to play optimally against our 4-bet, both of them will make sure they are defending with an optimal mix of value hands and 5-bet bluffs, so that they are defending correctly against getting exploited by a cold 4-bettor who is bluffing with any two cards. But in practice most players you meet will only shove a tight valuerange and almost never 5-bet bluff. So we should be able to exploit them by 4-bet bluffing more than we should be allowed to, if we pick good spots for it.

We’ll start by assuming Alice and Bob are using the same value ranges for 5-betting all-in that they would have used in a heads-up 3/4/5-bet war against each other (but they drop all 5-bet bluffs from their strategies after we come charging in with a cold 4-bet and make the pot multiway). So if Alice openraises from UTG, and gets 3-bet by Bob on the button, their value ranges are {QQ+,AK} and {KK+}, respectively, as shown in previous articles (see Part 1, Part 2). Later we’ll study the effect of allowing them to use other value ranges against our 4-bet.

3.2 The profitability of cold 4-bet-bluffing as a function of our opponents’ positions

We’ll investigate two scenarios:

– Alice openraises from UTG and Bob 3-bets from the button
– Alice openraises from CO and Bob 3-bets from the button

In both scenarios we’ll assume that Alice and Bob are using the optimal heads-up 3/4/5-bet strategies we have defined in previous articles. We’ll assume that we are in the small blind with a hand we elect to cold 4-bet bluff. Then we use Pokerazor to calculate the EV of our cold 4-bet bluff.

For each scenario we’ll cold 4-bet bluff with 4 different hands:

– 7 2
– A T
– A K
– Q Q

7 2 is a worthless bluff with no blocker effect. A T takes advantage of the blocker effect, since an ace in our hand reduces the probability that our opponents have the value hands AA and AK. A K and Q Q block the value hands AA, KK, QQ and AK, and they also have decent equity against our opponents value ranges. They are also borderline value hands for us in this situation, so it will be interesting to see if they can be played profitably as value hands (i.e. we 4-bet them, planning to call a 5-bet), even if we start out with a default valuerange of only {KK+}.

We’ll first find the EV for all 4 hands when we play them as bluffs (i.e. we 4-bet, planning to fold to a 5-bet), and then we’ll see if any of them can increase their EV by calling the 5-bet instead of folding. We suspect that AK and QQ might be profitable value hands for us when Alice starts with a wide 25% openrange in CO (which will cause Bob to 3-bet a wide range), but probably not when Alice starts with a tight 15% openrange in UTG (which will cause Bob to 3-bet a tight range).

These calculations will be very hard to do manually, but Pokerazor will do it for us in a few seconds. The program let us specify complete ranges and strategies for all players on all streets, and then it can find the EV for these strategies. Unfortunately, Pokerazor is no longer commercially available, but the developers seem to be working on a new and improved version to be released some time in the future. This means you don’t have the opportunity to use this fine poker software tool to verify my calculations or do similar modeling work on your own. That’s a pity, but you simply have to accept the numbers I present here, and focus on the results, not the computational method.

We won’t repeat all the optimal 3/4/5-betting ranges and strategies here, so look them up in the previous articles if you need to refresh them.

Scenario 1: Alice openraises from UTG and Bob 3-bets from the button
The complete list of strategies is:

  • Alice’s strategy in EP: Openraises to 3.5 bb with the ~15% EP default range, 5-bets the corresponding value range {QQ+,AK} all-in against our cold 4-bet and folds everything else
  • Bob’s strategy on the button: 3-bets to 12bb with an optimal {value range} + {bluff range} ={KK+, A5s,As4s,Ah4h,Ad4d} + {30% of “IP 3-bet air list”} ={KK+,A5s,As4s,Ah4h,Ad4d} + {A9s,As8s,Ah8h,K9s,Q9s,J9s,T8s,97s,65s}, 5-bets the corresponding value range {KK+} (where all 5-bet bluffs have been dropped) against a cold 4-bet from us, and folds everything else
  • Small blind: Folds a random hand
  • Our strategy in the big blind: Cold 4-bet bluff to 25 bb, folding to an all-in 5-bet

Note that we have replaced Bob’s 30% random 3-betting of bluff hands from “IP 3-bet air list” with 30 specific combos from the list (which has 100 combos). This makes it easier to construct the Pokerazor input.

We have also assumed that SB’s fold means he folds 100% of his hands, including AA and KK. To be exact, we should have taken into consideration the fact that small blind will sometimes wake up with a value 4-betting hand, but ignoring this won’t make much of a difference (since this range is very tight). Ignoring the small blind completely makes the calculations much simpler. In fact, including the small blind’s range when Alice and Bob are using wide ranges makes the calculations prohibitively complex for Pokerazor, since the number of possible hand combinations “explode”. But I checked this simplifying assumption in a set of separate calculations with tight ranges for Alice and Bob (where the calculations could be done taking into account small blind’s range), and the EV differences for our cold 4-bet bluffs were negligible (less than 0.05 bb difference between the exact and approximate calculations).

We start by computing the EV for our 4 candidate hands when we play them as pure bluffs, and always fold to a 5-bet. The EVs are given in big blinds, and computed as the difference between our final stack and our starting stack. For example, EV =+0.66 bb means our stack changed from 100 bb at the beginning of the hand (including the big blind we posted) to 100.66 bb when the hand was over.

The EV for cold 4-bet bluffing with each of the 4 candidate hands are:

– 7 2 : +0.66 bb
– A T : +2.35 bb
– A K : +5.51 bb
– Q Q : +0.83 bb

Against a tight EP openrange followed by a tight 3-betting range a random bluff with a trash hand is approximately break even under the assumptions made in our model. Using the blocker effect to our advantage increases the profitability, and A K performs best with an EV of more than +5 bb

The blocker effect for A T and A K is significant, particularly for A K which blocks both AA, KK and AK in our opponents’ value ranges, and it also has good equity against QQ. Q Q also has a small blocker effect against opponent value hands, but the hand is just barely performing better than the trash hand 7 2 . This makes sense, since Q Q only blocks other QQ in Alice’s value range {QQ+,AK} and no hands in Bob’s tight value range {KK+}.

Then we investigate how the EVs change when we play A K and Q Q as value hands heads-up those times one opponent folds and the other one 5-bets all-in. We call the 5-bet if Alice 5-bets and Bob folds, or if Alice folds and Bob 5-bets. If Alice 5-bets all-in and Bob calls, we will fold as before.

It makes sense to only call an all-in 5-bet when we are heads-up, since the probability of being up against AA or KK is huge when two players have gone all-in in front of us. In this particular case we of course know that Bob only can have {KK+} when he gets all-in (so we should fold against him heads-up as well), but we make things simple and assume we’re willing to get all-in with QQ and AK and take our chances, if we can do so heads-up.

– A K : +3.37bb
– Q Q : -6.38bb

We see that both hands perform worse as value hands than as bluffs after a tight open-raise and a correspondingly tight 3-betting range, even if the 3-bettor has a range full of 3-bet bluffs (60% of his 3-bets are bluffs in an optimal strategy). Calling a 5-bet all-in with A K when heads-up is not very bad, but we lose relative to folding (+5.51bb –> +3.37bb) and playing the hand as a 4-bet bluff. For Q Q calling an all-in is horrible, and this is due to a “double whammy” where we don’t block any of the hands AA/KK that beat us, and we have very bad equity against those hands (while AK is blocking both of those hands, and only has terrible equity against AA).

We conclude:

Against a tight openraiser from early position, followed by a 3-bet from an optimal (or near optimal) 3-betting range, cold 4-bet bluffing with a random trash hand is close to break even if the raiser and the 3-bettor use the value ranges {QQ+,AK} and {KK+}, respectively. We can increase the EV of our bluff by using the blocker effect, picking our bluffs from the best Ax hands (AK in particular). But we should not use a value range wider than {KK+} in this case. AK doesn’t suffer much from getting all-in heads-up, but QQ (and similarly all lower pairs) performs very poorly as a value hand).

Now we move Alice to CO and let her open her standard 25% CO range, while Bob attacks her with the corresponding optimal 3-betting range

Scenario 2: Alice openraises from CO and Bob 3-bets from the button
The complete list of strategies are:

  • Alice’s strategy in CO: Openraises to 3.5 bb with the ~25% default CO range, 5-bets the corresponding value range {TT+,AQ} all-in against our 4-bet, and folds everything else
  • Bob’s strategy on the button: 3-bets to 12 bb with an optimal range {value range} + {bluff range} ={QQ+,AK,A5s,A4s,A3s} + {70% of “IP 3-bet air list”} ={QQ+,AK,A5s,A4s,A3s} + {A9s-A6s,K9s-K6s,Q9s-Q6s,J9s-J8s,T8s-T7s,97s,6s5s,6h5h}, 5-bets the corresponding value range {QQ+,AK} (where all 5-bet bluffs have been dropped) against a 4-bet by us or by Alice, and folds everything else
  • Small blind: Folds a random hand
  • Our strategy in the big blind: 4-bet-bluff to 25bb, and fold to a 5-bet

The assumptions are the same as in the previous scenario. We have here specified 70 3-bet bluff combos from “IP 3-bet air list” for Bob to use instead of a randomizer, and we have assumed small blind is folding 100% of his hands to make the Pokerazor calculations practical.

The EV for playing our 4 candidate hands as 4-bet bluffs now becomes:

– 7 2 : -1.60bb
– A T : +0.27bb
– A K : +2.00bb
– Q Q : +0.02bb

The trend between the hands is the same as in the previous case (the blocker effect is significant), but now only AK has positive EV. However, none of the hands are losing big. Is this surprising? Not really. We would intuitively expect to make more from bluffing when Alice’s and Bob’s ranges are wide, but we have to remember that both of them are using the optimal HU strategies that defend them against being exploited by a any-two-cards-bluff. These strategies/ranges are not quite optimal in multiway scenarios, but they still do a pretty good job. So it’s not really surprising that our bluffs are close to break even, no matter where Alice is opening from, when both she and Bob are using the value ranges they would have used heads-up in a 3/4/5-bet war against each other. This illustrates an important property of optimal strategies: They are robust. Small changes in the situation don’t cause large changes in the optimal strategies, and playing near-optimally for any given situation is usually good enough.

The small changes in EV for our 4-bet bluffs when Alice moves from UTG to CO probably contains some “numerical noise” as well, so we won’t draw any strong conclusions from these changes. For example, our definitions of ranges for Alice and Bob are not perfect down to the last combo, and we also did some numerical rounding along the way when we defined these strategies. The most important observation is that our cold 4-bet bluffs with random trash is close to break even when both Alice and Bob defends with something close to the optimal heads-up 3/4/5-bet strategies they would have used against each other.

This means that the defense strategies for Alice and Bob work well against random 4-bet bluffing, and defending against random bluffing is partly what they were designed to do. If we had been able to exploit Alice and Bob hard by cold 4-bet bluffing any two cards, something would have been wrong.

Like in the previous case we now move on to see if A K and Q Q work as value hands in this scenario. Like in the previous case we now call a 5-bet if we can get all-in heads-up against either Alice or Bob, but not both (we’re assuming the risk of clashing against AA/KK is too high when this happens).

– A K : +8.29bb
– Q Q : +6.91bb

Not surprisingly both hands now perform well as value hands, and with a big increase in EV compared to bluffing with them. This is obvious when we look at some of the value hands Alice is now shoving: JJ, TT and AQ. All of these are dominated hard by AK and QQ. Also, when Alice is raising from CO, Bob is also shoving QQ and AK for value, so our AK and QQ are hurt much less when they get all-in against his tight value range.

Finally we’ll run a series of calculations where we let Alice tighten up her value range from {TT+,AQ} to {QQ+,AK} out of respect for our signal of strength when we 4-bet cold from the blinds. And we let Bob continue with his {QQ+,AK} value range.

The EV for 4-bet bluffing now increases (not unexpectedly) for all hands, since Alice folds a lot more:

– 7 2 : +0.97bb
– A T : +2.56bb
– A K : +4.76bb
– Q Q : +2.12bb

And it’s here we can gain EV by 4-bet-bluffing against wide ranges. We saw previously that there wasn’t much difference between cold 4-bet bluffing against wide and tight opponent ranges, if they defended close to optimally, and had the “guts” to keep 5-bet-shoving with the value ranges they would have used heads-up against each other. For Alice this means she has to be willing to continue to shove with both TT and AQ to avoid giving us and Bob an opportunity to exploit her 25% opening range (which is what will happen when she begins folding too much). This value range is designed to use heads-up against Bob, but if she deviates drastically from it, she will make herself vulnerable against our cold 4-bet bluffs.

When Alice “chickens out” by dropping 3 of her value hands (JJ, TT, AQ), she leaves “dead money” in the pot, and creates an opening for profitable cold 4-bet-bluffing with any two cards. Bob can now decide to take some of the defense responsibility that Alice refuses to take, but remember that Bob’s 3-bet strategy against a CO raiser starts with choosing {QQ+,AK} as value range and placing the next tier of good hands (JJ, TT and AQ) in the flatting range.

So Bob can’t increase his defense frequency against our 4-bet by 5-betting more for value since he doesn’t have any more value hands to use, only 3-bet bluff hands. So if he wants to defend more, he has to start 5-bet-bluffing. But bluff-shoving all-in with hands like K 9 and A 4 after bluff 3-betting and then getting cold 4-bet, takes a better understanding of the dynamics of the situation, better reads, and more guts than most players possess.

Note that folding JJ, TT and AQ against our 4-bet isn’t unreasonable for Alice. We’re signaling great strength, and she is in a squeeze between us and Bob who has 3-bet. But the mathematics dictates that if she doesn’t defend her 25% openrange often enough, we have to make a profit with our 4-bet bluffs if Bob doesn’t do anything to prevent it (and as we saw, this is hard for him to do).

Still, there is a balancing effect at work here, since Alice now pays off less to our value hands. The EV for playing A K and Q Q as value hands is now:

– A K : +7.81bb
– Q Q : +4.61bb

But if Alice’s folds of JJ, QQ and AQ have given us an opening for 4-bet bluffing any two cards profitably, she can’t make back this loss by folding the few times we have a value hand. After all, there are only 34 combos of {QQ+,AK} in our range, and 1292 other random hands we can now cold 4-bet bluff profitably.

Finally, if Alice should be scared enough to fold even QQ and AK against our cold 4-bet (for example, of she plays too tight against 4-bets to begin with, and if our table image is good) we can print money by cold 4-bet bluffing with any two cards:

– 7 2 : +2.92bb
– A T : +4.46bb
– A K : +6.42bb
– Q Q : +3.80bb

We conclude:

Cold 4-bet bluffing is not necessarily more profitable when the raiser and 3-bettor are using wide ranges, unless they begin to deviate significantly from optimal play by folding too many value hands against our seemingly strong 4-bet. If Alice and Bob are playing wide ranges, and they defend against our 4-bet using the optimal heads-up strategies they would have used against each other, our cold 4-bet bluffs are about break even But if they begin folding value hands against us that they would have played against each other, our bluffs become much more profitable. It’s the raiser in particular who is vulnerable to this, since she is “forced” to play many hands for value to defend correctly against getting exploited. 

3.3 Summary of cold 4-betting

We learned the following:

  • Cold 4-bet bluffing with random trash against a raiser and a 3-bettor who both defend optimally (or close to it) is about break even, regardless of their positions.
  • By using the blocker effect (first and foremost hands with an ace) we can improve the EV of our bluffs significantly.
  • The raiser and the 3-bettor can give us openings for profitable cold 4-bet bluffing if they begin to fold their weakest value hands (that they would have played against each other, but now decide to fold after getting cold 4-bet). For example, if a CO raiser folds his value hands JJ, TT and AQ
  • Using {KK+} as value range for cold 4-betting is a good default against tight ranges (for example against a ~15% raiser followed by an optimal 3-betting range). But against wide ranges (for example, a ~25% CO raise followed by an optimal 3-betting range) QQ and AK also become value hands

We have discussed cold 4-bet bluffing in isolation, but it’s important to see the cold 4-bet bluffing as “twin” to the value 4-bet we make in this situation. For example, if we always have {KK+} when 4-betting cold against two tight opponent ranges, it’s easy for the opposition to adapt. They can shove {AA} and fold everything else, and never let us get our stack in as big favorites preflop. But if we cold 4-bet bluff occasionally, it’s impossible for them to avoid paying us off one way or the other. They will either fold too often and make our bluffs nicely +EV, or they will be forced to pay off our A and KK with worse hands sometimes.

In theory one can calculate how to perfectly balance a {KK+} or {QQ+,AK} value ranges in a 3-way raised and 3-bet pot with cold 4-bet bluffs, but we won’t do that here. The point of our discussion was to illustrate what makes cold 4-bet bluffing work, how different types of hand perform, how we should choose our value range, and how the raiser and the 3-bettor make themselves vulnerable if they are not willing to felt their weakest value hands.

If they refuse to get all-in with hands weaker than QQ and AK after starting with wide ranges for openraising and 3-betting, we can cold 4-bet bluff any two cards profitably. At least until they adjust, but when they begin adjusting, our 4-bet value hands make more money, and now we can dial back on the bluffing and instead exploit their looseness. Note that when they have pegged us as a loose cannon who is willing to cold 4-bet bluff often, this impression will last a long time since these situations don’t come up often.

Using bluffs to guarantee ourselves action on our good hands is one side of optimal play that we haven’t talked much about, but it’s an important part of the equation.

By mixing value hands and bluffs in an optimal (or close to it) ratio, we’re making it impossible for the opposition to “escape” our value hands by folding a lot, and we guarantee ourselves a certain minimum profit. Then our opponents can choose whether they’d like us to get this guaranteed profit from our value hands (when they call or play back at us too much) or from our bluffs (when they fold too much). We’ll talk more about this side of optimal strategies in Part 6 and Part 7.

4. Summary

We studied two cases of 3/4/5-betting in multiway pots, namely “squeezing” and “cold 4-betting”

For squeezing we started with the heads-up strategies from previous articles and adjusted them to the new multiway scenario by taking the new pot size into account and then doing some simplifying assumptions. This let us estimate new optimal value/bluff ratios, and we used an example to illustrate how we can adjust our heads-up 3/4/5-bet strategies to use in squeeze scenarios.

For cold 4-betting we assumed that our opponents started with the heads-up optimal 3/4/5-bet strategies, and that they responded to our cold 4-betting by only 5-betting their value ranges from those strategies. We used the poker analysis software “Pokerazor” to study this scenario in detail, and we saw that the profitability of a cold 4-bet bluff is very dependent on the blocker effect, and our opponents’ value ranges.

What remains to be done before we end this article series on default preflop strategies in NLHE 6-max based on principles for game theory optimal play, is to do some numerical testing of the strategies we have designed. We shall do this in Parts 6 and 7. We’ll use Pokerazor again to do numerical simulations for various preflop scenarios. We’ll also discuss exploitative play versus optimal play, and when to use one or the other. In Part 7 we’ll also look at 3/4/5-betting in a blind vs blind scenario, which is a topic we haven’t looked at so far.

Good luck!
Bugs – See more at: http://en.donkr.com/forum/optimal-3-bet-4-bet-5-bet-strategies-in-nlhe-6-max—part-5-533565#sthash.ZwlU6ch6.dpuf