Tag Archives: postflop play

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

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 6

1. Introduction
This is Part 6 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.

I Part 1, Part 2, Part 3, and Part 4 we discussed postflop play heads-up in position after flatting preflop. Then in Part 5 we began working on postflop strategies for the preflop raiser out of position in this scenario.

We’ll continue this work in Part 6. Some of the things we’ll discuss are:

  • More about the consequences of choosing to bet a street
  • Show mathematically that the raiser’s optimal turn/river strategies defends her against any-two-cards floating
  • Study the effect of the raise’s opening range on her postflop strategies

We’ll warm up with a discussion of “follow through” when you have chosen to bet a street:

2. On the consequences of choosing to bet a street

The scenario we studied in Part 5 were based on the following set of assumptions:

  • Alice (100 bb) raises out of position and Bob (100 bb) flats in position
  • Alice’s standard bet sizing on the flop/turn/river (those times she chooses to bet) is 0.75 x pot/0.75 x pot/0.60 x pot
  • Alice c-bets 100% of her preflop range on the flop

Now Bob has position on Alice, and he defends postflop using the optimal strategies we built for him in Parts 1-4 of this article series. He will raise some hands, flat some hands, and fold some hands. The most interesting scenarios for us to study are the ones where Bob flats, so that Alice gets a bet/check-raise/check-call/check-fold decision to make on the next street. The reason these scenarios are the most interesting ones for us is that the rest of the hand will often be automatic when Bob raises anywhere (most of the time Alice will fold her weak hands and 3-bet her best hands for value, and there will be no real decisions to make.

So we will here focus on the postflop scenarios where Alice has the betting lead on the turn after c-betting the flop and getting called. As discussed in Part 5, it’s ow important for her to use a turn strategy adapted to her c-betting range. If she gives up too easily on the turn after c-betting the flop and getting called, Bob can exploit her by floating her c-bet with any two cards, planning to auto-bluff turns those times Alice checks.

To illustrate how this can happen, let’s warm up with some simple math. Let’s say that Alice openraises her default ~25% range from CO, and Bob flats on the button. The flop comes dry and without any possible draws:

Alice now decides to c-bet her entire range, since Bob’s preflop flatting range should mostly miss this flop. This is a reasonable assumption, since Bob’s default preflop flatting range in this case is:

IP flat list after ~25% CO openraise

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

140 combos
11%

We have the tools for checking this assumption, and we can use Pokerazor to calculate how often Bob’s flatting range has flopped a pair or better on this dry texture:

We see from the figure above that there’s only a 39.4% chance that Bob has flopped one pair or better (see the list “Cumulative frequency” to the right). Alice’s c-bet is 0.75 x pot, so if she picks up the pot more than 0.75/(1 + 0.75) =43% of the time, she profits from making a c-bet with any two cards. Therefore, if Bob only calls the c-bet with a pair or better (and there are no draws he can have), a c-bet will be automatically profitable for Alice, since Bob then folds 100 – 39.4 =60.6%.

But this does not mean that Alice’s any-two-cards c-bet is profitable against a good, thinking player that understands the situation. Bob knows that Alice knows that his range has missed the flop more than half the time. He also knows that if he folds more than 100 – 43 =57%, he is giving Alice a license to steal with any two cards. Therefore, Bob will also call the flop with some hands without a pair or a good draw, for example the overcard hands AK and AQ.

These calls that Bob makes with overcards and weak draws (when he has any), are floats. Bob bases this on a combination of several factors:

  • The chance of getting a bluffing opportunity on a later street
  • The chance of checking the hand down on the turn and river and winning a showdown unimproved
  • The chance of checking the hand down on the turn and river and winning a showdown after improving marginally (for example after making a low pair on the turn)
  • The chance of making the best hand on a later street and getting paid (in particular, having good implied odds when he floats with a good draw)

Note that may of the thin flop calls/floats Bob makes can’t be justified based on pot-odds alone, if Bob’s plan is to play strictly fit-or-fold on later streets. We’re only getting pot-odds 1.75 : 0.75 =2.33 : 1 on the flop, and we’re calling with a hand like AQ only to spike a pair, we need (47 – 6) : 6 =7 : 1 to call and draw to 6 outs on the turn (and not planning to sometimes steal the pot when we miss).

For a new NLHE player these thin flop calls might seem “incorrect”, since Bob seems to call with hands like AQ only to draw to two overcards, hoping to make a pair on the turn. But this is not the only reason why Bob calls. Keep in mind he is already beating many hands in Alice’s wide and weak c-betting range, and he will sometimes with unimproved against these hands. For example, he can win with ace high when the turn and the river goes check-check. Bob will also be able to steal some pots on the turn or river if he chooses to use AQ as a bluff when Alice checks to him.

Therefore, since Bob will often (and correctly) float the flop without a pair or a good draw on dry flops, Alice can’t check and give up on the turn every time her flop c-bet gets called. She knows that her 25% CO range is weak on the flop, and she knows that Bob knows this as well:

Alice’s Default 25% CO-range

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

326 combos
25%

Not only is Alice’s c-betting range (which is her entire preflop range) weak on this flop, but it’s weaker than Bob’s range!. We see from the figure above that the chance of Alice having flopped one pair or better is a measly 25.4%, compared to 39.4% for Bob’s range. The observant and optimally playing Bob can therefore easily make many light flop floats, hoping Alice will screw up on the turn or river and give him opportunities to steal profitably with any two cards.

If Alice also plays optimally postflop, Bob can’t expect to profit from any-two-cards floating (as we’ll see in a minute), but at the very least he can float enough to prevent her from c-betting any two cards profitably (and we showed that this was possible for him in Part 5).

The gist of it is that Alice knows that Bob’s range is weak, but her range is weak as well. She knows this, Bob knows this, and Alice knows that Bob knows this. Therefore Alice can expect Bob to call his optimal 57% on this flop (and as discussed in previous articles, Bob chooses to slowplay his few monster hands on thus type of flop). So Alice can’t c-bet her entire preflop range on this dry flop without having a plan for how to 2-barrel/check-call/check-raise on later streets to prevent Bob from floating her profitably with any two cards. A player that thinks he is exploiting the player in position by c-betting a lot on dry flop, expecting lots of folds, runs the risk of getting counter-exploited by the player in position if this player understands what is going on.

To illustrate what can happen without a good turn/river plan, assume that Alice gives up on 50% of turns (and she will have no pair/no draw far more often than this after c-betting her entire 25% CO range on a dry flop). Now Bob can call her 0.75 x pot c-bet on the flop and then auto-bet turns when Alice checks. If Alice never check-calls or check-raises, Bob knows that he wins when she checks. Bob’s risk when floating the flop was then only 0.75 x flop-pot, since his turn bluff has zero risk (Always check-folds).

So Bob risked 0.75 x flop pot (P) to win 1.75P (the flop pot + Alice’s c-bet). 50% of the time he loses 0.75P and 50% of the time he wins 1.75P. The EV for his any-two-cards flop float is then:

EV (float) =0.50(+1.75P) + 0.50(-0.75P) =0.5P

Bob’s EV for floating the flop with a random hand against a weakly playing Alice was 1/2 of the flop pot. Not bad! So what should Alice do? In Part 5 we designed the following defense equation for Alice’s turn play after c-betting the flop and getting called:

2-barrel% + 1.75 x check-continue% =70%

where check-continue =check-raise or check-call. And the same equation also applied to river play after 2-barreling the turn and getting called:

3-barrel% + 1.75 x check-continue% =70%

Note that the mathematics does not tell us whether or not it’s correct for Alice to c-bet her entire range on the flop. What it does tell us is that when she has chosen to do so she will be vulnerable to any-two-cards floating if she is not willing to play the next street according to these defense equations.

There’s a subtle point buried here:

If you bet a street, and the thought of barreling 70% (or an equivalent combination of barreling, check-calling and check-raising) on the next street will make you feel sick, almost regardless of which card falls, you are probably betting too much on the current street

An obvious example would be if Alice elected to c-bet her entire 25% CO range on a coordinated flop like this one:

This range hits Bob’s solid preflop flatting range hard, as shown below:

Bob’s flatting rage is full of pocket pairs and suited/coordinated medium/high cards, and this is a very good flop for him. There a whopping 63.9% chance he has one pair or better, and he also has lots of gutshots, open-enders and flush draws in his range. Alice’s range has also connected often with this flop, but rarely hard (wide ranges hits lots of flops in various ways, but often in weak ways), and she is out of position to boot.

Therefore, c-betting this coordinated flop, planning to 2-barrel/check-call/check-raise optimally on the flop seems like a very bad and unprofitable idea for Alice. She ha to respect the fact that Bob’s preflop range has hit this flop harder than her range, and that he also has advantage of position. So Alice should check some of her weak hands (hands like 22, A2s, etc) instead of c-betting her entire range.

By removing weak hands from her flop c-betting range, Alice is setting herself up for reaching the turn with a stronger range those times she chooses to c-bet and she gets called. When her turn range is stronger, it will be easier and much more comfortable for her to play the turn optimally, according to the defense equations, since a larger fraction of her turn range now will be strong enough to 2-barrel, check-calling, or check-raising without feeling sick about having to do so.

The main point is that if you often find yourself on the turn, out of position after having c-bet the flop and gotten called, and without a hand you feel comfortable 2-barreling, check-calling or check-raising, you have a problem. You might try to fix this problem by check-folding a lot of turns so that you don’t spew more chips, but you will probably (and correctly) feel that the player in position is bluffing you a lot.

And then you might conclude “Playing the turn out of position is hard, I need to get better at it” without realizing that the root of your problem is located in your flop c-betting strategy. You should fix the problem by starting with your flop c-bet decisions on textures that are bad for you and good for your opponent. Check and give up with more weak hands on these flops, and I can guarantee that your turn decisions will become easier and more pleasurable those times you do c-bet and get called.

The next step for us is to verify that the raiser’s optimal turn/river barreling/check-calling/check-raising strategies that we designed in Part 5 in fact do defend her sufficiently against any-two-cards floating.

3. Verifying mathematically that the preflop raiser’s turn/river strategies defend her against any-two-cards floating
We’ll now show that Alice’s turn/river strategies according to the defense equation protects her from getting exploited by a player who floats her with random weak hands in position.

In Part 5 we verified that Bob’s optimal calling with a bluffcatcher in position defended him correctly against any-two-cards barreling from Alice. She could not make money by c-betting a random worthless hand on the flop and then continuing to bet the turn or river when called.

Here we’ll use the same method to show that Alice’s optimal turn/river strategies defends her against Bob’s floating with random worthless hands. We calculate the probabilities associated with all possible outcomes, find Bob’s EV for each of them, and then write out the total EV equation for his float.

We’re assuming that Bob is floating with a worthless hand on a dry flop (where Bob is calling with all hands he defends with). To keep the math simple, we’ll assume that Bob’s only chance to win is when Alice checks and gives up on a later street (he has 0% pot equity, and will never win a showdown). Bob’s plan is to call the c-bet on the flop, and then auto-bluff the turn when Alice checks. Those times Alice 2-barrels the turn, Bob always folds.

Alice’s strategy is to play the turn and river in such a way that random floating is not automatically profitable for Bob. She does this by building barreling/check-calling/check-raising ranges that satisfy the defense equations defined previously.

Alice’s strategy on the flop
Let the pot size be P on the flop. We begin by assuming Alice c-bets 0.75P with 100% of her opening range on a dry flop. Bob calls with his worthless float, planning to bluff the turn if checked to. The pot grows to P + 2 x 0.75P =2.5 P, and both players have put 0.75P into the pot postflop.

Alice’s strategy on the turn
We’ll show that Alice can make Bob’s flop floats break even by playing the turn according to the defense equation:

2-barrel% + 1.75 x check-continue% =70%

First, assume that Alice defends by only 2-barreling, so that check-continue% =0 and 2-barrel% =70.

– Alice 2-barrels: 70%:
– Alice check-raises/check-calls: 0%
– Alice check-folds: 30%

Bob then folds his float to Alice’s 2-barrel 70% of the time and loses his 0.75P flop call. 30% of the time he gets the opportunity to bluff the turn. Alice always check-folds, and Bob makes +1.75P (the flop pot + Alice’s c-bet).

The EV equation for Bob’s flop float is:

EV (float)
=0.70(-0.75P) + 0.30(+1.75P)
=-0.525P + 0.525P
=0

So Alice’s 2-barrel strategy makes it impossible for Bob to profit from floating the flop with any two cards. Now we look at the more general form of the equation where Alice also check-calls and check-raises. For example, assume Alice 2-barrels 35% (Bob folds), check-raises 10% (Bob bets and folds to the check-raise), and check-calls 10% (Bob bets and gives up when called). Bob then folds to Alice’s 2-barrels, auto-bets the turn when checked to, and gives up with his worthless hand when check-called or check-raised.

– Alice 2-barrels: 35%:
– Alice check-raises: 10%
– Alice check-calls: 10%
– Alice check-folds: 45%

Note that this strategy satisfies the defense equation since:

35% + 1.75(10% + 10%) =70%

35% of the time Bob folds his float to Alice’s 2-barrel and loses his 0.75P flop-call. 10% + 10% =20% of the time he bluffs the turn with a 0.75 x turn-pot bet, gets check-raised or check-called and gives up. Hen then bets 0.75 x turn-pot =0.75 x 2.5P =1.875P, and loses this amount in addition to his 0.75P flop-call for a total loss of -0.75P – 1.875P =-2.625P. The remaining 100 – 35 – 20 =45% of the time he bluffs the turn successfully and picks up the 2.5P pot, where 1.75P is profit (the flop pot + Alice’s 0.75P flop c-bet).

The EV equation for Bob’s float now becomes:

EV (float)
=0.35(-0.75P) + 0.20(-2.625P) + 0.45(+1.75P)
=-0.2625P -0.525P + 0.7875P
=0

And we see that Alice can also defend optimally and make Bob’s random floats break even by going from a 2-barrel/check-fold strategy to a 2-barrel/check-raise/check-call/check-fold strategy. She builds her 2-barreling, check-calling, and check-raising ranges so that they satisfy the defense equation, and Bob’s random flop floats can not make money.

4. The effect of the raiser’s preflop range on her postflop strategies
We end this article with a new set of flop/turn/river strategies for the flop example we worked through in Part 5. In that example, Alice started out with her 15% EP range:

~15% UTG range

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

194 combos
15%

Bob called with his standard “IP flat list” against an UTG raiser:

IP flat list after ~15% EP openraise

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

162 combos

Flop/turn/river came

4.1 Alice’s postflop strategy after 15% UTG-raise
Alice began by c-betting 100% of her preflop range on the flop, and Bob called (we know that he will defend on this type of dry flop by only calling). Then Alice used turn/river strategies designed to prevent Bob from floating her profitably with any two cards on the flop or turn. Alice’s flop7turn/river bet sizing was 0.75 x pot/0.75 x pot/0.60 x pot, and Bob called flop and turn. We found the following turn/river strategies for Alice, based on this bet sizing and the defense equation we derived previously:

  • Check-raise:
    {JJ} =3 combos
    Value bet:
    {99,66,33,J9s,AA-QQ,AJ} =41 combos
  • Check-call
    {KJs,QJs,JTs,TT,A9s} =18 combos
    Bluff:
    {QTs,AK,AQ,KQs} =40 combos

 

  • Check-raise:
    {99} =3 combos
    Value bet:
    {66,33,J9s,AA-QQ} =26 combos
  • Check-call:
    {AJ} =12 combos
    Bluff:
    {10 AK-combos} =10 combos

We’ll now estimate Alice’s turn/river strategies after starting out with a 25% openraise in CO. Bob flats the same preflop range as before, except for 3-betting QQ/AK for value instead of flatting them.

4.2 Alice’s postflop strategy after a 25% CO-raise
Alice openraises:

Alice’s default 25% CO-range

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

326 combos
25%

And the flop comes as before:

As before, Alice begins postflop play by c-betting 100% of her preflop range, and Bob calls. We have to estimate her new turn/river strategies, based on her opening range, card removal effects and the requirement that the defense equation should be satisfied.

On the turn Alice’s range is reduced from 326 to 282 combos:

She must now play the turn so that:

2-barrel% + 1.75 x check-continue% =70%

If she only 2-barrels, this corresponds to barreling 70% of 282 combos, which is 0.70 x 282 =197 combos. We can rewrite the defense equation as:

2-barrel-combos + 1.75 x check-continue-combos =197

Below is a suggestion for a turn strategy for Alice that satisfies the defense equation. The corresponding turn strategy after a 15% UTG openraise is listed for comparison:

Turn strategy after 25% CO-raise

  • Check-raise:
    {JJ} =3 combos
    Value bet:
    {99,66,33,J9s,AA-QQ,AJ,KJ,QJ,JT} =77 combos
  • Check-call
    {J8s,TT,A9s,T9s,98s,97s} =21 combos
    Bluff:
    {QTs,T8s,AK,AQ,KQ,KT,87s} =76 combos

Test of defense equation:

(77 + 76) + 1.75(3 + 21) =195 (optimal =197)

Turn strategy after 15% UTG-raise

  • Check-raise:
    {JJ} =3 combos
    Value bet:
    {99,66,33,J9s,AA-QQ,AJ} =41 combos
  • Check-call
    {KJs,QJs,JTs,TT,A9s} =18 combos
    Bluff:
    {QTs,AK,AQ,KQs} =40 combos

Test of defense equation:

(41 + 40) + 1.75(3 + 18) =118 (optimal =118)

Compared to play after an UTG raise Alice is now forced to barrel and check-call much thinner in order to protect herself against Bob’s floats. We will not discuss whether or not these ranges are too loose, but keep in mind what we discussed previously about setting ourselves up for weak turn ranges by c-betting too wide a range on the flop. The defense equation does not mention the quality of our turn ranges, only that they should defend against random floating. The looser we c-bet the flop, the looser we have to barrel/check-call/check-raise the turn in order to avoid getting exploited by floating. If we’re not careful, we might take this too far.

The solution to this problem (if in fact it becomes a real problem for us) is to check more weak hands on the flop so that we get to the turn with a stronger range after c-betting and getting called. As discussed previously, this is very important on draw-heavy flops that hit the preflop flatters range hard.

Here we’ll simply assume that Alice has chosen to c-bet her entire range on the flop, and that she is willing to take the consequences of her flop actions on the turn. She 2-barrels the turn with the value/bluff ranges above, and Bob calls again. Alice now has the following range on the river:

99,66,33,J9s,AA-QQ,AJ,KJ,QJ,JT} + {QTs,T8s,AK,AQ,KQ,KT,87s} =77 + 76 =153 combos

The river card doesn’t touch these ranges, and Alice still has 153 combos in her range after accounting for card removal effects:

If Alice defends her turn betting range only by 3-barreling, she needs to defend 70% of 153 combos which is 0.70 x 153 =107 combos. Using the defense equation we get:

2-barrel% + 1.75 x check-continue% =70%

 

2-barrel-combos + 1.75 x check-continue-combos =107

With the bet sizing 0.60 x pot on the river, Bob is getting 1.60 : 0.60 on a call, so Alice uses 0.60/(1.60 + 0.60) =27% bluffs in her 3-barreling range to make it break even for Bob to call with a bluffcatcher. So she uses 27/73 =0.37 bluff combos per value combo.

Below is a suggestion for a river strategy for Alice that satisfies the defense equation. The strategy corresponding to a 15% UTG openraise is listed for comparison:

River strategy after 25% CO-raise:

  • Check-raise:
    {99} =3 combos
    Value bet:
    {66,33,J9s,AA-QQ,AJ} =38 combos
  • Check-call:
    {KJ,QJ,JTs} =28 combos
    Bluff:
    {AK} =16 combos

Test of defense equation:

(38 + 16) + 1.75(3 + 28) =108 (optimal =107)

River strategy after 15% UTG-raise:

  • Check-raise:
    {99} =3 combos
    Value bet:
    {66,33,J9s,AA-QQ} =26 combos
  • Check-call:
    {AJ} =12 combos
    Bluff:
    {10 AK-combos} =10 combos

Test of defense equation:

(26 + 10) + 1.75(3 + 12) =62 (optimal =57)

The widening of our postflop ranges that we observed on the turn is carried over to the river, and Alice is forced to value bet and check-call thinner on the river in order to prevent Bob from floating her turn bet profitably with random weak hands. But note that there should also be an adjustment for Bob.

We have let Bob flat the same preflop range in both cases (except that he 3-bets QQ/AK against the CO raise but flats them against the UTG raise). But an observant and optimally playing Bob should adjust his preflop flatting range to Alice’s position. When Alice moves from UTG to CO her opening range widens and more difficult to play out of position. This means more preflop flatting hands should become profitable for Bob.

And since Bob also needs to defend his preflop range enough against Alice’s postflop barreling, he will be forced to widen his postflop ranges as well, if he starts by widening his preflop range when Alice widens hers. So a certain symmetry should develop in this scenario where both players loosen up preflop, and as a result are forced to loosen up postflop as well. When both players are forced to play wider and weaker ranges postflop, Alice can value bet and check-call thinner.

So even if Alice’s two postflop strategies for the 15% UTG range and the 25% CO range seem very different, it’s not necessarily a big problem for Alice in practice. If Bob has started out with a wider preflop range as well, he will have to call and value bet weaker hands himself.

5. Summary
We have gone one step further with our study of optimal postflop strategies as the preflop raiser out of position. We started with a discussion of what it means to follow up a bet made on the current street. Simply put, we’re committing ourselves to a certain amount of betting, check-calling and check-raising on the next street. If we’re not willing to do this, we’re opening ourselves up for getting exploited by loose floating by a player with position on us.

Then we used mathematics to show that the turn/river strategies we designed for the raiser defended her optimally against random floating (by making them break even).

Finally, we studied the effect of the raiser’s opening range by building a new set of turn/river strategies for Alice, corresponding to her opening a 25% CO range instead of the 15% UTG range used in the previous example from Part 5. This resulted in significantly looser postflop strategies. We noted that starting postflop play by c-betting 100% of our preflop range on the flop leads to looser turn/river ranges, and that c-betting 100% of a wide preflop opening range perhaps isn’t optimal, even if the flop is dry and without draws.

In Part 7 we’ll talk about:

  • Optimal bet sizing for the raiser out of position on a dry flop, when he knows that the flatter in position has a weak range (he can use bigger bets to maximize value)
  • The effect of the player in position slowplaying his monster hands on dry flops (the raiser now must be a bit cautious when value betting big on the turn and river)
  • Some simulations of EV where we let the raiser’s and the flatter’s postflop strategies meet, and use Pokerazor to calculate EV for the raiser’s barreling line

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/optimal-postflop-play-in-nlhe-6-max—part-6-823#sthash.cVW6OWN8.dpuf

Optimal Postflop Play in NLHE 6-max – Part 5

1. Introduction
This is Part 5 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, Part 2, Part 3 and Part 4 we discussed postflop play heads-up in position after flatting preflop. This is an important postflop scenario for us, since our preflop strategies include lots of flatting in position.

When we have position on the raiser it’s important that we defend enough postflop to prevent her from c-betting any two cards profitably on the flop. When we flat on the flop, we have to defend enough against her turn bets to prevent her from 2-barreling any two cards as a bluff, and the same goes for river play after we flat the turn. How often we defend on each street depends on the raiser’s bet sizing. The smaller she bets, the more hands we defend. This makes sens intuitively, since smaller bets means the raiser is getting a better prize on her bluffs (we should defend more), while we’re getting better pot-odds to continue (so more of our weak hands are getting the right prize to see the next street).

We have used the following standard bet sizes in the postflop articles:

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

If Alice raises preflop and Bob flats in position, Alice is getting pot-odds 1 : 0.75 on her flop and turn bets. She then automatically makes a profit if Bob folds more than 0.75/(1 + 0.75) =43% , so Bob has to defend at least 100 – 43 =57% against Alice’s c-bets and turn bets. On the river Alice’s pot-odds on a 0.6 x pot river bet are 1 : 0.60. She automatically makes a profit if Bob folds more than 0.6/(1 + 0.6) =38%, so Bob should defend the river at least 100 – 38 =62% to prevent this.

Bob’s total postflop strategy in position after flatting preflop is made up of of value raising, bluff raising and flatting on each street. But as we discussed in previous articles, it will be better for him to only defend by flatting on the driest flops (like 2 6 6 ) to prevent his flatting range from being weak and easy for Alice to read and play against on later streets.

Bob did not have this problem when flatting on coordinated flops (like J 9 3 ), since these flops hit his preflop flatting range much harder and gives him many strong hands/strong draws that he can raise for value. Furthermore, his flatting hands on this type of flop will often improve to strong hands on the turn. So Alice can’t assume Bob’s turn range is weak on a coordinated board, just because he flatted the flop. Therefore, it is on the dry flops that we often have scenarios where the raiser c-bets the flop, 2-barrels the turn, and 3-barrels the river, while the raiser is calling down in position with a weak range.

In these scenarios both players rarely have anything better than one pair. Forcing the other player to fold his weak one pair hands and good overcards is therefor an important value component in both players’ postflop strategies. For example, if the raiser c-bets A A on a Q 8 4 flop and the flatter folds 2 2 , the raiser has gained a lot.

The raiser out of position tries to achieve this by c-betting a lot as a bluff, and then sometimes bluffing again on the turn when called, and again on the river when called on the turn. And the player in position tries to win pots by calling down a lot with his one pair hands, but also sometimes floating with very weak hands, planning to bluff with these hands if the raiser checks and gives up on a later street.

We define a float as a call done either with a weak hand that can’t win a showdown unimproved (so we plan to often bluff on later streets if we get the chance) or a hand with mediocre showdown value that we are hoping to take cheaply to showdown (but we are too weak to call down if the raiser bets all 3 streets). Using this definition, calling with both T 9 and 2:heart: 2:spade: on a Q 8 4 flop would be floats.

In previous articles we have studies Bob’s strategies in position. In this article we’ll turn the tables and study Alice’s strategies out of position. We’ll start with the following model:

– Both players begin with 100 bb stacks
– Alice openraises preflop and Bob flats in position
– Alice c-bets her entire preflop range on the flop

This creates a turn/river dynamic between the two players those times Bob calls the flop. In this article we’ll only look at dry flops, since this lets us use two simplifying assumptions:

1. Alice begins by c-betting her entire preflop range (reasonable, since Bob’s preflop flatting range will be weak on dry flops)
2. Bob never raises the flop (reasonable, since it makes sense for him to slowplay his best hands on dry flops for reasons previously discussed)

Whether or non Alice should c-bet her entire range on dry flops is not something we’ll discuss here, but it is reasonable on dry flops. We’ll use this as an assumption in our model, since it can never be a big mistakes when we are heads-up against a preflop flatter that will often have missed a dry flop. Furthermore, we’ll limit our discussion to scenarios where Bob never has a hand strong enough to raise for value on any street. This puts him in a situation where he is either calling or folding on each street. This creates a postflop dynamic where:

– Bob needs to defend enough against Alice’s barreling on all 3 streets
– Alice needs to defend enough against Bob’s floats on the flop and turn

Bob’s task is to prevent Alice from having an automatically profitable bet/bet/bet strategy (3-barreling) with any two cards. Alice’s task is to prevent Bob from having an automatically profitable float with weak hands on the flop and turn.

Bob starts by calling Alice’s c-bet with many medium/weak hands that are not strong enough to call down. Alice’s job on the turn and river is then to play these streets in such a way that Bob can’t call the flop or turn with any two cards and make a profit. For example, if Alice c-bets 100% of her range on the flop, but then check-folds 2/3 of her range on the turn without ever check-calling or check-raising, Bob can call her c-bet with any two cards, planning to auto-bet the turn as a pure bluff those times Alice checks and gives up.

If Bob can call a flop c-bet with automatic profit with a hand as weak as 2 2 on a J T 4 flop, Alice is probably doing something wrong on the turn and river. Note that when Alice checks the turn and gives up after getting floated on the flop, she has in reality lost the hand. If Bob has floated with a worthless hand, he will now bet and Alice will fold. If he has a hand with weak showdown value, as in the 2 2 hand above, he can choose between betting it as a bluff or checking it to showdown (we’re assuming Alice isn’t planning to bluff the river when the turn goes check-check). If the hand get checked down, Bob will usually win, since Alice on average will have few outs those times she checks and gives up on the turn.

At any rate, Alice can not allow Bob to sit behind her and call c-bet and turn bets profitably with any two cards, so she has to make sure she defends her betting range on the current street by not giving up too easily on the next street after getting called. In this article we’ll show how Alice can build turn and river strategies, based on pot-odds and simple theory, that prevents a player in position from floating her with any two cards on the turn or river.

Alice does this by betting, check-calling and check-raising enough on the next street after betting the current street and getting called. This prevents the player in position from getting enough profitable bluffing opportunities, or opportunities to get cheaply to showdown with weak hands that have some showdown value. Precisely how often Alice needs to continue on the next street after betting the current street and getting called is something we can estimate using mathematics and simple assumptions.

We’ll use theory borrowed from Matthew Janda’s excellent game theory videos at /Cardrunners.com. Then we adapt this theory to the “model game” we have designed throughout the NLHE preflop article series and this NLHE postflop article series. We’ll use our default preflop “core ranges” as a starting point for out postflop ranges.

Before we begin building Alice’s postflop strategy, we’ll warm up by verifying that Bob’s calling strategy in position (discussed in Parts 1-4 in this article series) does what it was designed to do, namely prevent Alice from c-betting/2-barreling/3-barreling profitably with any two cards those times Bob doesn’t have a hand strong enough to raise for value on any street.

2. How Bob’s calldown strategies makes Alice’s any-two-cards bluffs break even
Let’s quickly repeat an example from Part 4 where Alice c-bets the flop, 2-barrels the turn, and 3-barrels the river. We’re only looking at the region of possible outcomes where Bob only has a calling hand on each street.

Alice (100 bb) raises her default ~25% range from CO, Bob (100 bb) flats on the button with his standard flatting range in position (“IP flat list”):

IP flat list after a ~25% CO openraise

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

140 combos

The flop comes:

Bob’s preflop flatting range of 140 combos was reduced to 130 combos on this flop (card removal effects):

Bob then had to defend 57% against Alice’s c-bets on the flop, which is 0.57 x 130 =74 combos. We estimated Bob’s optimal flop strategy as:

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

Bob slowplayed all his strong hands on this very dry flop, and the reasons for this choice were discussed previously. Then the turn came:

The flop flatting range of 77 combos was reduced to 73 combos, given this turn card:

Again, Bob has to defend 57% of his range, which is 0.57 x 73 =42 combos. On the turn he will use a raising range of strong hands (some slowplayed monsters from the flop) and he balances this with bluffs in a 1 : 1 value/bluff ratio. The rest of the defense is done by flatting. We estimated his optimal turn strategy to be:

  • Raise for value
    {88,55,33} =9 combos
  • Flat
    {AQ,JJ,TT} =24 combos
  • Bluffraise
    {AJs,9 9 ,9 9 ,9 9 ,9 9 ,9 9 } =9 combos

we then moved on to playing the river after Bob had flatted the turn:

The river card had no effect on Bob’s range, and his 24 turn flatting combos were intact on the river:

Bob then had 24 combos in his river range, and he had to defend them optimally against Alice’s 0.60 x pot river bet. As calculated previously, Bob then has to defend 62% of his range to prevent Alice from bluffing profitably with any two cards. He has no hands strong enough to raise for value (he only has one pair hands to use as bluffcatchers), so he needs to defend 0.62 x 24 =15 combos by flatting them. We estimated Bob’s optimal river strategy to be:

  • Raise for value
    None
  • Flat
    {AQ,J J , J J , J J } =15 combos
  • Bluffraise
    None

What generally happens from street to street those times Bob finds himself inn a call-down process (those times he has medium/weak hands) on a dry flop texture is that he begins by flatting the flop with a wide range of almost any pair plus his best overcard hands. The overcard hands are floats that he doesn’t plan to call down with, but he has to call the flop with them in order to defend enough. Then he typically drops his overcards and lowest pairs to a turn bet when Alice bets again. And finally, he calls a 3rd bet with his best pairs on the river and folds his lowest pairs.

This makes sense intuitively, since Bob needs to balance two factors:

– He has to prevent Alice from often bluffing him out of the pot with any two cards
– But he has to avoid paying off her better hands too often

The optimal call-down strategy outlined above makes sure Bob isn’t giving Alice a big opening for bluffing profitably with any two cards on any street. He calls down enough to prevent this, but he also folds enough to prevent Alice’s strong hands from extracting a lot of value from his bluffcatchers.

We’ll now use mathematics to show that Bob’s optimal call-down strategy prevents Alice from running a profitable any-two-cards bluff against him. We’ll assume that:

– Bob has a bluffcatcher that always beats Alice’s bluffs
– Alice has a pure bluff that never draws out on Bob’s hand
– Alice decides to run a 3-barrel bluff with her worthless hand
– Bob calls down optimally

Bob’s defense on the flop
Let the pot size on the flop be P. Alice now c-bets 0.75P with her worthless hand. Bob calls 57% of the time with his bluffcatcher (he can use a randomizer to determine when he calls and when he folds) and folds 43% of the time. Those times he calls, the pot grows from P to P + 0.75P + 0.75P =2.5P. Both players have now put 0.75P into the pot postflop.

– % Bob folds the flop: 43%
– Alice’s profit when Bob folds the flop: P

Alice wins the flop pot when Bob folds.

Bob’s defense on the turn
The pot is 2.5P on the turn. Alice now 2-barrels 0.75x pot with her worthless hand. Bob calls 57% and folds 43% to this turn bet. When he calls, the pot again grows with a factor 2.5 and becomes 2.5 x 2.5 x P =6.25P. Both players have now put (6.25P – P)/2 =2.625P into the pot postflop.

– % Bob calls the flop and folds the turn: 0.57×0.43 =25%
– Alice’s profit when Bob folds the turn: P + 0.75P =1.75P

Alice wins the flop pot + Bob’s flop call when Bob calls the flop and folds the turn.

Bob’s defense on the river
The pot is 6.25P on the river. Alice now 3-barrels 0.60 x pot with her worthless hand. Bob calls (and wins against Alice’s bluff) 62% and folds 38%. When he calls, the pot grows from 6.25P to 6.25P + 2 x 0.6 x 6.25P =13.75P. Both players have now put (13.75P – P)/2 =6.375P into the pot postflop.

– % Bob calls flop and turn, and the folds river: 0.57×0.57×0.38 =12%
– Alice’s profit when Bob folds the river: P + 2.625P =3.625P

Alice wins the flop pot + Bob’s flop call + Bob’s turn call when Bob calls the flop + turn, and then folds the river.

– % Bob calls the flop + turn, and then folds river: 0.57×0.57×0.62 =20%
– Alice’s loss when Bob calls down: -6.375P

Alice loses her flop c-bet + turn bet + river bet when Bob calls down.

Total EV for Alice’s 3-barrel bluff
Below is a summary of all the possible outcomes, with Alice’s profit/loss for each of then in parentheses:

  • Bob folds flop: 43% (P)
  • Bob calls flop/folds turn: 25% (1.75P)
  • Bob calls flop/calls turn/folds river: 12% (3.625P)
  • Bob calls flop/calls turn/calls river: 20% (-6.375P)
  • Total: 100%

 

EV (3-barrel bluff)
=0.43(P) + 0.25(1.75P) + 0.12(3.625P) + 0.20(-6.375P)
=0

Bingo! Alice’s 3-barrel bluff project is exactly break even when Bob sits behind her with a bluffcatcher and calls down optimally. His call/fold percentages on each street are functions of Alice’s bet sizes on each street. If Alice had changed her bet sizes, Bob would have adjusted his call/fold percentages correspondingly (smaller bets =Bob calls more, bigger bets =Bob folds more). For example if Alice had bet the pot on each street, Bob would have called 50% and folded 50% on each street (since Alice’s pot-odds on a bluff are now 1 : 1 on each street). You can easily verify that Alice’s 3-barrel bluff EV would have been zero with this bet sizing scheme as well.

This verifies that when Bob is inn a call/fold scenario that stretches over multiple streets, his optimal postflop strategies will prevent Alice from running a profitable any-two-cards 3-barrel bluff against him. So Alice can’t exploit Bob by bluffing aggressively, but note that Bob isn’t doing anything to exploit Alice’s bluffing either.

To exploit Alice’s any-two-cards bluffing strategy (if she is in fact using such a strategy) Bob needs to call down more than optimally to exploit the opening Alice is offering him. For example, he can choose to call down 100% with his bluffcatcher if he believes that Alice is betting 100% of her range on every street in an attempt to bluff him off his weak hands.

This should be profitable for him, since there should be many more bluffs than value hands in Alice’s range on a dry flop. However, by doing so he is offering Alice an opening for exploiting him back by stopping to bluff and only betting her value hands. But Bob can always return to the optimal call-down strategy if he isn’t sure whether or not Alice is bluffing way too much, or if he suspects she will quickly adjust to his attempts to exploit her bluffing.

Now we have warmed up, and we move on to the main topic for this article:

3. Optimal 2- and 3-barreling heads-up and out of position
We’ll now look at the scenario where:

– Both players start with 100 bb stacks
– Alice raises preflop and Bob flats in position
– Alice c-bets her entire preflop range on a dry flop, and Bob flats
– Alice then uses a turn/river barreling strategy designed to prevent Bob from floating profitably with any two cards on the turn or river

We’ll do this in to steps:

1. Study a simple mathematical model
2. Implement the theory working through an example

3.1 Modeling barreling out of position
First, let’s define barreling. This is simply to keep betting on the next street after you have bet the current street and gotten called (and it doesn’t matter whether you’re weak or strong). So if Alice raises preflop, c-bets the flop, and then bets the turn, she has done a 2-barrel. If she also bets the river after getting called on the turn, she has done a 3-barrel.

When Alice is out of position versus Bob, c-bets the flop and gets called, it’s important for her to have a balanced strategy for turn play in order to prevent Bob from exploiting her by floating with any two cards on the flop (planning to steal the pot on later streets). If Alice checks and gives up on too many turns, it will be profitable for Bob to call her c-bet regardless of what he has, planning to auto-bluff the turn when checked to (for example if he floated the flop with a gutshot straight or overcards), or planning to check down a hand with marginal showdown value (for example, if he floated the flop with a low pair).

Alice can counter Bob’s floating strategy with random weak hands by 2-barreling enough on the turn and we’ll see how often she needs to do that in a minute). But Alice can’t only defend her flop betting range by 2-barreling, since this makes her turn checking range transparent and easy to exploit (since Bob then knows that Alice is always weak when she checks). So Alice needs to mix in some check-calling and check-raising on the turn as well.

The same logic applies to river play after Bob flats Alice’s turn bet. She has to 3-barrel/check-call/check-raise enough to prevent Bob from floating the turn with any two cards, planning to steal the pot on the river, or win a showdown with a weak hand that has showdown value (but not strong enough to call both the turn and the river.

We’ll use a simple model and a bit of math to estimate how often Alice needs to defend on the next street after betting the current street and getting called. We use our standard postflop bet sizing scheme:

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

When Alice c-bets 0.75 x pot on the flop, Bob is getting pot-odds (1 + 0.75) : 0.75 =1.75 : 0.75 on a call. If Alice never check-raises or check-calls the turn, Bob can float a random weak hand with automatic profit if Alice checks and gives up more than 0.75/(1.75 + 0.75) =30% on the turn. Therefore, if Alice defends against Bob’s flop floats by only 2-barreling, she needs to 2-barrel 100 – 30 =70% of her flop betting range on the turn. We can express this as:

2-barrel%=70%

This is a mathematically acceptable defense strategy against flop floats, but Alice can make things easier for herself by also check-calling and check-raising some on the turn. This makes it more expensive on average for Bob to steal the pot (which means Alice can get away with less 2-barreling). It also makes Alice’s turn checking range much harder to read, since she isn’t always ready to give up the pot when she checks.

Those times Alice 2-barrels the turn and Bob folds his random flop float, his loss is limited to his flop call of 0.75 x flop-pot. Now, assume Bob always bets his floats as a turn bluff when Alice checks to him. His plan is to fold to a turn checkraise, and give up his steal attempt if Alice check-calls Bob is then prepared to check down the hand and lose a showdown). Bob’s turn bet is 0.75 x turn-pot, and the turn-pot is 1 + 0.75 + 0.75 =2.5 x flop-pot. Bob then invests 0.75 x 2.5 =1.875 x flop-pot with his turn bluff.

Then his total risk for trying to steal the pot with a flop float + turn bluff is (0.75 + 1.875) =2.625 x flop-pot. When Alice check-calls or check-raises the turn, Bob’s expense is then 2.625/0.75 =3.5 x higher than when Alice 3-bets (so that Bob only loses his flop call of 0.75 x flop-pot).

To make Bob’s steal attempt break even, the following equation needs to be satisfied:

2-barrel%(-0.75P) + check-continue%(-2.625P)
+ (100 - 2-barrel% - check-continue%)(+1.75P) =0

In words:

The amount Bob loses by floating the flop and getting 2-barreled (-0.75P each time), plus the amount he loses by floating the flop and getting his turn bluff check-called or check-raised, plus the amount he makes when his turn bluff succeeds, should sum to zero. That makes his float flop + bluff turn strategy break even, which is what Alice’s wants her turn strategy to do for her.

We simplify this equation to get:

2-barrel%(-0.75P) + check-continue%(-2.625P)
+ 175P - 2-barrel%(1.75P) - check-continue(1.75P) =0
2-barrel%(-0.75P - 1.75P)
+ check-continue%(-2.625P - 1.75P) + 175P =0
-2.5P x 2-barrel% - 4.375P x check-continue% + 175P =0
2.5P x 2-barrel% + 4.375P x check-continue% =175P
P x 2-barrel% + 1.75P x check-continue% =70P

And the above equation for Alice’s turn defense strategy against flop floats can be generalized to:

2-barrel% + 1.75 x check-continue% =70%

The term check-continue is the label we use for all of Alice’s check-calling and check-raising. We have here assumed that Bob always loses the hand when he bets the turn and Alice doesn’t fold. Note that we are ignoring the equity of Bob’s hand, and we assume that he never wins a showdown after Alice check-calls the turn. Bob is always behind when this happens, he never improves to the best hand on the river, and he never bluffs the river. These are simplifying assumptions, but this is fine when we’re modeling a situation. Also, keep in mind that sometimes Alice bets or check-calls the worst hand, and then she draws out on the river. So as a first approximation we can assume that these two effects cancel out.

We’ll now put the above equation to work by studying an example scenario heads-up with the raiser out of position on a dry flop. On these flops we’ll often get a call-down scenario where the raiser c-bets any two cards on the flop, and then the preflop flatter sits in position with a medium/weak range of mostly one pair hands and overcards. usually the caller is not strong enough to raise anywhere along the way, so he will often be faced with a call/fold decision on every street those times the raiser fires multiple barrels.

What typically happens when two good, thinking players clash in this type of scenario is that both will be playing wide ranges on the flop (the raiser c-bets a lot and the player in position flats a lot). Then both players drop many (but not all) of their bluffs, floats and weak one pair hands on the turn, and then again on the river. And both players are trying to prevent the other player from bluff-barreling/floating profitably with any two cards on any street.

3.2 Example of optimal c-betting/2-barreling/3-barreling heads-up and out of position on a dry flop
Alice raises her default ~15% opening range from UTG:

~15% UTG-range

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

194 combos
15%

Bob flats on the button. At this moment we’re not particularly concerned with Bob’s flatting range or postflop strategy, but we can assume he uses his standard flatting range outside of the blinds (“IP flat list”):

IP flat list after ~15% EP openraise

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

162 combos

The flop comes:

We’ll now focus on Alice’s postflop strategy from street to street. She begins by c-betting her entire preflop range on this dry, low flop, since it mostly misses Bob’s preflop flatting range, and she expects him to fold a lot. We don’t know what Bob has, but we can assume his range is weak. Alice must now have a strategy ready for the turn, so that Bob can’t exploit her by floating the c-bet with any two cards.

We saw previously that Alice can achieve this by 2-barreling, check-calling and check-raising the turn so that the following equation is satisfied:

2-barrel% + 1.75 x check-continue% =70%

The turn comes

Before Alice builds a turn strategy, we take card removal effects into consideration and count the number of combos in her turn range, given the cards on the board. Since she c-bet her entire preflop range on the flop, her turn range equals her preflop range minus the combos that are eliminated due to card removal effects:

There are 168 combos remaining in Alice’s range. If she only 2-barrels and never check-calls or check-folds, she needs to bet 70% of these combos, which is 0.70 x 168 =118 combos. If she also check-calls and check-raises, we can rewrite the defense equation as:

2-barrel-combos + 1.75 x check-continue-combos =118

Alice now uses a turn strategy where she:

– Check-raises a few of her best hands
– Bets the rest of her best hands for value
– Check-calls with some medium strong hands
– Balances her value bets with some bluffs in a 1 : 1 ratio
– Check-folds the rest of her hands

Here we’ll not go into detail about which hands are good enough to check-raise, value bet or check-call, and we’ll use good poker sense when putting hands into different categories. Furthermore, we haven’t shown mathematically that 1 : 1 is the best value/bluff ratio to use for Alice’s 2-barrels, but we’ll assume this is reasonable (and it’s easy to remember).

Let’s estimate a reasonable total turn strategy for Alice and check whether or not this gives her enough protection against floats according to the defense equation:

  • Check-raise:
    {JJ} =3 combos
    Value bet:
    {99,66,33,J9s,AA-QQ,AJ} =41 combos
  • Check-call
    {KJs,QJs,JTs,TT,A9s} =18 combos
    Bluff:
    {QTs,AK,AQ,KQs} =40 combos

So Alice 2-barrels 41 + 40 =81 combos using an approximate 1 : 1 value/bluff ration, and she check-calls/check-raises 3 + 18 =21 combos. She makes things simple and choose top pair/top kicker or better as her value hands, check-calls with the remaining top pair hands + the best of the lower pairs, and bluffs with an open-ended straight draw and the best overcard hands.

The defense equation gives:

2-barrel-combos + 1.75 x check-continue-combos
=81 + 1.75 x 21
=118 (optimal =118)

Our estimate of Alice’s turn strategy satisfies the defense equation exactly. Now we can go back to our ranges and do some polish if we want to, particularly for the hands in between the obvious check-calling hands and the “air hands” (our 2-barrel bluffs). For example, we chose to check-fold T9s since we had enough better one pair hands to use for check-calling, and we preferred to bluff with draws and overcards, since these on average have more outs than marginal one pair hand.

Here it’s important to note that T9s will win some showdowns, since Bob will sometimes check the turn and give up. So check-folding these marginal one pair hands does not automatically mean they lose, since the player in position will sometimes be willing to check down weaker hands. And of the turn and river goes check-check we’d rather have T9s than two overcards. So it makes more sense to check turns with our weakest one pair hands, instead of turning them into bluffs. And then we pick our bluffs from hands that can’t win showdowns unimproved.

At any rate, when we’re building a mathematically sound total turn strategy this type of marginal decision making is not very important. For the moment we’re only concerned with building a reasonable turn strategy for Alice, and then we can polish it later.

Now we let Alice bet her turn 2-barrel range, which is a 1 : 1 mix of value hands and bluffs (note that top set JJ is not a part of our range, since we put it in our turn check-raising range):

  • Value bet:
    {99,66,33,J9s,AA-QQ,AJ} =41 combos
  • Bluff:
    {QTs,AK,AQ,KQs} =40 combos

Bob calls turn turn bet, and the river comes:

Alice’s 2-barrel-range of 40 + 41 =81 combos is unaffected by this river card, and she still has 81 combos on the river:

We remember that Alice’s bet sizing is 0.60 x pot on the river. Now there aren’t any more cards to come that can change hand strength, and only one round of betting remains. we can now calculate the exact optimal value/bluff ratio for Alice’s 3-barrels. When she bets 0.6 x pot, Bob is getting pot-odds (1 + 0.60) : 0.60 =1.60 : 0.6. Alice now wants to bluff exactly so often than Bob becomes indifferent to calling or folding with his bluffcatchers (those of his hands that can only win if Alice is bluffing, for example a marginal one pair hand).

The logic behind this is that if Alice bluffs less, Bob can exploit her by always folding his bluffcatcher and save money (since he isn’t getting the right pot-odds to call). But if she bluffs more than optimally, relative to her bet sizing, Bob can exploit her by calling with even more bluff catchers (since he is getting better pot odds than he needs), and Alice now loses money.

Therefore, Alice wants to bluff just enough to make Bob’s EV zero when he calls with a bluffcatcher. Then she has a guaranteed minimum profit from betting the river. If Bob tries to save chips by not paying off with his bluffcatchers, Alice will steal some pots with her bluffs. If he tries to snap off a possible bluff by calling with all his bluffcatchers, he will mostly be paying off Alice’s value hands.

Alice now makes her value/bluff ratio for the 3-barrel equal to the pot-odds Bob is getting, namely 1.60 : 0.60. Alice then bluffs 0.60/(1.60 + 0.60) =27% of the time and value bets 100 – 27 =73% of the time. She then needs 27/73 =0.37 bluff combos per value combo.

In addition to the requirement of optimal value betting/bluffing on the river, Alice needs to 3-barrel/check-raise/check-call the river so that:

3-barrel% + 1.75 x check-continue% =70%

This follows from her 0.75 x pot turn bet, which gives us the same mathematics as her 0.75 x pot flop bet, and the same defense equation (she has to play the river in such a way that Bob can’t flop her turn bet with any two cards). Alice’s 2-barrel-range on the turn had 81 combos and she still has 81 combos on the river. 70% of this is 0.70 x 81 =57 combos. The defense equation can be written as:

3-barrel-combos + 1.75 x check-continue-combos =57

The range Alice brought with her from the turn to the river after c-betting and 2-barreling is {99,66,33,J9s,AA-QQ,AJ} + {QTs,AK,AQ,KQs} =40 + 41 =81 combos. Below is a suggestion for a total river strategy that satisfies the defense equation and also has the optimal value/bluff ratio for her river 3-barreling range:

  • Check-raise:
    {99} =3 combos
    Value bet:
    {66,33,J9s,AA-QQ} =26 combos
  • Check-call:
    {AJ} =12 combos
    Bluff:
    {10 AK-combos} =10 combos

Alice check-raises one of her sets and value bets all other sets, two pair and overpairs. She check-calls top pair/top kicker, and bluffs with 10 of the 16 AK combos (for example all AKs and the 6 remaining A Kx and A Kx). She 3-barrels a total of 26 + 10 =36 combos with a bluff% of 10/(26 + 10) =28% (close to the optimal 27%), and she check-calls/check-raises 3 + 12 =15 combos.

We plug these numbers into the defense equation and get:

3-barrel-combos + 1.75 x check-continue-combos
=36 + 1.75 x 15
=62 (optimal =57)

We see that it’s easy for Alice to defend enough on this river when she starts out with a strong UTG range preflop and then gets called on the flop and turn on a dry board. She has enough sets and overpairs in her barreling range to comfortably get to showdown with only top pair or better, without giving Bob any opportunities to float her profitably with any two cards anywhere along the way. Note that Alice does not need to make crying check-calls on the river to defend optimally. But as we shall see in the next article, Alice’s opening range is an important factor. The tighter her opening range, the more of our range will be made up of top pair or better postflop, and the easier it becomes to defend optimally out of position.

For example, had we opened our default 25% range from CO and gotten the same flop, we would have had a much larger percentage of worthless hands in our postflop range after c-betting our entire range on the flop. Compared to a 15% UTG open range we would now be forced to defend with a much weaker range on the turn to satisfy the defense requirement of 70% 2-barreling (or the equivalent amount of 2-barreling, check-calling and check-raising, according to the defense equation). We would have carried this problem with us to the river, and we would have to defend a weaker range there as well. We’ll talk more about this in Part 6.

So what can we learn from the work done in this article? For example, we see that out one pair hands drop steadily in value from flop –> turn –> river. At the river top pair/top kicker became a check-calling hand in this example. Further more, all worse one pair hands (if we had had any) would have been put in the check-folding range, since we don’t need to check-call these hands to satisfy the defense equation.

Does all this make sense intuitively? Yes, since we can’t expect to win many pots by betting or check-calling a mediocre one pair hand after we have bet for value on the flop and turn and gotten called twice on a dry board. Villain will often have a better hand, and we will pay off a lot if we insist on taking all our mediocre one pair hands to showdown.

So the ranges we build based on pot-odds, mathematics and principles from game theory correlate well with our intuitive understanding of the situation. But of course, if you’re at the river in such a scenario and you expect Villain to bluff enough to make check-calling profitable with a mediocre one pair hand, by all means go ahead and check-call. The main point of the optimal strategy is that it gives us a good starting point for playing correctly.

If we follow the optimal strategy, Villain can’t exploit us by loose floating, that’s the big picture idea here. If we have additional information that tells us he is likely to bluff way too much if we check the river, we can exploit him by check-calling more than optimally.

Therefore, if the strategies above seem to loose or too tight to use as a default at the limits you are playing, you can view this as a sign that you usually have additional information that allows you to build exploitative strategies that are better than the optimal default strategy. But you will still benefit from training a good understanding of what the optimal strategies look like, so that you know where to start when you adjust to individual opponents’ mistakes. You will also have a solid default strategy to use against unknown players.

4. Summary
In this article we moved from postflop play heads-up in position after flatting preflop to postflop play out of position as the preflop raiser. We used simple mathematics and modeling to estimate an optimal c-bet/2-barrel/3-barrel strategy for the raiser.

We assumed that the raiser began postflop play by c-betting her entire preflop raising range on a dry flop texture, and the player in position called. On the turn and river the raiser used strategies that prevented the player in position from floating the flop or the turn profitably with any two cards- The raiser did this by barreling/check-calling/check-raising enough to make it mathematically impossible for the player in position to make a profit from floating a street with a weak hand, planning to bluff or sneak cheaply to showdown when the raiser checks the next street. We worked thoroughly through an example to illustrate how the theory can be implemented at the table.

In the next article we’ll continue with this topic. Some of the things we’ll talk about are:

  • More about the consequences of choosing to bet a street
  • Show mathematically that the raiser’s optimal turn/river strategies defends her against any-two-cards floating
  • Study the effect of the raise’s opening range on her postflop strategies

The plan for the rest of the article series is to move on from heads-up play in singly raised pots to heads-up play in 3-bet pots (which new players tend to find difficult). But before we move on to 3-bet pots we will gain a lot of insight from studying play in singly raised pots both as the raiser out of position and the flatter in position. The mathematics and models we use will come in handy when we learn about play in 3-bet pots later.

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/optimal-postflop-play-in-nlhe-6-max—part-5-821#sthash.0OkTQEBC.dpuf

Optimal Postflop Play in NLHE 6-max – Part 4

1. Introduction
This is Part 4 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.

I Part 1, Part 2 and Part 3 we introduced fundamental theory for flop play heads-up in position after flatting preflop. We then put the theory to use by working through several examples of flop play. We placed Bob as the preflop flatter heads-up in position against Alice’s openraise, and then we let him defend optimally against Alice’s c-bets.

The result of this work was a method for estimating defense ranges on the flop against the preflop raiser’s c-bet:

  • We found that we need to defend ~57% of the time, using a combination of value raising, bluff raising and flatting on the flop in order to prevent Alice from profitably c-betting any two cards as a bluff
  • Then we estimated Bob’s optimal value/bluff ratio for flop raising to be 1 : 1
  • We start by choosing our value range. Now we also know how many bluffs we need (number of bluffs =number of value hands), and also the total number of raising hands
  • Then we choose our flatting hands from the best hands not good enough to raise for value. We choose enough flatting hands to make the total number of valueraising + bluffraising + flatting hands equal to 57% of our total flop range
  • Then we choose the bluff combos we need from the best hands not good enough to raise for value or flat, and we fold everything else

We studied play on coordinated flops and dry flops separately. We found that coordinated flops (particularly those with medium/high cards) were easy to defend. On these flops we have many value hands and good flatting hands to use, and we defend with a combination of raising and flatting. On the dry flops we have few value hands, and we concluded that there are advantages to flatting everything on the flop (i.e. we slowplay our strongest hands, planning to raise for value on later streets).

By generating random flops (using Flopgenerator.com) and then building flop defense ranges for these flops, we can train our ability to quickly and accurately estimate optimal defense strategies. We don’t have to get it precisely right at the table. What we need is a sound qualitative understanding of how to play different types of hands on different types of flops.

We group our playable hands into 3 categories: Value hands, flatting hands, and bluffraising hands. As a start, using the simple classification scheme below will work well:

– Value hands: Two pair and better + monster draws
– Flatting hands: Good one pair hands and non-monster draws
– Bluffraising hands: Mediocre one pair hands, overcards, gutshots

Part 1, Part 2 and Part 3 gave us the necessary tools for defending in position on the flop against a c-bet after flatting preflop. But we did not talk about turn or river play. When we defend the flop against a c-bet from Alice, we have the following outcomes:

  • 1. We raise the flop, Alice 3-bets
  • 2. We raise the flop, Alice folds
  • 3. We raise the flop, Alice calls
  • 4. We flat the flop, Alice checks the turn
  • 5. We flat the flop, Alice bets the turn

The scenarios 3-5 leads to play on the turn. To complete our postflop strategies after flatting in position preflop we’ll now move on to play on later streets, after we have executed out optimal defense strategies on the flop. We’ll limit our discussion to the scenario where Bob flats preflop, flats the c-bet on the flop, and then Alice has the opportunity to bet again on the turn.

We can study this scenario from both sides. For Bob it’s important to play the turn in such a way that Alice can’t automatically profit from 2-barreling any two cards after getting called on the flop. For Alice it’s important to play the turn in such a way that Bob can not automatically profit from floating (calling with a weak hand, planning to steal the pot on later streets) any two cards on the flop.

If Alice checks and gives up on too many turns, Bob can exploit this by flatting any two cards on the flop, planning to bluff the turn when Alice checks and gives up. Alice prevents this by betting and check-calling/check-raising enough on the turn after c-betting the flop.

In this article we’ll look at Bob’s turn/river strategies and study them by working through two scenarios. We let Alice raise from CO, Bob flats on the button, and we get a heads-up postflop scenario. We use our default preflop ranges as a starting point, and we’ll do the work on the two flop textures we used in Part 3 (a coordinated flop and a dry flop).

2. Preflop ranges, flop textures and postflop model
We begin by defining our preflop ranges, the flop textures we’ll work on, and assumptions about postflop play.

2.1 Preflop ranger
We use our standard ranges. Alice opens her default ~25% range from CO, and Bob flats with the hands in “IP flat list” on the button:

Alice’s ~25% CO openrange

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

326 combos
25%

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

Here is a download link for this table in document form (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 is made up of the following 140 combos:

“IP flat list” after 25% CO openraise:

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

140 combos

2.2 Flop textures
We’ll look at two postflop scenarios, one on a coordinated flop, and one on a dry flop. We use the two flop textures we worked on in Part 3:

Coordinated flop

On this flop Bob defends with a combination of value raising, bluff raising and flatting.

Dry flop

On this flop Bob elected to defend only with a flatting range. He slowplayed all his strong hands, planning to raise for value on later streets.

2.3 Postflop model
In both scenarios we’ll let Bob face a 3-barrel from Alice those times he does not raise. When Bob flats the flop, Alice continues to bet the turn. If Bob flats again on the turn, Alice bets again on the river. So Bob has to make sure he defends enough on turn and river to prevent Alice from having an automatic profit by 3-barreling all 3 streets when Bob only calls (and signals a weak range which Alice might think she’ll be able to exploit by bluffing a lot).

The bet sizing scheme those times Alice bets all 3 streets is:

  • Alice raises 3.5 bb preflop, and the pot is 8.5 bb after Bob’s call
  • Alice c-bets ~3/4 pot on the flop, and the pot is ~21 bb after Bob’s call
  • Alice bets ~3/4 pot on the turn, and the pot is ~53 bb after Bob’s call
  • Finally, Alice bets 32 bb (06 x pot) into the 53 bb pot on the river

For both the coordinated and the dry flop we’ll use theory and assumptions from Matthew Janda’s brilliant Cardrunners video Visualizing your entire range, and we’ll apply the theory to our own default ranges for these scenarios.

3. Postflop play on coordinated flop
We first define Bob’s total flop strategy, and then we move on to turn play after Bob has flatted the flop:

3.1 Bob’s defense strategy against a c-bet on a coordinated flop

We remember from Part 3 that Bob defined the following flop strategy against Alice’s c-bet:

  • 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

We found that the original 140 combos in our preflop flatting range was reduced to 117 combos on this flop. Bob defends (17 + 33 + 17)/117 =57%, which is the optimal defense frequency. He defends 33/117 =28% by flatting and the rest by raising.

So the range we bring with us to the turn after flatting the flop is {AQs,AQo,KQs,KQo,JJ,JTs} =33 combos.

3.2 Bob’s turn strategy after flatting the flop
We generate a random turn card, and the turn board texture becomes: 9 Q T Q

Alice now fires a 2nd barrel for 3/4 of the pot. How should Bob play his {AQs,AQo,KQs,KQo,JJ,JTs} range on the turn?

We start by counting Bob’s turn range:

The flatting range Bob brought with him to the turn is reduced from 33 to 25 combos when this turn card falls. Since Alice bets 3/4 pot, she is getting pot-odds 1 : 0.75 on a any-two-cards bluff. She needs to succeed 0.75/(1 + 0.75) =43% to have an automatic profit bluffing the turn with any two cards, and Bob needs to defend 100 – 43 =57% on the turn to prevent this. Note that this optimal defense percentage is the same as the one we used on the flop, since Alice uses the same 3/4 pot bet sizing on both streets.

Defending 57% is trivial for Bob, given this turn card. 57% of 25 combos is only 14 combos. Bob can meet this requirement by only raising trips for value. The then defends 16 combos as shown below:

A bit more than we need, but that’s of course fine. In theory Bob should balance his value raises with an equal amount of bluffs, but that’s not necessary to get to 57% total defense. The simple turn defense job is a consequence of having a very simple defense job already on the flop. The flop hit our preflop range hard, which gave us a strong flop flatting range. And when the turn card makes our flop flatting range even stronger, we are left with mostly value hands in our range.

Let’s generate another random turn card so that Bob will have some decisions to make: 9 Q T 3

This turn card is a blank that doesn’t touch Bob’s flop flatting range, and he still has all the 33 flop flatting combos in his range on the turn:

In order to defend the required 57% against Alice’s turn bet, he needs to defend 0.57 x 33 =19 combos. Since his range did not improve noticeably between the flop and the turn (apart from picking up two flush draws with A Q and K Q), it’s obvious that the hands we classified as flatting hands on the flop are still (at most) flatting hands on the turn. So we go to the river with only a flatting range.

We then choose the 19 best combos, which ends up being a turn flatting range of only top pair hands. We then use all our AQs/AQo combos (12) plus 7 of the KQ combos. We obviously choose K Q , and then we add 6 more. For example K Q , K Q , K Q , K Q , K Q , K Q .

Bobs’ turn defense against Alice’s 2-barrel then consists of flatting the range {AQs,AQo,K Q , K Q , K Q , K Q , K Q , K Q , K Q } =19 combos.

Note that we have now begun the job of getting away from top pair hands in order to avoid paying off Alice better hands too much. We continue with all our top pair/top kicker hands, but we fold some of the weaker top pair hands. This intuitively makes sense, since always calling down with all top pair hands won’t be profitable against a competent opponent.

3.3 River strategy after flatting flop and turn
We continue with the second random turn card and assume that the board was 9 Q T 3 on the turn.

Bob flatted the turn with the range {AQs,AQo,K Q , K Q , K Q , K Q , K Q , K Q , K Q } =19 combos

We generate a random river card and get: 9 Q T 3 6

The river is a blank that doesn’t touch our turn flatting range, and we still have the 19 combos we flatted on the turn:

So by flatting the flop and the turn, given this board, we have gone to the river with a range of flatting hands that weren’t strong enough to raise for value at any point. Therefore, they are not strong enough to value raise on a blank river either. In other words, we have a range of bluffcatchers on the river.

Alice now bets 0.6 x pot (32 bb into a 53 bb pot), and she gets pot-odds 53 : 32 on a 3-barrel bluff. She needs to succeed more than 32/(52 + 32) =38% to automatically profit from bluffing any two cards on the river. Bob’s task on the river is therefore to defend at least 100 – 38 =62% to prevent this.

Since Bob has no value hands in his range on the river, he doesn’t have to think about balancing a raising range. He simply flats with enough of his bluffcatchers to prevent Alice from bluffing any two cards profitably, and then he folds the rest of his hands.

He needs to flat 0.62 x 19 =12 of the 19 top pair hands he brought with him from the flop to the turn to the river. We obviously choose the 12 AQs/AQo combos and fold all our KQ combos.

Again, we continue the process we began on the turn of getting away from our weaker one pair hands in order to avoid paying off too much to Alice’s better hands (she can have lots of straights, sets and two pair hands). But we call down with sufficiently many hands to prevent her from bluffing any two cards profitably anywhere along the way from flop to turn to river.

3.4 Summary of play on coordinated flop
Bob had an easy job on the flop 9 Q T when turn and river were blanks. On a flop texture that hits his preflop flatting range hard, he ends up with a strong flop flatting range with many good one pair hands. So when the turn and river brick off, Bob simply has to “peel off” his weaker one pair hands along the way, and the he calls the final river bet with his best bluffcatchers. In this example, this turned out to be his top pair/top kicker hands, which makes good sense (they quickly turn into bluffcatchers when our opponent keeps betting into us on a coordinated board).

4. Postflop play on dry flop
First we define Bob’s total flop strategy, then we move on to turn play after Bob has flatted the flop:

4.1 Bob’s defense strategy against c-betting on a dry flop

We remember from Part 3 that Bob defined the following flop strategy against Alice’s c-bet:

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

The original 140 combos in the preflop flatting range were reduced to 130 combos in this flop, so Bob defends 77/130 =59% (a bit more then the minimum 57%, which is fine). All defense is done by flatting for reasons discussed in Part 3.

So the range we bring with us to the turn after flatting this dry flop is: {88,55,33,JJ,TT,99,77,66,44,AQ,AJ} =77 combos

4.2 Turn strategy after flatting the flop

We generate a random turn card, and the turn texture becomes: 3 8 5 Q

Alice now fires a 2-barrel for 3/ of pot. What is Bob’s strategy on the turn with the range {88,55,33,JJ,TT,99,77,66,44,AQ,AJ}?

We begin by counting Bob’s turn range. The turn card hits a part of his flop flatting range, and we have 73 combos in our range on the turn:

To defend the required 57% against Alice’s 3/4 pot turn bet, Bob has to defend 0.57 x 73 =42 combos. We have some value hands in this range after slowplaying the flop, so we can raise the turn with a mix of value hands and bluffs.

We let our value hands be the 9 set combos {88,55,33} =9 combos Next we balance this range with some bluffs. We’ll make it simple and use the same 1 : 1 ratio we used on the flop. Note that this isn’t necessarily 100% correct, since the exact ratio we need depends on the equities of the hands involved, but we’ll assume that a 1 : 1 value/bluff ratio works well on both the flop and the turn. So we need 9 bluff combos, and end up with a total turn raising range of 9 + 9 =18 combos.

This means we have to defend 42 – 18 =24 combos by flatting to get to 57% total defense. We have many one pair hands to use, and it’s obvious to begin with the 12 AQ combos that made top pair on the turn. Then we add our best underpairs JJ/TT =6 + 6 =12 combos, and we are done. Our flatting range is then {AQ,JJ,TT} =24 combos.

Lastly, we pick 9 bluff raising combos for balance. We can choose from the remaining one pair hands (underpairs lower than TT) and our overcard hands. For example, we can choose the 4 AJs combos plus 5 of the 6 99 combos (for example 9 9 , 9 9 , 9 9 , 9 9 , 9 9 ). In other words, all AJs and all 99 with a spade or heart.

Our total turn defense is then:

– Value raise: {88,55,33} =9 combos
– Bluff raise: {AJs,9 9 ,9 9 ,9 9 ,9 9 ,9 9 } =9 combos
– Flat: {AQ,JJ,TT} =24 combos

When we raise for value, the rest of the hand plays itself. If Alice 3-bets, we get all-in on the turn with good equity. If she calls, we bet the rest of our stack on the river. After a bluff raise we fold to a 3-bet and have no decisions to make in that case. If Alice calls our bluff raise and checks the river, we choose between bluffing again or giving up and checking down. Note that the last decision is not a forced and tricky one, even if it can be hard to choose the best alternative.

Note that when Alice has checked the river after calling our turn bluff, this simply means we get one more chance to steal the pot (instead of having to fold to a turn 3-bet) without having risked any more money to get this opportunity. If we’re not sure about what to do, we can simply check down and not risk more chips, and when we see a good bluffing opportunity, we an take it. At any rate, we are under no pressure to make a difficult choice after raising the turn and getting checked to on the river. Additional bluffing opportunities on the river are simply gravy.

So now we look at what happens on the river after our turn flat with the range {AQ,JJ,TT} =24 combos.

4.3 River strategy after flatting the flop and turn
The board was 3 8 5 Q on the turn.

We assume Bob flatted Alice’s turn barrel with the range {AQ,JJ,TT} =24 combos. We then generate a random river card and let Alice fire a 3-barrel.

The river board texture becomes: 8 5 Q 2

A complete blank that doesn’t touch our turn flatting range, and we still have 24 combos in our range on the river:

Since none of the flatting hands improved between the turn and the river, and since none of them were value hands on the turn, we obviously have a range of bluffcatchers that we call or fold. In other words, the exact same river scenario we had on the second random river card for the coordinated flop previously.

When Alice bets 32 bb into thee 53 bb pot, we found previously that Bob needs to defend 62% to prevent a profitable any-two-cards river bluff. So Bob defends 0.62 x 24 =15 of the 24 bluffcatchers he brought to the river.

We call with the 12 AQ hands plus 3 of the best underpairs (3 JJ combos). We choose the 3 JJ combos with a spade: J J , J JJ J .

Bob’s river strategy is then to call with {AQ,J J , J JJ J } and fold his remaining JJ and TT hands.

As on the coordinated flop we ended up calling down with a range of medium strong bluffcatchers that failed to improved to value hands. But we folded many of them along the way, and only called all the way down with enough hands to prevent Alice from bluffing with any two cards anywhere long the way.

For practice, lets generate a river card that improves us enough to raise some hands. Assume that the river board texture now is 3 8 5 Q J .

This river card gives us a set, and the number of combos in our range is reduced from 24 to 21:

62% river defense means we defend 0.62 x 21 =13 combos. We now use the 3 JJ combos as value raising hands. We have about 74 bb left in our stack after calling 3/4 pot bets on the flop and turn. So when Alice bets 32 bb on the river, we shove to 74 bb, and Alice has to call 74 – 32 =42 bb to win a 53 (initial river pot) + 32 (Alice’s bet) + 74 (our shove) =159 bb pot. She gets pot-odds 159 : 42 =3.8 : 1 on this call.

We now want to balance our value raises (3 combos) with enough bluff combos to make Alice indifferent to calling of folding with the hands that can only beat us if we’re bluffing (her good-but-not-great hands like top pair and overpairs). When Alice is getting pot-odds 3.8 : 1, we need 1 bluff combo for every 3.8 value combos to make her calls with her bluffcatchers break even (she’s getting 3.8 : 1, and the odds against us bluffing are 3.8 : 1). So we need 3 x (1/3.8) =0.8 bluff combos, which we simply round to 1.

We choose one of the TT combos to use as a bluff: T T

Then we need 13 – 3 – 1 =9 flatting combos to get to 13 defense combos in total, and we then obviously choose 9 combos of AQ. For example A Q , A Q , A Q , A Q , A Q , A Q , A Q , A Q , A Q .

Bob’s river strategy when improving on the texture 3 8 5 Q J is then:

– Raise {JJ} =3 combos for value
– Raise {T T } =1 combo as a bluff
– Flat {A Q , A Q , A Q , A Q , A Q , A Q , A Q , A Q , A Q} =9 combos

We happily raise our 3 set combos for value, balance it out with one bluff combo so that Alice can’t save money by folding all her bluffcatchers, and then we do the rest of the defense by flatting with most (but not all) of our top pair/top kicker hands.

5. Summary
We have gone one step further in our study of optimal postflop play heads-up after flatting preflop. Previously, we have looked at defense strategies against a c-bet on the flop. In this article we have moved on to turn and river play after flatting the flop.

We worked on two flop textures (coordinated and dry) and looked at how the player in position has to play the turn after flatting the flop, and then play the river after flatting the turn. This gave us insight into how we defend against a preflop raiser that 3-barrels (bets all 3 streets) postflop when the player in position keeps calling.

We saw that our defense ranges grew stronger and stronger from flop –> turn –> river. This makes good sense, intuitively. In the cases where our flatting range on the turn and river only contained bluffcatchers (good one pair hands), we saw that we started the process of getting away from these (possibly losing) hands on the turn. Then we continued this process on the river those times we did not improve. We ended up calling down with just enough bluffcatchers to prevent the raiser from barreling any two cards as a bluff on any street.

This is an important mindset that will help you get away from situations where you are calling down way too much with so-so one pair hands in the hope that your opponent is bluffing. It can work well when you have a read on a very aggressive player, but if you call down mindlessly with all decent one pair hands against a good, thinking opponent who keeps betting into you, you will lose a lot. Against a player who keeps betting into you, even your good top pair hands quickly turn into bluffcatchers. But you should of course call down with enough of your bluffcatchers to prevent profitable bluff-barreling with any two cards.

And in the cases where we do improve along the way, we also defend by raising for value, and balancing this with some bluffs. The rest of the defense is done with our bluffcatchers, as before.

The strategies we have discussed here are of the type you need to train between sessions in order to use them in practice. You will not have the time to do all this thinking at the table, so train away from the table and aim for a good qualitative understanding of the principles involved. You don’t need to build perfect strategies at the table, and any reasonable approximate strategy will work fine.

The next topic in this article series is postflop play as the preflop raiser heads-up and out of position after getting flatted preflop. Now we’ll study how Alice should play postflop in order to prevent Bob from exploiting her by floating any two cards on the flop or turn, planning to steal the pot later.

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/optimal-postflop-play-in-nlhe-6-max—part-4-810#sthash.9WZ7jsWH.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 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