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PLO from scratch Part 1-12

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

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

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

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

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

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

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

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

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

3. The plan for the article series

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

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

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

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

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

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

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

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

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

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

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

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

4. Learning material and poker tools for PLO

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

– The video seriesPLO(Whitelime & Phil Galfond)

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

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

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

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

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

Pokersavvy.com
– Everything by LearnedFromTV (16 videos)

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

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

Continue reading PLO from scratch Part 1-12

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

1. Introduction

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

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

Part 7 will be about:

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

 

1.1 Introduction to Pokerazor simulations of flatting in position

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

IP flat list

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

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

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

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

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

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

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

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

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

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

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

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

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

1.2 Introduction to the scenario “blind vs blind”

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

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

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

2. Pokerazor simulations of flatting in position

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

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

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

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

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

2.1 A simple model for estimating preflop + postflop EV

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

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

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

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

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

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

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

194 combos
15%

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

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

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

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

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

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

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

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

We therefore conclude:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

We get:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The scenario is:

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

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

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

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

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

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

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

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

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

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

EV for flatting vs value 4-betting with AQ:

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

We have an easy conclusion:

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

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

3. Blind versus blind

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

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

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

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

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

3.1 Bob’s optimal defense percentage

We begin with the fundamental principle of defense:

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

Then we use some additional assumptions:

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

Our assumptions about stack sizes and bet sizes are:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Alice opens her default 35% button range:

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

458 combos
35%

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

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

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

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

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

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

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

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

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

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

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

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

Blind vs Blind flat list:

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

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

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

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

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

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

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

4. Summary:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

– Pokerazor
– StoxEV

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

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

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

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

Through this discussion we’ll also shed light on the difference between optimal play and exploitative play, and when we should use one or the other. Optimal strategies put a lot of weight on defense, and they are not necessarily the most profitable strategies against players with big leaks. One reason is that optimal strategies include defensive components (for example, 4-bet bluffing as a defense against light 3-betting) that are often unnecessary against weak players (for example, we don’t need to 4-bet bluff against an opponent who only 3-bets premium hands like {JJ+,AK}).

Against players with big and easily exploitable leaks, we’d rather deviate from optimal play and play exploitatively to take full advantage of these leaks. But we need to be aware that by doing so we are creating openings in our strategies that can be exploited by observant opponents. So we have to find a balance between optimal and exploitative play, and we should use different strategies against different opponents. We will do our best to exploit weak players’ big mistakes, but we can always fall back on optimal play against good opponents without big leaks. We can also return to optimal play if the player we’re trying to exploit with exploitative play suddenly changes his strategies to take advantage of the openings created by our exploitative strategies.

For example, let’s say we choose to never 4-bet bluff against a passive player who never 3-bet bluffs. He might now notice this, and adjust to our tight play by starting to 3-bet bluff us. Our exploitative adjustment against this particular opponent then runs the risk of getting counter-exploited if he starts 3-bet-bluffing us often with random weak hands. If this happens, we should return to our optimal optimal 3/4/5-bet strategy. Alternatively, we can make another exploitative adjustment to his adjustment by 4-bet bluffing him a lot (since we know he often is weak and have to fold). But the optimal strategy is always an alternative if we aren’t sure whether or not we can exploit his aggressive 3-betting.

In my opinion, this mindset is at the core of the thought processes of a strong NLHE player. He doesn’t have to use mathematics like we have done, but he will have a good feel for what an optimal (or near optimal) strategy is in the situation he is in. So he has a strong default strategy to fall back on against unknown players or known strong players, so that he can’t be easily exploited. But at the same time he knows how to deviate from optimal strategies to exploit his opponents’ systematic leaks. So he can adjust his play in a controlled manner against each individual opponent instead of being locked into a static strategy that he uses against everyone.

Rules of thumb such as “never 4-bet bluff against fish” or “don’t 3-bet hands that perform poorly when called” are then replaced by a dynamical mindset that gives is strong control over our choice of strategies. Using optimal play as a starting point (and as a strategy we can always fall back on regardless of who we’re playing against), we can move around freely in “strategy land” and exploit opponent leaks as we pick up information about how they play.

Optimal play is never bad play, but exploitative play is always better. But we need information about our opponents’ strategies before we can exploit them. If we don’t have this information, we can always fall back on optimal play as a good default.

2. Testing preflop strategies using the analysis software “Pokerazor”
In this part of the article we’ll use Pokerazor to study 2 things:

  • 1. How a tight openraise strategy without defense against light 3-betting is vulnerable to 3-bet bluffing with any two cards
  • 2. What the raiser can do to plug this leak, and how this leads to an optimal strategy pair for the raiser and the 3-bettor

2.1 ABC preflop strategies and how these can be exploited
Those of you who have played for a while probably remember the good old days (up to around 2007 or thereabouts) when micro and low limit NLHE was easily beatable by sticking close to the following rules of thumb for preflop play:
Those

  • Open tight from all positions (say, 10-12% from UTG/MP, ~20% from MP and ~30% from the button
  • 3-bet only for value with {QQ+,AK}, and possibly {JJ+,AQ+} against a loose raiser
  • When you get 3-bet and you are out of position, fold everything but {QQ+,AK} regardless of your position and where the 3-bet comes from
  • Defend the blinds very tightly (typically 10%)

Believe it or not, but this was more or less the standard getting-started preflop strategy recommended to beginning players at the micro and low limits up to $100NL or so. And it worked well, since the games were so loose and passive that it was correct both to openraise tight, and to fold a lot against 3-bets.

Those of you who have been members of Cardrunners for a while might remember Brystmar’s beginner video series “Small Stakes NL” in 6 parts (published during the spring of 2007). This series began with tight-aggressive preflop recommendations based on tight opening ranges and 3-betting only for value:

Brystmar’s preflop strategy for micro/low limit NLHE
Let’s take a trip down memory land and study Brystmar’s preflop recommendations given 3.5 years ago. Those who want to read discussion about his video series or download his preflop scheme can look at this Cardrunners forum thread.

Below is a summary of the default openraising ranges (note that “KTs” and “KTo” denote suited and offsuit hands, while “KT+” means both suited and offsuit:

  • UTG openraise:
    {22+,AJ+,KQs} =9.8%
  • MP openraise:
    {22+,AJ+,KQ} =11%
  • CO openraise:
    {22+,A7s+,A9o+,KT+,QTs+,QJo,J9s+,JTo,T9s} =19%
  • Button openraise:
    {22+,A4s+,A7o+,KT+,QTs+,QJo,J9s+,JTo,T8s+,
    T9o,98s,98o,87s,87o,76s} =26%
  • Raising from the small blind:
    Openraise the button range if it gets folded to you. In a limped pot, raise {JJ+,AK} for value and overlimp all other hands from your button range, plus all Axs and Kxs.
  • Raising from the big blind:
    If the small blind openlimps, raise the button range and otherwise check. Out of position in a limped pot, raise {JJ+,AK} for value and otherwise check

Tight opening ranges all around. This is of course not a leak our opponents can exploit, but we might perhaps say that we are exploiting ourselves by folding some profitable hands, particularly on the button.

But the strategies become easy to exploit when we get to playing against a raise:

  • In MP with position on a raiser:
    Reraise {JJ+,AK} for value and call with {TT-22,AJs+,AQo}
  • In CO with position on a raiser:
    Reraise {JJ+,AK} for value and call with {TT-22,AJ+,KQ,QJs,JTs}
  • On the button with position on a raiser:
    Reraise {JJ+,AK} for value (and AQo if the raise came from CO) and call with {TT-22,AJ+,KQ,QJs,JTs}. With callers between you and the raiser, also call with {JTo,T9s,98s,87s}
  • In SB after a raise:
    Reraise {QQ+,AK} for value (and also JJ/AQ if the raise came from CO or the button) and call with {TT-22,AJ+,KQ}
  • In BB after a raise:
    Reraise {QQ+,AK} for value (and also JJ/AQ if the raise came from CO or the button) and call with {TT-22,AJ+,KQ}. With callers between you and the raiser, also call with {QJs,JTs}

We note two systematic errors in these strategies:

– We’re 3-betting more or less the same range regardless of the raiser’s position
– We’re never 3-bet bluffing

We remember from Part 1 and Part 2 that an optimal 3-betting range on the button varied from 3.6% against a ~15% EP openraise to 8.7% against a ~25% CO openraise. And in all scenarios we used an optimal bluffing frequency of 60%. In other words, more than half of our 3-bets were bluffs- In Brystmar’s strategies the 3-betting range is a tight and static value range {JJ+,AK} =3.0%, which is sometimes widened to {JJ+,AQ+} =4.2% against a wide openraising range. We also note that Brystmar chooses to include AQ when loosening up. This is a hand we never 3-bet for value in position when we’re playing optimally (since AQ works better as a flatting hand in position).

Brystmar’s strategies don’t mention defense against 3-betting, but we can assume that default defense is to 4-bet a tight range {QQ+,AK} from all positions. Another significant leak is the squeaky tight blind defense. For example, of button openraises, the preflop scheme tells us to 3-bet {JJ+,AQ} =4.2% from the small blind and flat {TT-22, AJ,KQ} =7.4%. This results in a total defense of 11.6%, which is way lower than the optimal defense threshold of 16% that we estimated in Part 3.

So there are huge openings in Brystmar’s preflop recommendations, and these openings can be easily exploited by an aggressive and observant opponent. We’re also leaving money at the table because we’re openraising to tight, and the main reason for this is that Brystmar does not take full advantage of position. We can openraise a ton of hands on the button when it’s folded to us, and we can make life hell for a raiser by 3-bet bluffing him in position, but Brystmar chooses not to do so.

NB! Before we move on I want to point out that I am not trying to put Brystmar’s low limit preflop defaults from 2007 in a negative light. His preflop recommendations for beginning NLHE players were very useful back in the day, and gave many new players an easy start. His strategies are best viewed as “training wheels” for staying out of trouble (and he no doubt saw them as such himself) and they were tailored towards the micro/low limit conditions that existed at the time. They will probably still work okay at the lowest micro limits, but I would not recommend anyone to play $25NL and higher with such tight and easily exploitable preflop strategies.

It’s clear for everyone who plays $25NL and higher these days that common NLHE strategy has developed in leaps and bounds since Brystmar’s 2007 recommendations. Light 3-betting was rare in the “old days”, even at $100NL and $200NL. Today it’s common, even if you begin as low as $5NL.

The next step of the development of the average low limit NLGE regular back in the day was to add some light 3-bets in position (and Green Plastic’s 2006/2007 NLHE videos at Cardrunners inspired many to do so), call more raises and 3-bets in position, and in general get better at using positional advantage. A common mistake many aggressive players did was to 3-bet bluff with hands that were too strong to use as bluffs (for example, JTs). However, this did not cause man problems since most players defended poorly against 3-bets, particularly from out of position.

So the standard recipe in the good old days for an advanced low limit player who wanted to ramp up the aggression was to LAG it up in position. But not necessarily with balance in mind, and not necessarily with a good understanding of how to chose his value range, bluffing range and flatting range in a consistent manner. But this was not a big deal. He played tight out of position, opened a very wide range in position, and 3-bet something fierce in position against weak opponents. The 3-betting was very effective, since the raisers often did one of the following two mistakes:

– Folded a lot out of position and never 3-bet bluffed
– Called a lot with non-premium hands out of position

The first mistake lets the 3-bettor print money by giving him an opening to (in principle) 3-bet bluff any two cards. The second mistake occurs when the raiser tries to correct the first mistake, but he goes about it the wrong way. Defending against 3-bets by flatting weak hands out of position is ineffective, since the raiser now has to play postflop out of position in a scenario where it’s difficult for him to win without hitting the flop well. Playing weak starting hands well out of position against a good LAG player is hard, and often results in you losing more money postflop than if you had just folded to the 3-bet preflop.

The cure against light, positional 3-betting is of course to respond by 4-betting a correct value range (which follows from the size of our opening range), balanced with a correct amount of 4-bet bluffing. We have studied this in previous articles, and we have defined optimal strategies for the raiser from all positions. The 3-bettor uses similar thinking to design his 3-bet strategy so that the raiser can not 4bet bluff any two cards profitably. This way an equilibrium gets established.

This equilibrium is given by the optimal strategy pairs for the raiser and the 3-bettor defined in Part 1 and Part 2. We used mathematics to define these strategy pairs, but we can also think about them as a product of an evolutionary process.

The 3-better starts out by exploitative ant-two-cards 3-bet bluffing against a raiser that defends way too tight and folds too much. Then the raiser adjust by choosing a correct value range and introducing 4-bet bluffing. The 3-bettor responds by adjusting his value range and introducing 5-bet bluffing. To prevent the opponent from bluffing with any two cards anywhere, both players fine-tune their ranges until both are using an optimal set of ranges for 3/4/5-betting. “Optimal” here means that neither player can improve his EV by adjusting further. If one of them tries to do so, he is giving the other player an opportunity to increase his EV by making and exploitative adjustment.

We will now illustrate such an evolutionary process using Pokerazor simulations:

2.2 Numerical testing of optimal heads-up 3/4/5-bet strategies
We start with the following model:

  • Alice (100bb) openraises to 3.5bb with her standard 25% range from CO
  • Bob (100bb) is on the button and 3-bets to 12 bb or folds
  • Alice defends against 3-betting by 4-betting to 25 bb or folding
  • Bob defends against 4-betting by 5-betting all-in or folding
  • Alice defends against 5-betting by calling all-in or folding
  • The blinds always fold, no matter what Bob does

So we are studying a scenario where Alice openraises, Bob 3-bets or folds, and the blinds never get involved. Alice then makes 1.5 bb (the blinds) per raise when Bob folds, which is a win rate of 150 bb/100. This is her baseline EV for the simulation.

Before we begin the simulations, let’s repeat the ranges and optimal strategy pairs we defined in Part 2:

Alice’s 25% open-range from CO

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

326 combos

25%

The corresponding optimal strategy pair that’s being used when Bob 3-bets in position can be found from the summary of optimal strategy pairs in Part 2:

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

The optimal strategy pair is then:

  • Bob:
    3-bets {QQ+,AK,12 air} ={QQ+,AK,A5s-A3s} for value (including 5-bet-bluffing with Axs hands) and 70% av “IP 3-bet air list” as a bluff using a randomizer
  • Alice:
    4-bets {TT+,AQ+} for value and {AJ,AT,A9s-A7s} as a bluff

Pokerazor simulation 1 (Bob folds)
The baseline simulation is to let Bob fold 100%. Alice then picks up the blinds, and makes 1.5 bb each time (=150 bb/10):

– Alice openraises 25% from CO
– Bob folds

EV (baseline for Alice) =150 bb/100

Bob now begins 3-betting so that Alice ends up with EV < 150 bb/100. It’s obvious that Alice’s EV now becomes lower than the 150 bb/100 baseline, since she can not prevent Bob from making money by 3-betting only his best hands, for example his {QQ+,AK} default value range against her 25% CO openrange. Furthermore, it’s correct for Alice to never 4-bet bluff when Bob never 3-bet bluffs, so she has to fold the hands not strong enough to 4-bet for value against Bob’s tight value range and let Bob pick up the blinds.

We start by assuming that Bob never 3-bet bluffs and that Alice defends against the 3-bet by 4-betting the value range {KK+} after observing that Bob’s 3-betting range is {QQ+,AK}. This is correct since Alice does not want to get all-in preflop with QQ or AK against Bob’s {QQ+,AK} range (both are ~40% underdog) with only 3.5 bb invested.

Pokerazor simulation 2 (Bob 3-bets only for value)
Bob 3-bets his value hands. We let him use the pure value hands {QQ+,AK} that he would have used in an optimal strategy against Alice’s 25% CO openrange, and the drops his 5-bet bluffs for now:

– Alice openraises 25% from CO
– Bob 3-bets {QQ+,AK} for value and 5-bets all-in against a 4-bet
– Alice 4-better {KK+} for value

And the rest follows automatically. We get:

EV (Alice; KK+) =141 bb/100

If Alice also had 4-bet QQ/AK for value, the EV would have become:

EV (Alice; QQ+,AK) =139 bb/100

So Alice’s choice of value range against Bob’s extremely tight 3-betting range is correct. Note that Alice is now exploiting Bob by only 4-betting an extremely tight {KK+} value range! What happens here is that Bob mostly leaves Alice alone so that she can pick up the blinds almost every time. Bob only pops up with a value 3-bet the 2.56% of the time he has {QQ+,AK}, and Alice then responds by folding everything but {KK+}.

So Alice exploits Bob’s squeaky tight 3-betting by not paying off his value hands with weaker hands. And since Bob never 3-bet bluffs, there is never any doubt about his range. Alice can then play perfectly against a 3-bet and drop all her (now unnecessary) 4-bet bluffs as well as her weakest value hands (and we remember that Alice’s optimal value range in CO is {TT+,AQ+}). Exploiting someone by folding a lot to his aggression is not the first thing that comes to mind when poker players think about exploitative play. But avoiding paying off a strong range unnecessarily is just as profitable as playing aggressively against players that fold too much.

If Alice had responded with her optimal 4-bet strategy which is {TT+,AQ} for value and {AJ-AT,A9s-A7s} as bluffs, we get:

EV (Alice; optimal strategy) =126 bb/100

And we see that Alice’s optimal 3/4/5-bet strategy out of position loses relative to the best exploitative strategy (which gave her 141 bb/100). This is an example of a general rule: If we have an opportunity to exploit someone, we will make more money from this than from continuing to use an optimal strategy. The reason is that Alice now sacrifices EV by defending against a non-existing threat. She has 4-bet bluffs in her 4-betting range to defend against Bob’s 3-bet bluffs, but Bob never 3-bet bluffs.

So even if Alice’s optimal strategy can not be exploited by any-two-cards 3-bet bluffing from Bob, this defense is costing her EV relative to a strategy that exploits Bob’s very tight never-bluff 3-betting strategy. A good analogy would be a nations defense budget in peace time. There has been no warfare on Norwegian soil since 1945, but Norway has still had a defense budget in all the years since then. We are simply paying for a military defense so that other nations don’t get an opportunity to invade us without risk.

Now, let’s assume that Bob has been sitting there and 3-betting only {QQ+,AK} for value and folding everything else, while Alice has responded by 4-betting only {KK+} for value and folding everything else. Bob has observed that Alice almost always folds. It’s now easy for him to reach the conclusion that he could make a lot more money by throwing in some 3-bet bluffs, planning to fold then when Alice 4-bets. Since Bob does not have to worry about the blind players waking up with a hand, let’s allow him to 3-bet any two cards to maximally exploit Alice’s super tight defense against 3-bets:

Pokerazor simulation (Bob 3-bets any two cards)
Bob uses his read on Alice, and changes his strategy to exploit her. He keeps 3-betting {QQ+,AK} for value, and then he 3-bets all other hands as a 3-bet bluff. Alice, not knowing that Bob suddenly has changed his strategy, keeps 4-betting {KK+} for value and folding everything else:

– Alice openraises 25% from CO
– Bob 3-bets {QQ+,AK} for value + any two cards as a 3-bet bluff
– Alice 4-bets {KK+} for value

EV (Alice) =-286 bb/100

Alice now gets slaughtered by Bob’s any-two-cards 3-bet-bluffing and she actually loses money from trying to steal the blinds from CO. Her first adjustment is to return to the optimal value range {TT+,AQ+} associated with her 25% CO openrange. This gives us:

EV (Alice; optimal 4-bet value range) =-39 bb/100

It helps, but not enough. She will still lose money for every hand she openraises, unless she also starts 4-bet bluffing. Alice now returns to her complete optimal strategy in CO, designed to defend against any-two-cards 3-bet bluffing:

EV (Alice; optimal total 4-bet-range) =+179 bb/100

Bingo! Alice’s optimal 4-bet-strategy not only prevents Bob from exploiting her by 3-bet bluffing with any two cards, it also punishes him for it. Alice now makes 179 – 150 =+29 bb/100 more than if Bob had simply folded every time and let her pick up the blinds.

But is Alice’s optimal strategy the most profitable strategy against Bob’s any-two-cards 3-bet bluff strategy? No, and to find Alice’s most profitable strategy we use common sense. When Bob is 3-betting any two cards, but only continues with {QQ+,AK} after a 4-bet, he is extremely vulnerable for 4-bet bluffing. He only continues with 2.56% of his hands and folds the remaining 97.44% against Alice’s 4-bets

Alice can then maximally exploit Bob’s attempt to exploit her by 4-bet bluffing him with any two cards (or rather, any two cards in her original opening 25% opening range). We let Alice continue with her optimal {TT+,AQ+} value range. Then she deviates from the optimal strategy by widening her 4-bet bluffing range to the rest of her 25% CO opening range. Bob, unaware that Alice has suddenly changed her strategy to exploit his strategy, keeps 3-bet bluffing any two cards:

EV (Alice; 4-bet bluff any two cards) =+1261 bb/100

A huge increase in EV for Alice, and Alice now makes about 8 times more than if Bob had folded every hand. This is an extremely clear illustration of the difference between optimal and exploitative play. Alice’s optimal defense strategy against Bob’s 3-bets guarantees that he can not exploit her by 3-bet bluffing any two cards, but he optimal defense did nothing to counter-exploit Bob extreme strategy. Alice made a little bit more than if Bob had folded every hand, but not a lot ((179 bb/100 vs 150 bb/100) .

But when Alice responds by choosing the strategy that exploits Bob’s any-two-cards-bluffing strategy maximally, her EV explodes. Still, this does not come without risk for her, since Bob can take his exploitative strategy to the next level and begin 5-bet bluffing any two cards to exploit Alice’s any-two-cards 4-bet bluffing. Alice must then make a new exploitative adjustment (for example, widening her value range dramatically and calling Bob’s 5-bets with a very wide range of value hands) to stay one step ahead of Bob.

We can view this exploit/counter-exploit process as “strategic ping-pong” where both players zig and zag, using extreme strategy changes in different directions to maximally exploit their opponent. When one of them has made a big strategy change to exploit her opponent, she also creates an opening that the opponent can exploit by making an adjustment of his own. Then the first player has to make another big strategy adjustment, and the process repeats itself ad infinitum.

When we are playing optimally, we are using a different mindset. Instead of trying to stay one step ahead of our opponents by sudden “gear changes”, we can fall back on a strategy that performs more or less equally well no matter what our opponent does. If he uses an extreme strategy, our optimal strategy will win a bit from him, but maximizing our profit from his mistake(s) is not our main goal. Instead, we simply want to prevent him from exploiting us. In the simulations above we saw that this worked well for Alice, but not as well as the maximally exploitative strategy she can use against Bob’s any-two-cards 3-bet bluffing.

Okay Bob, now what? Bob can of course respond to Alice’s any-two-cards 4-bet bluffing by going to the next level and start 5-bet bluffing with any two cards. Alice then gets exploited for a while, until she realizes this and adjusts. But we will not take any-two-cards bluffing beyond what we did in the previous simulations, and we assume that Bob now adjusts by falling back on his optimal 3-betting strategy. Alice in turn falls back on her optimal 4-betting strategy:

Pokerazor simulation 4 (both players use optimal strategies)
Both Alice and Bob now uses the optimal strategy. As we have seen in previous articles, this optimal strategy pair follows from Alice’s openrange. Both players are now protecting themselves against any-two-cards bluffing from their opponent in all phases of the 3/4/5-bet war.

Alice’s EV now becomes:

EV (Alice; both players 3/4/5-bet optimally) =+129 bb/100

In the last round of simulations we will let one player stick to the optimal strategy while the other player is let “off the leash”, free to try anything to increase her or his EV against the other player’s optimal strategy. Then we compare the resulting EV with EV when both players play optimally (129 bb/100 for Alice). We start out with Bob playing optimally, while Alice is testing out some deviations from her optimal strategy:

Pokerazor simulation 5 (Bob plays optimally and Alice can do what she wants)
We remember that Alice makes 150 bb/100 when Bob doesn’t interfere, and Bob’s optimal 3-betting strategy reduces this to 129 bb/100 when Alice plays optimally too. We shall now see that Alice can’t do anything to increase her EV significantly when Bob sticks with his optimal strategy.

For example, let’s assume that Alice drops the weakest hands TT/AQ from her value range when she sees that Bob uses the strong value range {QQ+,AK} (and remember that he also 3-bets/5-bets with his 5-bet bluffs A5s-A3s). Alice then chooses to 4-bet TT/AQ as before, but she folds them when Bob 5-bets all-in:

EV (Alice; folds TT/AQ to 5-bet) =+128 bb/100

Alice’s EV drops a little, and she can’t increase her EV by playing a tighter value range against Bob’s strong value range. The reason is that Bob has put exactly so many 5-bet bluffs in his 5-betting range that Alice becomes indifferent to folding or calling with her weakest value hands. When Alice tries to avoid paying off Bob’s better value hands, his 5-bet bluffs makes more money.

Can Alice increase her EV by 4-bet-bluffing more than optimally? We test this by letting her 4-bet bluff any two cards against Bob’s optimal strategy, keeping everything else optimal:

EV (Alice; 4-bet-bluffs any two cards) =+139 bb/100

A small EV increase, but nothing comparable to the results of the extreme exploitative any-two-cards adjustments in previous simulations. Note that a perfectly optimal strategy for Bob should make it impossible for Alice to increase her EV, but our optimal strategy implementations probably contain some “numerical noise”, since we have made some approximations and rounding along the way.

At any rate, Alice’s attempt to exploit Bob’s optimal strategy with any-two-cards 4-bet bluffing only results in a small EV change. Bob’s optimal strategy therefore protects him well against exploitative 4-bet bluffing from Alice. And if Bob wanted to, he could probably fine-tune his strategy (for example, remove a couple of 3-bet bluff combos) to eliminate Alice’s small EV increase completely.

So we have seen that Bob’s side of the optimal 3/4/5-bet strategy pair seems robust against Alice’s extreme adjustments. Let’s now turn to Alice, and let her play the optimal strategy while Bob is allowed to do whatever he wants:

Pokerazor simulation 6 (Alice plays optimally, while Bob can do what he wants)
We saw previously that Bob’s attempts to exploit Alice’s optimal strategy by 3-bet bluffing any two cards didn’t work for him. He lost against her optimal strategy (her EV increased from the baseline 150 bb/100 to 179 bb/100), and he lost a lot when Alice counter-exploited him by 4-bet bluffing any two cards (her EV increased to 1261 bb/100). We shall now repeat this simulation by using Bob’s optimal strategy as a starting point, and then we make the adjustment that he 3-bet bluffs any two cards on top of that. All other ranges for value and 5-bet bluffing are as in the optimal strategy:

EV (Alice; Bob 3-bet-bluffs any two cards) =+179 bb/100

Alice’s EV increases with +50 bb/100 (from 129 bb/100) relative to Bob’s optimal strategy, and +29 bb/100 relative to the baseline EV when Bob always folds (150 bb/100). And we remember that if Alice wants to, she can exploit Bob hard by 4-bet bluffing any two cards and pocket more than 1200 bb/100 until Bob adjusts back. So Bob can’t increase his EV by aggressive 3-bet bluffing, which is what we expected.

Then we let Bob 3-bet bluff any two cards and also 5-bet bluff any two cards. In other words, we let him play like a complete maniac, where he 3-bets any two cards and then 5-bets any two cards if he gets 4-bet.

EV (Alice; Bob 3-bet-bluffs/5-bet-bluffs any two cards)
=+300 bb/100

This causes Alice’s EV to more than double relative to the 129 bb/100 she has against Bob’s optimal strategy. We conclude that Alice’s optimal strategy is waterproof against any-two-cards 3-bet bluffing, and that Bob only hurts himself if he tries.

The results from the last two simulations are worth noticing, since they can be uncomfortable scenarios to play when you don’t know whether or not you are defending correctly. But as we have seen, even against a total maniac who 3-bets you from position at every opportunity, you don’t have to do anything else than respond with your memorized optimal 4-bet strategy. If you do this, he will loose money relative to playing an optimal strategy himself, and he will probably end up losing money overall (so that 3-betting is worse for him than folding).

Note that an aggressive 3-bettor can still reduce our EV relative to the baseline EV (i.e. we will make less when he sometimes 3-bets than when he always folds), which is of course intuitively obvious since he has a lot of strong hands in his 3-betting range as well. For example, he could choose to 3-bet only AA, and we could not do anything to prevent him from making money in this situation and reduce our EV. ). So if we always respond with our optimal strategy, we can’t deny the 3-bettor some +EV.

For example, if Bob deviates from his optimal strategy by increasing his 3-bet bluffing from 70% of “IP 3-bet air list” to 100% of the list, we get:

EV (Alice; Bob 3-bet-bluffs all of "IP 3-bet air list")
=+130 bb/100

We still make a little bit more (129 bb/100 –> 130 bb/100) when Bob increases his 3-bet bluff percentage beyond the optimal percentage, but he still reduces our EV relative to our baseline EV when he always folds (150 bb/100 –> 130 bb/100). We can’t prevent Bob from making some money in this situation, and we just have to accept that a player in position has the right to make money by 3-betting us. Of course, our openraise will still be nicely profitable overall, just less profitable than if he had always folded behind us.

As we saw previously, we can exploit a complete maniac by deviating from optimal play to take advantage of the gaping holes in his strategy, particularly if he folds too much to 4-bets. If he lets us exploit him, we can make more money from an exploitative strategy than from our optimal strategy. But then we have to play guessing games with him, and we also run the risk of offering big openings to the other players at the table (they can deviate from optimal play to exploit our non-optimal play). Since an optimal strategy will protect us (and then some) from getting exploited by a wild 3-bettor, this trade-off might not be worth it

A couple of obvious adjustments we can use to exploit a very aggressive with position on us 3-bettor are:

  • 4-bet bluff more, if he folds easily to 4-bets (in other words, he defends his loose 3-betting range to tightly)
  • Drop 4-bet bluffing, but 4-bet more hands like AJ, AT, 99, 88, etc. for value, if he folds too little to 4-bets and calls and 5-bets a lot with weak hands

But we don’t have to make these adjustments to defend out of position against overly aggressive 3-betting. Our optimal strategy is more than enough. It might feel like we’re getting exploited, and some of the reason for that is that a strategy where we fold a lot (70% in the optimal strategy, as explained in Part 1) feels “weak”. But the reality is that a maniac 3-bettor in position ends up costing himself if he starts 3-betting any two cards against our optimal strategy. Keep this in mind every time you feel exploited by a 3-bettor in position.

3. Summary
We have tested optimal strategy pairs for heads-up 3/4/5-betting using the analysis software Pokerazor. We started with a discussion of ABC preflop strategies without 3-bet bluffing or 4-bet bluffing. We then used simulations to show how ineffective and vulnerable these strategies are against players who are capable of reraising as a bluff with any two cards. As a part of this simulation we looked at exploitative adjustments we can make against players with big leaks in their 3/4/5-bet strategies.

Then we tested the robustness of the optimal 3/4/5-bet strategies we defined in previous articles, with the raiser out of position. We concluded that both the raiser’s and the 3-bettor’s optimal strategies were robust, and that they did not give the opponent openings he could exploit by bluffing with any two cards.

In Part 7 we’ll do numerical simulations for flatting heads-up in position. Among other things we’ll compare EV for flatting versus 3-betting for value with hands that are in between clear value hands and clear flatting hands (for example QQ against a tight UTG raiser). In the last half of Part 7 we’ll adjust our heads-up 3/4/5-bet strategies for blind vs blind scenarios.

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/optimal-3-bet-4-bet-5-bet-strategies-in-nl-holdem-6-max—part-6-728#sthash.iNJOVogt.dpuf

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

1. Introduction
This is Part 4 in the series Optimal 3-bet/4-bet/5-bet-strategies i NLHE 6-max. In Part 1 and Part 2 we discussed 3-betting heads-up with the 3-bettor in position. In Part 3 we began working on the scenario where the 3-bettor is out of position (in other words, blind defense heads-up).

We then looked at the scenario where the raiser open-raises on the button, and then the 3-bettor is in the small blind, or in the big blind after the small blind has folded. We assumed that the raiser (Alice) opened our default button range, and then we defined default ranges for flatting and 3-betting for the player in the blinds (Bob). We also defined a defense strategy for Alice to use against Bob’s 3-bets.

In Part 4 we’ll continue this work and generalize the strategies from Part 3 to include openraising from other positions than the button. We’ll let Alice open-raise from UTG, MP, CO or button, and then Bob 3-bets heads-up from one of the blinds. In Part 3 we used a simple model where we assumed that the two players in the blind shared equally the responsibility of defending the blinds against Alice’s raise. Since Alice there raised from the button, the two players in the blinds had to do the whole job of defending the blinds enough to prevent Alice from raising any two cards profitably. We found that they had to defend a total of 30% to achieve this. And when this job is shared equally between them, they have to defend about 16% each.

In Part 4 we’ll let Alice open-raise from other positions. Then there will be some player(s) between her and the blinds. The total job of defending the blind 30% is then shared between the blinds and the player(s) between Alice and the blinds. This means the players in the blinds don’t have to defend as much as they had to against Alice’s button raises. Furthermore, it’s reasonable that most of the blind defense should be done by the players with position on Alice, especially the player on the button.

We’ll use a simple model to study the distribution of blind defense responsibility between the players left to act after Alice’s open-raise. When Alice raises, the remaining players have to defend at least 30% to prevent her from having a profitable raise with any two cards. The more players left to act, the less each of them have to defend for this to be achieved.

Our starting point will be the default ranges we have defined in the first 3 parts of this article series plus simple mathematical modeling. We want to study how often the blinds minimum have to defend heads-up against raises from various positions to prevent the raiser from having a profitable raise with any two cards. We’re mostly interested in qualitative trends, but we’ll also use the results to estimate reasonable blind defense ranges to use against raises from all positions.

When we have generalized Bob’s blind defense strategies to defense against raises from all positions, we’ll turn to Alice and generalize here defense strategies in all positions against Bob’s 3-bets from the blinds.

The structure for Part 4 is thus:

  • A generalization of heads-up 3-betting out of position (i.e. blind defense) against an openraise from any position
  • A generalization of the raiser’s defense heads-up in position against a 3-bet from the blinds

The work in this article will be somewhat abstract and mathematical in nature, and our purpose is first and foremost to learn how to think correctly about these topics. For example, we’ll learn that there is a huge difference in blind defense strategies against a button raiser and against an UTG raiser, and we’ll use mathematical modeling to quantify this difference.

We always want to play hands that are profitable and fold those that are not, but in practice we don’t know for sure which hands are the profitable ones in a given scenario. We know which hands are clearly profitable (e.g. the big pairs) and which hands are clearly unprofitable (e.g. 72o), but in all preflop scenarios there is a wide range of hands that are not clearly profitable or unprofitable (e.g 99, AJ, KQ, JTs and other medium strong hands).

Instead of thinking about how profitable a hand is, we can complement our understanding of the situation by attacking it theoretically from a different angle. Instead of asking “which hands are profitable?“we can ask “How many hands do I have to play to prevent my opponent from having a profitable bluffing opportunity with any two cards?“. Working along this line we can paint a picture of which hands we should be able to play profitably in a given scenario to prevent our opponent(s) from exploiting us. This is the kind of thinking we’ll use in the modeling work done in this article.

The original plan for Part 4 was to also talk about squeezing (3-betting in a multiway pot after the raise has been called before it’s our turn to act) and small blind vs big blind scenarios, but we’ll move these topics to future articles.

2. Generalization of the theory for 3-betting/blind defense heads-up and out of position
Let’s first quickly repeat the theory for blind defense against a button open-raise defined in Part 3:

We used the following model:

  • Both players start with 100 bb stacks
  • Alice open-raises pot (3.5bb) on the button
  • Bob defends against Alice’s raise by 3-betting pot (12 bb) with an optimal 40/60 ratio of value hands and 3-bet bluffs, plus flatting some medium strong hands
  • Alice defends against Bob’s 3-bets by 4-betting to 25 bb (a little less than pot) with an optimal 60/40 ratio of value hands and 3-bet bluffs, plus flatting some medium strong hands
  • Bob defends against Alice’s 4-betting by 5-betting his value hands and folding everything else

Alice’s open-range from the button was defined in Part 2:

Default button-range

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

458 combos
35%

Since Alice risks 3.5 bb to win 1.50 bb when she open-raises, she can steal profitably with any two cards if she succeeds more than (1.5 + 3.5) =70% of the time. The blinds can’t allow this, so they have to defend 30% of the time to make Alice’s weakest raising hands break even. We’ll assume they share this responsibility equally, and that they both defend some percentage x.

The probability that both of them fold is then (1-x)(1-x), so the chance that at least one of them defends is 1 – (1-x)(1-x). This expression should be 30% to prevent Alice from raising any two cards profitably, so we get:

1 - (1-x)(1-x) =0.30
1 - (1 -2x +x^2) =0.30
1 - 1 + 2x - x^2 =0.30
x^2 - 2x + 0.30 =0

This quadratic equation has the solutions x =1.84 and x =0.16 (you can use the online Quadratic Equation Solver), and we choose the solution x =0.16 =16%, since x is a probability (a number between 0 and 1). We then defined a 3-bet value range, a 3-bet bluffing range, and a flatting range for Bob so that his total blind defense was approximately 16%. The ratio of value hands to bluffs in his 3-betting range was the optimal 40/60 ratio that we have used throughout this article series. In addition Bob flats a range of medium strong hands that are not good enough to 3-bet for value, but too strong to fold or turn into 3-bet bluffs.

2.1 Minimum default blind defense heads-up against a button open-raiser
We estimated the following defense ranges for Bob against Alice’s button open-raise:

Value 3-bet-range OOP against a button open-raise

TT+
AQ+

62 combos

We remember that the weakest hands in this value range work as a “hybrid” between value hand and 5-bet-bluff. Alice will often flat Bob’s 3-bet with position, and TT/AQ have good equity against her flatting range (medium hands like 99, AJ, KQ, etc). When she 4-bets, we don’t expect TT/AQ to be favorites against her value-range (i.e. the hands she plans to call an all-in 5-bet with), so when we 5-bet these hands it makes more sense to think of them as 5-bet bluffs (that profit from folding out Alice’s 4-bet bluffs, but are underdogs when she calls). See Part 3 for a more thorough discussion of this topic.

In addition to the value hands Bob 3-bets, planning to 5-bet all-in, he uses a range of 3-bet bluffs (“OOP 3-bet air list”) and a range of medium strong hands that he flats (“OOP flat list”):

OOP 3-bet air list

66-22
A9s-A6s
K9s-K8s
QTs-Q9s
J9s-J8s
97s+
87s
76s
65s

98 combos

OOP flat list

99-77
AJs-ATs, AJo
KTs+ KQo
QJs
JTs

70 combos

Bob’s list of 3-bet bluffs to use out of position is stronger than the list of hands he 3-bet bluffed with in position (see Part 1 and Part 2). This is because Alice will sometimes flat the 3-bet in position, and then Bob will be forced to play postflop out of position. To make the most out of these scenarios it’s important for Bob to 3-bet bluff with the best of his worst hands, i.e. the range of hands just below his flatting range.

Bob defends a total of 62 + 70 + 98 =230 combos against a button steal raise, or 230/1326 =17% of the time (a bit more than the minimum 16% that we need, which is fine). We’ll now place Alice in CO, MP and UTG and estimate how often Bob minimum has to defend to prevent her from having a profitable raise with any two cards.

With players between Alice and Bob we also have to take into account the blind defense done by these players, and less of the total blind defense responsibility falls on the two players in the blinds. We’ll account for this by using a simple mathematical model.

Note that when Bob tightens up his blind defense against open-raises from earlier positions, hands should in theory move between ranges. When we drop some hands from our value range, these hands should be moved down to the flatting range, and some flatting hands should be demoted to 3-bet bluffs. This follows from the strength principle. But in practice we’ll simplify things by keeping our “OOP 3-bet air list” constant, so that we won’t have to memorize a range of 3-bet bluffing hands for each of the raiser’s positions. This is not quite optimal, but we accept this simplification in order to make it easy to build and memorize sound default ranges. When this work is done, we can always fine-tune our ranges later.

2.2 Blind defense heads-up against a CO open-raise
We assume Alice opens our default 25% CO range:

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

326 combos
25%

Button and the two players in the blinds now have a collective responsibility of defending the blinds at least 30%. We start by assuming button uses our optimal 3/4/5-bet strategy against a 25% CO open-raise, and also our default flatting range in position. So we start by estimating how often button defends against Alice’s CO raise.

In Part 2 we defined the following ranges for the 3-bettor in position:

IP 3-bet air list

A9s-A6s
K9s-K6s
Q9s-Q6s
J9s-J6s
T8s-T7s
97s-96s
87s-86s
76s-75s
65s

100 combos

IP 5-bet air list

A5s-A2s

16 combos

IP flat list

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

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

Button first defines a value range. Then he finds the percentage of the 3-bet bluff list he needs to use to get an optimal 40/60 value/bluff ratio. And then he chooses his flatting range. We’ll assume that a player on the button will flat all hands from “IP flat list” regardless of the raiser’s position.

In Part 2 we found the following optimal strategy to use in position behind a 25% CO raiser:

  • Flat the whole “IP flat list”: {22+,ATs+,AJo+,KTs+,KQo,QTs+,JTs,T9s,98s} =140 combos when {QQ+,AK} are 3-bet for value
  • 3-bet {QQ+,AK, 12 air} for value, planning to 5-bet all-in after a 4-bet
  • 3-bet 70% of “IP 3-bet air list”, planning to fold to a 4-bet

So button 3-bets {QQ+,AK} =34 combos for value, together with 12 Axs-combos that he 5-bet bluffs all-in if Alice 4-bets. This gives button a total value range (more correctly: all-in range) of 34 + 12 =46 combos. Button then needs 1.5 x 36 =69 3-bet bluff combos to get an optimal 40/60 value/bluff ratio.

So we use 69% of the “IP 3-bet air list”, which we round to 70% to keep things simple. To achieve this we use a randomizer every time we have one of the 3-bet bluff candidates from the list. We 3-bet bluff when the randomizer returns a number between 0 and 70, and otherwise we fold. Since there are are 100 combos total in “IP 3-bet air list”, this corresponds to 3-bet-bluffing 70 combos on average, which is what we want. Finally, we flat the 140 combos from “IP Flat List” that remain when {QQ+,AK} get 3-bet for value.

Button then 3-bets 34 + 12 + 70 =116 combos total (i.e. 116/1326 =8.7%) and flats 140 combos (140/1326 =10.6%). This means button defends 8.7 + 10.6 =19.3% total after a CO open-raise. Since button, small blind and big blind need to defend 30% combined, this means that button does most of job of defending the blinds. The two players in the blinds can therefore tighten up considerably compared to the ranges they had to defend with against a button open-raise.

We use the same mathematical model as before and assume button defends 19.3% as estimated above. Then the rest of the blind defense responsibility is shared equally between small blind and big blind, and both of them defend some percentage x those times button folds. The probability all 3 players fold is then (1-0.193)(1-x)(1-x), so the probability at least one of them defends is 1 – (1-0.193)(1-x)(1-x). This expression should be equal to 30%, so we get:

1 - (1-0.193)(1-x)(1-x) =0.30
1 - 0.807(1 -2x +x^2) =0.30
1 - 0.807 + 1.614x - 0.807x^2 =0.30
-0.807x^2 + 1.614x - 0.107 =0

We solve this expression with Quadratic Equation Solver, and get the solutions x =0.068 and x =1.93. We choose the solution between 0 and 1, and find that each of the players in the blinds need to defend x =0.068 =6.8%. We round this to 7%.

This is a very interesting result compared to the defense percentage of 16% against a button open-raise. When we get one player (button) between the raiser and the blinds, the minimum defense percentage for the two players in the blinds is reduced from 16% to only 7%!

So what does a ~7% defense range look like? As always, we start with a value range for 3-betting. Then we add 3-bet bluffs to get an optimal 40/60 value/bluff ratio. Finally we flat with the best hands not good enough to 3-bet for value.

We have some flexibility here. As discussed in Part 2, we are trying to do the big and important things correctly, and we don’t worry about the grey areas where the differences between the alternatives are small (for example, whether we should 3-bet for value or flat with a good-but-not-great hand like JJ). Our starting point is the optimal value/bluff ratio for our 3-betting range, and then we try to design a solid and reasonable total defense strategy.

Against a button open-range we used {TT+,AQ} as our value range. Against a default CO open-range we should tighten up our value range somewhat, let’s say to {JJ+,AK} or {QQ+,AK}. Let’s choose {QQ+,AK} =34 combos and see where this takes us. We now need 1.5 x 34 =51 3-bet bluff combos for an optimal 40/60 value/bluff ratio. Since there are approximately 100 combos in our “OOP 3-bet air list”, this corresponds to 3-bet bluffing all hands on the list 51% of the time, using a randomizer. We round this number to 50% to keep things simple.

So we 3-bet 34 + 50 =84 combos total for 84/1326 =6.3% of all hands. This means that almost all of the estimated minimal defense responsibility of 7% can be done by 3-betting {QQ+,AK} for value, together with the optimal number of 3-bet bluffs.

In addition, we can pick the best medium strong hands to flat with. Since the 3-betting range makes up almost everything we need, we can be picky and choose for example {JJ-TT,AQ} =28 combos (32/1326 =2.1%). This gives us a total defense percentage of 6.3 + 2.1 =8.4%, which is a bit more than the minimum 7% we need according to our model.

Note that what we’re doing here is to play with a mathematical model to estimate how often we minimum have to defend in the blinds against a CO open-raise to prevent him from open-raising any two cards profitably. We’re assuming button does his part of the job by following our optimal 3/4/5-bet + flat strategy in position, and then the small blind and big blind take care of the rest.

This is not the same as estimating which hands are profitable to play from the blinds after a CO steal raise when button has folded. But is gives us a starting point to build on. For example, if CO is passive postflop and often lets you get cheaply to showdown with marginal hands, this will make it easier to play postflop out of position with your medium strong flatting hands. So it would make sense to exploit his tendencies by adding more hands to the flatting range, for example 99-88,AJ,KQs. But our model indicates that you can’t get exploited by CO if you choose to play very tight, and fold these hands heads-up out of position (assuming button defends as actively as he should).

It’s also possible to define a minimum blind defense strategy against a CO open-raise without flatting. For example, we can choose to 3-bet {JJ+,AK} =40 combos for value, and then 1.5 x 40 =60 3-bet bluff combos (i.e. all hands from “OOP 3-bet air list” 60% of the time using a randomizer). This gives us a total 3-bet% of (40 + 60)/1326 =7.5% which is slightly more than the required 7%.

You can of course also use the looser 3-betting range {JJ+,AK} + {3-bet bluffs} together with a flatting range, as you wish. As we discussed in Part 3, what’s most important for us is to use strategies and ranges that are consistent and based on optimal 3/4/5-betting. Exactly what we choose in marginal spots (e.g. should JJ be flatted or 3-bet for value?) is less interesting for us when we’re defining a reasonable default strategy. Also, note that in practice we’ll often use reads to help us choose between similar alternatives at the table.

At any rate, we can use the following strategy as a minimum default blind defense strategy against a CO openraise heads-up from the blinds:

Minimum ~7% blind defense strategy heads-up against a CO open-raise
The minimum 7% blind defense is covered by:

  • 3-bet {QQ+,AK} for value together with 50% of “OOP 3-bet air list”
  • Flat {JJ-TT,AQ}
  • This gives ~8% total defense

Our starting point is to play minimum 34 + 50 + 28 =112 combos, i.e. 112/1326 =8.4% of all hands. Then we can add more hands when we think it’s profitable. Exactly what the range of profitable hands is for us in this scenario is partly a function of factors like CO’s postflop skills, our postflop skills, and the history/metagame between us, so this is impossible to determine exactly in practice. But we’ll talk more about this towards the end of this article series where we’ll discuss exploitative play versus optimal play, and adjustments based on reads and metagame.

We move on, and place Alice in MP. There are now two players (CO and button) between Alice and the blinds, and the blinds can now get away with defending even less.

2.3 Minimum default blind defense heads-up against an MP open-raise
We assume Alice opens our default 15% EP range from MP:

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

194 combos
15%

Button, CO and the two players in the blinds now share the collective responsibility of defending 30% against Alice’s open-raises. We start by assuming CO and button both use our optimal 3/4/5-bet + flat strategy in position against a 15% open-raise. From Part 2 we remember that the 3/4/5-bet strategy is:

  • 3-bet {KK+, 7 air} for value, planning to 5-bet all-in after a 4-bet
  • 3-bet 30% of “IP 3-bet air list”, planning to fold to a 4-bet

CO and button then both 3-bet {KK+} =12 combos for value, together with 7 Axs-combos as 5-bet bluffs for a total value range of 12 + 7 =19 combos. Then we need 1.5 x 19 =29 3-bet bluff combos for an optimal 40/60 value/bluff ratio, which we can round to 30. So we add 30% of the 100 combos in “IP 3-bet air list” using a randomizer. Both CO and button then 3-bets 12 + 7 + 30 =49 combos total, i.e. 49/1326 =3.7%.

Button flats the whole “IP flat list” as before. This is 162 combos when {KK+} is 3-bet for value, which gives 162/1326 =12.2%. For the flatting done by CO, we assume he will flat tighter than button because of poorer position (a reasonable assumption) and that he effectively flats with half the flat list. So MP flats 162/2 =81 combos, or 81/1326 =6.1%.

Under these assumptions CO defends 3.7 + 6.1 =9.8%, while button defends 3.7 + 12.2 =15.9%. As before we find that the probability of all players folding to Alice’s raise is (1-0.098)(1-0.159)(1-x)(1-x), so the chance of at least one of them defending is 1 – (1-0.098)(1-0.159)(1-x)(1-x). This should be 30%, so we get:

1 - (1-0.098)(1-0.159)(1-x)(1-x) =0.30
1 - 0.759(1 -2x +x^2) =0.30
1 - 0.759 + 1.517x - 0.759x^2 =0.30
-0.759x^2 + 1.517x - 0.059 =0

We plug this expression into Quadratic Equation Solver, and find the solutions x =0.040 and x =1.96. We choose the solution between 0 and 1 and find that each of the players in the blinds need to defend x =0.040 =4.0%.

As expected even less of the blind defense responsibility falls on the players in the blinds. We now have two players, CO and button, with position on Alice, and they do most of the defense. Button defends tighter against MP than he did against CO, since the 3-betting range becomes tighter against an MP open-range. But this is more than compensated for by the presence of CO, who also defends with 3-betting and flatting.

Defining a 4% minimal blind defense range is simple. We can use the value range {QQ+,AK} =34 combos as our starting point like we did against CO. We then used 50% of “OOP 3-bet air list” for optimal 3-bet bluffing, and landed on a total 3-betting range of 34 + 50 =84 combos, or 84/1326 =6.3%.

This means that we cover the minimum necessary blind defense (and then some) against an MP raiser by only 3/4/5-betting optimally with a {QQ+,AK} value range and no flatting. We don’t have to use a default flatting range to prevent MP from open-raising any two cards profitably when CO and the button defend optimally in position. So if you want to, you can play very tight and fold hands like JJ, AQ and KQ against a 15% open-raise from early position. The raiser can’t begin to exploit this by loosening up, even if it might feel like you’re being exploited when folding decent hands. Of course, if you think you have a profitable hand, you should play it, but the model indicates that we don’t have to play more than 4% of hands.

Note that the tighter the raiser’s range, the more 5-bet-bluff-like the hands QQ/AK become. QQ/AK have good equity against the range the raiser flats 3-bets with, but we don’t necessarily have good equity against the range of hands a tight player calls an all-in 5-bet with (for example, if he only calls a 5-bet with {QQ}). So in a sense, QQ/AK can be viewed as a value/bluff hybrid against a tight open-raising range, similar to how we played TT/AQ against a button open-raiser.

Alternatively, there’s nothing that prevents us from 3-betting an even tighter value range against MP and then adding a flatting range that includes QQ/AK. If we choose this, it’s obvious to reduce the value 3-bet range to {KK+} =12 combos, and then we add 1.5 x 12 =18 combos from “OOP 3-bet air list”, or 18% for all hands using a randomizer (and we can round this to 20%). Then we effectively 3-bet 12 + 20 =32 combos total, or 32/1326 =2.4%.

We then need a flatting range of at least 4.0 – 2.4 =1.6%, or 0.016 x 1326 =21 combos. This is covered pretty accurately by {QQ,AK} =22 combos. So we can easily defend the minimum 4.0% even with a super-tight strategy where we only 3-bet {KK+} for value together with an optimal number of 3-bet bluffs, and then we flat only {QQ,AK}.

We list both these alternatives:

Minimum ~4% blind defense strategy heads-up against an MP open-raise
The minimum 4% defense is covered by:

  • 3-bet {QQ+,AK} for value together with 50% of “OOP 3-bet air list”
  • This gives 6% total defense

Alternatively

  • 3-bet {KK+} for value together with 20% of “OOP 3-bet air list”
  • Flat {QQ,AK}
  • This gives ~4% total defense

Starting with one of these minimal default strategies we can then add more flatting hands if we think it’s profitable (hands like JJ, TT, AQ, etc). We can use reads to help us here.

It’s obvious that a minimal heads-up blind defense strategy against an UTG open-raise will be squeaky tight, based on our model. But let’s complete the modeling by also working through this case:

2.4 Minimum default blind defense heads-up against UTG open-raise
We assume Alice opens our default 15% EP range from UTG:

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

194 combos
15%

MP, button, CO and the two players in the blinds now share the collective responsibility of defending 30% total against the raise. We assume MP, CO and button all use our optimal 3/4/5-bet + flat strategy in position against the raiser. The 3/4/5-bet part of that strategy is:

  • 3-bet {KK+, 7 air} for value, planning to 5-bet all-in after a 4-bet
  • 3-bet 30% of “IP 3-bet air list”, planning to fold to a 4-bet

And all players between UTG and the blinds thus 3-bet 3.7% as shown previously. In addition button flats the whole flat list of 162 combos =12.2%, while CO flats half the list and 162/2 =81 combos =6.1%. We now assume (somewhat arbitrarily) that MP uses a tight flatting range of 1/4 of the list because of his poor position. In other words 162/4 =41 combos (rounded), or 41/1326 =3.1%.

So MP defends 3.7 + 3.1 =6.8% total, while CO and button defend 9.8% and 15.9%, like they did against MP previously. We set up the same equation as before and get:

1 - (1-0.068)(1-0.098)(1-0.159)(1-x)(1-x) =0.30
1 - 0.707(1 -2x +x^2) =0.30
1 - 0.707 + 1.414x - 0.707x^2 =0.30
-0.707x^2 + 1.414x - 0.0070 =0

We find the solutions x =0.0050 and x =1.995, and choose x =0.0050 =0.5%.

0.5% corresponds to 0.05 x 1326 =7 combos, which is basically {AA}. In other words: If MP, CO and button defend in position with a combination of optimal 3/4/5-betting and flatting, the blinds don’t have to defend with anything else than {AA} to prevent UTG from exploiting them!

The minimal defense percentage 0.5% is of course only meaningful within the context of our model. The number itself is much less interesting than what it represents. What the trend in our model (16% –> 7% –> 4% –> 0.5%) tells us is that you don’t have to worry about getting exploited if you should choose to play very tight from the blinds heads-up against an early position raiser. UTG and MP are handled effectively by the players with position on them, and the players in the blinds can basically just sit back and cherry pick hands they think are clearly profitable.

This is in strong contrast to the 16% default blind defense we were forced to do against a button steal-raise. There we had to 3-bet a wide range and also flat out of position with many so-so hands like 77, ATs, KTs, QJs, etc. All this to prevent button from running over us with loose open-raising. In other words, we were forced to do a lot of “dirty work” out of position with less-than-stellar hands.

On the other hand, our attitude heads-up in the blinds against a tight early position raiser should be more like this:

So should we only 3-bet {AA} from the blinds against an UTG raiser and fold everything else? Of course not, and that’s not what our model tells us. But what it does tell us is that we can’t be exploited by a loose UTG raiser, even if we should choose to defend extremely tight when it’s folded to us. In practice, let’s use the same minimum default defense range we used against MP:

Minimum ~0.5% blind defense strategy heads-up against an MP open-raise
The minimum 0.5% defense is covered by:

  • 3-bet {QQ+,AK} for value together with 50% of “OOP 3-bet air list”
  • This gives 6% total defense

Alternatively

  • 3-bet {KK+} for value together with 20% of “OOP 3-bet air list”
  • Flat {QQ,AK}
  • This gives ~4% total defense

And then we can add a range of flatting hands on top of this, when we think this is profitable.

2.5 Summary of the theory for 3-betting/blind defense heads-up and out of position
Based on simple mathematical modeling we defined the following minimum blind defense strategies heads-up from the blinds against an open-raiser from UTG, MP, CO and the button:

Minimum ~0.5% blind defense strategy heads-up against an MP open-raise
The minimum 0.5% defense is covered by:

  • 3-bet {QQ+,AK} for value together with 50% of “OOP 3-bet air list”
  • This gives 6% total defense

Alternatively

  • 3-bet {KK+} for value together with 20% of “OOP 3-bet air list”
  • Flat {QQ,AK}
  • This gives ~4% total defense

Minimum ~4% blind defense strategy heads-up against an MP open-raise
The minimum 4% defense is covered by:

  • 3-bet {QQ+,AK} for value together with 50% of “OOP 3-bet air list”
  • This gives 6% total defense

Alternatively

  • 3-bet {KK+} for value together with 20% of “OOP 3-bet air list”
  • Flat {QQ,AK}
  • This gives ~4% total defense

Minimum ~7% blind defense strategy heads-up against a CO open-raise
The minimum 7% blind defense is covered by:

  • 3-bet {QQ+,AK} for value together with 50% of “OOP 3-bet air list”
  • Flat {JJ-TT,AQ}
  • This gives ~8% total defense

Minimum ~16% blind defense strategy heads-up against a button open-raise
The minimum 16% blind defense is covered by:

  • 3-bet {TT+,AQ} for value together with 100% of “OOP 3-bet air list”
  • Flat the whole “OOP flat list”: {99-77,AJs-ATs,AJo,KTs+,KQo,QJs,JTs} =70 combos
  • This gives ~17% total defense

Understanding the trend is just as important as the ranges we have defined. There should be as dramatic change in mindset for the players in the blinds when the raiser moves from the button to UTG. Against a button raise we’re prepared to fight fiercely, but against an UTG range we’re content playing only our premium hands for value and avoiding trouble against a strong range with our medium strong hands and our weak hands. We should not try to outplay a tight UTG range, since he is effectively protected by his range (a tight range is easy to defend correctly), and he has position to boot.

We now turn to Alice and generalize the theory for her defense heads-up in position against Bob’s 3-bet from the blinds.

3. Generalizing the theory for heads-up defense in position against a 3-bet
In Part 3 we studied the scenario where Alice openraises a 35% open-range on the button, and then she gets 3-bet by Bob in the blinds. Mathematics dictates that Alice defends 30% of her opening range to prevent Bob from profitably 3-bet-bluffing any two cards. If Alice only 4-bets or folds (like she did when out of position) we can stick to 30%, but when Alice has position, it will also be profitable for her to defend some hands by flatting.

But when Alice defends partly by flatting, Bob’s 3-bet bluffs will sometimes get to see a flop (instead of having to fold to a 4-bet), and then he will sometimes outflop Alice’s better hands. To compensate for the fact that Bob now gets to freeroll flops this way, Alice needs to defend more than 30% total.

We defined the concept “call multiplier” in Part 3 to take into account that Bob freerolls flops with his 3-bet bluffs those times Alice defends against his 3-bets by flatting. We start by giving Alice a 4-bet value range, and then we add the optimal amount of 4-bet bluffs to get a 60/40 value/bluff ratio. This gives us a total 4-bet%, for example 10%. In that case Alice has to flat 30 – 10 =20% by flatting g to get to 30% total defense. But since this lets Bob freeroll flops, we scale the flatting percentage with a call multiplier, which is some number > 1. We elected to use 1.5, and with these numbers Alice now has to flat 1.5 x 20 =30% of her range in addition to the 10% she 4-bets.

3.1 Default defense in position against a 35% open-raise from the button
We defined this strategy in Part 3, and we started by choosing {QQ+,AK} =34 combos as our 4-bet value range for Alice. She balances this with (2/3) x 34 =23 combos of 4-bet bluffs for an optimal 40/60 value/bluff ratio. We pick the 4-bet bluffs from the hands not quite good enough to flat, and we chose {ATo,A9s-A7s} =24 combos.

This gives us a total of 34 + 24 =58 4-bet combos, which is 48/458 =13% of Alice’s 35% button range with 458 combos in it. She now needs 30 – 13 =17% flatting to get to 30% total defense, and we scale up this percentage with the call multiplier of 1.5 and get 1.5 x 17% =26% flatting. This corresponds to 0.26 x 458 =119 combos from her opening range, and we picked {JJ-88,AQ-AJ,ATs,KQ-KJ,KTs,QJ,QTs,JTs} =120 combos.

This gives us the following total default defense strategy on the button against a 3-bet from the blinds:

  • 4-bet {QQ,AK} for value and {ATo,A9s-A7s} as bluffs
  • Flat {JJ-88,AQ-AJ,ATs,KQ-KJ,KTs,QJ,QTs,JTs}
  • We then defend with 34 + 24 + 120 =178 combos total, or178/458 =39% of the button open-range. 13% by 4-betting and 26% by flatting

We now quickly repeat this process for open-raising from CO (25% open-range) and EP =UTG/MP (15% open-range for both). In all cases we choose to start with the value range {QQ+,AK}, and then we build the rest of the total defense strategy around this value range, using optimal 4-betting, 30% total defense, and a call multiplier for the flatting range.

3.2 Default minimum defense in position against a 3-bet after a 25% open-raise from CO
Alice’s default CO range is 25% with 326 combos in it. We 4-bet {QQ+,AK} =34 combos for value, and balance this with the 4-bet bluffs {AT,A9s-A8s} =24 combos. Note that this is a small change relative to the 4-bet bluffs we used on the button. As we shall see in a minute, we choose not to flat 3-bets with ATs in CO, so we demote this hand to the 4-bet bluffing range and kick out A7s.

So we 4-bet a total of 34 + 24 =58 combos, or 58/326 =18% of our 25% open-range with 326 combos in it. We flat 30 – 18 =12% to get to 30% total defense, and we scale this with the 1.5 call multiplier to get to 1.5 x 12% =18% flatting. This is 0.18 x 326 =59 combos from our opening range, and we can choose {JJ-99,AQ-AJ,KQs,QJs,JTs} =62 combos (using a few combos extra doesn’t matter)

Our total minimum defense strategy in CO heads-up against a 3-bet from the blinds becomes:

  • 4-bet {QQ,AK} for value and {AT,A9s-A8s} as bluffs
  • Flat {JJ-99,AQ-AJ,KQs,QJs,JTs}
  • We defend with 34 + 24 + 62 =120 combos total, or 120/326 =37% of our opening range. 18% by 4-betting and 19% by flatting

3.23 Default minimum defense in position against a 3-bet after a 15% open-raise from EP =UTG/MP
We treat these two positions as the same, since we open with the same default range in them. Alice’s default EP range is 15% with 194 combos in it. We 4-bet {QQ+,AK} =34 combos for value and balance this with the 4-bet bluffs {AQ,AJs-ATs} =24 combos (and we’ll see why in a minute).

So we 4-bet a total of 34 + 24 =58 combos, or 58/194 =30% of the 15% open-range with 194 combos in it. This means we don’t have to flat to defend sufficiently! We can flat medium strong hands like AQ, JJ, KQs if we want to, but we don’t have to in order to prevent Bob from exploiting us with loose 3-betting. So if we stick to a minimal strategy, we end up with the same defense against a 3-bet that we use out of position in EP.

The total minimal defense strategy in EP =UTG/MP heads-up against a 3-bet from the blinds then becomes:

  • 4-bet {QQ,AK} for value and {AQ,AJs-ATs} as bluffs
  • No flatting
  • We defend with 34 + 24 =58 combos total, or 58/194 =30% of our opening range. 100% by 4-betting and 0% by flatting

Does this make sense intuitively? Yes, since we have to expect the 3-bettor to have a tight range when he chooses to 3-bet our tight opening range heads-up and out of position from the blinds (and we saw that very tight 3-betting was correct previously in this article). Even if it feels overly tight to fold hands like JJ-TT, AJ and KQs in this case, the mathematics of the situation ensures that we don’t have to play them to defend optimally.

When we start with a 15% opening range, our tight range protects us, and all we have to do is to defend with a tight value range {QQ+,AK} balanced with an optimal number of 4-bet bluffs. There’s nothing anyone can do to exploit this defense strategy, even if we fold everything else.

Since 3-betting from out of position against an early position open-raise usually is done with a very tight range, I recommend that you stick to this minimal defense strategy from EP and don’t try to exploit the 3-bettor by flatting lots of medium strong hands (unless you have strong reads on him). If he 3-bets very loosely and then plays poorly postflop, you can of course flat hands like JJ, TT, AQ, AJ, KQ and use position to play them profitably postflop. But as a default, protect yourselves by using the unexploitable default strategy.

3.4 Summary of the theory for defense heads-up in position against a 3-bet
We elected to use a simple approach to this problem. We used the same value range {QQ+,AK} for all positions, balanced this with an optimal amount of 4-bet bluffs (picked from the best hands not good enough to flat) and the we did the rest of the defense with a flatting range. The flatting range started out very wide on the button, tighter in CO, and it disappeared in UTG/MP where we could defend sufficiently by only 4-betting.

These strategies should be simple to memorize if you already know the strategies for defending against 3-bets out of position (where we only 4-bet or fold) and the button defense we defined in Part 3. The defense from UTG/MP is identical both in and out of position, and we have already covered defense against 3-bets on the button. So the only new strategy to memorize is the CO strategy (and it’s not complicated).

We have now defined a complete set of defense strategies to use against 3-bets heads-up after open-raising from outside the blinds, both in and out of position. Memorize all of these strategies, and you will have a strong foundation to build your preflop game on. It will also be easy for you to adjust to your opponents when you spot opportunities to improve on the default strategies by deviating from optimal play.

4. Summary
We have generalized the theory for 3-betting heads-up out of position against an open-raise from an arbitrary position, and we have also discussed the raiser’s defense against this. We used a simple model to study trends in the minimum blind defense requirements for heads-up blind defense as a function of the raiser’s position. We found that we have to defend loosely and aggressively against a button open-raise, but we can play very tight against raises from early positions (assuming the players with position on the raiser defend as actively as they should).

The reason for this trend is that all players share the responsibility of defending the blinds against a raiser. And the players with position (particularly the button) should defend more than the players out of position. So heads-up against an early position open-raiser we can play tight, without worrying about being exploited. But against an open-raise from late position we have to defend very loosely to prevent the raiser from running over us with loose open-raising. Particularly when the raiser is on the button.

In Part 5 we’ll discuss two topics:

– Squeezing (3-betting in a multiway pot after a raise + call in front of you)
– Cold 4-betting (4-betting after a raise + 3-bet in front of you)

Good luck!

Bugs – See more at: http://en.donkr.com/forum/optimal-3-bet-4-bet-5-bet-strategies-in-nlhe-6-max—part-5-533565#sthash.ZwlU6ch6.dpuf

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

1. Introduction

This is Part 3 in the series Optimal 3-bet/4-bet/5-bet-strategies i NLHE 6-max. In Part 1 we outlined the necessary theory and mathematics by studying a simple model: A player (Alice) raises from some position outside the blinds, and she gets 3-bet by a player (Bob) with position on her. Both players start with 100bb stacks, both the raise and the 3-bet are pot-sized, and Alice’s options out of position were 4-bet or fold. We then used mathematics to estimate optimal strategy pairs for Alice’s and Bob’s 3/4/5-bet war.

In Part 2 we generalized this scenario by estimating optimal strategy pairs for a wide selection of possible opening ranges for Alice. We also gave Bob the option to call the raise (flat) in position. In Part 2 we also looked closer at how to implement these optimal strategies in practice, and we finished our discussion of the scenario where the raiser is heads-up and out of position against the 3-bettor.

The plan for Part 3 is to study the opposite scenario, namely when the raiser is heads-up with position on the 3-bettor. We’ll use a model where Alice openraises from outside the blinds, then Bob 3-bets from one of the blinds, and a 3/4/5-bet war arises. We’ll give this scenario the same systematic treatment as the scenario with Alice out of position, and we’ll specify a complete set of strategies for both players, flatting included.

As before, we assume:

– Both players start with 100bb stacks
– The raise and the 3-bet are pot-sized
– The 4-bet is 27bb (a bit less than pot-sized)
– The 5-bet is all-in

This means we can use the mathematics from the previous two articles. To avoid gaping over too much at once, we’ll narrow the scope for this article to the scenario where Alice openraises from the button, and then Bob is sitting in the small blind, or in the big blind after small blind has folded. We’ll use the standard ranges for openraising described in Part 2, and Alice will open our default 35% button range defined there. When Part 2 was published, the plan was to also study squeezing in Part 3, plus 3-betting from the blinds against openraises from other positions than the button. But due to space constraints we’ll move these topics to a future article.

Bob can both flat the raise and 3-bet from the blinds, and flatting the 3-bet now becomes an option for Alice because she has position on Bob. When Alice was out of position in Part 1 and Part 2, we chose to let he use a 4-bet-or-fold strategy. Her options were then to 4-bet her best hands for value, plus some 4-bet bluffs, and otherwise fold. The rationale behind this is that it’s difficult to defend against 3-bets by calling with medium strong hands out of position (i.e. hands not strong enough to 4-bet for value), trying to play them profitably postflop with 100bb stacks.

It’s easy for Bob to peg Alice with a range full of medium strong hands (KQ, AJ, JTs and the like) when she flats the 3-bet, since she 4-bets her best hands for value. Alice then gets a difficult job postflop, trying to play her medium strong hands profitably out of position. Bob both has position, and more information about Alice’s hand (medium strong) than she has about his hand (either a strong hand or a bluff). Bob’s range is easy to play postflop, since he polarized his 3-betting range preflop by 3-betting an optimal mix of value hands and 3-bet bluffs (and flatting his medium strong hands). Polarizing his 3-betting range into strong hands and bluffs makes it’s easier for Bob to know whether he’s strong or weak postflop when his 3-bet gets called, and this makes his postflop decisions easier. Therefore, with a range that’s easy to play well, and with position on Alice’s medium strong range, it’s relatively easy for Bob to outplay Alice postflop.

Therefore we denied Alice the option of flatting 3-bets out of position, even if there aren’t any “laws” that forbid it. But in Part 3, Alice has position, which makes it easier for her to play medium strong hands well postflop. So now she will flat 3-bets with a range of medium strong hands that are not good enough to 4-bet for value (hands like JJ-99, AQ, AJ, KQs, etc), and then use position to play them profitably postflop. Alice will still 4-bet her best hands for value, together with an optimal number of 4-bet bluffs, and then she flats with the next tier of hands that are good enough to play, but not strong enough to 4-bet for value, and she folds the rest. Note that “for value” in this context means hands we’re planning to go all-in with preflop, either by calling a 5-bet (when we’re the raiser), or by 5-betting all-in (when we’re the 3-bettor).

For example, if Alice raises KQs on the button, and Bob 3-bets from the small blind, we’ll see that this is an automatic call for Alice. She has a decent suited and coordinated high card hand, and she has position. But in Part 1 and Part 2, Alice folded this type of medium strong hands to avoid putting herself in difficult postflop scenarios out of position.

The mathematics and the theory behind optimal optimal 3/4/5-betting with the raiser in position is the same as when the raiser is out of position, but Alice’s option to call the 3-bet makes her total defense strategy after a 3-bet more flexible. Therefore, the theoretical work we do in Part 3 will involve a bit more sound poker sense than the very strict strategies we found when Alice could only 4-bet or fold out of position.

When Alice has position and the option to flat Bob’s 3-bets as well as 4-bet, we have to take two things into consideration:

  • Alice now has more strategic choices to make, when she has to define both a 4-betting range and a flatting range after a 3-bet
  • When Bob can get his 3-bets called, his choice of 3-bet bluffs and 5-bet bluffs becomes more important

We defined both an “IP 3-bet air list”, an “IP 5-bet air list” and an “IP flat list” for Bob in Part 2. These were the hands Bob picked his 3-bet bluffs, 5-bet bluffs, and flatting hands from, when he was in position (IP). These lists were based on the following simple principles:

  • Bob 3-bets his best hands for value, planning to 5-bet all-in if he gets 4-bet
  • Bob flats with the best hands not strong enough to 3-bet for value, but strong enough to play profitably
  • Bob 3-bet-bluffs with hands that are a bit too weak to flat (and he also 3-bet bluffs some Axs hands that he plans to use as 5-bet bluffs after a 4-bet)

Bob uses the same principles when out of position, but now he’ll use hands that are a bit stronger on average. The reason is that Alice’s positional advantage reduces the profitability of all of Bob’s hands that are forced to play postflop (after Alice chooses to flat his 3-bet). This means Bob should use stronger hands for both 3-bet bluffing and 5-bet bluffing. For example, in Part 2 we defined QTs as a candidate for flatting in position, but out of position Bob will use QTs as a candidate hand for 3-bet bluffing.

The fact that Bob often gets his 3-bets called also has consequences for his choice of 5-bet bluff hands. For example, we’ll not use Axs as dedicated 5-bet bluff hands for Bob when he is out of position, since these play poorly against the range Alice flats 3-bets with. When Alice only defended by 4-betting or folding, we picked Bob’s 5-bet bluffs from the region of hands not strong enough to flat Alice’s openraise. We chose the Axs hands, since these never have very poor equity against Alice’s value range (that she calls 5-bets with), no matter what it is (Axs has about 30% equity, no matter what Alice calls our 5-bets with). The reason for picking our 5-bet bluffs from this region, was that we didn’t want to use hands good enough to flat as bluffs, thereby wasting their postflop value. Keep in mind that when Alice never flats 3-bets out of position, our 3-bet bluffs will have to play postflop. They will either win the pot right there, or be folded to Alice’s 4-bets.

Since our 3-betting hands never gets to play postflop when Alice is out of position, we prefer to flat with the best non-value hands like AQ, rather than 3-betting them and turning them into bluffs. AQ has decent equity against Alice’s openraising range, but if we use it for 3-betting against a raiser who either 4-bets or folds, it becomes a bluff in practice. We can’t 5-bet it for value (unless Alice openraises an extremely wide range), so our response to a 4-bet is to either fold (and then we have wasted the hand’s decent postflop value), or 5-bet all-in (and only get called by better hands). The same logic can be applied to other medium strong hands like TT, 99, AJ, KQs, KJs, QJs, etc.

Thus, when we have position we assume that there is more value in flatting the raise with hands like AQ and play a pot postflop than turning it into a bluff. And when these medium strong hands are used as flatting hands, we pick our 5-bet bluffs from the next tier of hands, namely those slightly too weak to flat the raise profitably. From this class of hands we picked the Axs hand to use as 5-bet bluffs, since they always have decent equity against Alice’s value range that calls our 5-bet. In a future article we’ll study flatting versus 3-betting with medium strong hands in more detail and compare the EV for the two lines. Until then, we’re simply going to assume that is the best way to play medium strong hands in position.

Back to Bob’s choice of 5-bet bluffs out of position:

Instead of picking 5-bet bluffs from the region of hands slightly too weak to flat Alice’s raise, we’ll extend Bob’s 3-bet value range downwards. The value range will now include some of the best hands that Bob would have flatted in position. In other words, when Alice sometimes calls our 3-bets, we widen our 3-betting value range to include hands from the upper part of the range we would have flatted in position against a raiser who only 4-bets or fold out of position. These are hands that aren’t the favorite when they 5-bet all-in and get called, but they have decent equity when this happens, and they also have good equity when the 3-bet gets called. AQ/JJ/TT (all were flatting hands in position) are obvious candidates, and we’ll discuss this in more detail later in the article.

So we’ll define a new set of lists for 3-bet bluffing (“OOP 3-bet air list”) and flatting (“OOP flat list”) for Bob out of position (OOP). This means more ranges to memorize. But if you already have memorized the IP ranges for Bob’s 3-bet bluffing and flatting in position, his OOP ranges will be relatively easy to commit to memory. They are a bit different from the IP ranges, but not widely different. Just keep in mind that Bob needs stronger ranges out of position, and the differences become easy to understand.

We start Part 3 with Bob’s strategies for 3-betting and flatting from the blinds (e.g. blind defense) after a button steal raise from Alice. Then we turn to Alice, and study how she defends on the button against Bob’s 3-betting from the blinds.

In Part 4 we’ll generalize to scenarios where Alice has openraised from other positions. Later we’ll also generalize the 3/4/5-bet theory to squeezing (3-betting after Alice’s raise has been called by another player). The mathematics behind squeezing is the same as for 3/4/5-betting heads-up, but the percentages and ranges change a bit when the raise has been called in front of us, and we’ll use mathematics to explain why squeezing is so profitable. We’ll also look at the multiway scenario cold 4-betting, which is 4-betting after a raise and a 3-bet in front of us.

The structure of Part 3 is thus:

  • Blind defense heads-up against a button steal raise
  • The button raiser’s defense against a heads-up 3-bet from the blinds

Then well discuss squeezing and cold 4-betting in Part 5. This will be followed by the final Part 6, where we test our strategies using analysis software (Pokerazor), and also take a look at optimal postflop play.

2. 3-betting and flatting heads-up from the blinds

When Alice raises from outside the blinds and Bob is in one of the blinds, the discussion of optimal 3/4/5-bet strategies and flatting is equivalent to a discussion of blind defense. As we’ll see soon, we now have more things to think about than the corresponding scenario with Bob in position. Both in and out of position Bob has the 3 alternatives 3-bet/flat/fold, but when Bob had position, we didn’t specify how often Bob should flat.

We remember that the optimal 3/4/5-bet strategy pair for Alice and Bob with Bob in position followed from Alice’s open-range. Then we added a flatting range of medium strong hands for Bob, based on sound poker sense. But beyond the optimal 3/4/5-betting, we didn’t make any demands about how often Bob should get involved. We gave him a default flatting range, but he did not have to flat those hands.

Before we move on, let’s determine how often Bob gets involved in position after Alice’s openraises when he uses optimal 3/4/5-betting, and also flats with all the hands from “IP 3-bet flat list”. In Part 2 we designed the following set of optimal strategy pairs for various open-ranges for Alice, together with Bob’s lists of 3-bet-bluff hands and flatting hands in position:

Link to download(right-click and “Save as …”): IP_3-bet_summary.doc

For example, if Alice opens 25% from CO, Bob will 3-bet the value range {QQ+,AK,A5s-A3s} =46 combos (including 5-bet bluffs), and then he adds 1.5 x 46 =69 3-bet bluff combos from “IP 3-bet air list”, which is equivalent to 3-betting all hands on the list 69% of them time (rounded to 70% in the document above) using a randomizer.

Bob then uses a total of 46 + 69 =115 3-bet combos and this results in a a 3-bet% of 115/1326 =8.7%. Then he flats the 140 combos on “IP flat list”. Bob now plays a total of 115 + 140 =255 combos, or 255/1326 =19.2% of his hands on the button.

Calculating Bob’s total range for all of Alice’s open-ranges 15%, 20%, 25%, 30%, 35%, 40% used in Part 2, we get:

Alice opens 15%: Bob plays 15.8% (with 3-bet% =3.6%)
Alice opens 20%: Bob plays 17.0% (with 3-bet% =5.3%)
Alice opens 25%: Bob plays 19.2% (with 3-bet% =8.7%)
Alice opens 30%: Bob plays 19.5% (with 3-bet% =9.4%)
Alice opens 35%: Bob plays 20.3% (with 3-bet% =10.2%)
Alice opens 40%: Bob plays 20.3% (with 3-bet% =10.2%)

In practice, Bob should adjust his flatting range somewhat, according to Alice’s open-range, and fold the weakest hands (e.g. QTs, T9s, 98s, etc.) if Alice opens a very tight range. Regardless, his optimal strategy in position follow from two factors:

– Bob’s half of an optimal 3/4/5-bet strategy pair
– Flatting with the medium strong hands Bob considers profitable

2.1 How often do the blinds have to defend against a button steal raise?

Using the same philosophy as above (3/4/5-betting optimally, and otherwise flat profitable hands) when Bob is in the blinds is a start. But as we’ll see in a minute, we have more things to think about. We of course always want to play profitable hands and fold unprofitable ones (basic exploitative play). But with Bob heads-up in the blinds after a button steal raise, we can also formulate a mathematical requirement for how often the players in the blinds need to defend to prevent Alice from profitably stealing with any two cards.

When Alice openraises pot (3.5bb) on the button in a 1-2 blind structure (small blind =0.5 x big blind), she’s risking 3.5bb to win 1.5bb. The effective pot odds are 1.5 : 3.5, so if Alice succeeds more than 3.5/(1.5 + 3.5) =70% of the time, she can profitably open any two cards on the button. The two players in the blinds can’t allow this, so they have to defend combined at least 30% of the time.

We remember from the theory in Part 1 that an optimal 3/4/5-bet strategy pair is designed to make our opponent’s worst bluffing hands break even. So we start the process of finding an optimal(ish) blind defense strategy with the assumption that the blind players should defend 30% combined.

We use a simple model where we assume that the job of defending the blinds is shared equally between the small blind and the big blind. Both blinds defend a certain percentage x (where x is the same for both players), so that there’s a 30% chance of at least one of them defending.

The chance of one particular player folding is (1-x), so the chance of both folding is (1-x)(1-x). Thus, the chance that at least one of them isn’t folding is 1 – (1-x)(1-x), and we want this to equal 30%. We can formulate this as an equation:

1 - (1-x)(1-x) =0.30
1 - (1 -2x +x^2) =0.30
1 - 1 + 2x - x^2 =0.30
x^2 - 2x + 0.30 =0

This is a quadratic equation with solutions x =1.84 and x =0.16 (you can use Quadratic Equation Solver to compute this). Since we require x to be between 0 and 1 (x is a probability), we choose the solution x =0.16 =16%. Let’s check the solution before moving on. With a defense percentage x =16% for both blinds, the chance that at least one of them defends is:

1 - (1 - 0.16)(1 - 0.16) =0.30 =30%

And we conclude:

If the task of defending the blinds 30% against a button steal raise is shared equally between the small blind and the big blind, both players should defend about 16% of the time.

If we combine this with our general desire to defend with the hands that are profitable, we can say the following:

Heads-up after a steal-raise from the button, you want to defend with the hands that are profitable. But of this range is significantly tighter than 16%, you are probably doing something wrong, and/or you are exploiting your opponents’ mistakes.

Let’s pause for a bit and think about what this statement means. In practice you can often get away by defending the blinds tighter than optimal, without introducing a big leak into your game. This is typically the case in soft low limit games. There are two factors at work:

  • When your opponents don’t exploit tight blinds as hard as they should from the button
  • When your opponents also play too tight in the blinds, and/or give up too easily postflop when they choose to defend

In soft games with many passive players, this is more or less what happens. You don’t have to defend optimally (i.e. make it unprofitably for the button to steal with any two) because most players won’t try to exploit this opening if you offer it to them. And if you lose a bit by defending too little against some players, you can get it back when it’s your turn on the button, since your opponents often make the same mistakes as you in the blinds. So errors tend to cancel each other.

But then we’re in the realm of exploitative play where we’re profiting from opponent leaks, and not optimal play, which is the central topic for this article. We want to explore what optimal (or near-optimal) play is for this scenario, so that we can design a blind defense strategy to use against strong players who use aggressive button openrasising for what it’s worth.

If you think 16% blind defense is too loose for the limits you’re playing, and you don’t think you can defend such a range profitably, think ahead. Work on making this standard defense percentage profitable for you, and think of it as preparations for tougher games in the future. Also, you should replace the “fit-or-fold” mantra postflop with a more aggressive style.

The looser your preflop ranges, the more important it becomes to exploit steal opportunities postflop. Keep this in mind when you’re working on your blind defense. For example, when you flat a hand like KJs in the big blind, don’t always check-fold the flop when you miss. Look for profitable stealing spots, based on flop texture (e.g. sometimes checkraising dry flops like A 7 3 ), and based on your opponent’s tendencies (you can steal more against weak players).

Postflop play is not a topic for this preflop article series, but I’d like to point out the coupling between preflop play and postflop play, and that strict fit-or-fold generally isn’t a good strategy to use in heads-up postflop play when both players start with wide ranges. We’ll not go further into this, but I might write an article later about using principles for optimal play postflop.

Postflop play aside, our job is now to design Bob’s strategies for:

– 3/4/5-betting from the blinds heads-up after a button steal raise
– Flatting from the blinds heads-up after a button steal raise

And we want to end up with a total defense percentage of about 16% when Bob is in the small blind, or in the big blind after the small blind has folded. Both blinds defending the same percentage 16% is an approximation, since the big blind should defend somewhat more than the small blind. But it’s a good approximation, and we’ll use it throughout this article.

We build Bob’s strategy step-wise by giving him a total value range of value hands + 5-bet bluffs, a range of 3-bet bluffs (defined as an “OOP 3-bet air list”) and a range of flatting hands (“OOP flat list”). As always, we use the strength principle as a guideline:

  • We 3-bet the best hands for value
  • We flat with the best hands not strong enough to 3-bet for value
  • We 3-bet bluff some hands among those not strong enough to flat, and we fold the rest

Since we’re out of position, the hands we flat and bluff will be a bit stronger than the ranges we used in position (“IP 3-bet air list” and “IP flat list”) in the previous work done in Part 1 and Part 2.

2.2 Bob’s value range for 3-betting OOP against a button steal raise

Before we get into details, let’s look at the big picture, taking into consideration the difference between being in and out of position. Then we use the work done in Part 1 and Part 2 as a starting point for defining Bob’s strategies from the blinds.

We start with Bob’s value hands that he 3-bets for value, planning to 5-bet shove all-in if Alice 4-bets. When Bob had position on Alice, his value hands followed from Alice’s value range, which followed from her opening range (more precisely, the number of hands in her opening range), plus the requirement that Alice could only 4-bet or fold. But with Bob out of position, the strategies are more flexible, since Alice now has the option to flat Bob’s 3-bets.

Let’s start by assuming Alice opens our default 35% button range. According to the list of optimal strategy pairs defined in Part 2, Alice should use a 4-bet value range of {99+,AJ+} =84 combos if she only 4-bets of folds. She then uses {AT-A8, A7s-A6s} =56 combos as 4-bet bluffs to to get a 60/40 ratio of value hands to bluffs.

But now we have to take into consideration Alice’s positional advantage. When Alice has position on Bob, she doesn’t have to 4-bet all hands that are playable after Bob’s 3-bet. She can now choose between 4-betting or flatting. The weakest value hands Alice 4-bets out of position with a 4-bet-or-fold strategy are 4-bet because they gain enough EV from folding out Bob’s 3-bet bluffs, not because they are a favorite against Bob’s 5-bet-range. If the EV she gains from folding out Bob’s 3-bet bluffs is more than the EV she loses from getting 5-bet and being forced to call because of pot-odds, she has a profitable value 4-bet. But in position it might be more profitable to flat this type of hands and play postflop against Bob total 3-betting range.

For example, it’s seems reasonable to flat a 3-bet in position with one of the weaker OOP value hands like AQ, instead of 4-betting and planning to call a 5-bet. AQ should do well against Bob’s total 3-betting range (40% premium hands like AA-QQ,AK and perhaps a few more, and 60% 3-bet bluff hands like A9s, K9s, J8s, etc). So when Alice can play against this total range with position for the rest of the hand, this seems better than 4-betting, driving out most worse hands, and getting all-in against mostly better hands.

If Alice 4-bets AQ, she’ll probably get sufficient pot-odds to call a 5-bet against Bob’s total 5-bet range (we remember from previous articles that we need minimum 36% equity to call Bob’s shove). And since AQ gains a lot of EV from folding out Bob’s 3-bet bluffs, a 4-bet + call 5-bet might be profitable overall. But this doesn’t mean that 4-betting is the best way to play AQ when we have position. When Bob has to play his total 3-betting range out of position postflop (and 60% of this range consists of bluffs) he will get plenty of opportunities to make postflop mistakes that Alice can exploit. So it could very well be that Alice’s alternatives with AQ in position are ranked call > 4-bet > fold. We’ll look into this in more detail with analysis software in Part 5.

At any rate, by flatting 3-bets in position with the weakest hands she would have 4-bet for value out of position, it is reasonable to assume she’ll be able to extract more value than by playing for all-in preflop. After all, she has position and a hand that’s a favorite against the range that 3-bet her. We have talked about AQ here, but the same argument can be used for for AJ, JJ, TT and 99 (which would all be 4-bet value hands out of position after a 35% openraise). We can also flat 3-bets in position with various medium strong suited/coordinated hands like KQs, KJs, QJs, etc.. They have decent equity against Bob’s total 3-betting range, and our plan is to use position to play them profitably postflop through a combination of showdown equity (the ability to make hands) and steal equity.

So when Bob 3-bets, he can expect Alice to flat a lot with medium strong hands (medium pairs, high card hands of the type good-but-not-great, and the best suited/coordinated hands). This means two things for Bob:

  • His 3-bet bluffs should be stronger than in position, since they now often get called. Bob is then forced to play a weak hand postflop
  • The same goes for the hands Bob 3-bets, planning to 5-bet bluff

Compared to 3-betting in position, Bob should now drop the weakest 3-bet bluffs like K6s. And instead of using low Axs hands as 5-bet bluffs (they do poorly against Alice’s 3-bet flatting range), he should use hands that perform better when the 3-bet gets called.

Let’s start by estimating Bob’s value hands and see where this takes us. With value hands we mean the hands Bob 3-bets for value, planning to 5-bet all in, and where he expects to profit from getting called by Alice’s value hands. From the list of optimal strategy pairs we made in Part 2, we see that Bob will use the value range {JJ+,AK} in position against a 35% open-range. The same value range is also used against 30% and 40% open-ranges, so it seems reasonable to use {JJ+,AK} as our starting point for building a value range to use in the blinds against a button steal-raise (which is rarely tighter than 30%, and often looser than this).

What about the next tier of hands? If we move on to TT/AQ, we’re no longer favorites against Alice’s value range corresponding to 35% open-range, so we can’t define TT/AQ strictly as value hands. Remember that our definition of value hand for the 3-bettor (and this definition is mostly a conceptual tool to help us build ranges), is a hand that we 3-bet and 5-bet, expecting to be a favorite against the hands that call our 5-bet. If this is not the case, we define the hand as 5-bet bluff.

To see that TT/AQ can’t be value hands under this definition against a 35% button open-range, note that Alice optimal value range for a 35% open-range can’t be wider than {99+,AJ}. This is the value range she will use if she only defends against 3-bets by 4-betting or folding, and if she also can flat, her value range will be somewhat tighter. Against {99+,AJ}, both TT and AQ are small underdogs as shown below:

And in practice TT/AQ should be somewhat bigger underdogs against Alice ‘s actual value range in position, since she flats some hands, and therefore can 4-bet tighter than out of position when she defends optimally (30% defense when she 4-bets or folds, and a bit more when she 4-bets/flats/folds). For example, Alice might elect to flat with TT-99 and AJ in position. On the other hand, TT/AQ will have good equity against Alice’s flatting range, so TT/AQ can be viewed as value hands when the 3-bet gets called. For example, if Alice 4-bets {QQ+,AK} for value (plus some 4-bet bluffs) and flats a medium strong range {AQ,AJ,JJ-99,KQ,KJs,KTs,QJs,QTs,JTs}, TT and AQ have 60% and 55% equity against the hands that call the 3-bet, as shown below::

We can therefore view both TT and AQ as a “value/bluff hybrid” where we 3-bet for value against Alice’s flatting range, but when we get 4-bet, we turn them into 5-bet bluffs and 5-bet them all-in. Using these hands as 5-bet bluffs makes more sense equity-wise than using Axs hands as 5-bet bluffs. Axs are underdogs against Alice’s flatting range as shown below, and in addition they are difficult to play well out of position postflop in a 3-bet pot:

So we choose:

Bob’s value-range OOP against a button openraise

TT+
AQ+

62 combos

We landed on this range using a combination of theory from previous articles and sound poker sense. From the previous work it’s clear that {QQ+,AK} can always be used as value hands against a normal button range, and we chose to also include JJ based on the strategy pairs we estimated in Part 2. Then we concluded that TT/AQ work as “hybrids” between value hands and 5-bet bluffs. TT/AQ have good equity against the hands that call our 3-bet, so they can be viewed as value hands. But when we get 4-bet, we use them as 5-bet bluffs, so that we don’t have to use weak hands like Axs (poor equity against Alice’s flatting range) for this purpose.

We’ll also specify the hands Bob uses for 3-bet bluffing. First we’ll specify his flatting range, and then we pick his 3-bet bluffs from the hands a bit too weak to flat.

2.3 Bob’s range for flatting OOP against a button steal-raise

We now turn to Bob’s flatting range. We can use his flatting range in position (“IP flat list”) as our starting point and tighten it up a bit to compensate for Bob’s positional disadvantage. Before we list specific hands, we note that we need about 57 combos in the flat list to end up with a total blind defense range of 16%, which is the requirement we estimated previously.

Bob has 62 combos in his value range, and he wants to 3-bet 1.5 x 62 =93 bluff combos to get an optimal 60/40 ratio of value hands to bluffs. So he 3-bets a total of 62 + 93 =155 combos. Since a 16% total defense range contains 0.16 x 1326 =212 combos, Bob needs 212 – 155 =57 flatting combos.

In the Stoxpoker video series Optimal Preflop Play I-III (which we have used as background material for this article series), Matt Janda recommends the following OOP flatting range, which gives us a few more combos that we need:

OOP flat list

99-77
AJs-ATs, AJo
KTs+ KQo
QJs
JTs

70 combos

We have no reason to make any big changes here, so we’ll use this OOP flat list as standard from the blinds after a button steal-raise. There’s also a mathematical argument for flatting with a few more hands than we need to get to exactly 16% total defense. When we defend by 3-betting, Alice has to fold a lot of weak hands preflop, but when we flat, these weak hands get to see a flop. Therefore, Alice “freerolls” flops with many weak hands when we flat preflop, and she now gets an opportunity to flop something with these hands, or bluff us out postflop when both players miss the flop. We’ll return to this concept when we discuss Alice’s defense strategy against Bob’s 3-bet, where she will call a lot and therefore give Bob an opportunity to freeroll flops with his 3-bet bluffs.

2.4 Bob’s range for 3-bet-bluffing OOP against a button steal-raise

Bob has 62 combos in his value-range, so he needs 1.5 x 62 =93 bluff combos. Matt Janda recommends the following list of 3-bet bluff hands against a button steal-raise:

OOP 3-bet air list

66-22
A9s-A6s
K9s-K8s
QTs-Q9s
J9s-J8s
97s+
87s
76s
65s

98 combos

A bit more than we need, but that’s not a problem. Having an “OOP 3-bet air list” with about 100 combos will also come in handy for 3-betting against open-raises from other positions than the button. Then we want to 3-bet tighter, so we’ll use “OOP 3-bet air list” as a candidate list for 3-betting. With about 100 combos in the list we can easily convert between the number of bluff combos we need, and the corresponding bluff percentage that we can use together with a randomizer. For example, if we decide to 3-bet {QQ+,AK} =34 combos for value against an MP open-raise, we know that we need 1.5 x 34 =51 3-bet bluff combos for an optimal 60/40 ratio. So we need 51 combos from the list, and with ~100 combos in the list, this corresponds to 3-bet buffing the whole list 51% of the time (and we use a randomizer for this, as illustrated in Part 1 and Part 2).

We note that we should use all the hands in “OOP flat list” and “OOP 3-bet air list” when Alice openraises on the button, since these lists we designed to give a total defense percentage of about 16% for this case. But if Alice raises from an earlier position, Bob should tighten up a bit. Both because Alice’s open-range now is stronger, and because the players with position on Alice will do some of the job of defending the blinds. Therefore, the responsibility of denying Alice the opportunity to profitably openraise any two is now shared between the players behind Alice and the two players in the blinds. We’ll talk more about this concept in Part 4, and use a simple mathematical model to study the effect of having players with position on Alice when Bob is in the blinds.

Thus, against openraising from positions earlier than the button, we’ll use “OOP flat list” and “OOP 3-bet air list” as candidate lists, and then we use a bit of common sense to reduce the number of hands that we use, according to the raiser’s position. We’ll play somewhat tighter against a CO open-raise, and a lot tighter against an open-raise from early position. In principle, there are two ways to tighten up; the combo method (playing specific hands from OOP flat list” and “OOP 3-bet air list”) and the percentage method (playing all hands on the lists a certain percentage of the time, using a randomizer).

For the flatting hands, it’s obvious that we should fold the weakest hands on “OOP flat list” when we tighten up. Flatting implies we’ll always play postflop out of position, and this makes it important to always use the best possible hands. However, for the 3-bet bluffs this is less important, since we’ll often win the pot preflop. Therefore, to keep things simple I use “OOP 3-bet air list” with a randomizer in scenarios where I don’t need to use the whole list. But I choose my flatting hands by picking the best hands from “OOP flat list”, using common sense.

But we won’t look at blind defense against raises from other positions until Part 4. Here we’ll finish our work with Alice openraising on the button, and then we use all hands from “OOP flat list” and “OOP 3-bet air list”.

Before we move on to Alice’s strategies for defending in position against Bob’s 3-bets, here’s a “cheat sheet” for Bob’s blind defense strategies against a button steal-raise. You can download the document and have it open on the screen when playing for quick access. This strategy can be used as your default both from the small blind, and from the big blind after small blind has folded.:

2.5 Summary of Bob’s defense strategy heads-up OOP against a button openraise

We defend a total of 62 + 98 + 70 =230 combos. This is 230/1326 =17% of all hands, which is a bit more than the 16% we wanted. This is fine, since flatting lets Alice freeroll flops with her weakest raising hands, so in practice we should defend a bit more when we sometimes flat. If both blinds defend 17%, their total combined defense percentage is:

1 - (1-0.17)(1-0.17) =32%

Link to download (right-click and “Save as”): blind_defense_vs_button_summary.doc

3. Defense against OOP 3-bet after openraising on the button

In Part 1 we saw that when Alice open-raises pot and Bob 3-bets pot in position, Alice needs to defend 30% with a 4-bet-or-fold strategy to prevent Bob from exploiting her by 3-betting any two cards. This percentage changes slightly when Bob is in the blinds (it’s cheaper for him to 3-bet), but we’ll keep things simple and use 30% as our starting point.

3.1 Defending an openraise against a 3-bet using 4-betting and flatting with a “call multiplier”

When Alice defends with both 4-betting and flatting in position, she needs to defend a bit more than 30%, since her flatting lets Bob freeroll flops. For example, if Bob has 3-bet a hand like K9s as a bluff in position, he will never see a flop when Alice defends with only 4-betting or folding. So Bob’s 3-bet bluffs never get the opportunity to outflop Alice those times she defends with a better hand. For example when Alice 4-bets TT for value from CO after a 3-bet from Bob on the button.

But when Alice raises TT on the button, Bob 3-bets K9s from the blinds, and Alice defends by flatting, Bob gets additional ways to win. He can outflop her if the flop comes something like K 8 4 , or he might win with a bluff on flops that contain one or more overcards to Alice’s TT, for example A J 7 . Thus, Alice’s defense strategy in position gives Bob the opportunity to win some pots he would never have won had Alice used a 4-bet-or-fold strategy. When Bob can freeroll flops this way, Alice needs to defend a bit more than 30% in total.

We can adjust Alice’s strategy to compensate for this effect by using something Matt Janda calls a “call multiplier”. We know that Alice should defend at least 30% of her opening range against a 3-bet, so we start by choosing her 4-betting range, for example 10% of her openraising range. Then we must defend 30 – 10 =20% by flatting to get 30% total. But since flatting lets Bob freeroll flops, we scale up this flatting percentage with some constant factor > 1. Janda suggests using a factor 1.5. So we end up with 1.5 x 20% =30% flatting in addition to 10% 4-betting. We name this constant factor “the call multiplier”.

3.2 Alice’s total strategy for defending her button open-range against a 3-bet from the blinds

Alice openraises our default 35% button range that we defined in Part 2:

Default 35% open-range

22+
A2s+ A7o+
K2s+ K9o+
Q6s+ Q9o+
J7s+ J9o+
T7s+ T9o+
96s+
86s+
75s+
65s

458 combos
35%

When Bob 3-bets heads-up from the blinds, Alice knows that she will defend at least 30%, using a combination of 4-betting and flatting. Alice’s job is now:

  • Choose a value range of hands she 4-bets, planning to call an all-in 5-bet
  • Add an optimal percentage (60/40 ratio of value hands to bluffs) of 4-bet bluffs that she folds to a 5-bet
  • Find the percentage of her opening range she needs to flat to defend a total of 30%, then multiply this number with our call multiplier of 1.5 to find the total percentage of flatting

Alice’s value-range
When Alice was out of position, her value-range was uniquely determined from the requirement that she should 4-bet 30% of her opening range, using a 60/40 ratio of value hands to bluffs. But when she also has the option to flat the 3-bet in position, her value-range is no longer a simple percentage of her opening range, and we have to use some judgment.

If Alice hadn’t used flatting in position, she would have defended her 35% button range by 4-betting {99+,AJ+} for value and {AT-A8,A7s-A6s} as 4-bet bluffs, as we found in Part 2. So when she defends partly by flatting, she will obviously 4-bet tighter than this. Let’s use the value-range {QQ+,AK} =34 combos as a start, and see where this takes us. These hands are obviously strong enough to get profitably all-in against Bob’s 5-betting range as shown below (we remember from Part 1 and Part 2 that we need at least 36% equity to call the all-in 5-bet):

Alice’s 4-bet bluffs
Using the value-range {QQ+,AK} =34 combos, we need 34 x (2/3) =23 combos of 4-bet bluffs for an optimal 60/40 ratio of value hands to bluffs. We then pick the best hands not good enough to flat, for example {ATo,A9s-A7s} =24 combos. Alice’s total 4-betting range then becomes {QQ+,AK} + {ATo,A9s-A7s} =34 + 24 =58 combos. This is 58/458 =13% of her total opening range.

Alice’s flatting range
Alice 4-bets 13% of her 35% button opening range, and she needs to flat 30 – 13 =17% to get to 30% total defense. Then we scale up this percentage with the call multiplier 1.5 to compensate for the fact that Bob now can freeroll flops. We end up with a flatting percentage of 17 x 1.5 =26% of the opening range (35% =458 combos), which is 0.26 x 458 =119 combos. For example, we can use {JJ-88,AQ-AJ,ATs,KQ-KJ,KTs,QJ,QTs,JTs} =120 combos.

Summary
Our estimate of Alice’s optimal total defense strategy heads-up against a 3-bet from the blinds after openraising 35% on the button is:

– 4-bet {QQ+,AK} =34 combos for value
– 4-bet {ATo,A9s-A7s} =24 combos as a bluff
– Flat {JJ-88,AQ-AJ,ATs,KQ-KJ,KTs,QJ,QTs,JTs} =120 combos

Alice’s now defends 34 + 24 + 120 =178 combos. This is 178/458 =39% of her opening range, and top 178/1326 =top 13% of all hands. If you think this is loose for the games you’re playing in, keep in mind that we’re not trying to adjust to the tendencies of the 3-bettor here, we’re trying to make it impossible for him to exploit our button openraises by 3-betting any two cards from the blinds. If the 3-bettor is tight, feel free to tighten up yourself. For example, if you think he only 3-bets {JJ+,AK} for value and never 3-bet bluffs, you obviously should adjust by never 4-bet bluffing and fold your medium strong hands (unless you think you have the implied odds to call). But then we’re in the realm of exploitative play, not optimal play.

Note that even if you choose to not use exploitative play against a particular opponent, your knowledge about what optimal(ish) play is will make it easier to adjust to exploit him. You know that the strategy above is designed to prevent a loose 3-bettor from exploiting you, and then you can drop most of this defense against a tight 3-bettor when you have a strong read you can use to increase your profits. Drop all 4-bet bluffing, 4-bet a tight value-range, and only flat with the very best flatting hands when you think this is profitable.

Another thing worth mentioning is that if you don’t think you can play these button defense ranges profitably against a loose 3-bettor, you can take this as a sign that your postflop play needs improvement. Flatting hands like 88, ATs, and JTs to a 3-bet, and then playing them profitably postflop is not necessarily easy. You will get into many tricky spots postflop, but never forget that your position allows you to “turn the table” to some degree, and let your opponent get more than his fair share of postflop misery.

It’s important to realize that you should not flat 3-bets with medium strong hands in position and only plan to play fit-or-fold postflop. The weakest hands in our flatting range will probably be unprofitable for you, if you never use position to bluff and steal postflop. Aggressive and opportunistic postflop play is therefore a requirement when you try to defend optimally preflop, using a wide flatting range. Your postflop strategy for your flatting range should include a fair amount of (semi)bluff raising and floating.

Finally, note how loose you have to defend when trying to defend optimally, even when you’re starting with a relatively tight button range of 35%. Now think about how ultra-loose you would have to defend if you open something like 50% on the button, and you want to defend optimally against 3-bets. However, if you raise this loose on the button, it’s probably because you’re trying to exploit weak players in the blinds. If this is the case, it doesn’t make much sense to try to defend optimally against their 3-betting (since we don’t expect them to 3-bet very light, per definition).

Therefore it’s fine to use our estimated defense strategy for a 35% button core range also when you’re openraising a looser range in practice. You can think about the button openraising you do above and beyond 35% as “bonus raising”, based on an opportunity to exploit weak players in the blinds. If you’re trying to exploit the blinds, it’s fine to give them an opening to exploit (since we assume they won’t try to exploit us), and then you don’t worry about trying to defend the extra raising hands optimally. You just fold these additional weak hands when you get 3-bet, and you don’t worry about getting exploited until you notice the blinds have loosened up significantly against you. If they start fighting back by 3-betting a lot, it’s probably better to tighten up to something close to the 35% core range and defend this range optimally, than to try and defend a very loose opening range (e.g. 50%) optimally.

This will be discussed further in Part 6, where we’ll talk about optimal versus exploitative play. Until then, train optimal defense of our default 35% button range, and you will be a tough nut to crack for blinds trying to fight back against your button steals by 3-betting you a lot. If you can play well postflop after flatting 3-bets, a player 3-betting you often and light from out of position is likely to find himself in lots of trouble. He’ll often be faced with your optimal 4-betting range (mathematically impossible to exploit), and when you don’t 4-bet, he’ll often get called. When you flat your range of medium strong hands, the 3-bettor is forced to play postflop out of position, often with a worse hand than yours. This will be difficult for him when you play well postflop, including knowing when to steal.

As a thought experiment, think abut how you would like to sit in the blinds against a button player who plays this way. If you fold too much, he will rob you blind preflop. If you get feisty and try to defend with uncontrolled and overly aggressive 3-betting you will run into a wall of optimal 4-betting plus flatting followed by aggressive postflop play where you are out of position with a lot of weak hands in your range. The solution is of course to defend the blinds with a controlled mixture of optimal 3-betting and flatting, as discussed previously in this article, but this will be hard enough against a button player who plays close to optimal both preflop and postflop.

If button is a strong player, think “damage control”. Accept that his position + skills entitles him to make a profit in this scenario. Focus on limiting your losses, and don’t get fancy and try to outplay him from out of position. Stick close to the optimal strategies outlined here, and don’t spazz out. Spewy 3-betting and flatting out of position won’t do you much good against a strong player, but the mathematics of the situation guarantees that you can get away with some bluffing, and the optimal guidelines tells you how much. By sticking closely to a memorized optimal strategy, many of your preflop decisions become automatic, and you can direct more of your attention towards exploiting the weaker players at the table.

3.3 Questions that go away when we’re using optimal strategies

New players think a lot about how to play individual hands, and they can spend a lot of time mulling over relatively unimportant questions that they believe are important.

For example:

I raised JJ on the button, and an unknown player in the small blind 3-bet. Can I 4-bet? Should I 4-bet? What do I do if I 4-bet and get 5-bet? Is it perhaps best to flat the 3-bet?

A consequence of using a range-based way of thinking is that such specific questions about individual hands become less interesting. It’s obvious that JJ is a hand we can play profitably heads-up in position against a small blind who uses an optimal defense strategy, so our choice is between 4-betting for value and flatting. The question above can therefor be replaced by:

Do I want to 4-bet JJ for value or flat as a default? How does my choice affect the rest of my default defense strategy against small blind’s 3-bet?

Above we outlined a defense strategy against 3-bets where JJ was put in the flatting range, but there is nothing that forbids us from 4-betting it for value. We have 43% equity against small blinds estimated optimal value range {TT+,AQ} which we defined earlier in this article. So we have enough equity to call a 5-bet shove from this range (we need more than 36% equity against a 5-bet shoving range to call profitably, which you can easily verify for yourself).

Since JJ is an underdog against the range it gets all-in against (but we have to call the shove because of pot-odds), we see that JJ’s source of profit when used as a 4-betting hand is folding out small blinds 3-bet bluffs. And when we get 5-bet and have to call, we lose back a little bit of that money. However, since there are many 3-bet bluffs in small blind’s range, we might make money overall by 4-betting JJ, even if we’re an underdog against the range that 5-bets us. But even if this is the case (and we can verify whether this is the case with a little math) we might make more money by flatting JJ and playing a pot postflop in position against small blind’s total 3-betting range. This was our choice earlier in this article.

But let’s study 4-betting as an alternative default line for JJ heads-up in position against a small blind 3-bet. If we decide to use JJ as a value 4-betting hand, the rest follows automatically. We 4-bet and call a 5-bet (since this is what we do with all value hands). Then we adjust our 4-bet bluffing range to our new value range, so that we maintain the optimal 60/40 value/bluff-ratio. Finally, we also adjust our flatting range accordingly, so that we end up with an optimal overall defense strategy against small blind’s 3-bet (according to the principles of 30% total defense, adjusted with a 1.5 call multiplier for flatting).

We start with the previous value range for button, {QQ,AK} =34 combos, and then we add JJ and get {JJ+,AK} =40 combos. To get an optimal 60/40 ratio of value hands to 4-bet bluffs, we need 40 x (2/3) =27 bluff combos. We start with the previous bluffing range{ATo,A9s-A7s} =24 combos that we used with {QQ+,AK}. Then we add A6s and get {ATo,A9s-A6s} =28 combos. So we’re 4-betting 40 + 28 =68 combos, which is 68/458 =15% of our total button range.

We then have to flat 30 – 15 =15% of our button range to defend at least 30% total. This number is scaled up using the call multiplier 1.5, so we end up with a total flatting percentage of 1.5 x 15% =22.5%. This corresponds to 0.225 x 458 =104 combos from our 35% button opening range with 458 combos in it. We can choose {JJ-88,AQ-AJ,ATs,KQ-KJ,KTs,QJs,JTs} =104 combos (where we somewhat arbitrarily have removed QJo and QTs from the flatting range used previously).

Our new estimate of button’s optimal defense of a 35% opening range against a 3-bet from the blinds is then:

– 4-bet {JJ+,AK} =40 combos for value
– 4-bet {ATo,A9s-A6s} =28 combos as a bluff
– Flat {JJ-88,AQ-AJ,ATs,KQ-KJ,KTs,QJs,QTs,JTs} =104 combos

The original question How do I play JJ on the button after a 3-bet from an unknown small blind? has now disappeared. Instead we have the answer to how our total default button strategy against a 3-bet changes as a function of how we choose to play JJ in this situation (4-bet for value or flat).

Previously in this article we defined a button strategy where JJ was placed in the flatting range. But that doesn’t mean that you have to flat it. Feel free to experiment with other default defense strategies in position against 3-bets, based on the mathematical relations defined in this article:

– Optimal 60/40 value/bluff-ratio when you 4-bet
– As a starting point, use a total defense percentage of 30% of the opening range
– Adjust this percentage by using a call multiplier (we used 1.5) on your flatting range

Within these limits you can define your default strategies more or less as you please. But of course you should make sure that the hands you include in your value range are actually value hands (i.e. they can profitably call an all-in 5-bet against the 3-bettors 5-betting range). We’ll have more to say about this in a later article, but it’s easy to fall for the illusion that a hand is best played as a 4-betting hand, just because it makes money when we 4-bet it. It might be that the hand makes money from getting Villain to fold his 3-bet bluffs, and that we’re a small underdog when we call a 5-bet. If this is the case, we might make more money by putting the hand in our flatting range and playing a pot postflop with position.

We’ll return to this problem for JJ in a later article and analyze the EV for 4-betting versus flatting on the button after a 3-bet from the blinds. We’ll use the analysis software Pokerazor for these calculations, plug in small blind’s defense strategy, and compute the EV for the two ways we can play JJ after a 3-bet. We already know from ProPokerTools calculations that JJ is a small underdog against small blind’s 5-betting range (but we have pot odds to call the 5-bet). So we know that JJ’s profit after 4-betting comes from getting Bob’s 3-bet bluffs to fold, and then we lose back a small amount those times Bob 5-bets us all-in and we call for pot-odds.

But we’ll show that JJ makes money overall when we 4-bet it as a value hand. But we also know that JJ can be played profitably by flatting against small blind’s optimal 3-betting range (with 60% bluffs in it). So the question we want the answer to is what’s the most profitable way to play JJ in position after a 3-bet. The difference is probably not big, and in that case it’s impossible for us to make a big mistake. And when one alternative is about as good as the other, the decision is not all that important. What’s important is that we adjust the rest of our strategy accordingly, after we have made our choice. And then the what’s-the-best-way-to-play-JJ question simply evaporates.

We see that when we have a hand that works both as a 4-betting hand and a flatting hand, we have to use some judgment and try to choose the most profitable line for the hand. Note that when we’re the raiser out of position, the mathematics of the situation forces us to 4-bet and call a 5-bet all-in as an small underdog, since we now don’t have the option to flat the 3-bet (which is a choice we’ve made). In other words, we 4-bet and call a 5-bet as an underdog because this is more profitable overall than folding to the 3-bet. We have seen examples of this in Part 1 and Part 2. For example when we call an all-in 5-bet out of position with AK from UTG after having 4-bet against a button 3-bettor, even if we know that Villain only 3-bets {KK+} for value, plus some 5-bet bluffs. AK now becomes a small underdog against Villain’s total 5-betting range, but we have more equity than the minimum 36% we need to call, so we automatically go all-in after a 5-bet.

But when we’re the raiser in position, we have the option to flat hands that are small underdogs against Villain’s value range. So we can instead choose to play them postflop with position on his entire 3-betting range, which is heavy with weak 3-bet bluffing hands. In the scenario we studied above, JJ is a small underdog against Villain’s value range from the small blind after our button openraise. So even if we might make money by 4-betting it and calling a 5-bet as a small underdog, we have to think about what’s the most profitable line; 4-betting and ending the hand preflop, or flatting and playing postflop.

At any rate, the strategies we have defined in this article give you solid defaults. And I think even an optimal strategy without overly aggressive 3/4/5-betting will cause you to 3-bet and defend against 3-betting much more aggressively around the blinds than what’s common at the low limits. Train these strategies and play around with them, knowing that the mathematics behind them will protect you from getting exploited preflop.

Some of you might feel uncomfortable playing postflop in 3-bet pots, or after flatting preflop, using these strategies. Take this as a sign that you need postflop training. Stick to the optimal preflop strategies for 3/4/5-betting and flatting in blind stealing and blind defense, and force yourself to deal with the tricky postflop situations as they come. Getting better postflop is a matter of practice, and there are no shortcuts. Don’t be afraid to make mistakes, as long as you learn from them afterward. Keep in mind that when your preflop strategies are mathematically sound, you don’t have to worry about big preflop leaks, and you can focus on your postflop decision making when plugging leaks.

4. Summary

We have discussed 3-betting heads-up from the blinds against a button steal raise, and the raiser’s defense against this. We have designed default ranges for 3-betting and flatting in this scenario, both for the 3-bettor in the blinds and for the raiser on the button.

When flatting is an alternative for the raiser, the choice of 4-betting range becomes more ambiguous, and we therefore used more judgment than for the corresponding scenario with the raiser out of position (discussed in Part 1 and Part 2). Some of the raiser’s medium strong hands can be played profitably both by 4-betting them and flatting them in position after a 3-bet. For these hands we have to choose an alternative based partly on judgment.

In addition to 3/4/5-bet strategies with an optimal value/bluff ratio, we also took into consideration that the players in the blinds need to defend at least 30% to deny the button raiser the possibility of making a profit from stealing with any two cards. This mathematical requirement for minimum blind defense is not something we have discussed previously, but it’s always the case that the players sitting after the raiser have a collective responsibility for denying the raiser an opportunity to raise any two cards profitably. When button is the openraiser, all of this responsibility falls on the two players in the blinds. We used a simple assumption (both players in the blind defend the same percentage) to estimate an optimal defense percentage of 16% for the players in the blinds, heads-up against a button steal raise.

In Part 4 we’ll generalize the theory of 3-betting from the blinds to include scenarios where the raiser has opened from an arbitrary position (button, CO, MP or UTG). We’ll also talk more about the collective responsibility of defending the blinds sufficiently often. We’ll show that it’s mostly button openraising that forces the blinds to defend very aggressively, and that we can play much tighter against raises from earlier positions, without opening ourselves up from getting exploited by loose open-raising.

After that we’ll talk about two multiway 3-betting scenarios in Part 5, namely squeezing (3-betting after the raise already has been called), and cold 4-betting (4-betting after a raise and a 3-bet).

Then we’ll end this NLHE preflop article series with Part 6, where we test our strategies with the analysis software tool Pokerazor, and also discuss optimal play versus exploitative play, and when we should use one or the other. In Part 6 we’ll also discuss blind vs blind scenarios where the small blind openraises and the big blind defends by 3-betting and flatting, and we’ll use this scenario to give a taste of optimal postflop play.

So we’ll end up with a preflop series in 6 parts, and when we’re done, we’ll have touched upon most of the heads-up preflop scenarios, and some selected multiway scenarios.

Good luck!

Bugs – See more at: http://en.donkr.com/forum/optimal-3-bet-4-bet-5-bet-strategies-in-nlhe-6-max—part-5-533565#sthash.ZwlU6ch6.dpuf

Optimal 3-bet/4-bet/5-bet strategies in NLHE 6-max Part 2

1. Introduction

In “Optimal 3-bet/4-bet/5-bet strategies in NLHE 6-max – Part 1”, we discussed optimal 3/4/5-bet strategies for NLHE 6-max, based on principles from game theory. We only studied the scenario where one player (called Alice) open-raises from some position outside the blinds, and it’s folded to another player (Bob) who has position on Alice. We then used game theory principles to construct optimal strategy pairs for Alice and Bob when Bob elects to 3-bet.

The response for the article was good, and I got the impression that the the topic was interesting to many. I therefore decided to produce a mini article series (4 parts planned) about NLHE 6-max preflop play. The plan for the series is to discuss preflop standards based on a combination of sound poker sense and principles from game theory. In Part 2 we start with default ranges for openraising. Then we’ll generalize the theory from Part 1, and make a list of optimal 3/4/5-bet strategy pairs for opening ranges varying from 15% to 40%, with the raiser out of position.

We’ll also make some improvements in our implementation of the theory from Part 1. For example, we’ll use the blocker effect to Alice’s advantage when choosing her 4-bet bluffs, and we’ll give Bob a more balanced list of candidate hands for 3-bet bluffing. Finally, we’ll give Bob the option of calling raises (flatting) in position, and we’ll give him a default flatting range. Bob now has a complete set of “tools” to use when playing in position against Alice’s openraises.

When the generalized and improved strategies for openraising and heads-up 3/4/5-betting with the raiser out of position have been discussed in Part 2, we’ll move on to heads-up 3/4/5-betting with the raiser in position (e.g. 3-betting from the blinds) in Part 3. We’ll define a complete set of strategies to use both for Bob (folding, flatting the raise out of position, optimal 3/4-/5-betting), and for Alice (folding, flatting the 3-bet in position, optimal 3-/4-/5-betting). Our discussion of this scenario automatically provides us with strategies to use in blind defense.

Note that when the 3-bettor is out of position, flatting of 3-bets becomes a more profitable option for the raiser. Out of position, we elected to let Alice 4-bet or fold after getting 3-bet, since it’s problematic to flat 3-bets with medium strong hands and try to play them profitably out of position with 100bb starting stacks. But when Alice can have position on Bob for the rest of the hand, she can defend a wider range of hands profitably after a 3-bet.

Thus, position makes it easier for Alice to profitably flat 3-bets with a range of medium strong hands that are too weak to 4-bet for value (for example, JJ, AQo, KQs). In Part 3 we’ll look at the hands she should flat, and we’ll use principles for optimal 3/4/5-betting to construct a total defense strategy against 3-betting when Alice has position on Bob. Since optimal strategies come in pairs, our work will also produce a total blind defense strategy for Bob.

But before we place Bob out of position in Part 3, we’ll work through the implementation of all theory with Bob in position in Part 2. Since our goal with this article series is to construct a complete (or close to it) set of default preflop strategies, we’ll also define standard openraising ranges in this article. We start Part 2 with a review of the theoretical concepts from Part 1, and then we define default ranges for openraising. I expect most of the readers to have openraising under control, but defining a set of core ranges is useful, because:

– It makes it easier for us to make assumptions against unknown raisers
– It makes us more conscious about our own opening ranges

When we have defined default opening ranges and optimal strategies for 3/4/5-betting, both in and out of position, we’ll have defined a “model game” with strict preflop strategies. In reality we won’t follow the standards rigidly, since we obviously want to continually adapt to our opponents when we gather reads on them. But this default model game gives us solid standards to use against unknowns, and also against known strong players that we can’t easily exploit. A set of standard preflop strategies will also make it easier for us to model preflop scenarios mathematically, for example if we want to estimate the EV for some preflop line. Then we can plug in our default ranges and strategies (if we haven’t got other assumptions/reads) and do the math.

Defining solid preflop standards for openraising and 3/4/5-betting, based on game theory principles, will give us a much better understanding of what optimal (or at least near-optimal) preflop play is. In my opinion this is the most useful result of all this work. Exploiting weak opponents is (per definition) the same as moving away from optimal play to profit maximally from his mistakes. A good understanding of optimal play makes it easier to exploit our opponents. First, knowing what optimal play is makes it easier to spot opponent mistakes (e.g. their deviations from optimal play). Second, when we adjust to exploit these mistakes, we know both what we are adjusting away from and in which direction we should go.

The latter is something we’ll discuss further in Part 4. My plan is to use Part 2 and Part 3 to construct default strategies, and then we’ll talk more about how to use them in Part 4. There we’ll look more closely at the difference between default strategies/optimal play and exploitative play. We want to use exploitative strategies against weak opponents whenever we can, since this is generally more profitable than optimal play. But against strong opponents that are difficult to exploit, we want a set of solid standards to fall back on to prevent them from exploiting us. So we need to think about when to use one or the other, based on who we’re playing against. In Part 4 we’ll also test our strategies numerically, usingPokerazor simulations with various assumptions about our opponents.

Finally, Part 1 generated some interesting forum discussion, for example about our definitions of value hands/bluffs, and how these are chosen. We’ll return to this topic in Part 4, but first we’ll finish all the work and define our default preflop ranges and preflop strategies in Part 2 and Part 3.

The structure of Part 2 is:

  • Summary of the theory behind optimal strategy pairs for heads-up 3/4/5-betting with the raiser out of position. We’ll also point out areas where we’ll improve on our previous implementation of the theory (for example, using blocker effects when choosing 4-bet bluffs)
  • Default ranges for openraising
  • Generalization of the results from Part 1 for optimal 3/4/5-betting heads-up with the raiser out of position, using opening ranges varying from 15% to 40%

We’ll use heads-up scenarios with 100bb starting stacks, unless otherwise mentioned.

2. Summary of optimal 3/4/5-bet theory with the raiser out of position

We used the following model:

  • Both players start with 100bb stacks, both are outside the blinds, and Bob has position on Alice
  • Alice openraises pot (3.5bb) with some opening range
  • Bob 3-bets pot (12bb) with a mix of value hands and 3-bet bluffs
  • Alice responds by folding or 4-betting to 27bb (slightly less than pot =37.5bb) with a mix of value hands and 4-bet bluffs
  • Bob responds by folding his 3-bet-buffs and 5-betting all-in with a mix of value hands and 5-bet bluffs
  • Alice responds by calling with her value hands, and folding her 4-bet bluffs

The strategies we defined were based on the following mathematical relations (see Part 1 for details):

  • Alice needs to defend 30% of her opening range to prevent Bob from profitably 3-betting any two cards
  • The optimal ratio of value hands to bluffs in Bob’s 3-betting range is 40/60
  • The optimal ratio of value hands to bluffs in Alice’s 4-betting range is 60/40
  • Bob should have enough 5-bet bluffs in his 5-betting range to make Alice’s weakest value hands break even when they call the 5-bet (and we elected to use Axs hands as 5-bet bluffs)

Here we define “value hand” as a hand we plan to keep reraising until we get all-in. A “bluff” is a hand we fold to a reraise (when we’re 3-betting or 4-betting), or a hand we 5-bet bluff all in.

Then we defined optimal strategy pairs from Alice and Bob in two scenarios:

1. Alice raises a ~15% range from EP (UTG or MP)
2. Alice raises a ~25% range from CO

For both scenarios we did the calculations in full detail to show how you can do similar work on your own opening ranges. In this article we’ll generalize these results, and construct a list of optimal strategy pairs that you can use as a “cheat sheet” when 3-betting optimally against any opponent. We’ll list optimal strategy pairs for 15%, 20%, 25%, 30%, 35% and 40% opening ranges, which should cover all the opening ranges you will encounter in practice when engaging in 3/4/5-bet wars outside the blinds.

You can also use this list to estimate optimal 4-betting strategies for yourself when you are the raiser out of position (if you don’t feel like doing the calculations for the exact ranges you are using). For example, if you know that you’re openraising 32% from CO, you can use the strategy pair for 30% openraising when estimating your 4-betting range. In practice, this will be close enough.

Before we define the list of optimal strategy pair for 3/4/5-betting with the raiser out of position, we’ll make a couple of adjustments for Alice’s and Bob’s choice of 4-bet bluffs and 3-bet bluffs. In Part 1 we made some simplifications to make things easier to remember, but here we’ll improve on them:

2.1 Alice’s choice of 4-betting hands

Alice’s value hands follow directly from her opening range plus the requirement that she should defend 30% of this range, using a value/bluff ratio of 60/40. But Alice has a choice to make when selecting hands to 4-bet bluff. She can choose between two methods:

1. The combo method: Alice picks a set of hands to 4-bet bluff
2. The percentage method: Alice 4-bet-bluffs a fixed percentage with all her non-value hands

When Alice gets 3-bet out of position, her response is to never call, and 4-bet 30% of her opening range with a 60/40 ratio of value hands to 4-bet bluffs. So 0.30 x 0.60 =18% of her opening range should be 4-bet for value and 0.30 x 0.40 =12% should be 4-bet as a bluff. For example, when Alice opens a 15% range from early position, this corresponds to 4-betting 0.18 x 0.15 x 1326 =36 value combos and 0.12 x 0.15 x 1326 =24 bluff combos.

Alice then chooses her best 36 (or thereabouts) hands to use as value hands, and the obvious choice is {QQ+, AK} =34 combos. Then she can either choose 24 specific combos to use as 4-bet bluff hands, or she can 4-bet all non-value hands (the remaining 82% of her range when the 18% of value hands has been selected) a certain percentage of the time.

If she chooses the latter, she should use a percentage x that gives her a 60/40 ratio of value hands to bluffs. Formulated as an equation, we get:

0.18/0.82x =60/40
0.18/0.82x =1.5
0.18 =1.5(0.82x)
0.18 =1.23x
x =0.18/1.23
x =0.15 =15%

So Alice should 4-bet bluff 15% of the time with all hands not strong enough to 4-bet for value. This makes 4-bet bluffing easy to implement i practice. We only need to remember one number x =15%, and this 4-bet bluff percentage is the same regardless of Alice’s opening range. In Part 1 we therefore elected to use the percentage method for simplicity. We used a randomizer from random.org for this purpose, as illustrated below:

Example 2.1.1: Randomized 4-bet bluffing

$100NL
6-handed

Preflop:
Alice ($100) raises to $3.50 with K J from UTG, Bob ($100) 3-bets to $12. Alice uses the randomizer, planning to 4-bet bluff if it returns a number between 0 and 15, and otherwise she will fold.:

The randomizer returns 41, so Alice folds this time.

The percentage method works well and is easy to implement, but in practice the combo method will work better. The difference between the two is that the combo method gives us the opportunity to exploit the blocker effect to our advantage. If Alice uses her best hands not strong enough to 4-bet for value, this will ensure Bob’s range is poorer in value hands.

For example, assume Alice raises 15% from early position. In Part 1 we established that Bob’s value 3-bet range against this opening range should be {KK+} plus some Axs hands as 5-bet bluffs. Before the blocker effect is taken into account, there are 12 AA/KK combos in Bob’s value range. But when Alice has AQ (which she does not 4-bet for value), the chance that Bob has AA is reduced to half (there are 3 possible AA combos in his value range when Alice has an ace on her hand). So if Alice uses AQ and similar decent hands at the top of her non-value range, she ensures that her 4-bet bluffs go in those times it’s less likely that Bob has a value hand to continue with. The hand we fear most in Bob’s 3-betting range is AA, and it’s seems obvious that Alice should pick her 4-bet bluffs from the best Ax hands not strong enough to 4-bet for value. For example, AQ blocks the AA/QQ/AK hands in Bob’s range.

In Part 2 we’ll list specific 4-bet bluff combos for all of Alice’s opening ranges in order to exploit the blocker effect. This will require more memorization than with the percentage method, but on the other hand we save time and distractions when we don’t have to click the randomizer while playing.

2.2 Bob’s choice of value 3-bet hands

We defined Bob’s value hands as the range of hands he 3-bets, planning to 5-bet all-in after a 4-bet from Alice. His value range has two components:

– Value hands that profit from getting all-in against Alice’s value hands
– 5-bet bluffs

When building Bob’s total value range, our starting point was to find all hands with at least 50% equity against Alice’s value hands (which followed from her opening range). These are Bob’s value hands. Then we added Axs hands as 5-bet bluffs until it became break even for Alice to call a 5-bet with her weakest value hands. Note the difference between Bob’s value range an his value hands. The latter are the hands that profit when Bob’s 5-bets get called by Alice’s value hands, while Bob’s value range is the total range of hands he’s planning to 5-bet all-in (his value hands plus some 5-bet bluffs).

For example, in Part 1 we saw that when Alice raises a 15% opening range from UTG, Bob’s response is to 3-bet a total value range {KK+, A5s, A4s}, planning to 5-bet all-in after a 4-bet. AA/KK are clearly hands we want to get all-in against Alice’s value range {QQ+,AK}, and then we add Axs hands until it becomes break even for Alice to call a 5-bet shove with her weakest value hand (QQ).

Alice is getting pot odds 128.5 : 73 to call a 5-bet shove, and she needs minimum 73/(128.5 + 73) =36% equity to call profitably. We can make her call with QQ break even by using 7 Axs combos as shown below (we start with A 5 and work our way down towards A 2 ):

Thus, the exact number of 5-bet bluffs in Bob’s value range against Alice’s 15% opening range is {A5s,A 4 , A 4 , A 4 }. In Part 1 we simplified this to {A5s,A4s}, but in Part 2 we’ll use the exact number of 5-bet bluffs.

The next step for Bob is to define a 3-bet bluff range. He should use a 40/60 ratio of value hands to bluffs, so his 3-bet bluffing range should have 60/40 =1.5 times as many combos as his value range. For example, if he uses the value range {KK+,A5s,As4s,Ah4h,Ad3d} =19 combos against a 15% opening range for Alice, he should bluff with 1.5 x 19 =29 combos.

The number of 3-bet bluff combos Bob needs will chance with Alice’s opening range, and Bob can choose between two methods:

– Memorize a list of specific 3-bet bluff combos for each of Alice’s opening ranges
– Memorize a list of 3-bet bluff candidates, and use each of them a certain % for each of Alice’s opening ranges

We chose the latter, and defined the following list of 156 combos to use as 3-bet bluffs:

Candidate list for 3-bet bluffing

Ace high: A9s-A6s ATo-A8o (52 combos)
King high: K9s-K6s, KJo-K9o (52 combos)
Queen high: Q9s-Q6s, QJo-Q9o (52 combos)

The list is based on the fact that Bob wants to 3-bet bluff with the best hands not strong enough to 3-bet for value or call. If Alice always 4-bets or folds out of position, Bob’s choice of 3-bet bluffs doesn’t matter, but in practice we’ll sometimes get called. So we want to use the best possible hands in case the raiser calls our 3-bet and forces us to play postflop.

For example, if Bob needs 29 3-bet bluff combos to use against Alice’s 15% opening range, he can 3-bet each hand on the list above 29/156 =19% of the time. He uses a randomizer to achieve this, as illustrated below:

Example 2.2.1: Randomized 3-bet bluffing

$100NL
6-handed

Preflop:
Alice ($100) raises to $3.5, and it’s folded to Bob who has K 7 on the button. Bob knows from observation that Alice is opening a 15% range from UTG, and his response is to 3-bet a value range {KK+,A5s,As4s,Ah4h,Ad4d} =19 combos for value plus a range of 3-bet bluffs. He uses 1.5 x 19 =29 3-bet bluff combos, which corresponds to 3-betting all hands in the 3-bet bluff list 29/156 =19% of the time. Here he has one of these hands, so he uses the randomizer, planning to 3-bet when it returns a number between 0 and 19:

The randomizer returns 3, and Bob 3-bets pot to $12. Alice folds.

In this article we’ll make a change in Bob’s 3-bet bluff list. The list we designed in Part 1 was easy to remember, but it’s somewhat imbalanced. It only contains Ax/Kx/Qx-combos and few low cards, which makes it hard for us to connect with low flops. The list also contains lots of offsuit hands. We’ll replace this list with a more balanced list that has more suited hands, and also some medium/low connectors and 1-gappers. Designing Bob’s 3-bet bluff list is not an exact science, so we’ll use judgment and sound poker sense.

3. Open-raising

Before we move on to generalizing the theory for 3/4/5-betting, we’ll define a set of standard opening ranges for all positions. A good NL player should never feel like a “slave” to strictly defined preflop strategies, and ideally he should always try to play hands that are profitable, and otherwise fold. But there are good reasons for starting with a set of memorized opening ranges.

It’s obvious that the strongest hands like AA-QQ, AK, etc. are profitable raising hands from any position. But in practice it’s impossible to know exactly how profitable the weakest playable hands (for example A6o, 22, 76s) are in a given situation. Sometimes they will be profitable and sometimes not. Sticking to a default reasonable opening range for each position is probably just as good as trying to find exactly which weak hands can be openraised in a given situation and which can not. Sometimes we will be wrong, folding some playable hands and playing some hands that should have been folded, but usually it won’t matter much (close decisions don’t matter much). When we here say “scenario” we mean the combination of your position, the tendencies of your opponents, the history between you, and stack sizes for everyone involved.

Also, even if you are very flexible with respect to openraising, and always try to adapt to the situation, there will always exist a “core range” for a given situation. The core range is the range of hands you are always willing to open, regardless of the circumstances. For example, you might decide that you’ll never open less than 35% of your hands on the button, regardless of how the players in the blinds play. Starting with a well-defined set of standard opening ranges makes it easier to know what your effective core range is for a given position.

Starting with a standard set of opening ranges also makes it easier to defend against 3-betting, since we easily can memorize optimal defense strategies for our standard opening ranges. Against good players who 3-bet optimally of near optimally, we can fall back on our default optimal defense strategy. Against weak players who either 3-bet way too little or way too much, we can use the optimal defense strategy as a starting point, and then adjust as needed (folding more against a tight 3-bettor, and 4-betting or calling more against a loose 3-bettor). Note that exploitative adjustments against weak opponents become easier when we know what the optimal starting point is.

Thus, the point of starting with a set of standard opening ranges is not to “bind” you strategically, it’s quite the opposite. Building your preflop game on top of solid standard opening ranges will make it easier to adjust. By this I mean that it will be easier to adjust correctly to the situation when you start with a good understanding of what a good default strategy is, and this understanding starts with understanding the opening ranges we’re using. When solid default preflop strategies have been trained, you’ll spend less time at the table thinking about what the default play is, and more time thinking about how to adjust correctly away from default play.

The opening ranges listed below are on the tight side, and you can think about them as “core ranges” that you always open, regardless of the circumstances. You can elect to open looser, but unless the table conditions are extreme, you’ll probably not gain much by opening tighter.

Note that we can’t use game theory principles to find optimal opening ranges, since this is more or less equivalent to solving the game of NL Hold’em in it’s entirety. This is not possible in practice, so our choice of standard opening ranges is based on experience and sound poker sense.

Here is another way to look at it:

Instead of asking what the best standard opening range is for a given position, look at the range you’re using and ask yourself if your range is a good tool for the job. If you had given your opening ranges to the world’s best 6-max grinder and told him to grind your limits using your ranges, would he still be a big winner, even if he couldn’t play the way he wanted preflop? If you think the answer is “yes”, your ranges are probably fine.

3.1 Openraising from early position (EP =UTG and MP)

From the two earliest positions we should use tight default ranges, and I recommend you use a ~15% core range as a starting point from both UTG and MP. I’ll list an EP range here, and then you can choose for yourself how much to loosen up in MP (you should definitely not play tighter than this from MP), and how much to tighten up or loosen up from UTG.

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

194 combos
15%

From UTG you can drop the lowest suited connectors, if you think this range is too loose. From MP, don’t tighten up, but you can raise some more hands (e.g. K9s, Q9s, ATo, KJo, QJo, etc.) if you think 15% is too tight. But using something around 15% for both the UTG and MP positions is a good place to start.

If you feel uncomfortable having so many medium/low suited connectors in your EP range (they can be difficult to play well out of position), feel free to drop the lowest ones from the range, or replace them with high card hands. For example, you can drop 87s-65s (3×4 =12 combos) and play KJo (12 combos) instead. But don’t take this too far, and be cautious with the offsuit high cards hands (KQo, KJo, QJo, QTo, etc.). These hands are negative implied odds hands that often put you in tough spots postflop, especially out of position, even when you flop what you’re hoping for (top pair, mostly).

Keep in mind that domination is less of a problem for the suited connectors. Also, having a handful of them in your range makes it easier for you to credibly represent strength on low flops (which can be a problem when open-raising a strong range with lots of high card hands). So the suited connectors makes your range more balanced and more difficult to read.

Later, in the discussion about defense against 3-betting, we’ll for simplicity assume we’re using the 15% range above from both UTG and MP, and we’ll refer to both positions as “EP”. This simplification is acceptable, since the optimal strategies don’t change much when we move from, say, 15% –> 13% in UTG, or 15% –> 17% in MP.

3.2 Openraising from CO

Here I recommend a ~25% core range. In CO you’ll get more opportunities to exploit the players behind you, if they are playing too tight. For example, with a tight-passive button player and weak players in the blinds, it’s fine to open 30% and perhaps even more from CO. But start with a core range of 25% or so, and let this be the range you never tighten up from. And if the table conditions are right, you can loosen up.

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

326 combos
25%

3.3 Openraising from the button

From the button I recommend a ~35% core range. Button is the position with greatest flexibility with respect to openraising, and you should be willing to vary your range a lot. Openraise at least 35%, but be quick to loosen up if the blinds are weak. In this context, “weak” means blinds who either fold too much preflop, or who mostly defend by calling.

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

458 combos
35%

Don’t be afraid to openraise a loose range on the button against weak players in the blinds, even if they call a lot preflop. When they don’t punish your loose openraising with 3-betting, you’ll have 3 ways to win the pot:

  • They fold preflop
  • They call preflop, miss the flop, and you win with a c-bet
  • They call preflop, hit the flop, but you win anyway. Either by hitting the flop harder than them, or by drawing out on them later, or by bluffing them out of the pot (for example when a scare card falls on the turn or river).

3.3 Openraising from the blinds

Here we mean openraising from the small blind when everybody has folded, or raising from the big blind in a limped pot. Here I recommend starting with the following set of simple guidelines:

When it’s folded to you in the small blind, openraise your button range if the big blind doesn’t defend aggressively. If he is difficult to steal from, tighten up to the CO range. Don’t open-limp, unless you have specific reasons to think this will be more profitable than open-raising or folding.

When you’re in the big blind, it’s folded to the small blind, and he open-limps, raise the button range to punish his limps. If someone has limped in outside the blinds, raise a tight range for value. The more players have limped, the tighter you raise. The better the limpers play, the tighter you raise. This is logical, since you need a stronger hand to play out of position against many players, or against players who are competent postflop. Now you’ll win fewer pots without a showdown, and you compensate by basing your raises more on showdown equity and less on steal equity.

As a starting point, it’s a good idea to not raise much looser than {99+, ATs+, AJo+, KJs+, KQo} from the big blind out of position against limpers, unless you are heads-up and/or you expect the players behind you to give up easily.

3.4 Isolating limpers

As a starting point, if you have position on one or more limpers outside the blinds, raise with the same range you would have openraised. It’s of course fine to drop the very weakest hands (65s, Q2s and the like), especially against more than one limper, or against a very loose limper. Some of the hands you will be isolating with are technically too weak to play postflop against one or more limping ranges, but keep in mind that you base your raise partly on steal equity. Isolating a limper with a hand like T8s is a semibluff. Sometimes you win preflop, and when you don’t win preflop, you’ll often get heads-up with position on a weak player with a weak range. This will give you many opportunities to steal the pot postflop, and when this isn’t possible, you’ll sometimes make the best hand and win a a showdown.

But as mentioned previously, don’t make loose raises against limpers when you’re in the blinds. The exception is when you have reads telling you that a raise will make it easy to win the pot. Some limp and fold a lot to raises. Others limp and call a lot of raises, but then play fit-or-fold postflop. Pay attention to the players around you, and when you pick up information you can use to your advantage, use it. But against unknown players and loose players, raise a tight range out of position in the blinds in a limped pot.

3.5 Summary of standard opening ranges

I recommend that you memorize all these opening ranges, and also the number of combos in each range. For example, knowing that you’re opening 326 combos (25%) as a default from CO will make it easier for you to defend correctly against 3-betting.

For example, we know that the optimal defense percentage out of position is 30% of our opening range when we defend by 4-betting or folding out of position. 18% should be for value and 12% should be bluffs. If you know how many combos you have in your opening range, it will be easier to get a feel for how many hands you should defend, both for value and as bluffs.

4. Generalizing the theory for optimal 3/4/5-betting with the raiser out of position

We remember from Part 1 that optimal strategies for 3/4/5-betting come in pairs: A raise/4-bet/call 5-bet strategy for the raiser, and a 3-bet/5-bet strategy for the 3-bettor. In Part 1 we estimated these optimal strategy pairs after a ~15% EP-raise and a ~25% CO-raise

In Part 2 we’ll generalize this theory in two ways:

  • We’ll list strategy pairs based on 15%, 20%, 25%, 30%, 35% and 40% openraising
  • We’ll include flatting in position as a strategic option for the 3-bettor

As previously mentioned, we’ll also list specific hands that the raiser will use as 4-bet bluffs (the combo method) to exploit the blocker effect. We’ll always keep the strength principle in mind:

– With your best hands: Raise for value
– With your next best hands: Call
– With hands that aren’t good enough to raise or call: Fold or bluff

Which hands make up your “best hands”, and your “next best hands” will depend on the situation you’re in. For example, the value of a NLHE starting hand in position behind a raiser will vary with both our absolute position and the raiser’s range. But regardless of the situation, we’ll choose our value hands form our best hands that we are willing to get all-in with against our opponent’s value hands (for example {QQ+,AK}). Then we pick our calling hands from the best hands that aren’t strong enough to get all-in preflop. Finally, we pick the best of the remaining hands to use as bluffs.

Our 3-bettor Bob now will get a flatting range in position, and we start by listing all of the “tools” at Bob’s disposal against Alice’s openraise. As previously mentioned, we’ll make a new list of 3-bet bluffing hands (which we’ll name “IP 3-bet air list”), more weighted towards balance and suited hands than the simple list we used in Part 1. We’ll also list the 5-bet bluff hands Bob 3-bets, planning to 5-bet all-in as a bluff after a 4-bet (“IP 5-bet air list”). Finally, we list a range that Bob flats in position (“IP flat list”).

4.1 Ranges for 3-bet bluffing, 5-bet bluffing, and flatting in position

IP 3-bet air list

A9s-A6s
K9s-K6s
Q9s-Q6s
J9s-J6s
T8s-T7s
97s-96s
87s-86s
76s-75s
65s

100 combos

IP 5-bet air list

A5s-A2s

16 combos

IP flat list

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

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

Note that the flat list changes according to which hands we 3-bet for value. For example, against a 15% UTG raise, we use {KK+} as our value range. The flat list then becomes {QQ-22,AKs-ATs,AKo-AJo,KTs+,KQo,QTs+,JTs,T9s,98s} =162 combos.

The flat list is made up of hands that you can always flat as a default after a raise from UTG, MP and CO when you’re on the button. But you should flat somewhat tighter from earlier positions (for example, it’s probably best to fold 98s in MP after an UTG raise). You might also want to tighten up against a very tight raiser (folding hands like AJo, KTs, QTs and 98s after a tight 8% openraise from UTG is fine). Conversely, if you have position on a very loose and bad raiser, it’s allowed to sneak in more marginal flatting hands (e.g. 87s) if you think it’s profitable to see a flop with them in position. So use the flat list as a starting point, and exercise judgment.

The lists for 3-bet bluffing and 5-bet bluffing can be easily memorized to make them easier to apply at the table. I use this 3-bet list at the limits I play ($400NL to $1000NL), and I use it together with a randomizer. Of course we can also pick specific combos from the 3-bet bluff list, but in my opinion it’s much simpler to use a memorized “IP 3-bet air list” plus a randomizer than having to memorize a specific range of 3-bet bluffs for each opening range.

For the 5-bet bluffs I use a simpler method. In the summaries below I give the number of 5-bet bluffs Bob needs for the opening range he’s up against, and then he simply picks hands from the top of the list {A5s-A2s} and works his way from A 5 down towards A 2 . For example, if Bob needs seven 5-bet bluff hands, he picks {A5s, As4s, Ah4h, Ad4d} =7.

I’ve also made a simplifying little trick with the 3-bet bluff list. I put exactly 100 combos in the list, so that we easily can convert between percentages and combos. Let’s say you’re in position behind a raiser, and you know that your value range is {KK+} =12 combos, but you don’t remember how many % you should 3-bet bluff the hands on the list (you’re using a randomizer).

But you know that you should use a 60/40 ratio of value hands to bluffs, so with {KK+} =12 combos as your value range, you need 1.5 x 12 =18 bluff combos. Since the 3-bet bluff list has 100 combos, this corresponds to 3-bet bluffing all hands on the list 18% of the time. So if you pick up Q7s, you click the randomizer, planning to 3-bet if it returns a number between 0 and 18, and otherwise you fold.

You can either have the lists and percentages memorized, or have them in a document on your screen for easy access, but it’s a good idea to use structure/organization to keep things simple where you can. I have memorized everything, and I rarely have to pause and think about these strategies, but sometimes I slip up. Then it’s easy to start with the value range, then count the number of bluff combos you need, and with 100 combos in the 3-bet bluff list, you now also have the percentage to use with the randomizer.

A final word: These lists are not gospel. If you’d rather use hands like KJo, KTo and QTo instead of some of the lowest suited connectors on my 3-bet bluff list, go ahead and change things to your liking. I’ve chosen to use suited hands, since negative implied odds is less of a problem for suited hands when our 3-bet gets called and we’re forced to play postflop. Suited hands also pick up equity on the turn more often than offsuit hands, and this will give you more good spots to 2-barrel.

4.2 Optimal 3/4/5-bet strategy pairs with the raiser out of position

Bob’s lists of flatting hands, 3-bet bluffs, and 5-bet bluffs are given above. What remains to be done is to systematically find Alice’s 4-betting hands (value hands and bluffs) for various opening ranges between 15% and 40% in increments of 5%. Then we find the 3-bet bluff percentage that Bob should use with his “IP 3-bet air list”. Last we find the number of 5-bet bluffs Bob should use, and we’re done.

All calculations are done like in Part 1, so we simply present the results here. Note that we have done some rounding here and there. For example, when openrasing a 15% range, we 4-bet {QQ+,AK} =34 combos for value, while the theoretically optimal number of value combos is 0.18 x 0.15 x 1326 =36. We generally want to play all combos of a particular starting hand the same way. So we don’t pick 2 combos of JJ or AQ to get to exactly 36 value combos.

We have rounded the percentages to use with Bob’s “IP 3-bet air list” to the nearest 5% to get numbers that are easy to remember. It’s not critical to be accurate to the nearest percentage point, and rounding is fine. Another approximation we have done is to let Bob flat the same range of hands, regardless of his exact position behind the raiser (MP, CO or button). We’re assuming he can flat all hands from the flat list profitably from all positions behind the raiser, but in practice Bob should flat a bit tighter from MP and CO. When Bob flats from MP and CO, his position will be worse postflop, and the risk of a 3-bet behind him is higher, so he should flat a bit tighter than on the button.

We use the following notation:

– AJ =All AJ, suited and offsuit
– AJs =All suited AJ
– AJo =All offsuit AJ

Optimal 3/4/5-bet strategy pair with 15% openraising OOP
Alice’s strategy:

  • Openraise 15%
  • 4-bets {QQ+,AK} =34 combos for value and calls a 5-bet
  • 4-bets {AQ,AJs-ATs} =24 combos as a bluff and folds to a 5-bet

Bob’s strategy:

  • Flats the flat list: {22+,ATs+,AJo+,KTs+,KQo,QTs+,JTs,T9s,98s} =162 combos when {KK+} gets 3-bet for value
  • 3-bets {KK+, 7 air} for value, planning to 5-bet all-in after a 4-bet
  • 3-bets 30% of “IP 3-bet air list”, planning to fold to a 4-bet

For simplicity I have used the notation “x air” for Bob’s 5-bet bluffs. For example, “7 air” means that he picks the 7 best combos from “IP 5-bet bluff list” ={A5s-A2s}. So he picks {A5s, A 4 , A 4 , A 4 }. With this notation we only need to remember a number for the 5-bet bluffs, and then we pick the actual hands on the spot.

Optimal 3/4/5-bet strategy pair with 20% openraising OOP
Alice’s strategy:

  • Openraises 20%
  • 4-bets {TT+,AK,AQs} =50 combos for value and calls a 5-bet
  • 4-bets {AQo,AJ,ATs} =32 combos as a bluff and folds to a 5-bet

Bob’s strategy:

  • Flats the flat list: {22+,ATs+,AJo+,KTs+,KQo,QTs+,JTs,T9s,98s} =156 combos when {QQ+} gets 3-bet for value
  • 3-bets {QQ+, 10 air} for value, planning to 5-bet all-in after a 4-bet
  • 3-bets 40% of “IP 3-bet air list”, planning to fold to a 4-bet

Optimal 3/4/5-bet strategy pair with 25% openraising OOP
Alice’s strategy:

  • Openraises 25%
  • 4-bets {TT+,AQ+} =62 combos for value and calls a 5-bet
  • 4-bets {AJ-AT,A9s-A8s} =40 combos as a bluff and folds to a 5-bet

Bob’s strategy:

  • Flats the flat list: {22+,ATs+,AJo+,KTs+,KQo,QTs+,JTs,T9s,98s} =140 combos when {QQ+,AK} gets 3-bet for value
  • 3-bets {QQ+,AK, 12 air} for value, planning to 5-bet all-in after a 4-bet
  • 3-bets 70% of “IP 3-bet air list”, planning to fold to a 4-bet

Optimal 3/4/5-bet strategy pair with 30% openraising OOP
Alice’s strategy:

  • Openraises 30%
  • 4-bets {99+,AQ+,AJs} =72 combos for value and calls a 5-bet
  • 4-bets {AJo,AT-A9,A8s} =48 combos as a bluff and folds to a 5-bet

Bob’s strategy:

  • Flats the flat list: {22+,ATs+,AJo+,KTs+,KQo,QTs+,JTs,T9s,98s} =134 combos when {JJ+,AK} gets 3-bet for value
  • 3-bets {JJ+,AK, 10 air} for value, planning to 5-bet all-in after a 4-bet
  • 3-bets 75% of “IP 3-bet air list”, planning to fold to a 4-bet

Optimal 3/4/5-bet strategy pair with 35% openraising OOP
Alice’s strategy:

  • Openraises 35%
  • 4-bets {99+,AJ+} =84 combos for value and calls a 5-bet
  • 4-bets {AT-A8, A7s-A6s} =56 combos as a bluff and folds to a 5-bet

Bob’s strategy:

  • Flats the flat list: {22+,ATs+,AJo+,KTs+,KQo,QTs+,JTs,T9s,98s} =134 combos when {JJ+,AK} gets 3-bet for value
  • 3-bets {JJ+,AK, 14 air} for value, planning to 5-bet all-in after a 4-bet
  • 3-bets 80% of “IP 3-bet air list”, planning to fold to a 4-bet

Optimal 3/4/5-bet strategy pair with 40% openraising OOP
Alice’s strategy:

  • Openraises 40%
  • 4-bets {88+,AJ+,ATs+} =94 combos for value and calls a 5-bet
  • 4-bets {ATo,A9-A7} =60 combos as a bluff and folds to a 5-bet

Bob’s strategy:

  • Flats the flat list: {22+,ATs+,AJo+,KTs+,KQo,QTs+,JTs,T9s,98s} =134 combos when {JJ+,AK} gets 3-bet for value
  • 3-bets {JJ+,AK, 14 air} for value, planning to 5-bet all-in after a 4-bet
  • 3-bets 80% of “IP 3-bet air list”, planning to fold to a 4-bet

4.3 Summary of optimal strategy pairs for 3/4/5-betting with the raiser in position

We have now listed a set of strategy pairs that cover most of the opening ranges you will encounter as a 3-bettor in position. Note that we’re not claiming that a 35% or 40% range is a good opening range from positions earlier than the button, but we include it anyway. You might meet opponents that play this loose.

Those with good memory can memorize these strategy pairs once and for all. We’ll also organize them in a tidy document that you can keep on the screen while you play:

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

Here I have marked our own EP and CO standard opening ranges with grey, so that it will be easy for you to find your 4-betting ranges from the document when you’re out of position and get 3-bet. But the 4-betting ranges are easy to memorize, so this document will be most useful when you’re in position and need the optimal 3-betting strategy to use against an openraiser.

You can use the HEM stat “Raise 1st” to estimate Villain’s opening ranges for the different positions. You can include the positional “Raise 1st” stats in the HUD, or you can find them in the HEM pop-up when you click on the player.

Here is an example of using HEM stats when 3-betting in position:

Example 4.2.1: Optimal 3-betting based on HEM stats

$400NL
6-handed

MP ($400) raises to $14, CO folds, you ($526) are on the button with J 7 . This hand is on “IP 3-bet air list”, so it’s a candidate for a 3-bet bluff in position. You now need an estimate of MP’s opening range, so that you can estimate the bluff percentage to use with the 3-bet bluff list.

We have a large sample (¨~7k hands) on MP, and his HUD stats look like this:

The number we’re interested in is Villain’s “Raise 1st” from MP. The “Raise 1st” stats are located on the HUD’s last line: UTG, MP, CO, Button, SB, BB from left to right. We find that Villain raises 23% from MP

Then we turn to the summary document we made earlier and look up the strategy pair closest to an opening raise percentage of 23%. We choose the strategy pair for a 25% opening range. Our value range is then {QQ+,AK, 12 air} ={QQ+,AK,A5s,A4s,A3s}, and we use a bluff percentage of 70% for our 3-bet bluff list.

Next we click the randomizer, planning to 3-bet if it returns a number between 0 and 70, and otherwise fold:

The randomizer returns 34, so we 3-bet bluff to $48, and button folds.

4.3 HUD layout with positional “Raise 1st” stats

Most of the HUD layout used in Example 4.2.1 is a standard layout that can be downloaded from the HEM forums. Then I added a line with “Raise 1st” stats at the bottom. The original HUD with explanations of stats and color coding schemes can be found here: NL6max Layout: Optimization.

My modification of the HUD with positional “Raise 1st” stats on a separate line can be downloaded here (right click and choose “Save as”): nlsixmax.xml. The structure of the layout is shown below

5. Summary

In this article we have gone further with the theory discussed in Optimal 3-bet/4-bet/5-bet-strategies in NLHE 6-max – Part 1, and we have given more specific guidelines for how to implement the theory in practice.

We constructed a set of optimal 3/4/5-bet strategy pairs with the raiser out of position for a set of opening ranges between 15% and 40% in increments of 5%. We organized the strategies in a document for easy access during play. We also defined a HUD layout with the necessary stats for estimating an openraiser’s range. This enables us to quickly find optimal 3-betting strategies in position against an arbitrary openraising range. We can also use the table of optimal strategy pairs to estimate our own ranges (4-betting for value or as a bluff) for defending against a 3-bet when we’re the openraiser.

We also defined a set of standard opening ranges that we can use as “core ranges” for our preflop game. The purpose of these ranges is to give ourselves solid defaults, and to make future modeling and analysis simpler. For example, we’ll use the standard opening ranges when working on the theory for 3/4/5-betting with the raiser in position in Part 3. Then the raiser will defend against 3-bets using a combination of 4-betting and flatting, and we’ll find the 3-betting and flatting ranges from our standard opening ranges.

In Part 3 we’ll discuss 3/4/5-betting heads-up with the raiser out of position, for example when button openraises, and small blind 3-bets. The theory is based on the same mathematical principles used in Part and and Part 2, but the ranges will change a bit. The raiser can now defend against 3-bets by flatting in position, and not only defend by 4-betting or folding. Having the option to flat 3-bets in position makes the overall defense strategy more flexible, and we’ll use less mathematics and more good poker sense.

Good luck!

Bugs – See more at: http://en.donkr.com/forum/optimal-3-bet-4-bet-5-bet-strategies-in-nlhe-6-max—part-5-533565#sthash.ZwlU6ch6.dpuf

Optimal 3-bet/4-bet/5-bet strategies in NLHE 6-max – Part 1

1.1 Presenting the problem

Against weak low limit opposition, we can get away with playing an almost completely value-based game. We 3-bet/4-bet/5-bet mainly for value, and it’s not a big mistake to assume our opponents are doing the same. If we reraise as a bluff, we usually limit ourselves to the occasional 3-bet bluff. A value-based style with little bluffing works well at small stakes because our opponents use more or less the same strategy, and many of them execute it poorly. Of course, every now and then we run into aggressive players who are capable of reraising as a bluff, but there are plenty of fish that will pay off our straightforward game, even if we bluff much less than is game theoretically optimal.

But let’s say our Hero has built a bankroll by patiently grinding the low limits, and now he wants to take a stab at $200NL. He will now experience a lot more 3-betting, especially if he’s out of position.

For example:

Example 1.1.1: We get 3-bet out of position

$200NL
6-handed

Hero ($200) raises to $7 with J T from UTG, it’s folded to the button ($200) who 3-bets to $24, the blinds fold, and Hero folds.

Straightforward, and although Hero expects to get bluffed some of the time, he really doesn’t have any choice but to fold. It’s correct that his hand can no longer be played for value, but as we shall see later, it’s possible to turn it into a 4-bet bluff.

At any rate, Hero plays on. The players behind him keep 3-betting him frequently when he is out of position, and Hero keeps folding weak hands to 3-bets. After a while, this hand occurs:

Example 1.1.2: We get 3-bet out of position (again)

$200NL
6-handed

Hero ($200) raises to $7 with A J in MP, it’s folded to button ($200) who 3-bets to $24, the blinds fold, and Hero folds.

This is getting frustrating. Hero has a decent hand, but it’s not strong enough to defend against a 3-bet from out of position, so Hero folds. But he is starting to feel exploited. If only he could get dealt a good hand and punish these bastards!

What an inexperienced player now might do (as his frustration builds up more and more), is to make up his mind to fight back against the loose 3-bettors. But he doesn’t quite know what to do, and therefore he will often use poor strategies, and the wrong types of hands!.

Let’s look at two common (and sub-optimal) ways to defend against 3-betting, out of position with 100 BB stacks:

Example 1.1.3: We get 3-bet out of position (again) and we call

$200NL
6-handed

Hero ($200) raises til $7 with K Q in MP, button ($200) 3-bets to $24. Hero thinks for a bit, decides that this hand is too good to fold, but too weak to 4-bet, so he calls.

Flop: 944 ($51)
Hero ($176) checks, button ($176) bets $30, Hero folds.

Hero is frustrated, but he doesn’t see what else he could have done out of position with a hand of this type. Too strong to fold (at least in Hero’s mind) against a loose 3-bettor, but not strong enough to 4-bet. Or? Hmmmmm …. Hero contemplates his next move, and soon another 3-bet pot occurs:

Example 1.1.4: We get 3-bet out of position (again) and we 4-bet for value (or at least that’s what we think we are doing)

$200NL
6-handed

Hero ($200) raises to $7 with A J from UTG, MP ($200) 3-bets to $24. Hero decides to fight fire with fire, and he 4-bets pot to $75. Button 5-bets all-in, Hero calls. MP has K K . Hero screams in agony.

Flop: Q T 7 ($403)

Turn: Q T 7 Q ($403)

River: Q T 7 Q 4 ($403)

Hero tears his clothing and sprinkles ashes over his head. Damn!!

What happened throughout this sequence of hands?
OK, I made up this story, but it illustrates several of the problems an ABC low limit player faces when he moves up to tougher games. He will get 3-bet left and right, so he will have to fold a lot out of position (which is correct). He realizes he has to fight back to avoid getting run over (also correct), but he’s not quite sure how to do it. So his attempts to counter the aggression are often poorly executed, frustrating and tilt-inducing.

For example, Hero might start calling 3-bets out of position with hands he feels are too good to fold, but not strong enough to 4-bet for value. This leads to many miserable experiences like Example 1.3. Or he might start 4-betting medium/weak hands without a clear understanding of whether he is doing it for value (planning to call a 5-bet), or if he is bluffing (planning to fold to a 5-bet).

What our inexperienced Hero might not realize, is that his opponents’ loose 3-betting doesn’t necessarily mean they are willing to splash around with lots of weak hands in 4-bet and 5-bet pots. When two good and aggressive NLHE-players engage in 3-bet/4-bet/5-bet warfare preflop, this is what usually happens:

Both players operate with wide ranges, and all ranges have a significant percentage of bluffs in them, especially at the early stage (raising and 3-betting)
Both players are willing to fold most of their bluffs (but not all of them), when their opponent reraises them back

This results in ranges that start loose, but get more and more (but never completely) weighted towards value. And it’s usually plain wrong to assume you can 4-bet a medium hand like AJs for value against a loose 3-bettor, and expect to be a favorite when he 5-bets all-in. Yes, AJs is a decent hand against the range that 3-bet you, but it’s crushed by the range that 5-bets you, and it’s your opponent who decides when the 5th bet goes in (and that rarely happens unless he has the goods).

Therefore, if you decide on a frustrated whim to “take a stand” against an aggressive and competent 3-bettor with a hand like AJs, you will discover that in some mysterious way he almost always manages to come up with a better hand when you get all-in preflop.

This has lead many an inexperienced NLHE player to lose his stack, since these players:

Don’t understand the roles different types of hands have in different types of ranges. First and foremost: Do I have a value hand that wants to get all-in, or do I have a bluff hand that I will fold to further aggression?
Aren’t willing to fold hands that are strong at the early stages, but turn into weak hands when Villain keeps reraising

Let’s look at Example 1.4 again. Hero open-raised AJs (correctly), and he got 3-bet. He then decided that his AJs was a good hand against Villain’s 3-bet range (debatable, but not a big mistake), so he 4-bet for value (wrong!), planning to call a 5-bet all-in. Playing AJs for value after a 3-bet and going all-in with it was a big mistake. The 4-bet in itself was not a big mistake, since Villain has a lot of bluffs in his 3-betting range, and he will fold most of them to a 4-bet. So it’s not a problem to 4-bet AJs as a bluff against a range full of 3-bet bluffs. But when Villain comes over the top with an all-in 5-bet, our AJs crumbles to dust (if Villain knows what he is doing).

But our inexperienced Hero did not realize what had just happened when he got 5-bet, and he stuck with his plan of playing AJs for value against what he perceived to be a wide and weak range. The problem is that the range he faces after a 5-bet from a competent player isn’t wide and weak, it’s very narrow and very strong.

Note what the real mistake was in this hand. 4-betting AJs against a wide range was not a big mistake in isolation, and neither was calling a 5-bet getting 2: 1. But the combination of 4-betting AJs + planning to always call a 5-bet, now that was a big mistake against a competent opponent. It caused Hero to invest his remaining 96.5bb stack as a huge underdog. The problem was, as mentioned previously, that his opponent controlled when the 5th bet went in, and Villain made sure he had a hand.

Our goal for this article is to give Hero a set of tools he can use to comfortably counter preflop aggression when he is sitting as the raiser out of position. We’ll base our work on Hero’s opening ranges, and based on these, we can deduce defensive strategies against positional 3-bets. And we will use game theory to design these strategies in such a way that the 3-bettor can not exploit Hero in these scenarios. Our work on Hero’s game theory optimal defensive strategies also gives us a set of optimal 3-betting strategies for his opponent, so we kill two birds with one stone.

We have here talked mostly about the ills of getting 3-bet when sitting out of position, and this is what I feel inexperienced players find hardest to deal with. But the mirror image of this scenario, with us being the 3-bettor in position, is also worth discussing. These are easier scenarios to play, but we will benefit a lot from understanding optimal 3-bet/4-bet/5-bet dynamics also from this perspective. We’ll learn how to construct optimal 3-betting ranges, based on the raiser’s opening range, and we’ll learn how to play against a 4-bet.

Regardless of whether we’re the raiser or the 3-bettor, we want to understand which hands we can (re)raise for value, and which hands we (re)raise as bluffs. And above all else, we want it to be 100% clear which of these two things we are doing before we engage in a 3-bet/4-bet/5-bet war preflop.

1.2 Our model and overall philosophy

In this article we’ll design so-called optimal strategy pairs for the raiser and the 3-bettor in the following scenario:

– The raiser opens some range
– A player behind him 3-bets
– The raiser 4-bets or folds
– The 3-bettor 5-bets, or folds to a 4-bet Continue reading Optimal 3-bet/4-bet/5-bet strategies in NLHE 6-max – Part 1