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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

Reciprocality: The Cause of Profit at Poker

Before anything flows, there must be a difference. Between different elevations, water flows. Between different pressures, air flows. Between different poker players, money flows.

In the world of reciprocality, it’s not what you do that matters most, and it’s not what they do. It’s both. Reciprocality is any difference between you and your opponents that affects your bottom line. Reciprocality says that when you and your opponents would do the same thing in a given situation, no money moves, and when you do something different, it does.

You can mine for reciprocal gold anywhere in the poker universe. Pick a topic, any topic. It can be as general as “food selection” or as specific as “Ace-king in the big blind at limit hold’em.” You dig for gold by looking for things that you could do differently in the future, things that will create or increase advantageous differences between you and your opponents, and thereby cause theoretical money to flow from them to you.

Theoretical money doesn’t spend, but it does inspire. I remember when I first heard about it in the form of “expected value.” I learned that each wager has two results. There’s the expected result, based on analysis, and the actual result, based on events.

I was immediately and appropriately obsessed with theoretical money. All I wanted to know was my score. And I mean I wanted to know it now, as in, right after the hand. But I had no idea how to determine the actual expected value of a street, let alone a whole hand.

Without realizing it at the time, I borrowed from my prior life as a tournament bridge player — where my score was entirely dependent on the scores of others — and I came up with a way to analyze a hand of poker that satisfied my need.

After a hand was over, I’d trade places with my opponent. I’d give him my hole cards and my position, and I’d take his, and I’d do a reciprocal analysis. I would imagine how the play of the hand might have gone in the reversed scenario. Then I’d take the imaginary result and I’d compare it to what actually happened, and I’d get a sense of who really won the hand, in theory.

Sometimes I could not figure it out with much accuracy. But sometimes I could, especially if the hand had few variables and branches, and was against familiar opponents.

For example, let’s say one day I get pocket kings and Joe gets pocket aces. We play the hand, and Joe wins $100 from me. Right away I’d pretend it had been the other way around, me with the aces, and Joe with the kings. I’d play the streets out and I’d think through the most likely lines and I’d take the resulting probability wave and put a number to it.

In this example, let’s say I determined that had I had the pocket aces, I would have won $80. The equation would go like this. Joe won $100 in reality. I won $80 in reversed make-believe. So my final score on the hand is -$20. You can apply this method of review to any single street or group of streets.

Let’s hold on to that way of thinking and take a look at starting hands at hold’em. In reality, as we all know, the least profitable starting hand is 72, and the most profitable hand is pocket aces. In reciprocality, the least profitable hand is also 72, but not because 72 is the worst hand. 72 is the least profitable hand because it is the most similarly played hand.

So what is the most profitable hand, reciprocally speaking? Is it pocket aces? Nope. The hand that has the highest reciprocal potential must be a hand that gets played lots of different ways. It’s going to be somewhere between the hands that are rarely folded, and the hands that are rarely played. Aces are almost never folded before the flop, so we know they cannot be the most profitable hand. It seems most improbable that the most profitable hand would be exactly the same hand for everyone through all time and space, which means the answer will vary from player to player. And that means that any answer we produce is just an educated guess anyway. So what the heck. I’ll go first.

The hold’em hand I think I’ve made the most reciprocal profit on over the years is queen-ten. That’s the hand I think I have played most differently from my opponents most often.

So far, I have been answering the question, “What is reciprocality?” In the rest of this article, I will answer the questions, “Where do we find it, and how do we cash it in?” by examining reciprocality as it applies to information, position, bankroll, quitting, tilt, and betting. Bring your pick and pan. We’ll be mining for gold.

Information Reciprocality

“My secret is I keep secrets.”

I play poker on a need to know basis. I need to know the thoughts my opponents are thinking. I need to know the feelings they are feeling. And I need to know the cards they are playing. Meanwhile, I need them to know as little as possible about me. I call this relationship the information war.

The information war is fought on two fronts — sending and receiving. To win it, send less information than they send, while receiving more information than they receive. By controlling those differences, you control information flow. That’s where to mine for reciprocal gold.

On the internet, the information war is fought on a vast landscape made of statistics software, timing tells, chat boxes, forums, and more. Non-internet poker happens on a table, so I call it “table poker.” Table poker always comes with sights and sounds and smells and tells and it’s like an eternal orgy of information exchange. The rest of this section is about information reciprocality at table poker.

Muscles

Think of the human body as a communication device that uses muscles to broadcast information. It is not always obvious who is in charge of operating the muscles. Sometimes we are, and sometimes they are. The more control we can retain over our muscles, the more control we have over information reciprocality.

Face

Humans have twice as many facial muscles as any other animals. The favored explanation is that at some point in the past, increases in facial musculature made our ancestors better than their neighbors at silent communication. The better communicators had an advantage at surviving, and at getting laid, and that’s a genetic jackpot. So anytime a mutated gene gave mother nature a choice on this matter, more muscles in the face were naturally selected.

Now, many generations and mutations later, we’ve got these 40 muscles in our face, all wired up to send subtle silent signals, and we can’t unplug them. All we can do is try to talk them into keeping quiet when we need them to, for the sake of the team. During a poker hand, the brain can be saying “Holy Crap!” and then, just as the face is about to say the same thing, the brain will whisper urgently to the face, “Wait! Shhh! Don’t move a muscle!”

And when that happens, we see the poker face. The poker face is an instinctive reaction to situations in which the brain tells the body to stop sending information. Reciprocal gold goes to whoever is better at acting instinctively on purpose.

Hands

For the game to be played, chips and cards must move, and human hands must move them. And where there is motion, there is information. Sometimes a little hitch in the hands will tell me something. Sometimes it’ll be the way they handle their chips, sometimes it’ll be the way they handle their cards, sometimes it’ll be the way they check, sometimes it’ll be almost nothing, but there’s always something.

But the hand movement I get the most information from, by far, is the one where an opponent shows cards when he didn’t have to.

Mouth

Here we have a collection of muscles and parts that send information using not only expressions, but also sounds. And not just any old sounds. Words. Sentences. Information of the highest grade. This comes as great news for the reciprocality miner as there are no rules that require the muscles of the mouth to move while playing poker. You have the right to remain silent.

My Teachers

I cannot tell you that quieter is universally more profitable than louder. I cannot tell you that stillness always beats motion. I cannot tell you that less is always more. But I can tell you a story.

I used to shuffle chips until my hands got sore. My legs pulsed so much that my shoes had predictable wear patterns like a poorly aligned car. I have embedded myself at one casino for months or years at a time and talked so much at the table that I was a welcoming committee, table captain, and waitress translator all in one. And with all that movement, and all that talking, I was still able to support my food and rent habit from my poker winnings because I was still way, way ahead of my opponents in the information war, because of what I didn’t do, and didn’t say, and when.

I didn’t show hands. I didn’t talk about hands. I concealed elation and disappointment. When it came to information, I was wide open about everything, except the poker game. I learned to play that way because whenever I went to Vegas in the early years, I ran into two kinds of players: the ones I was afraid of and the ones I wasn’t. Naturally I paid most attention to the players I feared most. The conspicuous thing they all had in common was this uncanny way of looking like they didn’t give a shit. And it scared the shit out of me. So I copied them and I learned their skills. And the more I did what they did, the more I realized that what I had learned from my teachers was how to play what I call sixth street.

Sixth Street

Sixth street starts when the betting stops. Sixth street is when players relax, which is why it pays not to. Reciprocality.

Sixth street is when statues become fountains. While playing the turn and river, the players are stoic, doing their best to give up as little information as possible. And then, as soon as the betting stops, their parts start moving, broadcasting information about their thoughts, their feelings, and their cards. Sixth street is when players let their guard down, as if all of a sudden it’s safe to reveal classified secrets to the enemy. It’s like they don’t even know the war is still going on.

In the stream of information, sixth street is a reliable place to pan for gold.

Mum Poker

A military arms race results in bigger bombs and thicker bunkers. A zoological arms race results in exquisitely camouflaged prey, and predators who can see them anyway. The information war at poker has an arms race, and if one were to take it to its natural extreme — which I have — one would play a style of poker I call “mum poker” — which I do.

On the outside, mum poker is the traditional poker face, extended to the entire body, and maintained through sixth street. On the inside, mum poker is no complaining, no blaming, no regretting. Mum poker is stillness. Mum poker is readiness. If you wanted to go all the way with it, you could think of mum poker as being like absolute zero, the cessation of motion. It is knowable in theory, and forever approachable, yet unattainable.

Or you could just think of it as sit up and shut up.

Today, when I am playing purely for profit, I play mum poker. I wear a baseball cap, no sunglasses, and no lettering. I rarely make eye contact. I do not speak unless spoken to, and even then, I do not react to questions or comments about poker. I have found that the less information I send, the more I focus on the game. And when I am focused on the game, I send less information. When I employ mum poker, I fight for reciprocal gold on both fronts of the information war simultaneously.

Position Reciprocality

“The first shall be last and the last shall be first.” — Jesus

Think of every hand of poker. Think of the enormous number of hands played on the internet, and then add to that every hand played in home games and casinos. Now think of that sum total of all hands, broken down to street by street. All those streets. Millions, billions, whateverillions, it’s a lot. Now consider this. Every one of those streets has this in common: someone goes first, and someone goes last.

I agree with everyone who thinks that acting last is better than acting first. But we have to slow down here because this is delicate. Position reciprocality is not the difference between first and last. It’s the difference between firsts and lasts. When seen through the lens of reciprocality, positional advantage does not belong to the player who acts last. It belongs to the player who acts last most often. The advantage of acting last exists during every round of betting. It’s always there, at every moment, like home field advantage during a football game. At pro football, during the regular season, to keep everything fair, each team plays half their games at home and half on the road. The rules do not allow a team to create a home-game/away-game reciprocal advantage simply by folding their away games. But at poker, we are allowed to do exactly that. We can fold our “away games,” our bad positions, and thereby act last more often than we act first, and thereby create an advantage.

Bankroll Reciprocality

There’s your net worth, and after that it’s all just accounting. You can say you have a poker bankroll, but really what you have is an imaginary wall between some of your money and the rest of it.

Behind your main poker bankroll wall, there are two other walls on wheels that you construct and maneuver. There’s the money you partition off and put on the table to bet with. That’s one bankroll. And then there’s whatever other funds that are immediately available to you while you are playing, such as the money in your pocket, or maybe even the money in your buddy’s pocket. Wherever it is, if it’s money that is not on the table, and you can get to it without losing your seat, that’s another bankroll. So all together, you have three separate bankrolls when you play. That means you have three ways to run out of money. You can go table broke, pocket broke, and broke broke.

Let me ask you something. Do you play your best game when you are running out of money?

I sure don’t. The less concerned I am about funding, the better I play. And I believe the same is true for my opponents. So really, all I have to do is partition my money better than they do, and I make money. Reciprocality.

Quitting Reciprocality

“Walking away is easy. The hard part is standing up.” — me

I have always had very strict policies when it comes to quitting, even when I first started playing poker. Back then I had two main quitting rules that I never broke. I would always quit if I was out of money and nobody would lend me any, and I would always quit if everybody else did.

Eventually I quit all that stuff. I quit running out of money, and I quit being the last guy to quit. Nowadays I think of quitting as a skill set unto itself, with branching subsets of skills for each type of quitting situation. There’s knowing how to quit at limit games, and there’s knowing how to quit at no-limit. There’s knowing how to quit when you have a curfew, and when you don’t. There’s being able to quit when you’re ahead, and when you’re stuck. There’s quitting when you feel good, and for when that doesn’t happen, you need to know how to quit when you feel bad. There are many ways to outquit your opponents.

One thing about tournaments is nobody ever quits. That decision is done for you, or rather, to you. The good news is, it is impossible to make a bad quitting decision in a tournament. The bad news is, your opponents can’t screw it up either, which means there is no reciprocal gold to be found in tournaments by the superior quitter.

By one way of looking at it, I have made tens of thousands of terrible quitting decisions. Times when everything was wrong. When I was tired. And tilted. And the game was bad. But I’d play on. I’m talking situations where a panel of quitting experts would unanimously decree: “You are severely injured and you are bleeding all over the table. Quit. Quit now.”

But I wouldn’t. I’d take the next hand. And that’d be one bad quitting decision. After that hand, I’d have the option to quit, but no, I’d take another hand — I’d make another quitting mistake. That’s two quitting mistakes in four minutes. And I had just begun to not quit.

In time, my blood started to clot, and I got a little bit better at quitting, and then a little more better, and then one day I realized that every session of cash-game poker I ever play will end on a quit, so I really should continue forever to work on getting better at quitting, and a few years later I realized that if I wanted to quit well every session, then I’d have to be sharp at the very end of every session, since that’s always when the quitting happens, and a few years after that I realized that no action is an island, that everyone else’s sessions always end on a quit too, and that the real reason there is money to be made by quitting well is because sometimes my opponents don’t. Reciprocality.

Tilt Reciprocality

“To win at poker, you have to be very good at losing.” – me

During the first few years of my poker-playing career, I played almost entirely in home games that were almost entirely loose and reckless. All I had to do to win was play tight, which I had learned how to do. The trouble was, I had also learned how to tilt.

I was a great tilter. I knew all the different kinds. I could do steaming tilt, simmering tilt, too loose tilt, too tight tilt, too aggressive tilt, too passive tilt, playing too high tilt, playing too long tilt, playing too tired tilt, entitlement tilt, annoyed tilt, injustice tilt, frustration tilt, sloppy tilt, revenge tilt, underfunded tilt, overfunded tilt, shame tilt, distracted tilt, scared tilt, envy tilt, this-is-the-worst-pizza-I’ve-ever-had tilt, I-just-got-showed-a-bluff tilt, and of course, there’s the classics: I-gotta-get-even tilt, and I-only-have-so-much-time-to-lose-this-money tilt, also known as demolition tilt.

I’d tilt, and I’d look back on my tiltings, and I started seeing cycles, and then cycles within the cycles, and before long, I started to see my entire poker future as a ceaseless fluctuation between tight and tilt. I figured if I ever went broke at poker, it wouldn’t be because my best wasn’t good enough to keep me afloat. It’d be because my worst was bad enough to sink me.

A big day in my career was the day I realized that tomorrow I would still be a tilter. That there would be no quick fix. That any headway I made would be gradual. I realized that if I could somehow put progressively longer periods of time between my tiltings, and if I could somehow have them be progressively not quite as bad as the last time, then I’d have a chance to get some wind under my wings, and when I did, I’d soar indefinitely. Less often, less severe. Less often, less severe. That’s what I kept telling myself.

It is now fifteen years and thirty thousand hours of poker later. In that time I have gathered myself, and my thoughts�

On Tilt

Tilt has many causes and kinds, but it has only one effect. It makes us play bad. It makes us do things we wouldn’t do if we were at our very best. And that’s how I want to define it, exactly like that. Tilt is any deviation from your A-game and your A-mindset, however slight or fleeting.

There are two reasons to define tilt in this way. One is standardization. All A-games are identical. Anyone who is playing his A-game is making the best decisions he knows how, and his mind is as right as it ever is. That’s what A-game is. It’s our best. And we all have it. So by defining tilt from the top down, we can draw a line for any player that cleanly divides his tilt from his non-tilt.

The other reason is that we aren’t just playing with words here. We are using them as shovels to dig for gold. And by using the word tilt to focus on our best, instead of our worst, we hit a lode: Tilt is non A-game. Tilt is anything less than your utmost. Tilt is suboptimalness. Defining tilt in this way, everyone tilts. It’s just a matter of how often, how long, and how bad.

And so we arrive at the three dimensions of tilt: frequency, duration, and depth. How often do you deviate from your A-game? How long does it last? And how far below your A-game do you go? Revisit those questions.

Tilt is all about you. If you think you should have quit sooner, or if you think you should have played at different stakes, or if you think you made a bad call, then you tilted. Only you really know when you knew better.

Tilt reciprocality is your slippage matched up against everybody else’s. Tilt reciprocality recognizes that any reduction, however small, in the frequencies, durations, and depths of your own tiltings will always have the effect of favorably widening the gap between your tilt and theirs, thereby earning immediate reciprocal advantage. To make money from tilt, you don’t need to be tiltless. But you do have to tilt less.

Betting Reciprocality

“Seventy-five percent of all poker players think they play better than the other seventy-five percent.” — me

Betting reciprocality is the difference between your betting decisions — raise, bet, call, check, and fold — and theirs. There are two clean ways to think about betting reciprocality. I wrote about them in the opening of this article, and I’d like to expand on them now.

One way is to trade some parameters with your opponent, project the future in that reality, and compare. I call this reciprocal analysis.

The other way of thinking about betting reciprocality produced this conclusion: “The hold’em hand I think I’ve made the most reciprocal profit on over the years is queen-ten. That’s the hand I think I have played most differently from my opponents most often.”

Continuing from there, after queen-ten on my list of most-profitable-hands comes king-ten, queen-jack, jack-ten, king-nine, queen-nine, jack-nine, queen-eight, jack-eight, ten-nine, etc, not necessarily in that exact order, but thereabouts. The reason these hands cause the most amount of reciprocal motion is because these hands bring out the most consequential difference in how a hand gets played, which is, before the flop.

I am going to list the ways that two players can start a hand, starting with the least consequential, and moving toward the differences that make the most difference. If, in a given preflop situation:

1) Two players would both fold, then no reciprocal money moves between them on that hand. No-brainers are no-gainers.

2) Two players would both call before the flop, or if they would both raise, then still no money moves between them before the flop. There might be reciprocal motion on the hand after the flop, depending on how differently they would play it.

3) One player calls before the flop when the other would raise. Here we have reciprocal motion before the flop, with potential for more after the flop.

So far, either both players saw the flop, or both players didn’t. There are two other ways it can go:

4) One player folds before the flop when the other would call.

5) One player folds when another would raise.

If it is true that maximum potential reciprocal motion occurs when one player sees the flop when another player wouldn’t, then the most profitable hand is going to be the one that most often generates the play/don’t-play difference, which, for me, by my estimation, is queen-ten.

After the flop, no matter how anyone got there, we can focus the reciprocal lens on any single bet, or street, or combination of streets, and do a reciprocal analysis. For example, it’s on the river playing limit hold’em and you have the best hand. You bet and your opponent calls. If the situation was reversed, and your opponent bet the river, would you have called? If the answer is no, then you just won one bet. If the answer is yes, then you broke even.

At no-limit hold’em, the nature of all-in-ness narrows the reciprocal focus in a specific, recurring way. Let’s say Joe and Moe both hit the flop. At some point in the hand, they get all-in. In reality, Joe busts Moe. In reciprocality, the main question is, would Moe have busted Joe? If the answer is yes, then the hand is a tie. If the answer is no, then Joe wins the hand by however much money he has in front of him at the end of the hand in the imagined reality.

And that’s basically it on Betting Reciprocality. That is, if you like everything all neat and tidy. Or, you can follow me now, into the magnificently messy part of poker.

So far, all the reciprocalities have been black and white. For example, when tilt reciprocality causes money to flow, we know for certain which way it is flowing. The same is true with bankroll reciprocality, quitting reciprocality, and all the others. But there is one exception, and it’s a big one: Betting reciprocality. This is where black blends with white and leaves us in the gray. This is where uncertainty is certain. And thank goodness for that! It’s what makes poker poker.

Gray Area

In this image, black and white represent betting decisions that are definitely right or definitely wrong, and the various shades of gray represent all the others.

Think back to the very first hands of poker you ever played. Your gray area was almost everywhere, and your A-game stank. With every hand, with every round of betting, with every sixth street discussion, you gained significant experience and understanding. Your A-game improved at the same rapid pace that your gray area — your uncertainty — shrank.

As times passes, your rate of change slows. Your A-game improves more slowly, and your gray area shrinks more slowly. The main thing to realize is that no matter how good you get, you will always have a gray area. The gray is not part of you. It is part of the game.

Here are two examples of black and white decisions. 1) Playing limit hold’em, in a full game, you are under the gun with 72o. Should you raise, call, or fold? 2) Playing any poker game, it’s on the river, you are headsup, and you have the nuts. Your opponent checks. Should you check or bet?

As we move into the gray, the theoretical expectations of our options become more balanced. A decision might make us a 60-40 favorite, for example. Moving into the central gray region, we arrive at those decisions for which the expected outcome is 50-50 or nearly so. These are the decisions of little or no theoretical consequence, the decisions where each option is as good as the other. These are the decisions that matter least.

Also in the central gray — the land of closest decisions — we can expect disagreement to go up over which decisions are best. We can expect intelligent, elaborate debates with both sides insisting theirs is the right side. We can also expect to debate with ourselves and to second guess ourselves. In the central gray is where we are most likely to torture ourselves with the question: Did I get it right that time?

And that’s why I say: The decisions that trouble us most are the ones that matter least.

Let’s say you face a close betting decision, and afterwards, you want a definite answer. You want to know, one way or the other, if your play was right or wrong.

STOP!

That’s a mistake. Just by thinking like that, about right and wrong, you are making a mistake. If you play a hand and you face a close decision and then you write about it or talk about it, I think that’s great — seriously. Or if you talk about hands other people played, same thing. All good. But be careful. Don’t fall into the gray area’s trap. Don’t burn up valuable energy and waste precious sanity. Don’t assume that just because you have an answer, and just because someone else has a different answer, that one of you is right and the other is wrong.

Let’s say I’m on the button and everyone folds around to me. Depending on my cards, and my opponents, and other parameters, it might be obvious to me what the best choice is, or it might not be obvious at all. Should I assume that there is always a right answer? And even if there is a right answer, should I assume that I can always know what that answer is? I believe the answers to those questions are no and no.

Another example. It’s on the turn, playing limit hold’em. There are three players in the pot. I am second to act. The first guy bets out. Should I raise? Should I call? Should I fold? Okay, I’ll tell you more. I’ve got top pair. The guy who bet out might be on a draw, or he might have a monster. I can’t really tell. The guy behind me might be really weak, maybe drawing thin against my hand. But he’s acting so weak that maybe he’s strong and he’s about to raise it. Or maybe he is on a draw and I need to raise to either get him out or make him pay the maximum price. But the guy who bet out might have me beat. He might even have me drawing dead. If I raise, I open it up for him to reraise. Hmm. Tough one. Should I raise? Should I call? Should I fold?

I believe it is correct to believe in unknowableness. Analyze, evaluate, ponder, and then let it be. Resist the gray area’s mind-snaring entrapments. When you examine a betting decision, yours or someone else’s, at the table or away, on your own or with others, remind yourself that debates point to close decisions, and that close decisions matter least, and that the answer is sometimes unknowable. Dwell not on close decisions, and thus, when you play against dwellers, you will make reciprocal gains in energy conservation and sanity preservation.

What’s it take to max out at poker? Three things. You have to have an A-game that is good enough to beat somebody. You have to play against those people. And you have to play your A-game every hand. To play your A-game every hand, you must be paying attention every hand. You must be present. If you are thinking about the past, then you are, by definition, not present. If you want to be in the now, realize deeply that hey, I may never know if I played that hand right or wrong, so I might as well just play this hand!

Get good, and remain good while you play. That’s really all I said just now. That’s really all I ever say, to myself, or to anyone seeking the secret to poker, which, as it turns out, is not exactly a secret. So where does reciprocality fit in?

I don’t see reciprocality as a theory about how to win at poker. I see it as a way to see it. Reciprocality is a lens. When focused on a pair of choices, the resolution of reciprocality reveals differences large and small; it shows them as they are. When I look through the lens, I see every incremental improvement and deterioration, of mine and my opponents, at betting and everything else, as being immediately rewarded or punished in the currency of reciprocal gold. The profit is always flowing, and that motivates me.

But then, like I said, that’s how I see it. Now the lens is yours. Point it where you will, focus it as you please, and make your own discoveries.

© Tommy Angelo 2006


Various updates:

I started a blog in 2008 and it’s still going strong. I post about poker, mindfulness, and my life.

In 2011, I came out with my second book. It’s called A Rubber Band Story and Other Poker Tales. This book contains my best articles and blogs from the last 12 years, with new material too. You can buy it directly from me and get it personally inscribed if you like, here. Also available in eBook. Amazon reviews are here.

Also in 2011, I started a newsletter. Join my mailing list to receive the newsletter, and I’ll send you Episode 8 of my award-winning video series, The Eightfold Path to Poker Enlightenment.

It’s now 2012 and I am painlessly immersed in writing my third book: Painless Poker.

Tommy Angelo

http://tommyangelo.com/articles/reciprocality/

C-beting in NLHE 6-max- Part 2

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

Coordinated flop

Dry flop

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

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

326 combos
25%

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

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

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

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

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

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

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

Based on the assumptions above we reached the following conclusions:

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

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

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

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

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

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

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

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

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

2. Assumptions about ranges
Assume the following model:

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

We let Bob use 3 different preflop flatting ranges:

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

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

Tight 5% flatting range

JJ-55
AQs-AJs AQo
KQs

66 combos
5.0%

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

Loose 15% flatting range

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

200 combos
15.1%

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

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

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

3. C-betting on coordinated flop

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Non-optimal defense under weak-tight restrictions:

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

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

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

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

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

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

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

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

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

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

4. C-betting on dry flop

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

– Flatting with all his defense hands

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The number of combos after card removal is 126:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Based on our modeling, we conclude the following:

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

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

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

We summarize:

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

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

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

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

C-beting in NLHE 6-max- Part 1

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

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

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

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

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

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

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

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

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

2.1 Assumptions about preflop ranges

 

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

Alice uses our standard 25% opening range from CO:

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

326 combos
25%

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

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

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

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

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

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

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

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

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

– Very coordinated
– Very dry

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

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

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

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

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

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

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

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

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

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

Our coordinated flop is:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A range analysis with Pokerazor illustrates this with numbers:

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

We therefore conclude:

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

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

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

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

Our dry flop is:

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

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

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

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

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

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

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

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

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

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

Then we see how far he can go:

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

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

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

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

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

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

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

A range analysis with Pokerazor illustrates this with numbers:

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

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

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

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

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

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

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

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

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

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

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

Rational Bankroll Management – Part 1

1. Introduction
This is the first part of an article series about bankroll management (BRM) in poker. Using some form of bankroll management is vital for poker success. A serious poker player wants to maximize his profits in the long run. To achieve this, he has to protect his bankroll from ruin, and this principle of bankroll management is well known. What’s the best way to go about it is less well known.

In this article series we’ll study rational bankroll management. We define this as BRM based on the way we play, combined with rules derived from statistics. We want to use BRM adapted to our situation (the games we play, our win rate, our standard deviation, and our risk tolerance) instead of simple rules of thumb like “50 buy-ins for no-limit”. We’ll use mathematics to define a model for BRM that we can use to calculate the best BRM scheme, given our situation.

Our goal is make good decisions about which games to play and how high to play, given our bankroll, our skills, and our risk tolerance. Good BRM decisions will maximize our profit over the long run, which is what we strive for as serious poker players.

It’s been said that “all the best gamblers go bust sometimes”. Amateurs with gamble in their blood are perhaps attracted to the “romantic” notion of risking everything. But from a money making perspective, bankroll ruin is completely irrational and unnecessary.

In this series we’ll assume that we are rational gamblers who want to make the most money in the long run. Our only concern is to maximize profit and not to satisfy emotional urges. For us, variance and swings are obstacles that we want protection against. Our protection is our understanding of what variance is, mathematically speaking. Armed with this knowledge we can design plans for moving up and moving down in such a way that we both maximize our income and protect our bankroll.

1.1 Bankroll management in its simplest form
Before we begin our discussion of rational bankroll management, let’s talk about how most players practice BRM today. The most commonly used method is to choose some bankroll size that seems big enough to play our chosen game and limit, and then we play and hope we don’t bust.

BRM in its simplest form then means following certain rules that give you an acceptably low risk of ruin, based on your subjective assessment. Many rules of thumb exist for bankroll requirements, some better than others. For example:

  • 300 BB for fixed limit
  • 50 BI for NLE
  • 100 BI for PLO

And so on. All such rules have in common that they use some assumption about what is “enough” to play, and that you are “safe” if you stick with the scheme. This can work, especially when combined with rules for when to move down. For example:

  • 300 BB for fixed limit, but move down if the bankroll drops to 300 BB for the limit below
  • 50 BI for NLHE, but move down if the bankroll drops to 50 BI for the limit below
  • 100 BI for PLO, but move down if the bankroll drops to 100 BI for the limit below

To make this type of simple, rule-based BRM scheme complete, we can add a rule for when to move up:

  • 300 BB for fixed limit
    • Move down if the bankroll drops to 300 BB for the limit below
    • Take a 100 BB shot at the next limit if you have 300 BB for the current limit plus 100 BB for the next limit
  • 50 BI for NLHE
    • Move down if the bankroll drops to 50 BI for the limit below
    • Take a 10 BI shot at the next limit if you have 50 BI for the current limit plus 10 BI for the next limit
  • 100 BI for PLO
    • Move down if the bankroll drops to 100 BI for the limit below
    • Take a 10 BI shot at the next limit if you have 100 BI for the current limit plus 10 BI for the next limit

Most players who use BRM to guide their moving up and moving down between limits use some variation of the rule-based scheme outlined above. Not necessarily as strict, but it’s based on some idea about what is “enough” to play a certain limit. Then they add a lower threshold that tells them when to move down, and an upper threshold that allows them to start shotting at the next limit.

Here is an example of this kind of planning in practice:

Example 1.1: A rule-based BRM scheme for no-limit Hold’em
Bob begins his NLHE career at $25NL. He has decided to start with a 50 BI ($1250) bankroll, using the following BRM scheme:

  • Start with 50 BI ($1250) at $25NL
  • Move down to the limit below ($10NL) when the bankroll drops to 50 BI ($500) for $10NL.
  • Take a 10 BI shot at the next limit ($50NL) when the bankroll grows to 50 BI ($1250) for $25NL plus 10 BI ($500) for $50NL
  • Keep climbing in limits using the same system (50/50+10)

Bob uses the more compact notation 50/50+10 to describe his system. The first number, 50/50+10, is the number of buy-ins he wants in the bankroll before he considers himself established at the limit he is currently grinding. The second number, 50/50+10 is the threshold for moving down to the limit below. The third number, 50/50+10 is the minimum capital he needs, beyond his minimum 50 BI for the current limit, in order to take a shot at the next limit.

If everything goes smoothly, he will quickly grind in $500 and take a $50NL shot with this money. If the $50NL shot fails, he still has 50 BI ($1250) for $25NL, so he returns to this limit and tries again. If the $50NL shot succeeds, he keeps playing $50NL until he has 50 BI ($2500) for that limit, and then he considers himself to be established there. His next task is to grind in 10 BI ($1000) more for $100NL, so that he can take a shot at the next limit, rinse and repeat.

How will this bankroll management scheme work for Bob? When we try to answer this question, we run into the fundamental problem of this type of BRM planning:

1.2 Why simple, rule-based BRM schemes are problematic to use
The biggest weakness of a simple, rule-based BRM scheme is that it does not contain any information about the player’s win rate or the variance he expects to meet in the game.

Why is this a problem? Let’s think about what will happen to various player types, where we for now only consider their win rates. Let’s first assume that Bob is a steady low limit grinder who wins 10 bb/100. So he wins 1 BI per 1000 hands on average.

The probability that this player will face a -50 BI downswing is negligible, and even if that happens, he will be protected by his move-down rule. The 50/50+10 scheme will work well for this player, and he will quickly climb in limits to begin with. Still, his win rate might drop significantly when he reaches higher limits (for example, when he reaches $200NL, where he will play in games with many professional grinders).

If his win rate drops, but is still positive, Bob will still be protected by his BRM scheme. As long as he wins and has the discipline to move down when necessary, he will never experience bankroll ruin. But 50 BI might not be enough to properly deal with the swings he now experiences. He will survive them as long as he moves down when he should, but he might waste a lot of time bouncing up and down between limits.

For example, let’s say that the swings at $200NL are so big that Bob has to try 5 times before he establishes himself there. His move from $100NL to $200NL might have gone quicker if he had used a stricter scheme, say 100/100+20. It will take more effort at $100NL to get ready for a shot, but the chance of success has gone up. So the total effort to move up might be less.

The same problem would have occurred already at $25NL if Bob had started out as a small winner there, say 2 bb/100. A low win rate equals big swings, and he might have to pay a visit or two to $10NL before climbing up to $50NL for good. Having to make several attempts to move up takes time and effort, and can be bad for morale. So a stricter BRM scheme might be better overall.

But for both the 10 bb/100 winner and the 2 bb/100 winner, we can say this:

As long as we have a positive win rate and a reasonable scheme for moving between limits, we will not experience bankroll ruin. If we have the discipline to stick with the plan and move down when we should, we will eventually move up.

That’s a good start. During this article series we’ll see that we can do better than the simple rule-based systems by applying a bit of mathematics/statistics. Our goal is to design a bankroll management scheme tailor-made for us, based on our win rate and variance in the games we play. A unique BRM scheme for every player, not crude rules of thumb that tries to apply to everyone.

But the biggest problem associated with trusting the simple rule-based schemes is this:

Bankroll management does not protect losing players against ruin!

The simple schemes outlined previously circulate on poker forums and new players quickly get an idea about how big a bankroll should be to be “safe”. But all these rules are based on two things:

  • We have a positive win rate
  • We have the discipline to stick with the scheme during adversity

In practice, new players will often run into problems on one or both of the following areas:

  • They could be losing players, even when playing their best. Then no bankroll will be big enough.
  • They win when playing their A-game, but they struggle hard with tilt. The effect of tilt is to make them losing players overall.

This naturally causes some frustration if you believe bankroll management to be a “magical” recipe for success. BRM is for winners, not losers. BRM alone does not give you the right to survive, you need poker skills for that.

If you’re a losing player, you don’t need a system for BRM, you need a budget (how much you allow yourself to lose). Splashing around for fun as a losing recreational player is fine, if you can afford it and think the fun is worth it. But we will not consider this player type in our discussion of bankroll management.

2. Two components of bankroll management: Win rate and variance
To get to a rational system for bankroll management, we need some building blocks, first and foremost an understanding of win rate (also called “expected value”, “EV” or “expectation”) and variance.

Simply put:

Win rate
Our win rate in a game is what we expect to win (expressed as profit per game, per 100 games, or some other convenient unit). We distinguish between our true win rate (our hypothetical, fully converged win rate that we would have if we played infinitely long) and our observed win rate (the win rate we have now, given the amount we have played).

Win rate is a function of how well we play and how well the opposition plays. It’s the difference between our skills and their skills that determines our win rate. Good poker players actively seek out opponent that play worse than them, and then they play them. This is the skill of game selection, and it’s critical for your win rate.

Variance
Variance is a statistical property than tells us something about how far our results will deviate from the expected results. For example, let’s say our true win rate in a poker game is 10 bb/100. Does this mean we’ll have 10,000 x 10 bb/100 =1000 bb after playing 10,000 hands?

No, since poker is a game with a random component (the cards). In the long run our observed results will converge towards the expected results, but in the short run (say, some tens of thousands of hands, or some weeks) it’s very likely our observed results will deviate significantly from our expected results.

Variance is partly a function of the random component in the game (the randomness of the deck), partly a function of the specific rules of the game (some poker games have more variance than others), and partly a function of our playing style and our opponents’ playing styles. If you play tight and passive, you’re variance will be lower, while loose-aggressive play increases variance.

Since variance causes swings and emotional stress, it’s generally a good idea to take the low variance route if we have the choice between alternatives that are otherwise equivalent. But this doesn’t mean we should try to reduce variance wherever we can. If increased variance is accompanied by increased win rate, we should embrace the variance and make sure we’re properly bankrolled for it.

A good loose-aggressive player plays this way because it maximizes his win rate. A consequence is that he experiences more variance than a tighter player. More variance means bigger swings, so he probably needs a bigger bankroll (but not necessarily, more about this an a bit). This is a trade off he is willing to make in order to win as much as possible. And he protects himself by using a bankroll management scheme suited for his style.

A tight player has lower variance, which reduces his swings. Does this guarantee he can get away with a smaller bankroll? Maybe, but not necessarily, if his tight play keeps significantly reduces his win rate.

The mathematics behind the last statement will be discussed later, but for now let’s establish two important principles:

– Increased variance increases our bankroll requirements
– Decreased win rate increases our bankroll requirements

It follows that we want to maximize our win rate and minimize our variance. If these two considerations pull in opposite directions (and they usually do in poker) some compromise must be made. The obvious choice for a serious player is to maximize win rate, accept the variance, and use a BRM scheme that provides adequate protection against it.

There is a mathematical formula (the risk-of-ruin formula) that tells us how win rate, variance and bankroll requirement are related to each other. We’ll get there eventually, but before we do, we’ll build a better understanding of win rate and variance by playing around with a couple of “toy games”.

We start by defining win rate in a strict way, suited for our purpose:

2.1 Definition of win rate/EV
Assume we’re playing a game with n different outcomes. For example we can flip a coin, which gives us two outcomes (heads or tails). Furthermore, assume that each outcome is associated with some probability, expressed as a number between 0 (never happens) and 1 (always happens). For example, in a coin flip both heads and tails are associated with the probability 1/2 (50%). Finally, assume that each outcome is associated with some value. For example, we might wager $1 on a coinflip, win $1 on heads and lose $1 on tails.

The general definition of expected value (EV) which for our purpose is the same as win rate, can now be written as:

where the probabilities for the n outcomes are noted p, while the values are noted x.

In other words, for each outcome we multiply the probability and the value, then we sum over all outcomes. To illustrate the process, we’ll calculate the EV for a simple toy game:

EV for Dice Game 1
We wager $1 per throw in the following dice game:

  • Our wager is returned on 1, 2, 3 and 4
  • We lose our wager on 5
  • We win twice our wager on 6

This game has six outcomes (the die can land on 1, 2, 3, 4, 5, and 6). A fair die has equal probability for all six outcomes, so the probability for each outcome is 1/6. The value for the different outcomes are $0 for 1, 2, 3, 4, -$1 for 5, and +$2 for 6. Note that the payout of +$2 is the net payout (we get our $1 back, and then win $2 in net profit). We plug the numbers into the EV formula and get:

EV (Dice Game 1)
=(1/6)(0) + (1/6)(0) + (1/6)(0) + (1/6)(0)
+ (1/6)(-1) + (1/6)(2)
=(1/6)(1)
=0.1667

We profit from this game, and we should play it. Our EV is 1/6 times our wager, which becomes $1/6 =$0.1667 per throw.

One important property of EV is that it’s additive:

In other words, if we throw several times, the total EV equals the sum of the EVs for each throw. So if we play Dice Game 1 one hundred times, our total EV is $16.67:

EV (Dice Game 1, 100 throws)
=100(Dice Game 1, 1 throw)
=100(0.1667)
=16.67

When we know the EV for the game, we can calculate the variance:

2.2 Definition of variance
Win rate/EV tells us what we can expect to win (or lose) on average when playing a game. The variance tells us something about how far our result can deviate from the expected result. For us poker players the most interesting part of this is the size of the swings our bankroll will go through.

Let win rate/EV be defined like previously for a game with n different outcomes, each associated with a probability p and a value x. The variance (V) of the game can then be written as:

In other words, for each outcome, take the difference between its value and the EV. Then square the difference and multiply it with the outcome’s probability. Then sum over all outcomes. We can now calculate the variance for Dice Game 1:

V (Dice Game 1)
=(1/6)(0 - 0.1677)^2 + (1/6)(0 - 0.1677)^2
+ (1/6)(0 - 0.1677)^2 + (1/6)(0 - 0.1677)^2
+ (1/6)(-1 - 0.1677)^2 + (1/6)(2 - 0.1677)^2
=0.8056

Like EV, variance is additive. The total variance for a series of throws is found by summing the variance for each throw:

So the variance for 100 throws in Dice Game 1 is equal to 100 times the variance for one single throw:

V (Dice Game 1, 100 throws)
=100(0.8056)
=80.56

The last mathematical definition we need is standard deviation (SD):

2.3 Definition of standard deviation
The standard deviation is simply the square root of variance:

Standard deviation and variance are two ways of measuring the same, namely how wide the spread between observed results and expected results can be. Statistical formulas typically use the standard deviation, so we need this definition in our work.

The standard deviation for Dice Game 1 is then:

SD (Dice Game 1)
=sqrt(0.8056)
=0.8975

Where “sqrt” is the square root. Unlike EV and variance, the standard deviation is not additive. This follows immediately from the definition:

So if we throw 100 times in Dice Game 1, the total standard deviation is sqrt(100) =10 times the standard deviation for a single throw:

SD (Dice Game 1, 100 throws)
=sqrt(100)(0.8975)
=10(0.8975)
=8.975

We have now defined all the statistical concepts we’ll use throughout this article series when we discuss bankroll management for various games. How we use EV and standard deviation to calculate bankroll requirements will be discussed in detail in future articles. But before we do this, let’s study a variation of our dice game to illustrate the effect of variance. We define a new dice game with the same win rate as Dice Game 1, but different variance:

3. Illustrating the effect of variance
We define a new game, Dice Game 2, where we lose our wager on 1, 2, 3, 4, and 5, and win 6 times the wager on 6. EV, variance and standard deviation for the new game are easily calculated:

EV (Dice Game 2)
=(1/6)(-1) + (1/6)(-1) + (1/6)(-1) + (1/6)(-1)
+ (1/6)(-1) + (1/6)(6)
=(1/6)(1)
=0.1667

EV for Dice Game 2 is the same as for Dice game 1, namely +$0.1667. But the variance and standard deviation are much higher:

V (Dice Game 2)
=(1/6)(-1 - 0.1677)^2 + (1/6)(-1 - 0.1677)^2
+ (1/6)(-1 - 0.1677)^2 + (1/6)(-1 - 0.1677)^2
+ (1/6)(-1 - 0.1677)^2 + (1/6)(6 - 0.1677)^2
=6.8056
SD (Dice Game 2)
=sqrt(6.8056)
=2.6087

Let’s summarize and compare our two toy games:

Dice Game 1
We wager $1 per throw in the following dice game:

  • Our wager is returned on 1, 2, 3 and 4
  • We lose our wager on 5
  • We win twice our wager on 6
  • The expected value of the game is EV =0.1667
  • The variance of the game is V =0.8056
  • the standard deviation of the game is SD =0.8975

Dice Game 2
We wager $1 per throw in the following dice game:

  • We lose our wager on 1, 2, 3, 4, and 5
  • We win 6 times our wager on 6
  • The expected value of the game is EV =0.1667
  • The variance of the game is V =6.8056
  • the standard deviation of the game is SD =2.6087

The variance for Dice Game 2 is more than 8 times higher than for Dice Game 1, with a corresponding difference in standard deviation. What are the consequences when we play these two games?

The answer to this question pierces the core of rational bankroll management. Most readers will intuitively realize that if we have the option of choosing which game to play, we should choose Dice Game 1. Dice Game 2 exposes our bankroll to bigger swings without any additional reward. Bigger swings can cause problems for us, particularly if we’re playing with limited funding.

Le’s say we only have $5 in our pocket to play for. In Dice Game 1 there is only one outcome (rolling 5) that loses. 5 out of 6 times we’ll either get our wager back or win twice the amount. The risk of going broke before we have built our roll big enough to survive the expected swings in the game is fairly small.

But in Dice Game 2 we’ll lose our wager 5 times out of 6. We expect to make the same profit as in Dice Game 1 in the long run, but the most likely outcome is that we won’t reach the long run since we’ll go broke early. Another way of phrasing this is that both games are profitable, but it’s much less likely for us to realize the profit potential in the high-variance game (when we start out with a small bankroll).

Of course, if we’re loaded with cash the variance becomes irrelevant. We still choose Dice Game 1, since there is no reason to choose the higher variance game when the win rate is the same. But now our choice is more dictated by principle than by a need to protect our bankroll (since we are over-bankrolled for the game).

We end Part 1 with a graphical illustration of the effect of variance for our two dice games. We use a online variance simulator to generate graphs.

The simulator uses the units “per 100 hands” in the calculations, so we plug in EV and standard deviation expressed as “per 100 throws”. EV is additive, so the EV for 100 throws is 100 x EV for one throw, namely 100 x 0.1667 =16.67 per 100 throws. This is the EV for both games.

The standard deviation grows as the square root of the number of throws (see the previously defined formulas), so the standard deviation for 100 throws is sqrt(100) =10 times the standard deviation for one throw. We get SD =10(0.8975) =8.975 for Dice Game 1 and 10(2.6087) =26.087 for Dice Game 2.

Now we let 10 players (let “Num. of trials to run” =10) throw first 1000 times each (let “Num. of hands” =1000) and then 10,000 times each. We do this for both games and plot the profit graphs for all 10 players:

3.1 Variance simulation for Dice Game 1

The expected result after 1000 throws is 1000 x 0.1667 =+$166.70 . This is the dotted straight line on the graph. We see that the results of our 10 players are distributed around the expected result, with actual results ranging from about +$115 to +$200. The player with the largest deviation is about $52 below expected profit.

Even if all players expect to make +$166.70 on 1000 throws, there is a significant spread around the expected result. 6 players make more than they should and 4 make less (this is of course a random distribution, and a new simulation with the same input would distribute the players differently).

If we increase the number of throws to 10,000, we get:

All 10 players have solid profit. The observed profits range from about +$1550 to +$1950, where 10,000 x 0.1667 =+$1667 is the expected result. We see that the spread around the expected result is less, relatively speaking, than for the first simulation.

The player with the largest deviation is about $283 above expected profit, while the largest deviation for the 1000 throws simulation was $52 below expectation. But $283 relative to $1667 (283/1667 =17% deviation) is a smaller relative deviation than $52 relative to $166.70 (52/166.7 =31% deviation). This illustrates that the relative deviation from the expected result decreases as the number of throws increases.

This is the effect of the long run. The more we play, the more similar our observed results will be to the expected results. Mathematically, this stems from the fact that our profit grows linearly with the number of throws (EV is additive), while the standard deviation is non-additive and grows more slowly, as the square root of the number of throws (see the previously defined formulas).

Then we do the same simulations for Dice Game 2, with the same EV but far higher variance and standard deviation.

3.2 Variance simulation for Dice Game 2
First the simulation for 1000 throws:

The spread of the 10 players after 1000 throws is significantly larger than for Dice Game 1, from about $35 to $210. The largest deviation is then about $132 below the expected profit of $166.70. We can see clearly from the graph that it’s much more randomness in the final results than what we observed for the corresponding simulation for Dice Game 1.

Then we increase to 10,000 hands and get:

The spread is now from about +$1400 to +$2000, so the largest deviation is about $333 above the expected result. The corresponding spread for Dice Game 1 was from about $1550 to $1950, with the largest deviation about $283 over the expected result. The effect of the larger variance in Dice Game 1 is still significant, even after 10,000 throws.

3.3 Conclusion from variance simulations for dice games
The simulations we have done here with exactly known EVs and standard deviations for two toy games have given us a taste of things to come in future articles. We observe that games where we have the same win rate can have very different profit curves, and it’s the difference in variance that causes this.

In future articles we’ll look at win rate/EV and variance/standard deviation for poker. These quantities are far more complicated to calculate in real poker games than in our simple toy games, and in practice they can’t be solved for exactly. But we can estimate them from tracker software (such as HoldemManager and PokerTracker) and the estimates can be used for bankroll estimates.

Our next task is to learn how to use win rate and standard deviation for poker games to estimate bankroll requirements. We’ll use the risk-of-ruin formula for this, and that will be the topic for Part 3. But before we do that, we’ll spend Part 2 investigating how our definitions of EV, variance and standard deviation work in a simple poker toy game (the AKQ game).

4. Summary
We have started our discussion of principles for rational bankroll management. In this article we have defined the problem of bankroll management and gotten a basic understanding of win rate/EV and variance/standard deviation from exactly solvable toy games.

In Part 2 we’ll discuss how our definitions of win rate, variance and standard deviation can be applied to poker games. We’ll use a simple poker toy game (the AKQ game over 1/2 street with fixed-limit betting) to illustrate this.

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/rational-bankroll-management—part-1-406#sthash.Go52WHkR.dpuf

Optimal Postflop Play in NLHE 6-max – Part 7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Alice’s Default 15% UTG-range

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

194 combos
15%

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

IP flat list after ~15% UTG openraise

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

162 combos
12%

The flop comes:

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

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

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

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

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

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

The turn is:

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

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

The river is:

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

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

3. Defining Alice’s two bet sizing schemes

Standard bet sizing

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

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

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

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

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

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

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

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

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

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

We take the square root on both sides and get:

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

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

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

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

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

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

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

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

This is because Bob would have:

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

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

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

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

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

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

And this is because:

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

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

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

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

Flop defense against standard bet sizing, without slowplay:

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

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

Flop defense against standard bet sizing, with slowplay:

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

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

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

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

Turn defense against standard bet sizing, without slowplay:

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

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

Turn defense against standard bet sizing, with slowplay:

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

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

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

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

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

River defense against standard bet sizing, without slowplay:

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

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

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

River defense against standard bet sizing, with slowplay:

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

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

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

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

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

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

Turn defense against alternate bet sizing, without slowplay

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

Turn defense against alternate bet sizing, with slowplay:

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

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

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

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

River defense against alternate bet sizing, without slowplay

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

River defense against alternate bet sizing, with slowplay:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

We conclude:

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

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

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

We concluded that:

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

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

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

Optimal Postflop Play in NLHE 6-max – Part 6

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

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

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

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

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

2. On the consequences of choosing to bet a street

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

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

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

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

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

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

IP flat list after ~25% CO openraise

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

140 combos
11%

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

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

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

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

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

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

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

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

Alice’s Default 25% CO-range

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

326 combos
25%

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

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

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

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

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

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

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

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

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

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

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

There’s a subtle point buried here:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The EV equation for Bob’s flop float is:

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

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

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

Note that this strategy satisfies the defense equation since:

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

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

The EV equation for Bob’s float now becomes:

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

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

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

~15% UTG range

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

194 combos
15%

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

IP flat list after ~15% EP openraise

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

162 combos

Flop/turn/river came

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

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

 

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

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

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

Alice’s default 25% CO-range

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

326 combos
25%

And the flop comes as before:

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

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

She must now play the turn so that:

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

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

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

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

Turn strategy after 25% CO-raise

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

Test of defense equation:

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

Turn strategy after 15% UTG-raise

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

Test of defense equation:

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

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

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

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

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

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

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

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

 

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

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

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

River strategy after 25% CO-raise:

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

Test of defense equation:

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

River strategy after 15% UTG-raise:

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

Test of defense equation:

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

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

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

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

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

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

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

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

In Part 7 we’ll talk about:

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

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

Optimal Postflop Play in NLHE 6-max – Part 5

1. Introduction
This is Part 5 of the article series “Optimal Postflop Play in NLHE 6-max” where we’ll study optimal strategies for heads-up postflop play in NLHE 6-max.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

IP flat list after a ~25% CO openraise

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

140 combos

The flop comes:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Alice wins the flop pot when Bob folds.

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

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

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

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

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

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

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

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

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

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

 

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

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

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

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

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

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

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

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

We’ll do this in to steps:

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

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

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

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

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

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

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

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

2-barrel%=70%

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

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

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

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

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

In words:

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

We simplify this equation to get:

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

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

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

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

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

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

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

~15% UTG-range

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

194 combos
15%

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

IP flat list after ~15% EP openraise

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

162 combos

The flop comes:

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

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

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

The turn comes

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

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

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

Alice now uses a turn strategy where she:

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

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

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

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

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

The defense equation gives:

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

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

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

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

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

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

Bob calls turn turn bet, and the river comes:

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

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

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

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

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

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

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

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

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

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

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

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

We plug these numbers into the defense equation and get:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Optimal Postflop Play in NLHE 6-max – Part 4

1. Introduction
This is Part 4 of the article series “Optimal Postflop Play in NLHE 6-max” where we’ll study optimal strategies for heads-up postflop play in NLHE 6-max.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Alice’s ~25% CO openrange

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

326 combos
25%

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

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

With Alice in CO, Bob 3-bets {QQ+,AK} for value, so his flatting range is made up of the following 140 combos:

“IP flat list” after 25% CO openraise:

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

140 combos

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

Coordinated flop

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

Dry flop

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

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

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

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

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

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

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

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

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

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

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

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

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

We start by counting Bob’s turn range:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

4.2 Turn strategy after flatting the flop

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

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

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

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

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

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

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

Our total turn defense is then:

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

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

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

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

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

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

The river board texture becomes: 8 5 Q 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Optimal Postflop Play in NLHE 6-max – Part 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2.1. Coordinated flop

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

2.2 Uncoordinated flop flop

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

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

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

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

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

326 combos
25%

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

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

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

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

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

140 combos

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

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

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

35% button openrange:

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

458 combos
35%

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

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

Blind vs Blind flat list

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

362 combos

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

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

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

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

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

4.1 Play on the coordinated flop after flatting on the button

“IP flat list” after 25% CO openraise:

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

140 combos

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

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

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

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

– One pair hands
– Open-ended straight draws

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

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

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

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

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

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

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

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

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

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

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

4.2 Play on dry flop after flatting on the button

“IP flat list” after 25% CO openraise:

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

140 combos

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

5.1 Play on coordinated flop after flatting in the big blind

Blind vs Blind flat list

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

362 combos

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

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

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

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

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

For example:

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

This gives us 39 combos as shown below:

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

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

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

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

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

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

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

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

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

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

5.2 Play on dry flop after flatting in the big blind

Blind vs Blind flat list

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

362 combos

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

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

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

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

We have 93 combos of one pair or better:

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

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

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

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

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

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

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

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

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

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

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

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

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