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The Science of Poker Performance Emotions and Poker Poker is a game of small edges and big swings. Every hand is an investment into your edge. But the tidal waves of variance can easily overwhelm even the most seasoned players. In this article we’ll discuss the root causes of tilt, emotional awareness, and how to improve your mental game. Mental Game Matters! Performance in poker is not static – sometimes we play at a higher level, other times we play at a worse level. Playing your A-game more often and C-game less often is usually the fastest way to improve! Let’s demonstrate this with some simple math. We’ll pretend that we have three states, the A, B and C game, with the following win rates: A-game = 6 BB/100 (Zoned in!) B-game = 3 BB/100 (Solid but autopilot) C-game = -2 BB/100 (Tilted/Suboptimal) These categories are arbitrary, you could split it up using any number of letters. But the winrates assigned to them are extremely generous. Many players lose much more than 2BB/100 when tilted! Scenario 1: 30% of our volume is suboptimal Scenario 2: 10% of our volume is suboptimal Just shifting 20% of your volume from C to A nearly doubles your win rate! This is why mastering your emotions is an imperative skill in poker, and one of the fastest ways to improve quickly. Chapter 1: The Science of Performance So if maximizing performance is such low-hanging fruit, why is it so often neglected? Unlike poker, your mental game is not easy to measure. There are no solvers for tilt. Mental health is easy to neglect, and developing the emotional intelligence required to perform well is as much a skill as developing poker technique. It requires introspection, it requires addressing issues that we subconsciously ignore. Let’s dive into the science of it so we may better understand what drives optimal performance. The Zone Many performance specialists state that “the zone” is a state of flow whereby a person is fully immersed in what they’re doing. The zone is your A-game, so we need to try and define it. You are totally engrossed and in the flow of the action. You don’t experience any outside thoughts or distractions. You’re focused purely on execution and not results. Altered sense of time. Any thoughts of the past or future are purely strategic. How does one get into the zone though? You need the optimal level of stress. You need to be sufficiently challenged without becoming overwhelmed. Your physical needs have to be met. Stress levels Maximizing performance comes down to emotional control. Many players are too invested in their results, which creates tilt and prevents them from separating the process from the outcome. Placing too much emotion on the results of a poker hand is bound to end poorly. Emotional excess impedes logical thoughts and decreases quality of play. Some players try to counter this by turning into “poker zombies” and removing emotion from the equation entirely. This approach is not optimal either. Emotional disconnection makes motivation difficult and decreases quality of |
play. So what is the correct amount of emotion? 🤔 The Yerkes-Dodson Law proposes that there is an optimal level of stress between a zombie and a nervous wreck. The optimal level of stress depends on the complexity of the task. As always, there’s a middle ground. You need to be emotionally invested enough to retain your motivation and drive, yet relaxed enough to avoid tilt and maintain the ability to separate the process from the outcome. Degree of challenge Playing as an underdog (low skill) in a very challenging game causes anxiety. Conversely, a very skilled player against weak competition may lose flow due to overconfidence and boredom. Psychologist Mihaly Csikszentmihalyi argues that flow state of the zone is generated when your skills are sufficiently challenged without being overwhelmed. Experienced poker players may feel boredom when it comes to grinding. It may feel like a chore or a job. That feeling will hinder your performance, as you’re no longer able to stay in flow. It’s important to try and maintain focus to stay in the zone while you’re playing. Even when not in a hand, pay attention to what the other players are doing. The Vicious Cycle Accumulated stresses from work, relationships, results and life in general all contribute to emotional fatigue. These stressors can build up over time. Moreover, we often adopt poor habits as crutches to deal with these stressors. It becomes easy to neglect your sleep, routine, nutrition, and health, which only reinforces your long-term stress. This vicious cycle is especially common among poker players. The most dangerous part of this cycle is that you can become trapped in a state of perpetual fatigue. When you feel like crap for long enough, you’ll forget what being well is supposed to feel like . You’ll become accustomed to always feeling tired and worn out. You will get trapped in your “C-game”, without even realizing it. illustration Limit extraneous stress by optimizing routine, sleep, and nutrition. Establishing a daily routine is paramount! A routine can be as simple as your morning ritual, what time you wake up, work, eat, and go to bed. Sticking to a routine will improve your sleep, your eating habits, your stress, and your time effectiveness. Many studies have shown that a daily routine is strongly correlated with better mental and physical health. Building a Poker Training Routine In addition to maintaining a reasonably healthy lifestyle, you should build a poker training routine. A solid routine involves: Intentional, deliberate practice SMART goals A checklist of activities to complete Dedication over the long-term S.M.A.R.T. goals should be Specific, Measurable, Attainable, Relevant, and Time-bound. A lot of the goals we set for ourselves are V.A.P.I.D. goals. Vague, amorphous, pie in the sky, irrelevant, delayed. Here’s an example: Smart Goal : “I will practice 4bet pots, IP, for two hours every week, until I can regularly achieve a GTOW score of 95% in the trainer.” Vapid Goal : “I will memorize the strategy for every 4bet pot in every spot, I don’t care how long it takes!” If you |
want to achieve something, set SMART goals not VAPID goals. The four stages of competence The four stages suggest that individuals are initially unaware of how little they know, or unconscious of their incompetence. As they recognize their incompetence, they consciously acquire a skill, then consciously use it. Eventually, the skill can be utilized without it being consciously thought through: the individual is said to have then acquired unconscious competence. Unconscious incompetence (Ignorance) You don’t know what you don’t know. Therefore, you can’t recognize your own incompetence. Conscious incompetence (Awareness) You recognize your lack of competence. Conscious competence (Learning) You can use the skill with concentration and effort. Unconscious competence (Mastery) The skill has become second nature. As the philosopher Socrates once said, I know that I know nothing. The first step to mastery is recognizing that there are things you don’t even know you don’t know . If you want to improve at poker, you need to accept that there may be strategic facets to this game you had never even considered. Chapter 2: Identifying and Overcoming Tilt The Seven Types of Tilt (according to Jared Tendler) Tilt means playing suboptimally due to emotional reasons instead of logic. Everyone experiences tilt. The Mental Game of Poker outlines seven types of tilt . Learning to recognize these types of tilt and how to deal with them is crucial to maximizing your performance. (Credit: The Mental Game of Poker by Jared Tendler) Running bad tilt: Every human player experiences bad runs. This is a natural part of the game. But the tilt that comes from running badly is a form of accumulated tilt that develops over time. When variance makes you tilt, you play badly, which makes you tilt more often, repeating the cycle. To quote a section from The Mental Game of Poker: illustration A streak of bad variance doesn’t cause tilt. It reveals the underlying reasons you tilt. If you aren’t sure which of the other types of tilt running bad exposes, your thoughts can help you to implicate them. For example: “This isn’t fair, I’m due to run normal, I never win a flip!” = Injustice Tilt “How is it possible to lose like this again!” = Hate-losing Tilt “I knew that! Dammit, how can I play so bad?!” = Mistake Tilt “I’m too good of a player for this to happen. How can I keep losing to this donkey?” = Entitlement Tilt “I’m gonna 3-bet this idiot; who does he think he is!? I shove.” = Revenge Tilt “I’ll stay here for a week if I have to win!” = Desperation Tilt Injustice tilt Injustice tilt is triggered by coolers and bad beats. But its more than that. This type of tilt takes route when you believe the poker gods have cursed you. Injustice tilt is what happens when you believe that poker is simply not just or fair towards you. You believe your luck is worse than the math suggests is possible . Some players go so far as to accuse poker sites or dealers of rigging the |
deck against them. Injustice tilt often stems from being unable to conceptualize variance . This game is much swingier than you can imagine. Watch this video to learn more . All humans have an implicit bias where we remember losses more strongly than wins. That makes it easier to filter out good times, and pay extra attention to bad times. Another source of injustice tilt stems from not properly tracking results, or not recognizing when we’re getting outplayed. It’s easy to confuse bad luck with bad play if you can’t tell the difference. Identify the strengths and weaknesses of your game. Track your results, and evaluate your play using tools beyond intuition. Learn to recognize good and bad variance, and learn how to use a poker variance calculator to hone your expectations. Hate losing tilt Emotional response to losing triggers tilt even when understanding the variance at play A form of tilt found in highly competitive people who simple hate losing. This type of player hates losing more than they enjoy winning. If losing a pot on an otherwise good session, or having one losing session on an otherwise winning month, sends you over the edge, you have Hate-Losing Tilt. I HATE LOSING MORE THAN I LOVE WINNING. The root cause of this tilt is overcompetitiveness and placing too much of your selfworth onto your results. Remember, money lost at the table is an investment in your edge . If you lost making profitable plays then your losses are temporary so long as you keep playing well. Take a step back. Mistake tilt Mistake tilt is the frustration you feel when reflecting on poor plays you’ve made in the past. Frustration comes from reflecting on past suboptimal plays. Do you find yourself making ‘obvious errors’ or realising the right way to have played the hand immediately afterwards? Mistake tilt is routed in the unconscious faulty assumption that perfect play will always lead to perfect results. It stems from equating money with skill. The way to address mistake tilt is to first accept that perfect play won’t bring you perfect results. While self reflection is useful and necessary, obsessing over mistakes is not healthy. The root cause of mistake tilt is: Unrealistic expectations – Thinking or feeling that every cooler and crappy spot could have been avoided (this is obviously not true). Expecting Perfection – Being too hard on yourself for not playing like a poker god. Perfection cannot be attained all the time, and you need to recognize that. Inability to separate bad luck from bad play . Separate the process from the outcome. Just because you lost the pot doesn’t mean you made a bad play. Entitlement tilt Entitlement tilt occurs when you believe you “deserve” to win. For example, expecting your AA to always hold vs KK when this only occurs 80% of the time. Or thinking that you’re so much stronger than your opponent that you are entitled to the pot. Entitlement tilt is the idea that your skill or cards entitle you to win, when that’s not the |
case. The problem isn’t that you believe you’re better than your opponents. It’s that you believe this entitles you to the pot, leading to overconfidence, arrogance, riskier plays, and ultimately tilt. If you find yourself experiencing entitlement tilt, remember: Being more skilled does not entitle you to the pot. Having a stronger range does not entitle you to the pot. You may not recognize your opponent’s skill or strategy (maybe they’re deviating to exploit you) Address the lies you tell yourself to protect your ego. illustration Revenge tilt Revenge tilt tends to be aimed at specific players. Maybe you feel targeted by some opponent, or feel that they’re trying to disrespect or single you out with their play. So you go out of your way to attack them. This causes you to change your strategy and react sporadically. 👊 REVENGE! 👊 Remember: When you seek out revenge you surrender control! You’re no longer playing your A-game. You’re playing a strategy designed to attack someone. There’s a difference between calm logical exploitative tactics, and angry vengeful tactics. Check your ego. Stay in control. Weaponize patience. Identify specific situations or players who cause you to surrender control of your normal strategy. Resolve the flawed reasons and improve the technical mistakes that led you there in the first place. Desperation tilt Desperation is hard to recognize in the moment. This type of tilt is what causes you to chase losses, overplay your bankroll, and take on way more risk than you should. Desperation tilt is arguably the most dangerous type of tilt as it can destroy your bankroll rather than your session. This is driven by the typical gambler’s fallacy of gambling to break even by taking higher risk lines or gambling outside your roll to get unstuck. It’s exacerbated by an inherent human bias to focus on the negative that assuages our ego. “I’m good but unlucky” causes less damage to our ego than “I’m not good enough”. Desperation tilt is incredibly dangerous . If you regularly experience symptoms of desperation tilt you need to take immediate and non-negotiable steps to mitigate it: Have a sense of urgency – recognize desperation tilt before it becomes full blown. Set a strict stop loss – combine with other prevention methods for maximum effectiveness. Develop an emergency response – Shut down your session before it snowballs. Other types of tilt Below we cover other common types of tilt not addressed in the first Mental game book: Scared Money This is a conscious decision to play defensively out of fear of losing money. Fix this by using proper bankroll management. Winner’s Tilt More common than people realize. When you run extremely well you’re more likely to take on tremendous risk and overplay your hands. Learn to recognize when you’re running hot and check your overconfidence. Dissociation Tilt A very common zombie-like autopilot state that leads to random button clicking and a lack of caution. Distraction Tilt Not giving your full attention to the game. Caused by multitasking (e.g. watching videos or on social media while playing). This prevents |
you from getting in the zone or playing your A-game. Give poker your full concentration! Implicit Bias Humans remember losses more strongly than wins. It’s much easier to recognize bad variance than good variance. This risk aversion is built into our subconscious mental framework. It affects all of our decisions. In poker, this often leads to placing too much mental energy on bad runs, and not recognizing good runs. General Tips for dealing with tilt Circle of Control The Circle of control argues that you should focus your energy on things within your sphere of influence. Stop wasting energy on things out of your control. When it comes to poker: You cannot control the deck, but you can control your strategy. You cannot control your luck, but you can control how you react to luck. You cannot control your emotions, but you can take away the power they have over you. Injecting Logic There is no “one size fits all” solution to control emotions at the poker table. Jared Tendler recommends using the following strategy to overcome tilt: Recognize when you’re straying from logical thoughts Inject logic into your thought process Rationalize a sound strategy Failing this, quit your session. Edges are small and a technically gifted player playing their c-game is unlikely to be winning. Separate the process from the outcome The most important takeaway is that your goal is to separate the process from the outcome. Did you play the hand well? Yes? That’s what matters. Do NOT be results oriented in poker. Weaponizing Your Mental Game In poker you never play your hand. You play the man across from you. – James Bond, Casino Royale A strong mental game provides just as much an edge if not more than a strong technical game. Once you learn to recognize what causes tilt you can also recognize when your opponents are tilting and how to capitalize on it. Make no mistake, they are doing the same to you! Example 1: Your opponent nits up after losing a big pot. This player fears losing more money Great target to run a big bluff against as their fear may cause them to overfold. Aggression beats scared money, just don’t overdo it. Example 2: Your opponent lost a big pot and is playing way too many hands trying to win back their money. This player will continue too wide as they chase losses Against this opponent you should bet for thinner value and avoid bluffing, as they will call too often. Remember your bluffs won’t get enough fold equity, and your value will overperform. Patience! If they’re overly aggressive, tighten up, stop bluffing, and let them maniacly build the pot for you. Don’t call with static bluff-catchers unless you intend to call future streets as well. Conclusion Tilt affects all players. There’s no easy solution. The first steps to improving your mental game is learning to recognize the different types of tilt and addressing the underlying emotional fallacies that cause them. Only once you’ve dealt with your own tilt can you begin to weaponize |
your mental game. Start bringing your A-game more often than your C-game. Below is a list of resources for further reading: The Mental Game of Poker – Jared Tendler The Mental Game of Poker 2 – Jared Tendler Peak Poker Performance In summary Establish a lifestyle and training routine Do not neglect your mental or physical health Aim to get in the zone every time Learn to separate the outcome from the process Identify the emotional root-cause of different types of tilt Improve your ability to conceptualize variance Recognize your implicit biases Weaponize your mental game GTO Wizard the #1 App for Poker players Study any spot imaginable Practice by playing vs. GTO Analyze your hands with 1-click START CRUSHING NOW JOIN DISCORD Author Tombos21 Tom is a long time poker theory enthusiast, GTO Wizard coach and YouTuber, and author of the Daily Dose of GTO. Latest article Navigating Nasty Rivers Out of Position Playing out of position (OOP) is hard. It’s fundamentally disadvantageous to act on each street… Crack the Shell of Nut Draw Strategy Nut draws hold obvious appeal as betting and raising candidates. They benefit from fold equity,… Preflop Raise Sizing: Examining 2 Key Factors It amazes me that, after nearly twenty years of playing no-limit hold ‘em professionally, I… How to Handle Loose-Passive Limpers Solver charts and presolved solutions do not offer much in the way of explicit guidance… Top 5 Mistakes in Spin & Gos As a Spin and Go poker coach, I regularly encounter common mistakes that players make,… An Introduction to Spin & Gos I was playing Tournaments for the first decade of my career. But after my first… Relationships and Poker In my more than 15 years of poker coaching, I’ve helped hundreds of clients improve… Do Multiple Sizes Matter? Should I work out the other tools or just stick to the crowbar? Solver results… The Science of Learning Applied to GTO Wizard The way we were taught to study in schools was flawed and has negatively impacted… Check-Raising a Single Pair In this article, I will address a question raised by Twitter/X user ’Matt Riley’, who… The Turn Probe Bet Imagine yourself defending as the BB caller vs BTN preflop raiser in a 100bb cash… When To Encourage Multiway Pots in PKOs I was recently shown this fascinating PKO hand from the GTO Wizard PKO library. Somebody… Protect Equity and Prosper When we think about the best hands to bet in a given situation, we tend… Do Solvers Have Targets? Targeting has long been a staple of exploitative poker thinking. When you’re value betting against… Disciplining Big Blind in Limped Pots When the BTN open-limps off a short stack, the postflop play against the BB should… I’d Rather Be Drawing Which is the better hand on a K♥8♦5♦ flop: K♠T♣ or 7♥6♥? OK, I’ll admit… The Curious Case of Open-Limping Buttons Before the advent of solvers, open-limping was considered the province of passive, gambley recreational players… Cleaning Out Sticky IP Callers A “sticky” player on your left, one who calls your preflop raises |
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BB In poker, overcalling refers to calling a bet or raise after another player has also… Understanding Which Mistakes Cost You the Most Money I am fortunate enough to have worked for many years with the mental game coach… Overbetting The Flop in Cash Games Although most commonly analyzed on turns and rivers, flop overbetting can be a devastating weapon… Using New Skills at the Poker Table How many times has it happened to you that you make a mistake only to… Flatting Ace-X Hands Shortstacked Playing out of position is tricky, but it’s an essential part of tournament play. With… Playing Limped Pots as the BB in MTTs Success as the BB in limped pots begins before the flop. The BB’s equilibrium strategy… The Absurd Game Theory of Chopped Boards A “chopped board” in poker is one where the community cards make up the best… Playing Limped Pots as the SB in MTTs MTT confrontations between the small blind and big blind force no-limit players into uncomfortable situations,… The Art of Learning Poker In his book The Art of Learning, chess and Push Hands champion Josh Waitzkin describes… Is Donk Betting for Donkeys? “Donk betting”, or betting from out of position into the player who was the aggressor… Delayed C-Betting This is a companion piece to Probe Betting. It stands on its own, but understanding… Probe Betting As far as rules of thumb go in poker, “check to the raiser” is a… ICM and Blind Battles: The Big Blind This is a companion piece to this article on the SB’s strategy for blind vs… ICM and Blind Battles: The Small Blind ICM incentivizes more conservative play, which means that at stages of the tournament where risk… The Initial Bettor’s Advantage The mathematics of poker holds a little-known secret: the first bettor enjoys better bluffing odds… Mastering the Chiplead: The Scalpel and the Sledgehammer When you have a lot of chips at a final table or in another situation… Flop Heuristics: IP C-Betting in Cash Games Our never-ending quest for money and fame leads us to the flop. We open the… How ICM Impacts Restealing From The Blinds When I first started working with preflop solves, one of the features I found most… The ICM Benefits of Late Registration in Poker Tournaments Why do so many professional poker players register late for tournaments when they clearly would… Flop Heuristics: OOP C-Betting in MTTs Playing against an in-position cold-caller is dramatically different–and dramatically more difficult–than playing against a caller… Counterintuitive Calls Solver outputs such as those provided by GTO Wizard are models of no-limit hold ‘em,… How To Negotiate Final Table Deals Most people’s first introduction to the Independent Chip Model (ICM) is not from studying bubble… How does ICM impact PKO strategy? If you have read The Theory of Progressive Knockout Tournaments you already know why we… Understanding Blockers in Poker Often when looking through a solver’s meticulously crafted grid of deception, one of the following… Flop Heuristics: IP Cold-Caller in MTTs A Cold-call refers to the act of calling a preflop |
raise when you don’t close… Flop Heuristics: IP C-Betting in MTTs In the world of poker, the flop is a crucial stage of the game that… Flop Heuristics for Defending the Blinds in MTTs The field of Game Theoretically Optimal poker is extensive, and thus, to simplify our strategy,… Multiway strategy in Progressive Knockout Tournaments Progressive Knockout (PKO) tournaments are much more complex than traditional MTTs. The fact that players… How To Calculate Raises In Poker One of the most frequently asked questions in poker is how to calculate the size… Principles of River Play The river offers the least complicated strategy of any betting street. With no further cards… Exploitative Dynamics GTO solutions are designed to be unexploitable, performing reasonably well no matter how your opponents… How ICM Impacts Postflop Strategy Most players understand how the Independent Chip Model (ICM) in poker impacts their strategy preflop,… 10 Tips for Multiway Pots in Poker While lots of work has been placed into research and explanation of heads-up pot strategy,… What is the Bubble Factor in poker tournaments? The Independent Chip Model (ICM) shows us the real money implications of tournament chip decisions.… The Theory of Progressive Knockout Tournaments In poker, a PKO (Progressive Knockout) tournament is a type of tournament where a portion… How To Analyze Turn Textures In Poker Master the art of turn analysis in poker. Understand how the turn card changes the… Principles of Turn Strategy Learn the principles of turn strategy in poker and discover what makes the turn unique.… Poker Strategies: Tournaments vs. Cash Games Tournament poker is not as different from cash game poker as people sometimes assume. Most… How Solvers Work A Game Theory Optimal solver is an algorithm that calculates the best possible poker strategy.… The Most Powerful Play in Poker The central strategic implication of stack depth is the question of how much equity a… When does ICM become significant in MTTs? The Independent Chip Model (ICM) was invented in 1987 by M. Malmuth and is one… How to Study GTO Solutions Tools like GTO Wizard and other solvers should not be used to memorize specific strategies.… Principles of GTO Game theory, or GTO (for Game Theoretically Optimal), is best understood not as a rigid… Interpreting Equity Distributions GTO Wizard’s range vs range equity distribution graphs are an excellent tool for visualizing which… What is Leverage in poker? Leverage refers to the risk of future betting. When you call a bet on the… Reasons for value betting in poker You’re probably accustomed to thinking of bets as either value or bluff. In this taxonomy,… The Science of Poker Performance Poker is a game of small edges and big swings. Every hand is an investment… What is Valor in poker? Can a poker hand be courageous? How does your ability to value bet future streets… Pot Geometry “Pot Geometry” refers to betting an equal fraction of the pot on each street, such… A Beginner’s Guide to Poker Combinatorics Combinatorics is a fancy term for evaluating the number of possible “combinations” (combos) of any… Stack-to-pot |
ratio Stack-to-pot ratio, or SPR, is a way of measuring how deep the effective stack is… Equity Realization In poker jargon, equity expresses how much of the pot a hand will win, on… Variance and Bankroll Management Variance and Bankroll Management. Vital Soft Skills for Every Poker Player. How to Become a GTO Wizard Welcome to the wonderful world of game theory optimal poker! This article will serve as… The Three Laws of Indifference Indifference in poker is one of the most misunderstood concepts. The word “indifference” means that… Poker subsets and abstractions An “abstraction” is a way to simplify the game of poker. This game is so… Mathematical Misconceptions in Poker Pot Odds and MDF are foundational mathematical formulas in poker. How to solve toy games Poker is a complicated game. It’s impossible to calculate GTO strategies on the fly. By… The Value of Fold Equity – Experiment The term “fold equity” is ambiguous in the poker community. Firstly, it’s an EV equation,… MDF & Alpha MDF & Alpha. Minimum Defence Frequency and Alpha are metrics on poker that determine how… What are Pot Odds in poker? Pot odds are a fundamental calculation in poker. In the simplest terms, pot odds tell… Does your range affect your strategy? Advanced players will often tell you that your range affects your strategy. Visualizing implied odds Implied odds are the value you expect to gain over and above your raw equity.… ICM Basics As a tournament player, you may have heard about the concept of “ICM”, but what… What is Equity in Poker? The term “Equity” is ubiquitous throughout poker theory. Equity refers to your chances of winning… Range Morphology “Morphology” is the study of form or structure. You’ve probably heard terms thrown around such… What is Expected Value in Poker? Expected value (EV) is the most fundamental metric in poker. Every decision you make is… Why doesn’t my solution match GTO Wizard? So, you’ve got your own solutions to some spot, compared it to GTO Wizard, and… Understanding Nash Distance Understanding Nash Distance. One of the most common questions when studying with solvers is “Why… Combinatorics Poker Puzzle Today we have a special puzzle for you that will test your intuition as a… The 10 Most Important Concepts for New Poker Players The 10 Most Important Concepts for New Poker Players. Fundamentals are one of the most… What is GTO in Poker? The meaning of GTO in poker is Game Theory Optimal. Game theory is a study… What does GTO aim to achieve? So what exactly is a GTO strategy? What makes it “unexploitable”? What does it try… |
What is Valor in poker? CASH , MTT , Theory Can a poker hand be courageous? How does your ability to value bet future streets impact your… 02/12/2022 7 min. / 46 sec. |
ICM and Blind Battles: The Small Blind ICM incentivizes more conservative play, which means that at stages of the tournament where risk premiums are highest, action is more likely to fold around to the blinds Blind bet To bet pre-flop without looking at your cards, or to bet postflop before the next street has been dealt. Typically, not allowed in online poker. . Consequently, it is especially important to learn how to navigate blind versus blind confrontations under ICM pressure. Even when one player covers the other, both are incentivized to realize their equity without provoking a big confrontation. There are two ways to do this: Limp Limp To call the big blind preflop, before anyone has made a raise. or check to keep the pot small. Move all-in, forcing your opponent to fold or risk losing a big pot. The higher the risk premiums, the stronger the incentives for both players to pursue one of these two strategies. The all-in option becomes viable only with relatively shallow stacks. The following chart shows the SB’s strategy when the action folds to them at various stages of the tournament as well as the BB’s response to a limp. In this simulation, all players have precisely 40bb, so neither covers the other. The most dramatic trend here is that, as the risk premium increases, the SB does less open raising , arriving at a nearly pure limp-or-fold strategy when under the extreme pressure of the bubble or a nine-handed final table. Facing the limp, the BB’s raising frequency mostly decreases as risk premium mounts, but the change is not nearly so dramatic, and they retain a robust raising strategy at all stages of the tournament. Growing the pot is bad for both players, but it is especially bad for the out of position SB, who will have more trouble realizing equity and controlling the pot size after the flop. Even after facing a raise from such a wide range, SB responds primarily by calling rather than raising. Growing the pot is bad for both players, but it is especially bad for the out of position SB. An extreme example of this effect is on display in this final table scenario , where the SB is second in chips and the BB first by a wide margin. Only if we give the SB a massively covering stack at the final table are they incentivized to raise aggressively. Which Hands Does the SB Raise? Despite increased incentives for limping, raising remains an integral part of the SB’s strategy under all but the most intense ICM scenarios, so it’s worth learning a few heuristics for which hands to raise. Here’s the strategy with 40bb stacks and 25% of the field remaining . SB’s raising range is narrow and polar , with the value region consisting of only the biggest pairs, suited Aces, and suited Broadways . Only the strongest offsuit Broadway Broadway A ten through ace straight. “Broadway cards” can also be used to describe any 10 , J , Q , K , or A card. |
hands raise, with smaller offsuit broadways and medium suited hands limping heavily to preserve their equity and avoid playing a big pot. The bluff portions of the raising range are a diverse mix of lower suited hands and higher offsuit hands . It may seem arbitrary at first, but there’s a method to the madness: the BB responds to this raise with a robust calling range and some polarized three-betting. This mix of calls and three-bets prevents the SB from following a simple heuristic of raising exclusively small suited hands or higher offsuit hands. When the BB calls, SB prefers to have the suited hands, as they realize equity better from out of position. When the BB raises, however, SB regrets having suited hands, as they give up more EV by folding. The resulting equilibrium is that the SB bluffs with a mix of offsuit hands that do reasonably well when called but don’t mind folding to three-bets and suited hands that have the least to lose by folding to three-bets . This is why the SB treats T3s as a better opening candidate than T6s: the latter performs slightly better when called but also gives up more when it folds to a three-bet. The BB’s mix of calls and three-bets prevents the SB from following a simple heuristic. BB Response to SB Open, 40bb stacks, 25% of Field Remaining The same patterns persist as the bubble approaches, but they are harder to discern as the SB raises so rarely. Shallower Stacks With shallower effective stacks, the SB has a new tool for avoiding big confrontations: open shoving. It’s a bit counter-intuitive, but this is actually a low-risk play because the BB’s own risk premium incentivizes them to fold quite often, even with hands that are well ahead of the SB’s shoving range. As the risk premium increases, the SB’s strategy shifts away from calls and smaller open raises and toward shoves and folds. The following chart shows how SB’s strategy changes as the tournament progresses, 20bb deep. This pattern persists even when the SB is covered. In this final table scenario , the BB has 30bb to SB’s 19, but SB shoves aggressively nevertheless, as calling remains a significant risk for the BB. The composition of these ranges also matches that of the symmetric stack simulations: the SB shoves a blocker-heavy range of medium-strength hands too strong to raise-fold but not so strong they want to induce action. Their 3bb raising range is more polar , with big pairs and suited Aces balanced by small suited hands and larger off-suit hands. SB’s limping range is also somewhat polar, with stronger hands limping to call or raise and weaker ones limping to fold. Note that AA is a poor candidate for limping as a trap because it blocks so much of the BB’s raising range. When the SB covers the BB , they mostly shove or fold, with just a narrow, polar raising range. They fold much more often than when covering with deeper stacks, as the BB will find more hands |
worth risking their tournament life when shallow. Conclusion By treating these as the default plays and using the heuristics in this article to determine whether to make an exception for a particular hand, you can turn ICM pressure into a tool for taking chips from your opponent rather than a danger to yourself. Blind vs. blind confrontations are among the trickiest spots in hold ‘em, as they incentivize both players to contest the pot with wide ranges. This conflicts directly with the pressures of ICM, which incentivize players to avoid big confrontations. As the SB, your primary tools for navigating these spots are open limping with deeper stacks and open shoving with shallower stacks. GTO Wizard the #1 App for Poker players Study any spot imaginable Practice by playing vs. GTO Analyze your hands with 1-click START CRUSHING NOW JOIN DISCORD Author Andrew Brokos Andrew Brokos has been a professional poker player, coach, and author for over 15 years. He co-hosts the Thinking Poker Podcast and is the author of the Play Optimal Poker books, among others. Latest article Preflop Raise Sizing: Examining 2 Key Factors It amazes me that, after nearly twenty years of playing no-limit hold ‘em professionally, I… How to Handle Loose-Passive Limpers Solver charts and presolved solutions do not offer much in the way of explicit guidance… Top 5 Mistakes in Spin & Gos As a Spin and Go poker coach, I regularly encounter common mistakes that players make,… An Introduction to Spin & Gos I was playing Tournaments for the first decade of my career. But after my first… Relationships and Poker In my more than 15 years of poker coaching, I’ve helped hundreds of clients improve… Do Multiple Sizes Matter? Should I work out the other tools or just stick to the crowbar? Solver results… The Science of Learning Applied to GTO Wizard The way we were taught to study in schools was flawed and has negatively impacted… Check-Raising a Single Pair In this article, I will address a question raised by Twitter/X user ’Matt Riley’, who… The Turn Probe Bet Imagine yourself defending as the BB caller vs BTN preflop raiser in a 100bb cash… When To Encourage Multiway Pots in PKOs I was recently shown this fascinating PKO hand from the GTO Wizard PKO library. Somebody… Protect Equity and Prosper When we think about the best hands to bet in a given situation, we tend… Do Solvers Have Targets? Targeting has long been a staple of exploitative poker thinking. When you’re value betting against… Disciplining Big Blind in Limped Pots When the BTN open-limps off a short stack, the postflop play against the BB should… I’d Rather Be Drawing Which is the better hand on a K♥8♦5♦ flop: K♠T♣ or 7♥6♥? OK, I’ll admit… The Curious Case of Open-Limping Buttons Before the advent of solvers, open-limping was considered the province of passive, gambley recreational players… Cleaning Out Sticky IP Callers A “sticky” player on your left, one who calls your preflop raises too often and… Defending Against Tiny 3-Bets Tournament poker is about precision. |
Cash games, where stacks typically run into the hundreds of… Heads up! Exploiting SB’s Preflop Mistakes Heads up (HU) tournament play presents unique challenges to the average player. Many tournament players,… C-Betting As the OOP Preflop Raiser Position is an extremely valuable advantage in poker. Most players would describe the comfort zone… Overchoice: Making Sense of Multiple Sizings Solver solutions with multiple bet sizes often involve a lot of mixing between those sizes,… River Refinements: When Your Opponent Won’t Check-Raise One of the biggest strategic differences between playing in versus out of position involves betting… Turn Barreling in 3-Bet Pots There are two articles on the GTO Wizard blog that make good prerequisites to this… The Right Way To Think About Protection Suppose Paul the Protector is in a 200bb cash game and opens A♠A♣ in the… The Importance of Board Coverage The first public version of the GTO Wizard announced AI upgrade already has some useful… How To Adjust When Villain Has No Bluffs “They always have it!” is a common, semi-joking expression in poker to describe the tendency… Adjusting on Later Streets After Exploiting Flop The key to exploiting opponents profitably is knowing when to stop. I’m not referring to… C-Betting IP vs Passive Poker Players A preflop raiser enjoys a significant range advantage over a BB caller, enabling them to… C-Betting Against ICM Preflop Ranges You raise from early position (EP) in a multi-table tournament and the BB calls. Both… The Art of Bluff Catching Bluff catching—calling a bet with a modest hand which you expect to win only if… Exploiting Excessive C-Betting by IP On some flops, it is simply not possible for the preflop raiser to continuation bet… Is Limping Pimping? In a past Discord post, one of our members noted that an upside of implementing… Exploiting Excessive C-Betting by OOP A preflop raiser who continuation bets too often is always a potential target for exploitation,… Preflop Button Mastery in PKOs Playing from the Button presents many unique opportunities in any tournament format, especially in PKOs.… The Worst Turn Card Greetings from GTO land. In today’s article, we will use the Turn Reports feature to… Navigating Range Disadvantage as the 3-Bettor Picture this: We’re playing a NL500 6-max cash game, and we’ve won a few pots,… Mastering Three-Bet Pots In Position in MTTs Playing in position against a three-better is a lesson in playing to your advantages. As… Mastering Three-Bet Pots Out of Position in MTTs Three-bet pots are scary. Playing out of position is scary. Put them together, and you’ve… The Greatest Final Table in Triton History Today, we’ll examine one of the most epic final table battles I’ve ever witnessed! Our… Mastering Thin Value Bets in Checked-Down Pots Valuing your hand appropriately—determining whether it’s strong enough to value bet or so weak you… Maximizing Value on Monotone Flops Monotone flops tend to provoke great anxiety among poker players. It’s easy to imagine that… How to Defend Against Turn Donk Bets Responding effectively to donk bets on the turn is tricky;, there’s no getting around that.… |
How and Why You Should Use Turn Donk Bets Donk betting–taking the betting lead away from the previous street’s aggressor–on the flop is rarely… How Stack Sizes Change Your Range In this series, we have looked at factors that can influence ranges in otherwise identical… How to Leverage GTO Wizard’s EV Comparison Tool With the advent of solvers, some troubling new turns of phrase have entered the poker… Should You Ever Cold Call a 3-Bet? When there’s been a raise and re-raise before the action is on you, you will… When ICM Breaks Down If you have read my previous GTO Wizard articles and my books, you’ll know I… When Is It Correct To Fold AA Preflop? One of the criticisms of ICM is that it is boring and robotic. In my… OOP C-betting vs Loose Cold-callers In a previous GTO Wizard article, I asserted that “Playing against an in-position cold-caller is… Short-Stacked Play in MTTs Playing a short stack well is an essential tournament skill. It is extremely rare to… When To Ladder in Tournaments In my last two articles, I wrote about the effect field size and payout structure… How Payout Structures Impact ICM Last time we discussed the ICM considerations in game selection, we looked at field size,… C-Betting IP in 3-Bet Pots When you three-bet before the flop and the original raiser calls from out of position,… C-Betting OOP in 3-Bet Pots Continuation betting from out of position can be a dicey proposition in single-raised pots. Cold… Dynamic Sizing: A GTO Breakthrough Dynamic Sizing is a revolutionary new poker algorithm that automatically simplifies your strategy with the… Dynamic Sizing Benchmarks Dynamic Sizing is a revolutionary new poker algorithm that automatically simplifies your strategy with the… Understanding The Impact Of Field Size On ICM In Poker When people talk about game selection in multi-table tournaments (MTTs), they invariably talk about the… Defending vs BB Check-Raise on Paired Flops This is a companion piece to Attacking Paired Flops from the BB; you will get… Mastering PKO Final Tables Final tables can be the most exciting and rewarding phase of a tournament, and being… Attacking Paired Flops From the BB Paired flops present unique challenges to both a preflop raiser and a BB caller. The… KK vs QQ vs JJ at WSOP Main Event With only fourteen players remaining in the 2023 WSOP Main Event and blinds of 400K/800K/800K,… Responding to BB Squeezes You open raise, someone calls, and the BB squeezes. How should you construct your strategy?… Crushing a Top HUNL Poker Bot In 2022, Philippe Beardsell and Marc-Antoine Provost, a team of Canadian programmers from Quebec, developed… GTO Wizard AI Explained We are excited to introduce GTO Wizard AI, formerly known as Ruse, the world’s best… GTO Wizard AI Benchmarks GTO Wizard has combined the power of artificial intelligence with traditional solving methods to bring… Table Management in PKOs Perhaps the most important strategic consideration in Progressive Knockout Tournaments (PKOs) is making sure you… Overcalling From the BB In poker, overcalling refers to calling a bet or raise after |
another player has also… Understanding Which Mistakes Cost You the Most Money I am fortunate enough to have worked for many years with the mental game coach… Overbetting The Flop in Cash Games Although most commonly analyzed on turns and rivers, flop overbetting can be a devastating weapon… Using New Skills at the Poker Table How many times has it happened to you that you make a mistake only to… Flatting Ace-X Hands Shortstacked Playing out of position is tricky, but it’s an essential part of tournament play. With… Playing Limped Pots as the BB in MTTs Success as the BB in limped pots begins before the flop. The BB’s equilibrium strategy… The Absurd Game Theory of Chopped Boards A “chopped board” in poker is one where the community cards make up the best… Playing Limped Pots as the SB in MTTs MTT confrontations between the small blind and big blind force no-limit players into uncomfortable situations,… The Art of Learning Poker In his book The Art of Learning, chess and Push Hands champion Josh Waitzkin describes… Is Donk Betting for Donkeys? “Donk betting”, or betting from out of position into the player who was the aggressor… Delayed C-Betting This is a companion piece to Probe Betting. It stands on its own, but understanding… Probe Betting As far as rules of thumb go in poker, “check to the raiser” is a… ICM and Blind Battles: The Big Blind This is a companion piece to this article on the SB’s strategy for blind vs… ICM and Blind Battles: The Small Blind ICM incentivizes more conservative play, which means that at stages of the tournament where risk… The Initial Bettor’s Advantage The mathematics of poker holds a little-known secret: the first bettor enjoys better bluffing odds… Mastering the Chiplead: The Scalpel and the Sledgehammer When you have a lot of chips at a final table or in another situation… Flop Heuristics: IP C-Betting in Cash Games Our never-ending quest for money and fame leads us to the flop. We open the… How ICM Impacts Restealing From The Blinds When I first started working with preflop solves, one of the features I found most… The ICM Benefits of Late Registration in Poker Tournaments Why do so many professional poker players register late for tournaments when they clearly would… Flop Heuristics: OOP C-Betting in MTTs Playing against an in-position cold-caller is dramatically different–and dramatically more difficult–than playing against a caller… Counterintuitive Calls Solver outputs such as those provided by GTO Wizard are models of no-limit hold ‘em,… How To Negotiate Final Table Deals Most people’s first introduction to the Independent Chip Model (ICM) is not from studying bubble… How does ICM impact PKO strategy? If you have read The Theory of Progressive Knockout Tournaments you already know why we… Understanding Blockers in Poker Often when looking through a solver’s meticulously crafted grid of deception, one of the following… Flop Heuristics: IP Cold-Caller in MTTs A Cold-call refers to the act of calling a preflop raise when you don’t close… Flop Heuristics: IP C-Betting in MTTs In |
the world of poker, the flop is a crucial stage of the game that… Flop Heuristics for Defending the Blinds in MTTs The field of Game Theoretically Optimal poker is extensive, and thus, to simplify our strategy,… Multiway strategy in Progressive Knockout Tournaments Progressive Knockout (PKO) tournaments are much more complex than traditional MTTs. The fact that players… How To Calculate Raises In Poker One of the most frequently asked questions in poker is how to calculate the size… Principles of River Play The river offers the least complicated strategy of any betting street. With no further cards… Exploitative Dynamics GTO solutions are designed to be unexploitable, performing reasonably well no matter how your opponents… How ICM Impacts Postflop Strategy Most players understand how the Independent Chip Model (ICM) in poker impacts their strategy preflop,… 10 Tips for Multiway Pots in Poker While lots of work has been placed into research and explanation of heads-up pot strategy,… What is the Bubble Factor in poker tournaments? The Independent Chip Model (ICM) shows us the real money implications of tournament chip decisions.… The Theory of Progressive Knockout Tournaments In poker, a PKO (Progressive Knockout) tournament is a type of tournament where a portion… How To Analyze Turn Textures In Poker Master the art of turn analysis in poker. Understand how the turn card changes the… Principles of Turn Strategy Learn the principles of turn strategy in poker and discover what makes the turn unique.… Poker Strategies: Tournaments vs. Cash Games Tournament poker is not as different from cash game poker as people sometimes assume. Most… How Solvers Work A Game Theory Optimal solver is an algorithm that calculates the best possible poker strategy.… The Most Powerful Play in Poker The central strategic implication of stack depth is the question of how much equity a… When does ICM become significant in MTTs? The Independent Chip Model (ICM) was invented in 1987 by M. Malmuth and is one… How to Study GTO Solutions Tools like GTO Wizard and other solvers should not be used to memorize specific strategies.… Principles of GTO Game theory, or GTO (for Game Theoretically Optimal), is best understood not as a rigid… Interpreting Equity Distributions GTO Wizard’s range vs range equity distribution graphs are an excellent tool for visualizing which… What is Leverage in poker? Leverage refers to the risk of future betting. When you call a bet on the… Reasons for value betting in poker You’re probably accustomed to thinking of bets as either value or bluff. In this taxonomy,… The Science of Poker Performance Poker is a game of small edges and big swings. Every hand is an investment… What is Valor in poker? Can a poker hand be courageous? How does your ability to value bet future streets… Pot Geometry “Pot Geometry” refers to betting an equal fraction of the pot on each street, such… A Beginner’s Guide to Poker Combinatorics Combinatorics is a fancy term for evaluating the number of possible “combinations” (combos) of any… Stack-to-pot ratio Stack-to-pot ratio, or SPR, is a way of measuring how deep |
the effective stack is… Equity Realization In poker jargon, equity expresses how much of the pot a hand will win, on… Variance and Bankroll Management Variance and Bankroll Management. Vital Soft Skills for Every Poker Player. How to Become a GTO Wizard Welcome to the wonderful world of game theory optimal poker! This article will serve as… The Three Laws of Indifference Indifference in poker is one of the most misunderstood concepts. The word “indifference” means that… Poker subsets and abstractions An “abstraction” is a way to simplify the game of poker. This game is so… Mathematical Misconceptions in Poker Pot Odds and MDF are foundational mathematical formulas in poker. How to solve toy games Poker is a complicated game. It’s impossible to calculate GTO strategies on the fly. By… The Value of Fold Equity – Experiment The term “fold equity” is ambiguous in the poker community. Firstly, it’s an EV equation,… MDF & Alpha MDF & Alpha. Minimum Defence Frequency and Alpha are metrics on poker that determine how… What are Pot Odds in poker? Pot odds are a fundamental calculation in poker. In the simplest terms, pot odds tell… Does your range affect your strategy? Advanced players will often tell you that your range affects your strategy. Visualizing implied odds Implied odds are the value you expect to gain over and above your raw equity.… ICM Basics As a tournament player, you may have heard about the concept of “ICM”, but what… What is Equity in Poker? The term “Equity” is ubiquitous throughout poker theory. Equity refers to your chances of winning… Range Morphology “Morphology” is the study of form or structure. You’ve probably heard terms thrown around such… What is Expected Value in Poker? Expected value (EV) is the most fundamental metric in poker. Every decision you make is… Why doesn’t my solution match GTO Wizard? So, you’ve got your own solutions to some spot, compared it to GTO Wizard, and… Understanding Nash Distance Understanding Nash Distance. One of the most common questions when studying with solvers is “Why… Combinatorics Poker Puzzle Today we have a special puzzle for you that will test your intuition as a… The 10 Most Important Concepts for New Poker Players The 10 Most Important Concepts for New Poker Players. Fundamentals are one of the most… What is GTO in Poker? The meaning of GTO in poker is Game Theory Optimal. Game theory is a study… What does GTO aim to achieve? So what exactly is a GTO strategy? What makes it “unexploitable”? What does it try… |
Understanding Nash Distance One of the most common questions when studying with solvers is “Why does the solver take this action when that action is higher EV” ? For example, in this scenario below we can see checking is higher EV than betting, yet the solution wants to bet this hand: Example of solver noise The understand why this happens, we first need to understand a fundamental principle of game theory: The law of mixed actions In perfect equilibrium, mixed actions should always have the same expected value. That is to say, when a hand mixes between two or more actions, those actions should have the same EV . Think about it; why would you intentionally choose a worse strategy? A truly perfect strategy would never “sacrifice EV for balance”, so to speak. This is a hard law of Nash Equilibrium. In the example above we see the solver mixing its strategy with A7o between checks, bet 27%, 73%, and 127%. Yet these actions are not the same EV. So why do we see this happening in our solutions? Why is checking with A7o considered an “inaccuracy” when it’s the highest EV action? Solver Noise GTO Solutions are imperfect. In practice, GTO solutions are not solved to perfect accuracy, they are solved to a certain exploitability threshold. The less exploitable the entire solution is, the higher its accuracy. We define the accuracy of a solution with a metric called “ Nash Distance ” or dEV. For reference, GTO Wizard typically solves to about 0.2%-0.3% of the pot. How exploitable is this? Well imagine we have a BTN vs BB scenario, the pot is 5.5BB. That means the best possible strategy can exploit this solution for at most 0.3% of 5.5, or 0.017BB per hand. This is well beyond the level of human play. This is what we call “solver noise”. Hands don’t always take the highest EV action. Solved to perfect accuracy this noise would disappear. All mixed actions would have the same EV. In situations where you take a line that’s literally never played, you may notice massive EV discrepancies. That’s because the solver stops calculating dominated (0%) lines early on during the solve process to boost efficiency. This is normal, although the strategies and EVs in those lines are less accurate. Exploitability Ok, so does that mean the highest EV action is always the best move? Not necessarily. Against this exact strategy , it happens to be the best move. However, if you always check A7o then theoretically BB might be able to adjust their strategy such that checking becomes lower EV! Remember, the solver mixes to remain unexploitable. What would happen if we solved the example above to perfect accuracy? Would checking still be the highest EV move? One of two things will happen to A7: Checking will converge to a lower EV and never be used; or Checking will converge to the same EV as the other betting options and may continue to be used at some frequency. There’s no way to know with certainty without solving to |
perfect accuracy. Generally speaking, low-frequency actions will disappear and become lower EV, whereas actions taken at a reasonable frequency will remain part of the strategy. This is why actions taken at less than 3.5% frequency are marked as an “inaccuracy”. Let’s take a look at another example: Example Example 2: A7o mixing calls despite calling being higher EV Here we see A7o mixing between a call and a fold; however, calling appears to be significantly higher EV than folding. Calling is about 1.7BB higher EV than folding. So why does it mix folds? Well, we need to put this into perspective . The pot after you call will be 200.05 BB. So a 1.8 BB error is only about 0.9% of the pot. In reality, this is much closer than it looks. A 1% margin of error looks much bigger when the pot gets bigger. The problem with calling everything is that you would become exploitable. Imagine calling all of these borderline bluff-catchers. Now all of a sudden you’re overcalling and can be exploited by a value-heavy opponent. Why not just solve to perfect accuracy? Perfectly accurate solutions are simply not feasible at the scale of mass production. The problem is that solvers converge much more slowly as it approaches equilibrium. It takes about as much time to go from completely unsolved to 0.5% dEV as it does to go from 0.5% to 0.25% dEV. Doubling the accuracy doubles the time it takes to solve. And there are diminishing returns . A solution that’s accurate to 0.3%dEV is almost identical to a solution solved for 0.15% dEV, and both would still have solver noise. The alternative is to create very simple trees that are easier to solve. This creates its own problems, as oversimplifying your game tree leads to artificial distortion caused by the solver exploiting the limitations of that tree. So at the end of the day, it doesn’t make a lot of sense to obsess over incredibly high accuracy and microscopic EV. The solution is well beyond human levels of exploitability anyway. Here’s an example of one of our solves. As you can see, progress happens quickly at the start, then gets slower and slower as it approaches equilibrium. This one solved down to 0.3% of the pot. What to take away When you see the solver mixing between actions, instead of looking for the highest EV action, you should instead look at the highest frequency actions. This is because any EV discrepancy you see is due to noise in the solution. Treat mixed actions as if they were the same EV. Any discrepancy you see can be thought of as the margin of error, (all hands are roughly +- that EV). The point of GTO is to find the highest EV strategy that cannot be exploited. The marginal errors you see are found in every solver solution. Your goal is to abstract higher-level strategy and develop GTO reasoning, not to memorize frequencies. GTO Wizard the #1 App for Poker players Study any spot imaginable Practice by playing vs. GTO |
Analyze your hands with 1-click START CRUSHING NOW JOIN DISCORD Author Tombos21 Tom is a long time poker theory enthusiast, GTO Wizard coach and YouTuber, and author of the Daily Dose of GTO. Latest article Preflop Raise Sizing: Examining 2 Key Factors It amazes me that, after nearly twenty years of playing no-limit hold ‘em professionally, I… How to Handle Loose-Passive Limpers Solver charts and presolved solutions do not offer much in the way of explicit guidance… Top 5 Mistakes in Spin & Gos As a Spin and Go poker coach, I regularly encounter common mistakes that players make,… An Introduction to Spin & Gos I was playing Tournaments for the first decade of my career. But after my first… Relationships and Poker In my more than 15 years of poker coaching, I’ve helped hundreds of clients improve… Do Multiple Sizes Matter? Should I work out the other tools or just stick to the crowbar? Solver results… The Science of Learning Applied to GTO Wizard The way we were taught to study in schools was flawed and has negatively impacted… Check-Raising a Single Pair In this article, I will address a question raised by Twitter/X user ’Matt Riley’, who… The Turn Probe Bet Imagine yourself defending as the BB caller vs BTN preflop raiser in a 100bb cash… When To Encourage Multiway Pots in PKOs I was recently shown this fascinating PKO hand from the GTO Wizard PKO library. Somebody… Protect Equity and Prosper When we think about the best hands to bet in a given situation, we tend… Do Solvers Have Targets? Targeting has long been a staple of exploitative poker thinking. When you’re value betting against… Disciplining Big Blind in Limped Pots When the BTN open-limps off a short stack, the postflop play against the BB should… I’d Rather Be Drawing Which is the better hand on a K♥8♦5♦ flop: K♠T♣ or 7♥6♥? OK, I’ll admit… The Curious Case of Open-Limping Buttons Before the advent of solvers, open-limping was considered the province of passive, gambley recreational players… Cleaning Out Sticky IP Callers A “sticky” player on your left, one who calls your preflop raises too often and… Defending Against Tiny 3-Bets Tournament poker is about precision. Cash games, where stacks typically run into the hundreds of… Heads up! Exploiting SB’s Preflop Mistakes Heads up (HU) tournament play presents unique challenges to the average player. Many tournament players,… C-Betting As the OOP Preflop Raiser Position is an extremely valuable advantage in poker. Most players would describe the comfort zone… Overchoice: Making Sense of Multiple Sizings Solver solutions with multiple bet sizes often involve a lot of mixing between those sizes,… River Refinements: When Your Opponent Won’t Check-Raise One of the biggest strategic differences between playing in versus out of position involves betting… Turn Barreling in 3-Bet Pots There are two articles on the GTO Wizard blog that make good prerequisites to this… The Right Way To Think About Protection Suppose Paul the Protector is in a 200bb cash game and opens A♠A♣ in the… The Importance of Board Coverage The first |
public version of the GTO Wizard announced AI upgrade already has some useful… How To Adjust When Villain Has No Bluffs “They always have it!” is a common, semi-joking expression in poker to describe the tendency… Adjusting on Later Streets After Exploiting Flop The key to exploiting opponents profitably is knowing when to stop. I’m not referring to… C-Betting IP vs Passive Poker Players A preflop raiser enjoys a significant range advantage over a BB caller, enabling them to… C-Betting Against ICM Preflop Ranges You raise from early position (EP) in a multi-table tournament and the BB calls. Both… The Art of Bluff Catching Bluff catching—calling a bet with a modest hand which you expect to win only if… Exploiting Excessive C-Betting by IP On some flops, it is simply not possible for the preflop raiser to continuation bet… Is Limping Pimping? In a past Discord post, one of our members noted that an upside of implementing… Exploiting Excessive C-Betting by OOP A preflop raiser who continuation bets too often is always a potential target for exploitation,… Preflop Button Mastery in PKOs Playing from the Button presents many unique opportunities in any tournament format, especially in PKOs.… The Worst Turn Card Greetings from GTO land. In today’s article, we will use the Turn Reports feature to… Navigating Range Disadvantage as the 3-Bettor Picture this: We’re playing a NL500 6-max cash game, and we’ve won a few pots,… Mastering Three-Bet Pots In Position in MTTs Playing in position against a three-better is a lesson in playing to your advantages. As… Mastering Three-Bet Pots Out of Position in MTTs Three-bet pots are scary. Playing out of position is scary. Put them together, and you’ve… The Greatest Final Table in Triton History Today, we’ll examine one of the most epic final table battles I’ve ever witnessed! Our… Mastering Thin Value Bets in Checked-Down Pots Valuing your hand appropriately—determining whether it’s strong enough to value bet or so weak you… Maximizing Value on Monotone Flops Monotone flops tend to provoke great anxiety among poker players. It’s easy to imagine that… How to Defend Against Turn Donk Bets Responding effectively to donk bets on the turn is tricky;, there’s no getting around that.… How and Why You Should Use Turn Donk Bets Donk betting–taking the betting lead away from the previous street’s aggressor–on the flop is rarely… How Stack Sizes Change Your Range In this series, we have looked at factors that can influence ranges in otherwise identical… How to Leverage GTO Wizard’s EV Comparison Tool With the advent of solvers, some troubling new turns of phrase have entered the poker… Should You Ever Cold Call a 3-Bet? When there’s been a raise and re-raise before the action is on you, you will… When ICM Breaks Down If you have read my previous GTO Wizard articles and my books, you’ll know I… When Is It Correct To Fold AA Preflop? One of the criticisms of ICM is that it is boring and robotic. In my… OOP C-betting vs Loose Cold-callers In a previous GTO Wizard article, I |
asserted that “Playing against an in-position cold-caller is… Short-Stacked Play in MTTs Playing a short stack well is an essential tournament skill. It is extremely rare to… When To Ladder in Tournaments In my last two articles, I wrote about the effect field size and payout structure… How Payout Structures Impact ICM Last time we discussed the ICM considerations in game selection, we looked at field size,… C-Betting IP in 3-Bet Pots When you three-bet before the flop and the original raiser calls from out of position,… C-Betting OOP in 3-Bet Pots Continuation betting from out of position can be a dicey proposition in single-raised pots. Cold… Dynamic Sizing: A GTO Breakthrough Dynamic Sizing is a revolutionary new poker algorithm that automatically simplifies your strategy with the… Dynamic Sizing Benchmarks Dynamic Sizing is a revolutionary new poker algorithm that automatically simplifies your strategy with the… Understanding The Impact Of Field Size On ICM In Poker When people talk about game selection in multi-table tournaments (MTTs), they invariably talk about the… Defending vs BB Check-Raise on Paired Flops This is a companion piece to Attacking Paired Flops from the BB; you will get… Mastering PKO Final Tables Final tables can be the most exciting and rewarding phase of a tournament, and being… Attacking Paired Flops From the BB Paired flops present unique challenges to both a preflop raiser and a BB caller. The… KK vs QQ vs JJ at WSOP Main Event With only fourteen players remaining in the 2023 WSOP Main Event and blinds of 400K/800K/800K,… Responding to BB Squeezes You open raise, someone calls, and the BB squeezes. How should you construct your strategy?… Crushing a Top HUNL Poker Bot In 2022, Philippe Beardsell and Marc-Antoine Provost, a team of Canadian programmers from Quebec, developed… GTO Wizard AI Explained We are excited to introduce GTO Wizard AI, formerly known as Ruse, the world’s best… GTO Wizard AI Benchmarks GTO Wizard has combined the power of artificial intelligence with traditional solving methods to bring… Table Management in PKOs Perhaps the most important strategic consideration in Progressive Knockout Tournaments (PKOs) is making sure you… Overcalling From the BB In poker, overcalling refers to calling a bet or raise after another player has also… Understanding Which Mistakes Cost You the Most Money I am fortunate enough to have worked for many years with the mental game coach… Overbetting The Flop in Cash Games Although most commonly analyzed on turns and rivers, flop overbetting can be a devastating weapon… Using New Skills at the Poker Table How many times has it happened to you that you make a mistake only to… Flatting Ace-X Hands Shortstacked Playing out of position is tricky, but it’s an essential part of tournament play. With… Playing Limped Pots as the BB in MTTs Success as the BB in limped pots begins before the flop. The BB’s equilibrium strategy… The Absurd Game Theory of Chopped Boards A “chopped board” in poker is one where the community cards make up the best… Playing Limped Pots as the SB in MTTs MTT confrontations |
between the small blind and big blind force no-limit players into uncomfortable situations,… The Art of Learning Poker In his book The Art of Learning, chess and Push Hands champion Josh Waitzkin describes… Is Donk Betting for Donkeys? “Donk betting”, or betting from out of position into the player who was the aggressor… Delayed C-Betting This is a companion piece to Probe Betting. It stands on its own, but understanding… Probe Betting As far as rules of thumb go in poker, “check to the raiser” is a… ICM and Blind Battles: The Big Blind This is a companion piece to this article on the SB’s strategy for blind vs… ICM and Blind Battles: The Small Blind ICM incentivizes more conservative play, which means that at stages of the tournament where risk… The Initial Bettor’s Advantage The mathematics of poker holds a little-known secret: the first bettor enjoys better bluffing odds… Mastering the Chiplead: The Scalpel and the Sledgehammer When you have a lot of chips at a final table or in another situation… Flop Heuristics: IP C-Betting in Cash Games Our never-ending quest for money and fame leads us to the flop. We open the… How ICM Impacts Restealing From The Blinds When I first started working with preflop solves, one of the features I found most… The ICM Benefits of Late Registration in Poker Tournaments Why do so many professional poker players register late for tournaments when they clearly would… Flop Heuristics: OOP C-Betting in MTTs Playing against an in-position cold-caller is dramatically different–and dramatically more difficult–than playing against a caller… Counterintuitive Calls Solver outputs such as those provided by GTO Wizard are models of no-limit hold ‘em,… How To Negotiate Final Table Deals Most people’s first introduction to the Independent Chip Model (ICM) is not from studying bubble… How does ICM impact PKO strategy? If you have read The Theory of Progressive Knockout Tournaments you already know why we… Understanding Blockers in Poker Often when looking through a solver’s meticulously crafted grid of deception, one of the following… Flop Heuristics: IP Cold-Caller in MTTs A Cold-call refers to the act of calling a preflop raise when you don’t close… Flop Heuristics: IP C-Betting in MTTs In the world of poker, the flop is a crucial stage of the game that… Flop Heuristics for Defending the Blinds in MTTs The field of Game Theoretically Optimal poker is extensive, and thus, to simplify our strategy,… Multiway strategy in Progressive Knockout Tournaments Progressive Knockout (PKO) tournaments are much more complex than traditional MTTs. The fact that players… How To Calculate Raises In Poker One of the most frequently asked questions in poker is how to calculate the size… Principles of River Play The river offers the least complicated strategy of any betting street. With no further cards… Exploitative Dynamics GTO solutions are designed to be unexploitable, performing reasonably well no matter how your opponents… How ICM Impacts Postflop Strategy Most players understand how the Independent Chip Model (ICM) in poker impacts their strategy preflop,… 10 Tips for Multiway Pots in Poker While lots |
of work has been placed into research and explanation of heads-up pot strategy,… What is the Bubble Factor in poker tournaments? The Independent Chip Model (ICM) shows us the real money implications of tournament chip decisions.… The Theory of Progressive Knockout Tournaments In poker, a PKO (Progressive Knockout) tournament is a type of tournament where a portion… How To Analyze Turn Textures In Poker Master the art of turn analysis in poker. Understand how the turn card changes the… Principles of Turn Strategy Learn the principles of turn strategy in poker and discover what makes the turn unique.… Poker Strategies: Tournaments vs. Cash Games Tournament poker is not as different from cash game poker as people sometimes assume. Most… How Solvers Work A Game Theory Optimal solver is an algorithm that calculates the best possible poker strategy.… The Most Powerful Play in Poker The central strategic implication of stack depth is the question of how much equity a… When does ICM become significant in MTTs? The Independent Chip Model (ICM) was invented in 1987 by M. Malmuth and is one… How to Study GTO Solutions Tools like GTO Wizard and other solvers should not be used to memorize specific strategies.… Principles of GTO Game theory, or GTO (for Game Theoretically Optimal), is best understood not as a rigid… Interpreting Equity Distributions GTO Wizard’s range vs range equity distribution graphs are an excellent tool for visualizing which… What is Leverage in poker? Leverage refers to the risk of future betting. When you call a bet on the… Reasons for value betting in poker You’re probably accustomed to thinking of bets as either value or bluff. In this taxonomy,… The Science of Poker Performance Poker is a game of small edges and big swings. Every hand is an investment… What is Valor in poker? Can a poker hand be courageous? How does your ability to value bet future streets… Pot Geometry “Pot Geometry” refers to betting an equal fraction of the pot on each street, such… A Beginner’s Guide to Poker Combinatorics Combinatorics is a fancy term for evaluating the number of possible “combinations” (combos) of any… Stack-to-pot ratio Stack-to-pot ratio, or SPR, is a way of measuring how deep the effective stack is… Equity Realization In poker jargon, equity expresses how much of the pot a hand will win, on… Variance and Bankroll Management Variance and Bankroll Management. Vital Soft Skills for Every Poker Player. How to Become a GTO Wizard Welcome to the wonderful world of game theory optimal poker! This article will serve as… The Three Laws of Indifference Indifference in poker is one of the most misunderstood concepts. The word “indifference” means that… Poker subsets and abstractions An “abstraction” is a way to simplify the game of poker. This game is so… Mathematical Misconceptions in Poker Pot Odds and MDF are foundational mathematical formulas in poker. How to solve toy games Poker is a complicated game. It’s impossible to calculate GTO strategies on the fly. By… The Value of Fold Equity – Experiment The term “fold equity” is ambiguous in |
the poker community. Firstly, it’s an EV equation,… MDF & Alpha MDF & Alpha. Minimum Defence Frequency and Alpha are metrics on poker that determine how… What are Pot Odds in poker? Pot odds are a fundamental calculation in poker. In the simplest terms, pot odds tell… Does your range affect your strategy? Advanced players will often tell you that your range affects your strategy. Visualizing implied odds Implied odds are the value you expect to gain over and above your raw equity.… ICM Basics As a tournament player, you may have heard about the concept of “ICM”, but what… What is Equity in Poker? The term “Equity” is ubiquitous throughout poker theory. Equity refers to your chances of winning… Range Morphology “Morphology” is the study of form or structure. You’ve probably heard terms thrown around such… What is Expected Value in Poker? Expected value (EV) is the most fundamental metric in poker. Every decision you make is… Why doesn’t my solution match GTO Wizard? So, you’ve got your own solutions to some spot, compared it to GTO Wizard, and… Understanding Nash Distance Understanding Nash Distance. One of the most common questions when studying with solvers is “Why… Combinatorics Poker Puzzle Today we have a special puzzle for you that will test your intuition as a… The 10 Most Important Concepts for New Poker Players The 10 Most Important Concepts for New Poker Players. Fundamentals are one of the most… What is GTO in Poker? The meaning of GTO in poker is Game Theory Optimal. Game theory is a study… What does GTO aim to achieve? So what exactly is a GTO strategy? What makes it “unexploitable”? What does it try… |
How To Analyze Turn Textures In Poker In community card games like hold ‘em, the face-up cards on the board are an important source of information about the hands you are likely to see at showdown. The board’s texture determines which pairs are likely and whether straights, flushes, or full houses will be possible . Each time a new card is revealed, you learn new information about these variables and how likely you are to hold the winning hand 🏅 The flop reveals the most information of any street. We can categorize flops based on their high card, the number of suits they contain, whether they are paired, how many straights or straight draws they make possible, and more. These are all ways of describing how the board cards interact with each player’s range of likely hole cards. When it comes to analyzing the new board card on the turn, the question we are most interested in is: What has changed? Which new hands have become possible that were not possible on the flop? Even blank turn cards, those unlikely to interact with any player’s range significantly, convey important information. Hands that were already strong on the flop are stronger by virtue of fading some threats on the turn, while draws and unpaired hands are weaker with only one remaining opportunity to improve. Range Advantage Range advantage is a way of talking about how different players’ ranges interact with one another. There are two main kinds of range advantage: Equity advantage – How is equity distributed between the ranges? Which player has the most, and how large is the difference between them? Equity advantage tends to correlate with betting frequency. The stronger a player’s range, the more incentive they have to bet . Nuts advantage – How likely is each player to have the strongest possible hands? Who is most likely to have them, and how large is the difference? Nuts advantage tends to correlate with bet size . The more likely a player is, relative to their opponents, to have nutty hands, the more incentive they have to grow the pot with large bets. A player’s equity on the flop is the average of their equity on all possible turns. Once a specific turn card is revealed, their equity will increase or decrease based on how much that card helps them relative to their opponent, and their betting frequency will increase or decrease along with it. Nuts advantage is more binary. One player has the nuts advantage after the flop action, and they will either retain or lose that advantage depending on what the turn brings. We tend to see big bets from this player on blank turns that do not significantly change the board texture by making new nutty hands possible and smaller bets (sometimes accompanied by more frequent betting from a player who has gained the nut advantage) on turns that change the board texture. Prior Action When analyzing turn texture, it’s important to keep in mind the pre-flop and flop action, which determines which player enjoyed the range |
advantage(s) prior to the turn. A player who comes into the turn with a big advantage may retain that advantage even on a slightly unfavorable turn that makes some new nutty hands possible for their opponent. Aggression correlates with nut advantage. A player who has been betting or raising is more likely to have nutty hands than one who has been checking and calling. When a player raises pre-flop, continuation bets the flop, and gets called, they will have the nuts advantage going into the turn. On the average turn card, they will bet big or check; texture changing cards are, by definition, exceptions. The preflop aggressor should hold fewer nutted hands if they checked the flop. Depending on the magnitude of their nut advantage on the flop, they may maintain that advantage on the turn, but it will be diminished. Example We can see these principles in action in a simple example from an NL500 100BB cash game. UTG opens, BB calls, UTG bets 33% pot on a K ♠ 8 ♠ 7 ♥ flop and BB calls again. At this point, before the turn card is revealed, UTG has the nuts advantage but not the equity advantage. BB had a lot of incentive to raise their best hands and to fold their worst ones, so their call indicates a relatively strong but condensed range, whereas UTG’s bet is more consistent with very strong and relatively weak hands. Let’s remind ourselves what this will mean for UTG’s turn strategy: On blank turns, we expect a modest amount of betting for a large size. On turns that negate UTG’s nuts advantage, we expect smaller bets and perhaps a lower betting frequency (depending on how the turn changes the equity distribution). On turns that maintain UTG’s nuts advantage and also improve their equity, we expect more frequent betting, often for a smaller size. Now here’s the report summarizing UTG’s strategy on all possible turns , if BB checks to them: UTG vs BB Strategy by turn card on K ♠ 8 ♠ 7 ♥ after bet 33% call A few patterns appear: UTG consistently uses smaller bets on spade turns , which negate their nuts advantage by making many flushes possible for BB. The only exception is the 7s, which also pairs the board. UTG has more quads and full houses in their range and so can do some overbetting on this card. Cards that complete straights function similarly , though the effect is less dramatic because BB will have fewer straight combos. Fives are the exception because BB would have folded 96o pre-flop and mostly raised 96s on the flop, making straights possible but quite unlikely for them on these turns. UTG has the highest betting frequency on non-spade A, K, and Q turns . These cards do not make straights or flushes possible and improve UTG’s equity more than BB’s, as UTG is more likely to hold unpaired Broadway cards after the flop action. This correlates with a smaller bet size, especially when an A turns. These cards help UTG’s equity |
so much they actually run out of good bluffing candidates to support an overbetting range, and they are incentivized to use a smaller size in order to make thinner value bets with their many Ax and Kx hands. The blankest turns are non-spade wheel cards, which neither complete draws nor improve the unpaired hands in UTG’s range. On these cards, UTG mostly checks or overbets . The lower board pairs, 7 and 8, are not nearly as good as K for UTG as BB is more likely to turn trips . Consequently, UTG rarely bets these turns, though as with the 7 ♠ , their full house and quads advantage still facilitates a small overbetting range. BB almost exclusively checks the turn after check-calling the flop. The only exceptions are the cards we’ve already identified as introducing significant new nut hands into BB’s previously capped range: BB’s strategy by turn card after check-calling 33% UTG bet on K ♠ 8 ♠ 7 ♥ BB has some incentive to bet these cards precisely because they are bad for UTG, who can not be expected to bet as often when checked to. We have been using Aggregate reports a lot in this article. Here is a guide on how to use them. When the Flop Checks Through Now think about what happens when UTG checks the flop, which they do with about 25% of their range. It’s important to note that while UTG’s checking range is somewhat weaker than their betting range, it is not at all weak. In fact, UTG is incentivized to bet many of their weakest hands on the flop, leaving their checking range rather condensed . Their checking range is concentrated around pairs: UTG flop strategy by made hand category, filtered for checking range UTG’s pre-flop range is much stronger than BB’s, so despite a slight weakening effect, they retain both the equity advantage and some degree of nut advantage after checking. That means that in general, we should expect a low betting frequency from BB on the turn and mostly small bet sizes. Here’s the report after the flop checks through, this time showing the strategy from BB’s perspective: BB Probe strategy by turn card on K ♠ 8 ♠ 7 ♥ There is indeed a low frequency of betting. The best cards for BB to bet are, once again, those that change the board texture. In fact, these cards are even better for BB after UTG checks behind the flop! This is because action on the flop concentrates ranges around cards that coordinate with the board. UTG had a lot of incentive to bet their flush and straight draws on the flop, so they are less likely to make flushes and straights when those cards turn. Action on the flop concentrates ranges around cards that coordinate with the board Here’s a breakdown of the draws in UTG’s flop checking range: UTG flop strategy by draw hand category, filtered for checking range Declining to bet the flop is an indication that UTG is unlikely to hold a draw. BB’s |
flop check does not convey this same information, as they check their entire range. Thus, when a draw completes after the flop checks through, BB acquires a nut advantage and can do some overbetting. This is true even though UTG checks some draws on the flop–they do not check enough to make overbetting undesirable on these turns. If BB checks a second time, we see mostly small bets from UTG, a consequence of their less nutty range. The main exceptions are the Broadway cards that most improve them, introducing some new nutty hands into their previously capped range. UTG delayed c-bet strategy by turn card on K ♠ 8 ♠ 7 ♥ Conclusion Range disparities are most pronounced in UTG vs BB confrontations, and this is an especially good flop for UTG. These patterns are not always as clear and pronounced as they appear here, but the underlying principles should inform your evaluation of the turn card in any situation. Start with a clear understanding of how the equities and nutty hands are distributed prior to the turn card, then consider how much the turn changes those distributions . GTO Wizard the #1 App for Poker players Study any spot imaginable Practice by playing vs. GTO Analyze your hands with 1-click START CRUSHING NOW JOIN DISCORD Author Andrew Brokos Andrew Brokos has been a professional poker player, coach, and author for over 15 years. He co-hosts the Thinking Poker Podcast and is the author of the Play Optimal Poker books, among others. Latest article Preflop Raise Sizing: Examining 2 Key Factors It amazes me that, after nearly twenty years of playing no-limit hold ‘em professionally, I… How to Handle Loose-Passive Limpers Solver charts and presolved solutions do not offer much in the way of explicit guidance… Top 5 Mistakes in Spin & Gos As a Spin and Go poker coach, I regularly encounter common mistakes that players make,… An Introduction to Spin & Gos I was playing Tournaments for the first decade of my career. But after my first… Relationships and Poker In my more than 15 years of poker coaching, I’ve helped hundreds of clients improve… Do Multiple Sizes Matter? Should I work out the other tools or just stick to the crowbar? Solver results… The Science of Learning Applied to GTO Wizard The way we were taught to study in schools was flawed and has negatively impacted… Check-Raising a Single Pair In this article, I will address a question raised by Twitter/X user ’Matt Riley’, who… The Turn Probe Bet Imagine yourself defending as the BB caller vs BTN preflop raiser in a 100bb cash… When To Encourage Multiway Pots in PKOs I was recently shown this fascinating PKO hand from the GTO Wizard PKO library. Somebody… Protect Equity and Prosper When we think about the best hands to bet in a given situation, we tend… Do Solvers Have Targets? Targeting has long been a staple of exploitative poker thinking. When you’re value betting against… Disciplining Big Blind in Limped Pots When the BTN open-limps off a short stack, the |
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C-Betting IP in 3-Bet Pots Having invested a considerable number of chips in your three-bet and faded several worst-case scenarios, you can expect to enjoy both an equity advantage and a positional advantage post-flop. As a result, you will get to play aggressively and over-realize your equity on most flops. When you three-bet before the flop and the original raiser calls from out of position, you have already enjoyed some good luck. True, much of your three-betting range would have preferred to take it down immediately. However, getting to play heads up after the flop in position is a nice consolation prize . For all but your strongest hands, it is a better outcome than facing a four-bet from the original raiser or, horror of horrors, a cold four-bet. In this article, we will look at how best to do that: which bet sizes to use, when you should not bet at all, and more. As mentioned above, positions do not make a big difference. The trends are the same for most scenarios, regardless of position . When the BTN three-bets the CO, they will play most flops similarly to how they would if they had three-bet against UTG. This is because the three-better modulates their preflop raising strategy based on the strength of the original raise , and the original raiser similarly modulates their calling strategy. When the open raise comes from the CO, that’s a much wider range than when it comes from UTG. The BTN compensates by three-betting a wider range than they would against UTG, and CO continues to that three-bet with a wider range. So, the relative differential between their ranges remains similar. An early position three-bet takes on additional risk that a late position three-bet does not. When the three-bet comes from earlier position, that makes a bigger difference, because an early position three-bet takes on additional risk that a late position three-bet does not. UTG1 must three-bet a tighter range against an UTG open than BTN would because they have more players behind them who would wake up with four-betting hands. Again, UTG compensates for this by calling a bit tighter against UTG1 than they would against BTN, so the difference is not huge. Stack sizes, however, make a big difference. The three-better mostly bets smaller and more often as stack sizes get shallower. In addition to their positional advantage, the three-better will enjoy a nuts advantage on most flops. This is because they are more likely to hold big pocket pairs and big broadway cards like AK that can flop strong pairs . Their three-bet mostly serves to get the SPR for the flop below 4, at which point they can treat these strong pairs as the nuts on most boards. Despite the low SPR, they want to take advantage of leverage by spreading their bets across three streets. This both sets them up to access stacks on the river when they wish to do so (whether for value or as a bluff) and puts tremendous pressure on their opponent’s medium-strength hands, which must |
account for the risk of that river shove when responding to bets on earlier streets. With shallow stacks, that means betting tiny–often 25% pot or less–on the flop so they have room for small bets on the turn and river as well. With deeper stacks, it requires slightly larger flop bets–50% starts to become more common–to get all the money in with similarly sized turn and river bets. Blank Flops The three-better arrives at the flop with a significantly stronger range than the original raiser, so flops that miss both players effectively cement that preflop equity advantage . We will take, for our example, a 552 r flop. BTN will leverage their big pair advantage to bet a polar range on the turn and river and so benefits from growing the pot to set up a river shove. Even with 100bb starting stacks , a BTN three-better doesn’t have to worry about running into many trips or better from an UTG opener and so can bet with abandon: Notice the heavy use of the 50% and 75% sizes. On most runouts, BTN will leverage their big pair advantage to bet a polar range on the turn and river and so benefits from growing the pot to set up a river shove with these effectively nutty hands. A CO opener will have more 5x and 22 in their calling range, so a BTN three-better proceeds with more caution against them : Against CO, BTN checks more often and moves away from the 75% and 50% bets toward the 25% pot option. This remains a very favorable scenario for them, however. The equities are split roughly 50/50, but BTN realizes 10.78bb in EV to CO’s 7.72bb. This is leverage and a polarized range in action: With 40bb stacks , the presence of trips or better in CO’s range is no longer a concern. That doesn’t mean they aren’t there, just that BTN can more comfortably treat their pocket pairs as the nuts at this lower stack depth . They continuation bet their entire range, just as in the 100bb BTN v UTG scenario, but now prefer a smaller size because of the shallower stacks: The hands that bet 50% pot (the larger size, in this scenario) are not BTN’s strongest. In other words, this is not a nuts-or-air bet , as larger bet sizes sometimes are. With such shallow stacks, relatively invulnerable nut hands like KK and AA want to bet smaller to induce additional action, as they will be strong enough to continue betting for value on later streets and can get stacks in easily by the river despite the small flop bet. The hands betting 50% pot are strong but vulnerable hands: JJ , TT , even A2 s, and AKo fall in this category. These hands benefit from denying equity but still expect to be ahead when the flop is called. However, on many turns, they will not be able to continue value betting, so they prioritize putting in more money on the flop . Ace-High Flops Ranges in three-bet pots |
tend to be heavy on Ax hands, making strategy for Ace-high flops tricky. If you don’t have an Ace, there’s a good chance your opponent does. Additionally, Ace-high flops are among the most static because no higher card can come. As a result, the three-better tends to do a lot of checking or tiny betting , depending on the extent of their range advantage. Ace-high flops are quite favorable for them; it’s just that a big chunk of their range–hands like KK, QQ, and weaker Ax–don’t have much to gain from betting. Ace-high flops are among the most static. For example, here is the 100bb strategy for a BTN v UTG confrontation on A93 r: BTN has 55% of the equity yet checks quite often and sticks to a small size when they do bet. Notice that these checks are mostly pocket pairs and weaker Ax , the hands that can’t deny much equity and don’t want to play large pots. For strong hands like AQ , AK , and especially AA , it’s hard to shovel money in because of the blocking effect on UTG’s call-down range. In fact, the hand BTN bets most consistently is 99, which is their only extremely strong hand that unblocks UTG’s Ax region . In a 100bb UTG1 v UTG confrontation, the three-better’s equity advantage exceeds 60%. They don’t check much in this scenario, but the check is mostly replaced by a 10% pot bet , so they aren’t exactly shoveling money in, either: With shallower stacks, the small bets can be even smaller. With 40bb starting stacks , the BTN three-better bets 10% pot with more than half their range into an UTG caller: Two Broadway Flops Big cards on the flop tend to make hand values more static, and this is especially true in three-bet pots, where big cards predominate in both players’ ranges. So, continuation betting strategy on these boards looks similar to Ace-high flops, with many small bets and checks from medium pairs. Here is BTN v CO on KQ5r with 40bb effective stacks : Deeper stacks accentuate the three-better’s nuts advantage, enabling them to bet their modest hands and start sizing up with a more polar range. Here is BTN v CO with 100bb stacks: Deeper stacks accentuate the three-better’s nuts advantage. Depending on the runout, some medium-strength hands that bet flop can turn into bluffs on later streets. The deeper stacks enable BTN to leverage their nuts advantage for a very high bluffing frequency on flop and turn, allowing for bluffs even with unideal candidates like small pocket pairs. The same scenario with 40bb effective stacks sees more checking from these hands: Checking medium-strength hands is also safer because shallower stacks limit the pressure CO can apply with polarized bets on the turn and river. Medium Connected Flops Medium connected flops like 876tt have a lot in common with low paired boards like 552r. Equities are split roughly 50/50, but the prevalence of unpaired overcards in the three better’s range and the risk of nutty hands in the caller’s |
range compels a lot of checking back. Here is the 100bb BTN v CO strategy on 876tt, which is quite similar to the BTN v UTG strategy: There’s a lot of mixing, but some heuristics stand out. Straights are pure bets for the same reason so many medium hands check: this is an easy flop for the caller to check-raise . The other frequently bet hands are draws and strong-but-vulnerable pairs, especially when they are also draws ( TT , 99 ). With 40bb stacks , BTN doesn’t have many straights, but this doesn’t really change their betting frequency. They compensate by betting more overpairs , which can more comfortably stack off with a low SPR: This highlights some of the reasons why many players consider these coordinated flops a nightmare (as the three-better, anyway): With shallower stacks, they see too many monsters under the bed and are reluctant to play their overpairs strongly. With deeper stacks, they construct their three-betting ranges poorly . When stacks are too deep to treat overpairs as the nuts on many flops, you need to three-bet a more diverse range so that you can flop nuts on boards like 876tt. Conclusion The stack depth should also determine the size of these bets. When possessing a nuts advantage, the three-better structures the betting so they have the option to put stacks in across all three streets. With shallow stacks, this can mean continuation bets as small as 10% of the pot to leave room for betting the turn and river. With deeper stacks, they may prefer larger (but still rarely more than half-pot) bets, so they will not need to overbet later streets to get stacks in. Strong-but-vulnerable hands are the best candidates for these larger bets, as they won’t be able to keep value betting on many runouts, and so should prioritize putting more money in sooner. An awareness of the stack-to-pot ratio and the composition of your own range is essential to playing well in three-bet pots. The biggest difference between the three-better’s range and the caller’s is generally the prominence of big pairs and the biggest unpaired hands. When circumstances permit playing these hands strongly, the three-better should continuation bet more frequently. GTO Wizard the #1 App for Poker players Study any spot imaginable Practice by playing vs. GTO Analyze your hands with 1-click START CRUSHING NOW JOIN DISCORD Author Andrew Brokos Andrew Brokos has been a professional poker player, coach, and author for over 15 years. He co-hosts the Thinking Poker Podcast and is the author of the Play Optimal Poker books, among others. Latest article Preflop Raise Sizing: Examining 2 Key Factors It amazes me that, after nearly twenty years of playing no-limit hold ‘em professionally, I… How to Handle Loose-Passive Limpers Solver charts and presolved solutions do not offer much in the way of explicit guidance… Top 5 Mistakes in Spin & Gos As a Spin and Go poker coach, I regularly encounter common mistakes that players make,… An Introduction to Spin & Gos I was playing Tournaments for the first |
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The Initial Bettor’s Advantage The mathematics of poker holds a little-known secret: the first bettor enjoys better bluffing odds than someone who raises by the same amount . In this article, we’ll explore the calculations behind this phenomenon and discuss how this plays into theoretically optimal strategies. Let’s start with an example to illustrate this principle. Blind vs. Blind, Single-raised pot: The pot is 6 bb SB bets 3 bb (50% pot-sized bet) BB raises to 9 bb (50% pot-sized raise ) Let’s calculate the bluffing odds for both players. This calculation is also known as Alpha, or 1 – MDF. You can read more about that here , but to recap: Breakeven Bluff = Risk / (Risk + Reward) Where: Risk is the amount risked with a pure bluff. Or how much you would lose if your bluff fails. Reward is the amount gained if your opponent folds, compared to if you had given up. SB needs 3 / (3 + 6) = 33% fold equity to profitably bluff the initial bet. BB needs 9 / (3 + 6 + 9) = 50% fold equity to profitably bluff-raise the bet. Both players risked the same fraction of the pot. Therefore, both the bet and the raise lay the same pot odds. Despite that, the initial bet required less fold equity for a bluff to be profitable ! We can also consider the inverse calculation, MDF. The Minimum Defense Frequency describes how often players must defend to prevent the opponent from profitably bluffing. SB facing the raise is only required to defend, at most, half their range. BB facing the initial bet is required to defend two-thirds of their range. However, the pot odds requirements have not changed. In both cases, either player should continue with hands that can claim more than 25% pot share. In summary, facing a raise you don’t need to defend as wide compared to facing an initial bet of the same size, while at the same time, the equity required to profitably call does not change. Visualizing This Effect The following table shows how much extra fold equity you need when raising, compared to if your raise was the initial bet. For example, a 50% bet needs 33.3% fold equity. But a 50% raise over a 30% bet requires 45.83% fold equity, resulting in a required surplus 16% fold equity ! Put another way, the half-pot raise over the 30% bet requires your opponent to fold 16% more often before your bluffs break even. We can see the effect is strongest at the minimum raise size, where you’d normally get the best price on a bluff. This often ties into block-betting theory – but more on that later. The reason behind this effect is actually fairly intuitive. The initial pot is a prize that incentivizes bluffing and calling. As a series of raises occurs, the size of the initial pot shrinks relative to the “reward” part of our calculation. To make it more intuitive, imagine you’re facing a 100bb 5bet shove preflop. The initial 1.5bb unopened pot |
now has little bearing on the risk/reward ratio of this shove. The same concept applies here. As a series of bets and raises fly in, the initial pot which had originally incentivized betting becomes less relevant. It should be noted that just because you get a slight mathematical advantage when bluffing, that doesn’t give you a free license to start leading every spot! There are still many spots where you should check range. Practical Example Let’s return to our previous example . SB opens 3, BB calls. Flop’s J ♥ 6 ♦ 3 ♣ . SB c-bets half-pot, BB raises half-pot, SB calls. Turn is the K ♠ . BB barrels 75%, SB calls. River is the Q ♣ . The Queen of clubs is a decent card for SB here. It turns many of their weak one-pair hands into two pair. For that reason, SB develops a donking strategy Donk bet Any bet made by a player before the previous street aggressor has had an opportunity to act. A donk bet can only be made by a person out of position relative to the last aggressor on the previous street. – leading into the aggressor. The SB called a flop raise and then a big turn bet, so their range is mostly condensed towards made hands with few draws at this point. The BB, however, arrives at this unfortunate river with too many bluffs. This wasn’t a mistake in their strategy, this is just the nature of Hold’em. Some runouts favor you; others do not. We can see by observing the equity distribution that no player really has the nut advantage, but SB has an equity advantage by virtue of not having trash in their range. The SB therefore donk leads 35% pot representing a range of two pair+ for value, along with some marginal top pair. The following chart displays the donk bet construction: There are several reasons for this bet. One is to push their middling equity advantage. Another is that, if they had range-checked, BB could check back more to exploit them. But one of the deeper reasons, buried within the mathematics of the game, is that the SB’s lead grants them favorable bluffing odds compared to check-raising. Facing this donk bet, the BB has a 48% pot shove behind, which they utilize about ⅓ of the time. However, despite their raise being bigger than the donk, they are required to bluff less often. When the BB does shove the last half-pot, SB only needs to defend 50% of their range, as opposed to the 67% that would be required if we had instead checked and faced a half-pot bet. Summary When comparing the mathematical consequences of a 50% bet against a 50% raise we see that, unlike pot odds, MDF and Alpha do not scale with pot%. This leads to four key takeaways: The initial bettor gets a better price on a bluff than a raise of the same size. You should defend less often facing a raise than an initial bet of the same size. Facing |
a raise, the defender has a slight advantage, as they are not obligated to defend as wide (compared to a bet of the same size), but the required pot odds to call haven’t changed. This often ties into block-betting theory, especially on the river. GTO Wizard the #1 App for Poker players Study any spot imaginable Practice by playing vs. GTO Analyze your hands with 1-click START CRUSHING NOW JOIN DISCORD Author Tombos21 Tom is a long time poker theory enthusiast, GTO Wizard coach and YouTuber, and author of the Daily Dose of GTO. Latest article Preflop Raise Sizing: Examining 2 Key Factors It amazes me that, after nearly twenty years of playing no-limit hold ‘em professionally, I… How to Handle Loose-Passive Limpers Solver charts and presolved solutions do not offer much in the way of explicit guidance… Top 5 Mistakes in Spin & Gos As a Spin and Go poker coach, I regularly encounter common mistakes that players make,… An Introduction to Spin & Gos I was playing Tournaments for the first decade of my career. But after my first… Relationships and Poker In my more than 15 years of poker coaching, I’ve helped hundreds of clients improve… Do Multiple Sizes Matter? Should I work out the other tools or just stick to the crowbar? Solver results… The Science of Learning Applied to GTO Wizard The way we were taught to study in schools was flawed and has negatively impacted… Check-Raising a Single Pair In this article, I will address a question raised by Twitter/X user ’Matt Riley’, who… The Turn Probe Bet Imagine yourself defending as the BB caller vs BTN preflop raiser in a 100bb cash… When To Encourage Multiway Pots in PKOs I was recently shown this fascinating PKO hand from the GTO Wizard PKO library. Somebody… Protect Equity and Prosper When we think about the best hands to bet in a given situation, we tend… Do Solvers Have Targets? Targeting has long been a staple of exploitative poker thinking. When you’re value betting against… Disciplining Big Blind in Limped Pots When the BTN open-limps off a short stack, the postflop play against the BB should… I’d Rather Be Drawing Which is the better hand on a K♥8♦5♦ flop: K♠T♣ or 7♥6♥? OK, I’ll admit… The Curious Case of Open-Limping Buttons Before the advent of solvers, open-limping was considered the province of passive, gambley recreational players… Cleaning Out Sticky IP Callers A “sticky” player on your left, one who calls your preflop raises too often and… Defending Against Tiny 3-Bets Tournament poker is about precision. Cash games, where stacks typically run into the hundreds of… Heads up! Exploiting SB’s Preflop Mistakes Heads up (HU) tournament play presents unique challenges to the average player. Many tournament players,… C-Betting As the OOP Preflop Raiser Position is an extremely valuable advantage in poker. Most players would describe the comfort zone… Overchoice: Making Sense of Multiple Sizings Solver solutions with multiple bet sizes often involve a lot of mixing between those sizes,… River Refinements: When Your Opponent Won’t Check-Raise One of the |
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is a way to simplify the game of poker. This game is so… Mathematical Misconceptions in Poker Pot Odds and MDF are foundational mathematical formulas in poker. How to solve toy games Poker is a complicated game. It’s impossible to calculate GTO strategies on the fly. By… The Value of Fold Equity – Experiment The term “fold equity” is ambiguous in the poker community. Firstly, it’s an EV equation,… MDF & Alpha MDF & Alpha. Minimum Defence Frequency and Alpha are metrics on poker that determine how… What are Pot Odds in poker? Pot odds are a fundamental calculation in poker. In the simplest terms, pot odds tell… Does your range affect your strategy? Advanced players will often tell you that your range affects your strategy. Visualizing implied odds Implied odds are the value you expect to gain over and above your raw equity.… ICM Basics As a tournament player, you may have heard about the concept of “ICM”, but what… What is Equity in Poker? The term “Equity” is ubiquitous throughout poker theory. Equity refers to your chances of winning… Range Morphology “Morphology” is the study of form or structure. You’ve probably heard terms thrown around such… What is Expected Value in Poker? Expected value (EV) is the most fundamental metric in poker. Every decision you make is… Why doesn’t my solution match GTO Wizard? So, you’ve got your own solutions to some spot, compared it to GTO Wizard, and… Understanding Nash Distance Understanding Nash Distance. One of the most common questions when studying with solvers is “Why… Combinatorics Poker Puzzle Today we have a special puzzle for you that will test your intuition as a… The 10 Most Important Concepts for New Poker Players The 10 Most Important Concepts for New Poker Players. Fundamentals are one of the most… What is GTO in Poker? The meaning of GTO in poker is Game Theory Optimal. Game theory is a study… What does GTO aim to achieve? So what exactly is a GTO strategy? What makes it “unexploitable”? What does it try… |
How Solvers Work A Game Theory Optimal solver is an algorithm that calculates the best possible poker strategy. But how exactly do these solvers work? What makes their strategy “the best”? This article will take a deep dive into how solvers work, what they achieve, and their limitations. The Goal Before we dive in, start with these two articles . This article is meant to provide a deep understanding of how solvers work and may feel overwhelming without some basic prerequisite knowledge. If you’re new to solvers, read these first, then return to this article. As discussed in What is GTO in Poker? And What does GTO aim to achieve? The purpose of GTO is to achieve an unexploitable Nash Equilibrium strategy. Nash Equilibrium is a state where no player can do better by unilaterally changing their strategy . This means that if each player were to publish their strategy, no player would be incentivized to change their strategy. This is often described as the “Holy Grail” of poker strategies. But that’s not actually what solvers are designed to do. In fact, solvers have no idea what “Nash Equilibrium” means . Solvers are simply EV-maximizing algorithms. Each agent in a solver represents a single player. That player has one goal, and one goal only: To maximize money . The problem is the other agents play perfectly. When you force these agents to play against each other’s strategies, they iterate back and forth, exploiting each other’s strategies until they reach a point where neither can improve. This point is equilibrium ☯︎ GTO is achieved by making exploitative algorithms fight each other until neither can improve further. How to solve GTO Assign each player a uniform random strategy (each action at each decision point is equally likely). Compute the regret (EV loss against the opponent’s current strategy) for each hand throughout the game tree. Change one player’s strategy to reduce their regret , assuming the opponent’s strategy remains fixed. Go back to step 2, recalculate regret, then change the opposing player’s strategy to reduce regret. Repeat until Nash. Each of these cycles is called an iteration. The number of iterations required varies from a few thousand to a few billion, depending on the size of the tree and sampling method. Step 1) Define the gamespace Inputs Poker is far too complex to solve directly; we need to reduce the gamespace using subsets and abstractions to make it computable. In general, to run a solver, you need to define the following parameters: The betting tree. Required accuracy. Starting pot and stack sizes. Starting ranges (postflop solvers). Board cards (postflop solvers). Postflop card abstractions such as card bucketing or NNs (preflop solvers). Modifications to the utility function such as rake or ICM . Betting tree complexity We need to define the available bet sizes to reduce the size of the game tree. Before we get to the algorithm, we need to understand what a “Betting Tree” looks like. The solver operates within the parameters you provide. If you give a solver a very simple game tree, |
you produce less complex strategies, but keep in mind the solver will exploit the limitations of your game tree. The solver will generate a “ tree ” containing all possible lines within the given betting structure. Each decision point throughout that tree contains a “node”. For example, OOP facing a ⅓ pot-sized bet is a single “ node ”. The number of nodes in a tree defines how big the tree is. Each node needs to be optimized. An extremely simple tree like this one has 696,613 individual nodes that must be optimized. A more complex tree like the type GTO Wizard uses contains ~87,364,678 nodes. As you can see, complexity grows the tree exponentially. The complex tree above is using 4-5x as many sizes per node, yet it’s 125x bigger and harder to solve. And this is still a major simplification of the true game space . One of the most difficult problems with solvers is optimizing betting trees to produce solid strategies within the constraints of current technology. We can only make a tree so big before it becomes unsolvable due to its size. We can only make a tree so small before the solver starts exploiting the limitations of that tree. We’re working on algorithms that find simplified betting trees to make it easier to study poker! These solutions will find the best sizes for every spot. GTO Wizard has many different types of solutions, such as complex solutions with up to 19 sizes and Simple solutions with only a few sizes. Ultimately, we find that it’s better to start with a complex tree, then trim it to a smaller, more manageable tree by removing infrequent lines. Nodelocking GTO Wizard plans to add nodelocking and real-time solving in 2023! This powerful feature will be used to explore exploitative strategies and underlying cause > effect principles. Nodelocking is the process of fixing one player’s strategy at some node in the gametree. We force that player to play a specific way. Nodelocking is commonly used to develop exploitative strategies! For example, if you force it to rangebet the flop, the opposing player will maximally exploit that strategy. It’s important to keep in mind, however, that both players will adjust before and afterwards to accommodate that locked node. Turn and river strategies will change. If you force a solver to play badly, it will course-correct prior and later nodes to minimize the damage. The process of locking a single node and letting the solver work around that deficit is known as a “ minimally exploitative strategy ”. We are not modeling some leak throughout the entire tree, but rather, just a specific point in the tree. More complex nodelocks are possible. For example, some solvers let you lock the strategies for specific combinations at one node (while letting other combos adjust their strategy). It’s also possible to lock many nodes to recreate and exploit larger trends in your opponent’s strategy – but modern tools don’t accommodate multistreet nodelocks effectively. Step 2) Solve the game tree! So we’ve defined the game tree. Now |
it’s time to solve it! First, we need to understand how solvers calculate the expected value of strategies. How do solvers calculate EV? Let’s picture the game tree above. Each dot represents a node or decision point. How do we know how much EV each hand generates at each node? The process is simple ( for computers ). Firstly, we define the terminal nodes ( AKA leaf nodes ). These are points where the hand terminates, either because someone folded or because the hand went to showdown. Each terminal node is assigned a probability (p) and a value (x). Each hand (i) in our range generates a separate value and probability of reaching each terminal node. We multiply the value and probability of each terminal node and sum them to find the total expected value. The value of each node is defined as the total pot we win minus how much we invested into that pot Start with a strategy pair. Based on our strategy and our opponent’s strategy, our hand will reach each terminal nodes this often ( p ). The value of each terminal node is x . The sum of x * p for each terminal node gives us our expected value (EV). Do this calculation for every single hand at every single node. E [ X ] = Σx i p ( x i ) x i = The values that X takes p(x i ) = The probability that X takes the value x i Solvers can make this calculation almost instantaneously. Before solvers came along, programs like CREV were used to calculate EV. The limitation being that users had to define the strategies at each node. So typically, we reduced later nodes to pure equity calculations, which was extremely inaccurate. We can observe the expected value of every hand at every point in GTO Wizard . For example, here’s a BTN vs BB SRP in a NL50 cash game . We can see the overall expected value of each hand (use the strategy dropdown), and hovering over any individual hand reveals the EV of various actions we could take: Regret Start by assuming our opponent’s strategy is fixed (unchanging) . Then run the EV calculation described above for every potential action our hand could take at each node throughout the game tree. Then we select the highest EV decision at each point and work backward from the terminal nodes to calculate the EV of different actions from the first decision point. Ok, so we know the value of each hand at each node. How do we improve the strategy? This is where the concept of “regret” comes into play. Minimizing regret is the basis of all GTO algorithms. The most well-known algorithm is called CFR – counterfactual regret minimization. Counterfactual regret is how much we regret not playing some strategy . For example, if we fold and find out that calling was a way better strategy, then we “regret” not calling. Mathematically it measures the gain or loss of taking some action compared to our overall |
strategy with that hand at that decision point. Regret = Action EV – Strategy EV For example, if our current hand’s strategy is to fold, call, and shove ⅓ of the time, and the EV of each of those actions is (Fold = 0bb, Call = 7bb, Shove = 5bb), then the EV of our current strategy is: (⅓ x 0) + (⅓ x 7 bb) + (⅓ x 5bb) = 4bb Folding has negative regret, meaning it loses more than our average strategy. Calling and shoving have positive regret, meaning they outperform our current strategy. The next step is to change our strategy to minimize regret The most obvious approach is to simply choose the highest EV action at each decision point with every hand (AKA a maximally exploitative strategy). In our above example, that would mean always calling. The problem is that our opponent can change their strategy, and this can get us stuck in a loop . For example, Player A under-bluffs the river, player B folds all their bluff-catchers, then player A always bluffs the river, then player B calls all their bluff-catchers, then player A stops bluffing the river, repeating forever. Instead of switching all the way to the best response on each iteration, each player can gently adjust their strategy one step at a time in that direction. This resolves the issue of getting stuck in loops and converges more smoothly when the strategy pair is close to equilibrium. We can instead use CFR to update our strategy. Let’s return to our previous example. Any actions with negative regret stop getting played. Any actions with positive regret use the formula: New strategy = Action Regret / Sum of positive regrets In our example, calling ->3/(3+1) = 75%, and Shoving ->1/(3+1) = 25%, while folding becomes 0% because it has negative regret. Current Strategy EV = 4.0 New Strategy EV = 6.5 Note that this is just one iteration. As we repeat this process many times, the strategy will approach a point where neither player can improve, achieving Nash Equilibrium . Accuracy The accuracy of a solution is measured by its Nash Distance . We start with one question: How much could player A win if they maximally exploited Player B’s current strategy? This is easy for a computer to calculate as it already knows the regrets. The difference between the EV of player A’s current strategy and the EV of their maximally exploitative strategy represents their nash distance. The smaller that number, the less exploitable and more accurate the strategies are . We take the average of all players’ Nash distances to find the accuracy of the solution. These exact nash distance measurements only work if you’re enumerating the entire strategy each iteration. Most preflop solvers use abstractions and sampling methods which render these calculations impractical and inaccurate. Solvers like HRC or Monker estimate convergence by measuring how much strategies/regrets change every iteration. GTO Strategies start to converge at a nash distance below 1% pot. Beyond this threshold, strategies are extremely mixed and tend to be |
unreliable. Most pros consider anything worse than 0.5% pot to be unacceptable. GTO Wizard solves to an accuracy of 0.1% to 0.3% of the starting pot depending on the solution type. The more complex your game tree, the more accuracy is required to differentiate between similar bet sizes. Similar sizes result in similar payoffs, so more complex game trees with many bet sizes require higher accuracy to converge. Convex payoff space How do we know this iterative approach works? Can we get stuck in a local maximum? Poker, in general, can be described as a “ bilinear saddle point problem ”. The payoff space looks something like this: Each point on the x-axis and y-axis represents a strategy pair . Each strategy pair contains information about how both players play their entire range in every spot across every runout. The height (z-axis) represents the expected value of the strategy pair , with higher points representing an EV advantage for one player, and lower points representing an advantage for the other player. Most solvers use a process called Counterfactual Regret Minimization (CFR) . This algorithm was first published in a 2007 paper from the University of Alberta by Martin Zinkevich. That paper proves that the CFR algorithm will not get stuck at some local maximum, and given enough time, will reach equilibrium. The center of this saddle represents Nash Equilibrium. the point(s) on this graph that have no curvature – meaning neither player can change their strategy to improve their payoff. Further Reading We highly recommend reading this article to learn more about CFR in poker. This website offers a great tutorial and step-by-step instructions for building a simple CFR algorithm. Academic resource on using CFR in poker . Deep CFR – Applying neural networks to speed up CFR calculations. Improving the original CFR algorithm with “Discounted” regret minimization. Conclusion Phew, that was a lot of information! Hopefully, by now, you have a much better understanding of how solvers actually work. Let’s recap the main points: For all practical purposes, the main takeaway is that solvers are EV maximizing algorithms that take advantage of the game tree we provide them. Solver algorithms generate max-exploit strategies . Pitting these algorithms against each other produces unexploitable equilibrium strategies. Calculating the expected values of a pair of strategies is the easy part ( for computers ). Nudging the strategy in the right direction and iterating this process countless times is the hard part. Poker is too complex to solve directly , so we simplify the gamespace using abstraction techniques like limiting the bet sizes. Solvers are only as accurate as the abstract game you give them . Too much complexity is impossible to solve and difficult for humans to learn from. Too much simplicity results in the solver exploiting the limitations of that game tree. GTO Wizard the #1 App for Poker players Study any spot imaginable Practice by playing vs. GTO Analyze your hands with 1-click START CRUSHING NOW JOIN DISCORD Author Tombos21 Tom is a long time poker theory enthusiast, GTO Wizard coach and |
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Model (ICM) shows us the real money implications of tournament chip decisions.… The Theory of Progressive Knockout Tournaments In poker, a PKO (Progressive Knockout) tournament is a type of tournament where a portion… How To Analyze Turn Textures In Poker Master the art of turn analysis in poker. Understand how the turn card changes the… Principles of Turn Strategy Learn the principles of turn strategy in poker and discover what makes the turn unique.… Poker Strategies: Tournaments vs. Cash Games Tournament poker is not as different from cash game poker as people sometimes assume. Most… How Solvers Work A Game Theory Optimal solver is an algorithm that calculates the best possible poker strategy.… The Most Powerful Play in Poker The central strategic implication of stack depth is the question of how much equity a… When does ICM become significant in MTTs? The Independent Chip Model (ICM) was invented in 1987 by M. Malmuth and is one… How to Study GTO Solutions Tools like GTO Wizard and other solvers should not be used to memorize specific strategies.… Principles of GTO Game theory, or GTO (for Game Theoretically Optimal), is best understood not as a rigid… Interpreting Equity Distributions GTO Wizard’s range vs range equity distribution graphs are an excellent tool for visualizing which… What is Leverage in poker? Leverage refers to the risk of future betting. When you call a bet on the… Reasons for value betting in poker You’re probably accustomed to thinking of bets as either value or bluff. In this taxonomy,… The Science of Poker Performance Poker is a game of small edges and big swings. Every hand is an investment… What is Valor in poker? Can a poker hand be courageous? How does your ability to value bet future streets… Pot Geometry “Pot Geometry” refers to betting an equal fraction of the pot on each street, such… A Beginner’s Guide to Poker Combinatorics Combinatorics is a fancy term for evaluating the number of possible “combinations” (combos) of any… Stack-to-pot ratio Stack-to-pot ratio, or SPR, is a way of measuring how deep the effective stack is… Equity Realization In poker jargon, equity expresses how much of the pot a hand will win, on… Variance and Bankroll Management Variance and Bankroll Management. Vital Soft Skills for Every Poker Player. How to Become a GTO Wizard Welcome to the wonderful world of game theory optimal poker! This article will serve as… The Three Laws of Indifference Indifference in poker is one of the most misunderstood concepts. The word “indifference” means that… Poker subsets and abstractions An “abstraction” is a way to simplify the game of poker. This game is so… Mathematical Misconceptions in Poker Pot Odds and MDF are foundational mathematical formulas in poker. How to solve toy games Poker is a complicated game. It’s impossible to calculate GTO strategies on the fly. By… The Value of Fold Equity – Experiment The term “fold equity” is ambiguous in the poker community. Firstly, it’s an EV equation,… MDF & Alpha MDF & Alpha. Minimum Defence Frequency and Alpha are metrics on poker that |
determine how… What are Pot Odds in poker? Pot odds are a fundamental calculation in poker. In the simplest terms, pot odds tell… Does your range affect your strategy? Advanced players will often tell you that your range affects your strategy. Visualizing implied odds Implied odds are the value you expect to gain over and above your raw equity.… ICM Basics As a tournament player, you may have heard about the concept of “ICM”, but what… What is Equity in Poker? The term “Equity” is ubiquitous throughout poker theory. Equity refers to your chances of winning… Range Morphology “Morphology” is the study of form or structure. You’ve probably heard terms thrown around such… What is Expected Value in Poker? Expected value (EV) is the most fundamental metric in poker. Every decision you make is… Why doesn’t my solution match GTO Wizard? So, you’ve got your own solutions to some spot, compared it to GTO Wizard, and… Understanding Nash Distance Understanding Nash Distance. One of the most common questions when studying with solvers is “Why… Combinatorics Poker Puzzle Today we have a special puzzle for you that will test your intuition as a… The 10 Most Important Concepts for New Poker Players The 10 Most Important Concepts for New Poker Players. Fundamentals are one of the most… What is GTO in Poker? The meaning of GTO in poker is Game Theory Optimal. Game theory is a study… What does GTO aim to achieve? So what exactly is a GTO strategy? What makes it “unexploitable”? What does it try… |
How To Negotiate Final Table Deals Most people’s first introduction to the Independent Chip Model (ICM) is not from studying bubble play or endgame ranges; it is their first time making a big final table. This is when deal-making is proposed, and ICM is first introduced. Understanding ICM is particularly important when making final table deals , and making an error in this department can be very costly. Why Make A Deal At Final Tables? ICM was first introduced to poker to negotiate final table deals. It began life as the Mason-Harville Formula in 1987. David Harville created a model for making horse racing predictions in 1973, and Mason Malmuth adapted it for final table deals. The model would estimate how likely each remaining player was to finish in every position based on the percentage of chips they had. The chances of each player coming 1st was the percentage of total chips they held, but every other position was much more complex to calculate. Only much later in the modern poker era was ICM used to guide strategic decisions. There are lots of excellent reasons to deal at a final table, including: To save time by not playing the final table at all To guarantee a specific prize amount To guarantee you win the trophy To reduce variance The final reason is at the core of why most people deal. We make deals to reduce variance when we reach big final tables. There are opportunities in poker that don’t come around very often, like if you make a final table with many runners and/or you qualified via a satellite . You don’t know when you’ll make the next 5,000+ runner final table. If third place prize money would double your bankroll, get you out of makeup, allow you to move up stakes, pay for a house or otherwise change your life positively, there is no need to needlessly subject your current equity to variance. If you are happy with the prize, there is no such thing as a bad deal, but when you understand the mechanics of final table deals, you put yourself in a better position to negotiate for your own self-interest. Chip Chop Deals A Chip Chop deal simply takes the remaining prize pool and awards money to the players based on the percentage of chips they have . As you can imagine, a Chip Chop deal is usually the one proposed by the chip leader . Let’s look at an example to highlight this type of deal. Below is a final table of five players, their chips, and what their prize would be if they maintained the position they are currently in: There are two ways to potentially Chip Chop, the first one is literally just to take the remaining prize pool and split it between the players based on their chip stacks. There is $23,300 remaining in the prize pool. Chopping it up that way would look like this: You can probably spot the immediate issue with this type of deal. Not only has Bob managed to secure |
a prize more than first place prize money, Sergei, the short stack, takes home less than half of what he was guaranteed to take with no deal in place . This type of Chip Chop is clearly a bad idea, so what typically happens is a Chip Chop deal based on the percentage of chips, but after everyone has locked up the min-cash. So in this example, everyone would take home $2,300 and deal based on the remaining prize pool, which would be $11,800. Such a deal would look like this: On the surface, this looks like a favorable deal for all parties. The bottom three stacks have all secured more than their projected finish position would pay them, Jane barely sacrifices any of her 2nd place prize money to lock it up, and Bob takes the biggest hit but still locks up a lot more than 2nd place. ICM deals The ICM Deal is the alternative to the Chip Chop deal and is the gold standard for final table deals. ICM has limitations as a strategy and also in final table deals: It doesn’t account for skill. ICM assumes all the players are of equal ability. ICM ignores blinds increasing. If you know the blinds are about to increase, that can affect the optimal strategy, especially with shortstacks involved. ICM doesn’t factor in table dynamics. For example, being seated next to an aggressive chip leader can be a nightmare to play. The only thing an ICM deal considers are the payouts and the stacks of the remaining players. Let’s go back to our first example and look at it through the lens of ICM instead. Below is the exact same final table but with the ICM value of each stack. We have also added the Chip Chop value from the previous table so you can compare the two: You can see why a chip leader favors the Chip Chop deal, it is worth $706.29 more to them. Jane, second in chips, also does slightly better with Chip Chop. That money comes from our three short stacks, who are all between $200-$300 worse off with the Chip Chop deal. Two things always strike players who first encounter ICM when they are making final table deals: The big stack is always worth less than most players think, the short stack is always worth more than most players think. The key takeaway here is that ICM deals favor shorter stacks. This is a core principle of ICM, the fewer chips you have, the more each one is worth. In this example, one big blind is worth $299.11 to the short stack, but only worth $64.91 to the chip leader . If these payouts seem wrong or unfair to you, let’s explore them further. Below are the projected finish distributions based on the ICM model for this final table, if no deal was struck: Bob wins it more than anyone else, but 64% of the time he doesn’t come 1st, making a Chip Chop deal for more than 2nd place prize money unfair. Sergei busts |
next most of the time, but 33.3% of the time he manages to ladder to a prize of at least $3,100, so he needs to get something to encourage him not to play on to try and spin up. Only the big stacks benefit from a Chip Chop, and it is at the expense of everyone else. A short stack is worth more and ladders more often than a Chip Chop deal would suggest. A Chip Chop deal makes sense only when the final two players are heads-up . There is no ICM heads-up and you could, therefore, give each player 2nd place prize money and then chop up the remaining money equally based on their stacks at the time. Skill deals The biggest flaw with the ICM model in terms of final tables is it does not factor in skill. Few professionals would want to give a bad player a deal assuming equal skill. Likewise, elite players would not want to sacrifice edge. ICM deals work because the edges are quite small at most final tables, especially with shallow stacks or a fast structure. The difference between two good regulars will be minimal, so unless there is a big gulf in class between the players, an ICM deal is better than subjecting oneself to variance. When one player shines above the rest, they can negotiate a final table payout based on their perceived edge . This would involve using ICM as a baseline after the elite player has taken a markup from the remaining prize pool. In our example above, let’s assume that our chip leader Bob is a professional player with an edge over the others. He estimates his edge is 10%. What he might propose is that he takes a 10% markup on the ICM value of his stack, while everyone else at the final table negotiates amongst themselves based loosely on ICM. Usually, it would be the other big stacks taking the hit to secure a payout because the short stack has less to lose by refusing. This sort of deal is part science, part negotiation, with some art thrown in. Such a deal might look like this: Getting better than 1st place prize In rare cases, an excellent player can negotiate a deal where they take more than the projected first prize . They either have such an edge that can dictate these terms, or the other players are so scared of busting next that they agree to poor terms. A great example was when online beast ‘€urop€an’ won the 2016 WCOOP Super Tuesday. He had both an enormous chip lead and is also regarded as one of the best online tournament players of all time. This is how things stood when the deal was proposed, with the potential payouts as they were (the chip stacks are not exact but the last way each one was reported during the event): However, €urop€an managed to negotiate this deal: He secured $5,268 more than the official first place payout ! In exchange for this, ‘XingMaster’ secured $32,901 more than |
what 3rd would have got, but $24,283 less than what he would have won for finishing second. ‘reno8’ won $19,016 more than he would have for his third place finish. If this were an equal skill matchup then amazingly €urop€an only had $277,205 in equity despite a 10-to-1 chip lead. Assuming equal skill this is what ICM says would be their likely finish position chances: Then you have the matter of edge which is not insignificant here . It actually doesn’t look like a terrible deal on the surface. XingsMaster and reno8 essentially are playing a pseudo heads-up match for $57,184 (the difference between 2nd and 3rd place prizes) and have paid €urop€an $5,268 for the privilege of splitting $51,916 amongst themselves. €urop€an only winning 82.1% of the time might surprise some people and would lead them to argue that it is worth the other two players not dealing. The problem for the other two players is that both of them don’t win 90% of the time. Deal-making mindset issues Beyond the numbers, there are several mindset issues that can lead a player to reject a good deal or accept a terrible deal. First of all, there is the anchoring effect of the money itself. Some players might have looked at what 3rd place prize money would do for them personally (a new car, moving up stakes, a holiday, etc.) and are unwilling to look beyond that. Others will look at their position in the tournament and be unwilling to give up what their potential prize would be if they maintained their position. There is also a social pressure element to final table deals. It might be that you do not want to give the arrogant professional player $500 more than ICM because he has suggested he has an edge on you. On the flipside, it is easy to get bullied into a bad deal for fear of being ostracized by the other players. You will have a sense of where your own mindset issues will lie in this regard, but perhaps the best way to ensure you do not get pressured into a bad deal is by understanding ICM in the first place. If the social pressure is too much, an easy hack to push for a good deal or refuse a bad deal, without social stigma, is to just tell the other players you have a backer who has insisted on your terms. You can make this fictional backer the bad guy while still enjoying the final table camaraderie with the other players. Conclusion Even if you play on a site that doesn’t facilitate final table deals, it is a good use of your time to study them. Studying final table deals and understanding the value of your stack will help with strategic decisions, particularly where Bubble Factor is concerned. Understanding the ICM implications of final table deals puts you on a good footing for that once-in-a-lifetime final table you may find yourself on one day in your career . Key Takeaways There are lots of good reasons to deal, |
most notably reducing variance Chip Chop deals benefit big stacks at the detriment to short stacks Short stacks are worth more than people think in ICM deals Big stacks are worth less than people think in ICM deals ICM deals assume equal skill Edge deals are rare and there should be a big gulf in skill to consider them GTO Wizard the #1 App for Poker players Study any spot imaginable Practice by playing vs. GTO Analyze your hands with 1-click START CRUSHING NOW JOIN DISCORD Author Barry Carter Barry Carter has been a poker writer for 16 years. He is the co-author of six poker books, including The Mental Game of Poker , Endgame Poker Strategy: The ICM Book , and GTO Poker Simplified . Latest article Preflop Raise Sizing: Examining 2 Key Factors It amazes me that, after nearly twenty years of playing no-limit hold ‘em professionally, I… How to Handle Loose-Passive Limpers Solver charts and presolved solutions do not offer much in the way of explicit guidance… Top 5 Mistakes in Spin & Gos As a Spin and Go poker coach, I regularly encounter common mistakes that players make,… An Introduction to Spin & Gos I was playing Tournaments for the first decade of my career. But after my first… Relationships and Poker In my more than 15 years of poker coaching, I’ve helped hundreds of clients improve… Do Multiple Sizes Matter? Should I work out the other tools or just stick to the crowbar? Solver results… The Science of Learning Applied to GTO Wizard The way we were taught to study in schools was flawed and has negatively impacted… Check-Raising a Single Pair In this article, I will address a question raised by Twitter/X user ’Matt Riley’, who… The Turn Probe Bet Imagine yourself defending as the BB caller vs BTN preflop raiser in a 100bb cash… When To Encourage Multiway Pots in PKOs I was recently shown this fascinating PKO hand from the GTO Wizard PKO library. Somebody… Protect Equity and Prosper When we think about the best hands to bet in a given situation, we tend… Do Solvers Have Targets? Targeting has long been a staple of exploitative poker thinking. When you’re value betting against… Disciplining Big Blind in Limped Pots When the BTN open-limps off a short stack, the postflop play against the BB should… I’d Rather Be Drawing Which is the better hand on a K♥8♦5♦ flop: K♠T♣ or 7♥6♥? OK, I’ll admit… The Curious Case of Open-Limping Buttons Before the advent of solvers, open-limping was considered the province of passive, gambley recreational players… Cleaning Out Sticky IP Callers A “sticky” player on your left, one who calls your preflop raises too often and… Defending Against Tiny 3-Bets Tournament poker is about precision. Cash games, where stacks typically run into the hundreds of… Heads up! Exploiting SB’s Preflop Mistakes Heads up (HU) tournament play presents unique challenges to the average player. Many tournament players,… C-Betting As the OOP Preflop Raiser Position is an extremely valuable advantage in poker. Most players would describe the comfort |
zone… Overchoice: Making Sense of Multiple Sizings Solver solutions with multiple bet sizes often involve a lot of mixing between those sizes,… River Refinements: When Your Opponent Won’t Check-Raise One of the biggest strategic differences between playing in versus out of position involves betting… Turn Barreling in 3-Bet Pots There are two articles on the GTO Wizard blog that make good prerequisites to this… The Right Way To Think About Protection Suppose Paul the Protector is in a 200bb cash game and opens A♠A♣ in the… The Importance of Board Coverage The first public version of the GTO Wizard announced AI upgrade already has some useful… How To Adjust When Villain Has No Bluffs “They always have it!” is a common, semi-joking expression in poker to describe the tendency… Adjusting on Later Streets After Exploiting Flop The key to exploiting opponents profitably is knowing when to stop. I’m not referring to… C-Betting IP vs Passive Poker Players A preflop raiser enjoys a significant range advantage over a BB caller, enabling them to… C-Betting Against ICM Preflop Ranges You raise from early position (EP) in a multi-table tournament and the BB calls. Both… The Art of Bluff Catching Bluff catching—calling a bet with a modest hand which you expect to win only if… Exploiting Excessive C-Betting by IP On some flops, it is simply not possible for the preflop raiser to continuation bet… Is Limping Pimping? In a past Discord post, one of our members noted that an upside of implementing… Exploiting Excessive C-Betting by OOP A preflop raiser who continuation bets too often is always a potential target for exploitation,… Preflop Button Mastery in PKOs Playing from the Button presents many unique opportunities in any tournament format, especially in PKOs.… The Worst Turn Card Greetings from GTO land. In today’s article, we will use the Turn Reports feature to… Navigating Range Disadvantage as the 3-Bettor Picture this: We’re playing a NL500 6-max cash game, and we’ve won a few pots,… Mastering Three-Bet Pots In Position in MTTs Playing in position against a three-better is a lesson in playing to your advantages. As… Mastering Three-Bet Pots Out of Position in MTTs Three-bet pots are scary. Playing out of position is scary. Put them together, and you’ve… The Greatest Final Table in Triton History Today, we’ll examine one of the most epic final table battles I’ve ever witnessed! Our… Mastering Thin Value Bets in Checked-Down Pots Valuing your hand appropriately—determining whether it’s strong enough to value bet or so weak you… Maximizing Value on Monotone Flops Monotone flops tend to provoke great anxiety among poker players. It’s easy to imagine that… How to Defend Against Turn Donk Bets Responding effectively to donk bets on the turn is tricky;, there’s no getting around that.… How and Why You Should Use Turn Donk Bets Donk betting–taking the betting lead away from the previous street’s aggressor–on the flop is rarely… How Stack Sizes Change Your Range In this series, we have looked at factors that can influence ranges in otherwise identical… How to Leverage GTO Wizard’s EV |
Comparison Tool With the advent of solvers, some troubling new turns of phrase have entered the poker… Should You Ever Cold Call a 3-Bet? When there’s been a raise and re-raise before the action is on you, you will… When ICM Breaks Down If you have read my previous GTO Wizard articles and my books, you’ll know I… When Is It Correct To Fold AA Preflop? One of the criticisms of ICM is that it is boring and robotic. In my… OOP C-betting vs Loose Cold-callers In a previous GTO Wizard article, I asserted that “Playing against an in-position cold-caller is… Short-Stacked Play in MTTs Playing a short stack well is an essential tournament skill. It is extremely rare to… When To Ladder in Tournaments In my last two articles, I wrote about the effect field size and payout structure… How Payout Structures Impact ICM Last time we discussed the ICM considerations in game selection, we looked at field size,… C-Betting IP in 3-Bet Pots When you three-bet before the flop and the original raiser calls from out of position,… C-Betting OOP in 3-Bet Pots Continuation betting from out of position can be a dicey proposition in single-raised pots. Cold… Dynamic Sizing: A GTO Breakthrough Dynamic Sizing is a revolutionary new poker algorithm that automatically simplifies your strategy with the… Dynamic Sizing Benchmarks Dynamic Sizing is a revolutionary new poker algorithm that automatically simplifies your strategy with the… Understanding The Impact Of Field Size On ICM In Poker When people talk about game selection in multi-table tournaments (MTTs), they invariably talk about the… Defending vs BB Check-Raise on Paired Flops This is a companion piece to Attacking Paired Flops from the BB; you will get… Mastering PKO Final Tables Final tables can be the most exciting and rewarding phase of a tournament, and being… Attacking Paired Flops From the BB Paired flops present unique challenges to both a preflop raiser and a BB caller. The… KK vs QQ vs JJ at WSOP Main Event With only fourteen players remaining in the 2023 WSOP Main Event and blinds of 400K/800K/800K,… Responding to BB Squeezes You open raise, someone calls, and the BB squeezes. How should you construct your strategy?… Crushing a Top HUNL Poker Bot In 2022, Philippe Beardsell and Marc-Antoine Provost, a team of Canadian programmers from Quebec, developed… GTO Wizard AI Explained We are excited to introduce GTO Wizard AI, formerly known as Ruse, the world’s best… GTO Wizard AI Benchmarks GTO Wizard has combined the power of artificial intelligence with traditional solving methods to bring… Table Management in PKOs Perhaps the most important strategic consideration in Progressive Knockout Tournaments (PKOs) is making sure you… Overcalling From the BB In poker, overcalling refers to calling a bet or raise after another player has also… Understanding Which Mistakes Cost You the Most Money I am fortunate enough to have worked for many years with the mental game coach… Overbetting The Flop in Cash Games Although most commonly analyzed on turns and rivers, flop overbetting can be a devastating weapon… Using New Skills |
at the Poker Table How many times has it happened to you that you make a mistake only to… Flatting Ace-X Hands Shortstacked Playing out of position is tricky, but it’s an essential part of tournament play. With… Playing Limped Pots as the BB in MTTs Success as the BB in limped pots begins before the flop. The BB’s equilibrium strategy… The Absurd Game Theory of Chopped Boards A “chopped board” in poker is one where the community cards make up the best… Playing Limped Pots as the SB in MTTs MTT confrontations between the small blind and big blind force no-limit players into uncomfortable situations,… The Art of Learning Poker In his book The Art of Learning, chess and Push Hands champion Josh Waitzkin describes… Is Donk Betting for Donkeys? “Donk betting”, or betting from out of position into the player who was the aggressor… Delayed C-Betting This is a companion piece to Probe Betting. It stands on its own, but understanding… Probe Betting As far as rules of thumb go in poker, “check to the raiser” is a… ICM and Blind Battles: The Big Blind This is a companion piece to this article on the SB’s strategy for blind vs… ICM and Blind Battles: The Small Blind ICM incentivizes more conservative play, which means that at stages of the tournament where risk… The Initial Bettor’s Advantage The mathematics of poker holds a little-known secret: the first bettor enjoys better bluffing odds… Mastering the Chiplead: The Scalpel and the Sledgehammer When you have a lot of chips at a final table or in another situation… Flop Heuristics: IP C-Betting in Cash Games Our never-ending quest for money and fame leads us to the flop. We open the… How ICM Impacts Restealing From The Blinds When I first started working with preflop solves, one of the features I found most… The ICM Benefits of Late Registration in Poker Tournaments Why do so many professional poker players register late for tournaments when they clearly would… Flop Heuristics: OOP C-Betting in MTTs Playing against an in-position cold-caller is dramatically different–and dramatically more difficult–than playing against a caller… Counterintuitive Calls Solver outputs such as those provided by GTO Wizard are models of no-limit hold ‘em,… How To Negotiate Final Table Deals Most people’s first introduction to the Independent Chip Model (ICM) is not from studying bubble… How does ICM impact PKO strategy? If you have read The Theory of Progressive Knockout Tournaments you already know why we… Understanding Blockers in Poker Often when looking through a solver’s meticulously crafted grid of deception, one of the following… Flop Heuristics: IP Cold-Caller in MTTs A Cold-call refers to the act of calling a preflop raise when you don’t close… Flop Heuristics: IP C-Betting in MTTs In the world of poker, the flop is a crucial stage of the game that… Flop Heuristics for Defending the Blinds in MTTs The field of Game Theoretically Optimal poker is extensive, and thus, to simplify our strategy,… Multiway strategy in Progressive Knockout Tournaments Progressive Knockout (PKO) tournaments are much more complex |
than traditional MTTs. The fact that players… How To Calculate Raises In Poker One of the most frequently asked questions in poker is how to calculate the size… Principles of River Play The river offers the least complicated strategy of any betting street. With no further cards… Exploitative Dynamics GTO solutions are designed to be unexploitable, performing reasonably well no matter how your opponents… How ICM Impacts Postflop Strategy Most players understand how the Independent Chip Model (ICM) in poker impacts their strategy preflop,… 10 Tips for Multiway Pots in Poker While lots of work has been placed into research and explanation of heads-up pot strategy,… What is the Bubble Factor in poker tournaments? The Independent Chip Model (ICM) shows us the real money implications of tournament chip decisions.… The Theory of Progressive Knockout Tournaments In poker, a PKO (Progressive Knockout) tournament is a type of tournament where a portion… How To Analyze Turn Textures In Poker Master the art of turn analysis in poker. Understand how the turn card changes the… Principles of Turn Strategy Learn the principles of turn strategy in poker and discover what makes the turn unique.… Poker Strategies: Tournaments vs. Cash Games Tournament poker is not as different from cash game poker as people sometimes assume. Most… How Solvers Work A Game Theory Optimal solver is an algorithm that calculates the best possible poker strategy.… The Most Powerful Play in Poker The central strategic implication of stack depth is the question of how much equity a… When does ICM become significant in MTTs? The Independent Chip Model (ICM) was invented in 1987 by M. Malmuth and is one… How to Study GTO Solutions Tools like GTO Wizard and other solvers should not be used to memorize specific strategies.… Principles of GTO Game theory, or GTO (for Game Theoretically Optimal), is best understood not as a rigid… Interpreting Equity Distributions GTO Wizard’s range vs range equity distribution graphs are an excellent tool for visualizing which… What is Leverage in poker? Leverage refers to the risk of future betting. When you call a bet on the… Reasons for value betting in poker You’re probably accustomed to thinking of bets as either value or bluff. In this taxonomy,… The Science of Poker Performance Poker is a game of small edges and big swings. Every hand is an investment… What is Valor in poker? Can a poker hand be courageous? How does your ability to value bet future streets… Pot Geometry “Pot Geometry” refers to betting an equal fraction of the pot on each street, such… A Beginner’s Guide to Poker Combinatorics Combinatorics is a fancy term for evaluating the number of possible “combinations” (combos) of any… Stack-to-pot ratio Stack-to-pot ratio, or SPR, is a way of measuring how deep the effective stack is… Equity Realization In poker jargon, equity expresses how much of the pot a hand will win, on… Variance and Bankroll Management Variance and Bankroll Management. Vital Soft Skills for Every Poker Player. How to Become a GTO Wizard Welcome to the wonderful world of game theory optimal |
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Cleaning Out Sticky IP Callers A “sticky” player on your left, one who calls your preflop raises too often and doesn’t give up easily after the flop , can be a major headache. Part of what makes them so annoying is that your options are limited. Being out of position is a big disadvantage, sometimes so compelling that you just have to give up. Their stickiness is not necessarily a mistake. But there are players out there who fight too hard, who think every flop that doesn’t obviously favor the raiser’s range is theirs for the stealing. This article will examine how to exploit them, turning their stubbornness into a liability. Test Scenarios These scenarios start with 30bb CO vs BTN single-raised pot (SRP) Single-raised pot (SRP) A pot where one player open-raised and was called preflop. ranges. These positions are where you are most likely to play pots OOP as the preflop raiser (PFR), and this stack depth is one where excessive stubbornness will be easier to exploit. The deeper the stacks, the more correct it is for your IP opponent to get sticky after the flop, and so the fewer opportunities you will have to exploit them. We will look at two flops: 9 ♥ 6 ♥ 6 ♦ , CO has a significant continuation betting range at equilibrium, and T ♥ 3 ♥ 2 ♦ , where CO checks their entire range. I’ve chosen these flops because they are of the kind where a cold-caller is most likely to get stubborn. Ace- and King-high flops are generally perceived as favoring the raiser and being easier to play from out of position. Hence, you are less likely to encounter opponents who float and bluff-raise exploitatively on those boards. For each board, we will compare CO’s equilibrium flop c-betting strategy with one where BTN is nodelocked to fold half as often to all bet sizes (except all-in, against which a sticky opponent could not justify a light call with the plan of outplaying on later streets). We will also examine how CO’s turn strategy should change after betting the flop into a stubborn opponent. Flop bet size options for both players are 20%, 50%, or 80% of the pot or all-in. Later street options are set using the Custom Solving ‘Automatic’ feature . Ace-High C-Bet Flop Perceptions aside, CO actually enjoys slight equity and nuts advantages on a 9 ♥ 6 ♥ 6 ♦ flop. They have more 6-x in their range and more overpairs, which are also very strong on this board. Only the dynamism of the board favors BTN, whose position will enable more informed decisions when the board texture changes on later streets. But with both players having such wide ranges, few turn cards will dramatically change the dynamics. As a result, CO continuation bets roughly half their range for half the pot. At equilibrium, BTN folds about 28% of the time to this bet. Let’s see how CO’s strategy changes when we nodelock BTN to fold half as often to any of these bet sizes: CO bets |
much more often and stops using the half pot size entirely . The 80% pot bet takes priority, but the 20% pot bet also sees significant use. The Manhattan graphs provide better insight into what’s going on. At equilibrium, CO’s bets come from all over the spectrum, with the only discernible pattern being a slight preference for betting stronger hands. This is typical for out of position play, especially on a dynamic board. Information hiding is of utmost importance, so CO includes many different types of hands in both their betting and checking ranges. When CO can predict that BTN will call too often, they mostly lose the incentive to hide information. They consistently bet big with their strongest hands (“strongest” here includes even some hands with less than 50% equity), bet small with their next-best hands, and check their weakest hands. The reason to hide information is to prevent your opponent from playing correctly against you. If you believe they will not play correctly, you can mostly forgo information hiding in favor of taking advantage of the imbalance you anticipate. In this case, that mostly means big value bets. If you can predict that your opponent will call or raise too often, you can forgo information hiding in favor of taking advantage of their imbalance. There are no pure bluffs here—you don’t want to bluff an opponent who calls too much—but there are plenty of semi-bluffs from hands with 25-50% equity. Why does CO bet hands that are not favored into an opponent whom they know will rarely fold? Two reasons: Most of these hands have too much equity to check-fold , so they will put chips in against BTN’s best hands whether they bet or check. Betting enables them to also put chips in against some of BTN’s weaker hands, the ones that should fold but which a sticky player will continue with. Against that part of BTN’s range, these hands may well be favored. Calling too wide on the flop will lead to BTN having many weak hands on the turn. One way to exploit that is to bet strong hands on the flop, as we see CO doing here. Another way is to barrel aggressively on the turn to force folds from those weak hands. Many of CO’s lower-equity flop bets will be good candidates for barreling turns. Barreling the Turn Although we can’t run reports on Custom Solutions ( yet ), we can “spot check” a few turn cards to get a sense of how CO’s barreling strategy changes against a sticky BTN. As we consider these scenarios, let’s keep a few things in mind: The only imbalance we are assuming in BTN’s strategy is not folding often enough to flop bets. That means that, in these simulations, BTN is assumed to play optimally on the turn . They will still be at a disadvantage due to arriving at the turn with an overly weak range, but these simulations do not rely on any assumptions about what BTN will do wrong on the turn. A real-life opponent |
who floats the flop aggressively might as well bet too often when checked to on the turn, which would increase CO’s incentive to check-raise, but that kind of assumption is not reflected here. Comparisons of CO’s equilibrium to exploitative turn strategies are not apples-to-apples. In fact, the exploitative scenarios often involve bet sizes CO uses rarely or never at equilibrium. So, they reflect changes not only in BTN’s flop calling range but also in CO’s betting range . We’ll start by looking at a blank turn. In an unlocked scenario, had CO bet around 80% pot on the flop, here’s how they would follow through on a 2 ♣ turn : And here’s their strategy against a BTN who does not fold enough on the flop: The major difference, in terms of frequencies, is less checking and more use of the smaller 45% pot bet size. This reflects the incentive we identified above to barrel into BTN’s overly weak calling range . These bets are small because they target weak hands that struggle to continue even getting 3:1 odds in position. On a blank turn, the weakest part of BTN’s range consists of backdoors, which did not improve, and hands like A♠8♠ or K ♥ J ♠ . BTN sometimes floats with these hands at equilibrium, but not as often as when they are nodelocked to fold less. This leads to an imbalance on blank turns, where they have too many such hands and so are vulnerable to small bets . Exploitative opportunities are fewer on a T ♣ turn , which connects better with BTN’s floating range. Here’s the equilibrium strategy: And the exploitative one: We see the same increased frequency of smaller bets , but the difference is less dramatic. Do keep in mind, however, that CO’s range for seeing the turn is wider in the exploitative scenario, so, for instance, shoving 18% of their range in both scenarios means shoving more hands in the exploitative one. The same kinds of shoving hands like top pair A9 , and draws, but CO has more of this type of hands because they bet them more often on the flop. They are extra incentivized to bet the flop with AT , for instance, because it performs so well against backdoor draws like KT , QT , and JT which are disproportionately represented in BTN’s expanded calling range. When the T turns, AT is a great shove because it gets called by those dominated Ten-x hands while denying a lot of equity to QJ and KJ , which could profitably call smaller bets. So, CO’s EV is dramatically improved in the exploitative scenario, even though their strategy does not appear wildly different. They earn 8.9bb in EV, as opposed to 7.4bb at the unlocked version of this same node. Even on a 2 ♥ turn, which completes the most obvious draw, BTN simply has too many low-equity hands and is vulnerable to small bets. CO’s equilibrium strategy bets 45% pot with 16% of hands on this card, while the exploitative one bets 52% |
of hands for that same size. And that’s despite the CO having a wider range themselves in the exploitative scenario! BTN faces the familiar problem of too many backdoors that whiffed. Hands like A7 , A8 , and Q ♠ J ♠ cannot withstand even modest pressure on this turn. Gutshots like T ♠ 8 ♠ also purely fold. Not even all of BTN’s turned heart draws can continue! Some weaker overcard floats fold despite picking up the draw: You might wonder why BTN would call the flop with these hands in the first place if they have to fold to a small bet on a turn card that improves them. The answer is they wouldn’t, at least not so often, at equilibrium. Their exploitable tendency to call too often on the flop is what got them into this mess. Now, holding a range that’s too wide on the turn, the best they can do is fold away their many weak hands. Keep in mind: that’s the best they can do. Even if BTN folds these weak hands, CO’s exploitative strategy shows a profit . But these folds aren’t nodelocked. That means the solver considered other options, such as bluff-raising or floating a second time, and found them to be, at best, no more profitable. So, the exploit of barreling frequently here does not rely upon your opponent folding these weak hands on the turn for its profitability. If they refuse to fold, your bluffs may lose money, but you will more than make up for it with your value bets. Once BTN arrives on the turn with too wide a range, they have no good options . That’s what CO’s expanded betting range is exploiting. Low C-Bet Flop CO’s equilibrium strategy on T ♥ 3 ♥ 2 ♦ is to check their entire range, yielding an EV of 2.45bb. Even though continuation betting is not part of CO’s equilibrium strategy, BTN’s response to bets of various sizes is still an important part of their equilibrium strategy. Remember: an equilibrium strategy does not assume the opponent will also play an equilibrium strategy. Rather, it is an unexploitable strategy for responding to any action the opponent might take. So, we can still see how often BTN would fold to various bets and then nodelock them to fold half as often, just as in the above example. Rope-a-Doping an Aggressive Raiser When we nodelock BTN to under-fold this flop, they adjust by raising hands that would otherwise fold, providing an opportunity to investigate how to exploit an over-aggressive raiser. Against a BTN who bluff-raises excessively , CO improves their EV to 2.53bb by developing a betting range: Naturally, this range is weighted toward strong hands , but the strategy is not as straightforward as “bet all your best hands.” Indeed, CO never bets the nuts, which has a strong blocking effect on BTN’s raising range, and they only intermittently bet other strong hands like AA and KK . The hands CO most wants to bet are medium-strength , those strong enough to get all-in |
on the flop but also vulnerable enough to benefit from fold equity. This includes a lot of top pairs, but also some lower pocket and board pairs, and the best unpaired hands. If these hands benefit from fold equity, why does CO want to bet them into an under-folding opponent? BB still has a folding range. It’s weaker than it would be at equilibrium, but it does include hands with live equity against AK and KT . Unlike TT or AA , these hands are vulnerable enough that they benefit enough from folding out even the weakest part of BTN’s range. But they’re also strong enough that they welcome the extra action from an opponent who under-folds. As in the 966 example, CO even bets a few quite weak semi-bluffs like QJ with a heart. In addition to their blocking effect, these will make good barreling candidates for pressuring BTN’s overly weak range on many turns. If raised, CO plays a virtually pure 3-betting strategy: If you know your opponent will under-fold and over-raise, you don’t want to bet hands that can’t stand up to the raise. And because CO favors betting strong but vulnerable hands, they’d rather 3-bet to deny equity than call and play turns out of position. Essentially, what they are doing is betting to induce a raise from an over-aggressive opponent, then 3-betting to lock up their equity . Hence, the ‘rope-a-dope’. Floaters Excessive calling is much less exploitable than excessive raising, as it forces CO to play out of position on future streets rather than allowing them to deny equity and end the action on the flop with a reraise. In fact, the ease with which BTN can float the flop and outplay them on later streets is the main reason CO does not continuation bet at equilibrium. Excessive calling (in position) is much less exploitable than excessive raising (IP). We can test this by nodelocking BTN to fold half as often while leaving their frequency of raising unchanged: CO can exploit this imbalance for just 2bb/100 by developing a narrow betting range: As we’ve seen with previous exploitative betting ranges, this one is disproportionately but not exclusively strong . It includes some semi-bluffs that can continue applying pressure on the turn. Mostly, however, CO still checks. Being out of position on such a dynamic board is simply a huge disadvantage. BTN is incentivized to put in a lot of money at future decision points after CO checks, so even when CO knows they will be called excessively on the flop, they still check many strong hands. Conclusion Stubborn, “sticky” opponents come in many varieties . Some are loose passive calling stations that just want to see more cards. Some love to bluff-raise, while others prefer floating to bluff turns. With a more specific read, we could craft more nuanced exploits for each type. However, it’s important to not let the perfect be the enemy of the good . You don’t need a laser-sharp read or even a prediction of how your opponent will respond to an |
exploit in order to profit from that exploit. We, for example, found a powerful, profitable exploit by identifying that our opponent was not folding enough—without taking a stance on whether they would call more or raise more or how they would fold the turn. When your opponent does not fold enough, you should bet more often , including in situations where you would not ordinarily bet at all. Your betting range should be heavy on value , but your best hands are not necessarily the ones that gain most from betting. The best bets are hands that welcome a wider calling range (this includes strong made hands but also some semi-bluffs) but also benefit from folding out weak holdings . Then, on the next street, you continue the aggression with both value bets and bluffs. This enables you to profit from your opponent’s overly weak range no matter how they respond. GTO Wizard the #1 App for Poker players Study any spot imaginable Practice by playing vs. GTO Analyze your hands with 1-click START CRUSHING NOW JOIN DISCORD Author Andrew Brokos Andrew Brokos has been a professional poker player, coach, and author for over 15 years. He co-hosts the Thinking Poker Podcast and is the author of the Play Optimal Poker books, among others. Latest article Navigating Nasty Rivers Out of Position Playing out of position (OOP) is hard. It’s fundamentally disadvantageous to act on each street… Crack the Shell of Nut Draw Strategy Nut draws hold obvious appeal as betting and raising candidates. They benefit from fold equity,… Preflop Raise Sizing: Examining 2 Key Factors It amazes me that, after nearly twenty years of playing no-limit hold ‘em professionally, I… How to Handle Loose-Passive Limpers Solver charts and presolved solutions do not offer much in the way of explicit guidance… Top 5 Mistakes in Spin & Gos As a Spin and Go poker coach, I regularly encounter common mistakes that players make,… An Introduction to Spin & Gos I was playing Tournaments for the first decade of my career. But after my first… Relationships and Poker In my more than 15 years of poker coaching, I’ve helped hundreds of clients improve… Do Multiple Sizes Matter? Should I work out the other tools or just stick to the crowbar? Solver results… The Science of Learning Applied to GTO Wizard The way we were taught to study in schools was flawed and has negatively impacted… Check-Raising a Single Pair In this article, I will address a question raised by Twitter/X user ’Matt Riley’, who… The Turn Probe Bet Imagine yourself defending as the BB caller vs BTN preflop raiser in a 100bb cash… When To Encourage Multiway Pots in PKOs I was recently shown this fascinating PKO hand from the GTO Wizard PKO library. Somebody… Protect Equity and Prosper When we think about the best hands to bet in a given situation, we tend… Do Solvers Have Targets? Targeting has long been a staple of exploitative poker thinking. When you’re value betting against… Disciplining Big Blind in Limped Pots When the BTN open-limps |
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edges and big swings. Every hand is an investment… What is Valor in poker? Can a poker hand be courageous? How does your ability to value bet future streets… Pot Geometry “Pot Geometry” refers to betting an equal fraction of the pot on each street, such… A Beginner’s Guide to Poker Combinatorics Combinatorics is a fancy term for evaluating the number of possible “combinations” (combos) of any… Stack-to-pot ratio Stack-to-pot ratio, or SPR, is a way of measuring how deep the effective stack is… Equity Realization In poker jargon, equity expresses how much of the pot a hand will win, on… Variance and Bankroll Management Variance and Bankroll Management. Vital Soft Skills for Every Poker Player. How to Become a GTO Wizard Welcome to the wonderful world of game theory optimal poker! This article will serve as… The Three Laws of Indifference Indifference in poker is one of the most misunderstood concepts. The word “indifference” means that… Poker subsets and abstractions An “abstraction” is a way to simplify the game of poker. This game is so… Mathematical Misconceptions in Poker Pot Odds and MDF are foundational mathematical formulas in poker. How to solve toy games Poker is a complicated game. It’s impossible to calculate GTO strategies on the fly. By… The Value of Fold Equity – Experiment The term “fold equity” is ambiguous in the poker community. Firstly, it’s an EV equation,… MDF & Alpha MDF & Alpha. Minimum Defence Frequency and Alpha are metrics on poker that determine how… What are Pot Odds in poker? Pot odds are a fundamental calculation in poker. In the simplest terms, pot odds tell… Does your range affect your strategy? Advanced players will often tell you that your range affects your strategy. Visualizing implied odds Implied odds are the value you expect to gain over and above your raw equity.… ICM Basics As a tournament player, you may have heard about the concept of “ICM”, but what… What is Equity in Poker? The term “Equity” is ubiquitous throughout poker theory. Equity refers to your chances of winning… Range Morphology “Morphology” is the study of form or structure. You’ve probably heard terms thrown around such… What is Expected Value in Poker? Expected value (EV) is the most fundamental metric in poker. Every decision you make is… Why doesn’t my solution match GTO Wizard? So, you’ve got your own solutions to some spot, compared it to GTO Wizard, and… Understanding Nash Distance Understanding Nash Distance. One of the most common questions when studying with solvers is “Why… Combinatorics Poker Puzzle Today we have a special puzzle for you that will test your intuition as a… The 10 Most Important Concepts for New Poker Players The 10 Most Important Concepts for New Poker Players. Fundamentals are one of the most… What is GTO in Poker? The meaning of GTO in poker is Game Theory Optimal. Game theory is a study… What does GTO aim to achieve? So what exactly is a GTO strategy? What makes it “unexploitable”? What does it try… |
A Beginner’s Guide to Poker Combinatorics Combinatorics is a fancy term for evaluating the number of possible “combinations” (combos) of any given hand: the combination of 2 cards of certain ranks and suits. It allows us to answer questions like how many different versions of AK you can hold in a specific spot, what hands make for better bluff-catchers and many more. Combo Counting Basics The Hand Matrix There are 1326 possible combinations of starting hands in Texas Hold’Em. They are frequently represented by a grid that is commonly referred to as a hand matrix. Simply looking at the hand matrix, it would be easy to mistakenly assume that pocket pairs comprise 13/169 of possible hands and suited and offsuit hands each make up 78/169. By looking at the grid and summing the squares that correspond to each hand class we would arrive at the aforementioned answer. However, this relies on all squares being equal to each other. Despite this being the case geometrically, it is not the case combinatorically. There are in fact: 6 combos of each pocket pair A ♠ A ♥ , A ♠ A ♦ , A ♠ A ♣ , A ♥ A ♦ , A ♥ A ♣ , A ♦ A ♣ 4 combos of each suited hand A ♠ K ♠ , A ♥ K ♥ , A ♦ K ♦ , A ♣ K ♣ 12 combos of each unpaired, offsuit hand A ♠ K ♥ , A ♠ K ♦ , A ♠ K ♣ , A ♥ K ♠ , A ♥ K ♦ , A ♥ K ♣ , A ♦ K ♠ , A ♦ K ♥ , A ♦ K ♣ , A ♣ K ♠ , A ♣ K ♥ , A ♣ K ♦ This leads us to the conclusion that there are 78 pocket pairs ( 6 combos times 13 squares), 312 suited hands ( 4 combos times 78 squares) and 936 unpaired, offsuit hands ( 12 combos times 78 squares). The Card Matrix Dead Cards The above ratios are applicable preflop but falter when in the presence of “dead cards”. Dead cards refer to cards that are known to not be in a player’s hand. The most common example occurs when a flop is dealt. The flop cards are dead as it is impossible for any player to hold them. The simplest way to count combinations is to multiply the number of unseen cards . Let’s go over some examples to familiarize you: BTN opens, BB calls, flop is AK2: How many combinations of AQ does the BTN have? Note that 3/12 of those are AQ s, and 9/12 are AQ o. Suited: A ♦ Q ♦ A ♥ Q ♥ A ♣ Q ♣ A ♠ Q ♠ Offsuit: A ♦ Q ♥ A ♦ Q ♣ A ♦ Q ♠ A ♥ Q ♦ A ♥ Q ♣ A ♥ Q ♠ A ♣ Q ♦ A ♣ Q ♥ A ♣ Q ♠ A ♠ Q ♦ A ♠ Q ♣ |
A ♠ Q ♥ How many combinations of AA does the BTN have? Pocket pairs use a slightly different formula. We start with 6 combos of AA . If the flop comes A ♠ K ♥ 2 ♦ , the A ♠ is no longer in our range. Our combos of AA are reduced from 6 to 3 : A ♠ A ♥ , A ♠ A ♦ , A ♠ A ♣ , A ♥ A ♦ , A ♥ A ♣ , A ♦ A ♣ . We can calculate the number of available combos of a pocket pair using the following formula: Where a = the number of available cards of the pocket pair’s rank. In this scenario, we would input a = 3 since there are 3 available aces (A ♥ , A ♦ and the A ♣ ). (3 * (3-1))/2 = 3 combinations of AA remaining. Another way of saying this is “3 choose 2”. How many combinations of AKs does the BTN have? Suited hands can be counted intuitively and do not require a formula. On A ♠ K ♥ 2 ♦ , the number of available combos of AK s is reduced from 4 to 2 due to the A ♠ and K ♥ on board: A ♠ K ♠ , A ♥ K ♥ , A ♦ K ♦ , A ♣ K ♣ . We can calculate the total number combos of AK using the following formula: Where a = the number of available cards of 1 card’s rank and b = the number of available cards of the other card’s rank On A ♠ K ♥ 2 ♦ , there are 3 available aces (A ♥ , A ♦ , A ♣ ) and 3 available kings (K ♥ , K ♦ , K ♣ ). We would therefore input a = 3 and b = 3, giving us a result of 9. Knowing that 2 of these combos are suited, we are left with a remainder of 7 combos of AK o. Other Card Removal Examples Dead cards exist when public information makes certain hands impossible to hold. There exist other, more subtle versions of card removal, however. One notable version is blockers. This is when one player uses the private information provided by their hand, or private dead cards if you will, to combinatorically narrow their opponent’s range. Simple Blocker Example Blockers are a vital part of poker strategy that can make or break close decisions. By eliminating specific combos from players’ ranges, you can shift their action frequencies and value to bluff ratios. Let’s examine a simple example where we can make use of this effect: 🎲 [6Max Cash, 500NL, 100BB] BTN opens to 2.5BB and BB defends. BB check-calls a 33% flop c-bet, a 125% turn c-bet and faces an 85% river c-bet on J ♠ 5 ♥ 2 ♠ K ♦ 8 ♠ . Which of the following hands can profitably call? K ♠ 9 ♥ K ♥ 9 ♠ K ♣ T ♣ A ♥ |
5 ♦ 🎲 Answer K ♠ 9 ♥ BB needs at least 31.5% equity to profitably call ( calculate your pot odds ). By “blocking” value hands, K ♠ 9 ♥ has more equity against BTN’s range than the other answers and can make this call. BTN value bets two pair and better on this river and bluffs ace high and worse so all the listed hands are bluffcatchers. The ratio of value and bluffs in our opponent’s range changes depending on the cards we hold! BB only has more than 31.5% equity vs BTN’s range when holding K ♠ 9 ♥ , making it the only profitable (+EV) call. View this spot in GTO Wizard . Strategic vs Actual Frequencies Despite blockers changing range action frequencies, solvers (including GTO Wizard ) do not display the impact of this effect in their strategic frequencies. This leaves a discrepancy between strategic and actual frequencies. This is clearly demonstrated on flush boards: [6Max Cash, 500NL, 200BB] HJ opens to 2.5BB and BB defends. BB check-calls a 130% c-bet, the turn checks through and BB check-shoves (for 407%) vs a 61% river bet on a board of Q ♠ J ♠ 4 ♥ 7 ♦ 2 ♠ . Despite the strategic frequency window declaring that HJ calls 19.8% of the time (1.07 combos out of a total 5.51 combos), they only call 13.8% in reality (0.52 combos out of a total of 3.78 combos). The 19.8% frequency includes “ghost” combos: ones that cannot exist due to our opponent’s blockers. BB only shoves hands that contain the A ♠ , however, HJ’s 19.8% calling frequency includes many A ♠ hands. Once removed, the calling frequency becomes 13.8% . This doesn’t mean the strategy is incorrect. The expected value calculations account for this effect. It’s just a matter of interpreting and displaying data. Do we want to examine the strategic frequency or the actual frequency? Some solvers have a switch that allows you to alternate between both views. Check out this spot in GTO Wizard . Card Bunching The final card removal effect we will examine is card bunching. Players tend to fold low cards more often than high cards preflop. So when many players have folded, the remaining players and deck are skewed towards high cards. At a 3-handed table, BTN acts first, and every combo in their range is equally likely. This is not the case at a 6-handed table when several players have already folded. This card removal affects the ranges and strategy of all players to act, frequencies, and runout probabilities. UTG RFI range at a 6-handed table This means that BTN is more likely to hold higher cards when it has folded to them. This can change the BTN’s actual opening frequency even if they don’t change their strategic opening frequency . Let’s visualize this effect. The following animation displays the density of cards from 2 through Ace in the deck after several players fold. Remember that they’re more likely to fold low cards, so the remaining cards in the deck (and |
player’s ranges who are yet to act) are more top-heavy. Generally speaking the card removal due to bunching is not very significant, although it depends on the spot. As this is a beginner article, we’ll not cover this effect in great detail. Check out this article by HRC if you’d like to learn more about the bunching effect. Conclusion There are 1326 possible combinations of starting hands in Texas Hold’Em made up of 78 pocket pairs, 312 suited hands and 936 unpaired, offsuit hands. Some of these combos can be blocked by the board or players’ hands and ranges, further decreasing the number of available combos. This should be accounted for and optimized for when forming a strategy. There are 3 key card removal effects: dead cards , blockers and card bunching ; each with their own unique properties. Try this combinatorics poker puzzle if you feel like challenging your skills! If there’s one thing you take away from this article, it’s to think about your hands a little less linearly and more in relation to the range you are facing and how this dictates your use of blockers. Good luck at the tables! GTO Wizard the #1 App for Poker players Study any spot imaginable Practice by playing vs. GTO Analyze your hands with 1-click START CRUSHING NOW JOIN DISCORD Author Tombos21 Tom is a long time poker theory enthusiast, GTO Wizard coach and YouTuber, and author of the Daily Dose of GTO. Latest article Preflop Raise Sizing: Examining 2 Key Factors It amazes me that, after nearly twenty years of playing no-limit hold ‘em professionally, I… How to Handle Loose-Passive Limpers Solver charts and presolved solutions do not offer much in the way of explicit guidance… Top 5 Mistakes in Spin & Gos As a Spin and Go poker coach, I regularly encounter common mistakes that players make,… An Introduction to Spin & Gos I was playing Tournaments for the first decade of my career. But after my first… Relationships and Poker In my more than 15 years of poker coaching, I’ve helped hundreds of clients improve… Do Multiple Sizes Matter? Should I work out the other tools or just stick to the crowbar? Solver results… The Science of Learning Applied to GTO Wizard The way we were taught to study in schools was flawed and has negatively impacted… Check-Raising a Single Pair In this article, I will address a question raised by Twitter/X user ’Matt Riley’, who… The Turn Probe Bet Imagine yourself defending as the BB caller vs BTN preflop raiser in a 100bb cash… When To Encourage Multiway Pots in PKOs I was recently shown this fascinating PKO hand from the GTO Wizard PKO library. Somebody… Protect Equity and Prosper When we think about the best hands to bet in a given situation, we tend… Do Solvers Have Targets? Targeting has long been a staple of exploitative poker thinking. When you’re value betting against… Disciplining Big Blind in Limped Pots When the BTN open-limps off a short stack, the postflop play against the BB should… I’d Rather |
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poker? Can a poker hand be courageous? How does your ability to value bet future streets… Pot Geometry “Pot Geometry” refers to betting an equal fraction of the pot on each street, such… A Beginner’s Guide to Poker Combinatorics Combinatorics is a fancy term for evaluating the number of possible “combinations” (combos) of any… Stack-to-pot ratio Stack-to-pot ratio, or SPR, is a way of measuring how deep the effective stack is… Equity Realization In poker jargon, equity expresses how much of the pot a hand will win, on… Variance and Bankroll Management Variance and Bankroll Management. Vital Soft Skills for Every Poker Player. How to Become a GTO Wizard Welcome to the wonderful world of game theory optimal poker! This article will serve as… The Three Laws of Indifference Indifference in poker is one of the most misunderstood concepts. The word “indifference” means that… Poker subsets and abstractions An “abstraction” is a way to simplify the game of poker. This game is so… Mathematical Misconceptions in Poker Pot Odds and MDF are foundational mathematical formulas in poker. How to solve toy games Poker is a complicated game. It’s impossible to calculate GTO strategies on the fly. By… The Value of Fold Equity – Experiment The term “fold equity” is ambiguous in the poker community. Firstly, it’s an EV equation,… MDF & Alpha MDF & Alpha. Minimum Defence Frequency and Alpha are metrics on poker that determine how… What are Pot Odds in poker? Pot odds are a fundamental calculation in poker. In the simplest terms, pot odds tell… Does your range affect your strategy? Advanced players will often tell you that your range affects your strategy. Visualizing implied odds Implied odds are the value you expect to gain over and above your raw equity.… ICM Basics As a tournament player, you may have heard about the concept of “ICM”, but what… What is Equity in Poker? The term “Equity” is ubiquitous throughout poker theory. Equity refers to your chances of winning… Range Morphology “Morphology” is the study of form or structure. You’ve probably heard terms thrown around such… What is Expected Value in Poker? Expected value (EV) is the most fundamental metric in poker. Every decision you make is… Why doesn’t my solution match GTO Wizard? So, you’ve got your own solutions to some spot, compared it to GTO Wizard, and… Understanding Nash Distance Understanding Nash Distance. One of the most common questions when studying with solvers is “Why… Combinatorics Poker Puzzle Today we have a special puzzle for you that will test your intuition as a… The 10 Most Important Concepts for New Poker Players The 10 Most Important Concepts for New Poker Players. Fundamentals are one of the most… What is GTO in Poker? The meaning of GTO in poker is Game Theory Optimal. Game theory is a study… What does GTO aim to achieve? So what exactly is a GTO strategy? What makes it “unexploitable”? What does it try… |
Is Donk Betting for Donkeys? “Donk betting”, or betting from out of position into the player who was the aggressor What is Aggression in poker? Any bet or raise is considered an aggressive action. A player who frequently tends to bet or raise more than they check or call can be considered aggressive. on the previous street, is a derogatory term originating in poker’s younger days, when such bets were considered the hallmark of a bad player, or “donk” (short for donkey). That stereotype is not without basis in reality: betting into the previous street’s aggressor is rarely correct, so when it is done, it is often done incorrectly. The following chart, based on data from GTO Wizard’s Flop Reports, shows the BB’s Big Blind The minimum forced betting increment in a game of poker (bb). Can also refer to the position ‘big blind’ (BB), which is forced to put in the largest blind preflop, typically after the small blind. The big blind position acts last preflop and ats first postflop. average flop donk betting frequency at equilibrium at a variety of MTT Multi Table Tournament (MTT) A poker tournament with at least two tables of players. stack sizes and facing opponents in early, middle, and late positions. There appears to be slightly more donk betting as the opener’s position gets later, but even that is not true across the board. By far the most clear trend here is simply that the BB should almost never donk bet unless doing so with a clear exploitative purpose . Donk betting should not be your top study priority, and until you are studied on it, you are better off sticking to a simple rule of never donk betting. That doesn’t quite tell the full story, though. Here’s another chart, showing the BB’s donk betting frequency against a BTN opener, sorted by the high card on the flop. Against the HJ And against UTG With the data sorted in this way, we can see there are a few flops on which donk betting is in fact a significant part of the BB’s strategy. Low card flops rarely occur , which is why the global average is so low, but when a 6- or 5-high flop does come along, it is not necessarily donkey behavior for the BB to come out betting. That said, it is nevertheless true that donk betting is rarely correct and easy to misapply . It should not be your top study priority, and until you are studied on it, you are better off sticking to a simple rule of never donk betting. This rule will rarely lead you astray, whereas donk betting haphazardly can lead to big mistakes and exploitability. When is Donk Betting Correct? If you’re still reading, you’re either advanced and well-studied or stubborn (like a donkey…) . Virtually all the BB’s donk betting occurs on low card flops. The thing about 6- and 5-high flops is they are almost always connected. Low-connected boards are among the worst for the preflop raiser to continuation bet, which is where the |
BB gets their incentive to take the betting lead . Donk betting occurs exclusively in situations where the BB should not expect their opponent to continuation bet often. On boards where the preflop raiser is likely to bet anyway, there is little reason for the BB to pre-empt them. Whether the BB wishes to bet for value or as a bluff, they’d prefer to let their opponent put money into the pot with a wide , weak range first. So, donk betting occurs exclusively in situations where the BB should not expect their opponent to continuation bet often if checked to. Two factors make a flop less desirable for continuation betting: The preflop raiser loses their equity advantage . A preflop raiser sees the flop with a much stronger range than a BB caller, and so has a significant equity advantage on the average flop, which enables them to bet at a high frequency and still force a lot of folds from the BB. Only very specific flops, usually low or medium-connected cards, bring the BB up to close to 50% equity. BB flops the nuts advantage . This doesn’t mean the literal nuts Nuts Describes the strongest possible hand on a certain board. For example, on K 2 7 8 9 , any player holding any combination of JT is holding the nuts. . In fact, with very shallow stacks, any top pair and even some second pair can be strong enough to play as the nuts. But boards where the BB is more likely to flop sets, straights, and two pairs are harder for the preflop raiser to continuation bet because of the risk of a check-raise. For example, here’s the range breakdown for 20bb UTG vs. BB spot on a 654r flop: This is a spot where BB donk bets 60% of their range. Notice how the equities are almost exactly 50/50, with the BB holding more of the Best Hands and Good Hands. For comparison, here’s the same configuration on a JT9r flop, which BB never donk bets: Which Hands Should Donk Bet? Even after you identify a good flop for donk betting, there remains the thorny problem of which hands belong in your betting range. The following chart shows BB’s donk betting range on a 764r flop facing an UTG opener with 20bb effective stacks. This is one of the highest frequency donk betting spots across all positions and stack sizes, so it provides a clearer picture of what makes specific hands appealing for betting. Part of what makes donk betting so challenging to implement correctly is that many candidates are not obvious or intuitive, even in this relatively high frequency betting spot. We can start by noting some trends among BB’s betting hands, however: Top pair is an almost pure bet, regardless of kicker. Second pair is also a very high frequency bet. Most open-ended straight draws bet. There’s a lot of betting from BB’s better A x and K x hands. We can also look at what kinds of hands do not bet. BB’s most |
consistent checks are: Smaller offsuit overcards ( QJ o, JT o). A2 o and K2 o, which not only have a weak kicker but also lack the straight draw that A3 and K3 have. Their very worst pairs, especially if they also have a straight draw ( 33 mostly checks but 22 mostly bets) The nuts. BB mostly checks 85 , which blocks important continuing hands from UTG like 98 , 88 , and 87 , but mostly bets 53 . Slowplaying sets and two pairs is less desirable because they could easily lose action or even the entire pot to a scary turn card. Articulating these principles provides some insight into what exactly this donk bet accomplishes. Note that it’s a small bet, just 33% of the pot. The only other bet size included in this simulation is all in, which BB never uses, so we can’t say for sure the BB would never use, say, a 75% pot bet if given the option. But a small bet from out of position ought to result in UTG rarely folding, which means BB is mostly betting hands with good equity when called but which still benefit to some degree from denying equity . In other words, there are few airball bluffs in this range. A2 o and K2 o are among BB’s lowest equity hands, and those are pure checks. This betting range is mostly about thin value and protection . AT o and KT o are not simply bluffs. While they sometimes cause dominating hands like AJ o and KQ o to fold, they also get called by some hands they dominate, like QT s and K9 s. And they deny equity to QJ o, which would have not only 8 outs but also the benefit of position on later streets. By betting, these hands clean up their equity, setting them up to keep betting for value when they turn a T or to check and induce bluffs when they turn an A or K . The danger of betting so many medium-strength hands is that it incentivizes UTG to raise. Thus, BB balances these hands that hate to get raised with hands that do very well when raised. This includes not just the sets and straights but also the 75 , 65 , and 54 , which can shove to deny equity to UTG’s bluffs while maintaining robust equity even against the strong hands that call the shove. Most of BB’s betting range consists of hands that benefit from denying equity but also retain good equity and playability when called. Many of these hands are vulnerable to raises, so the nut hands and strong draws serve to make UTG indifferent to raising with many hands that would otherwise be good candidates. The danger of betting so many medium-strength hands is that it incentivizes UTG to raise. Deeper Stacks The shallower stacks make this play more feasible for BB. Getting raised holding T7 is not so scary with an SPR of 3. BB does not actually have more of the Best |
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