Poker Theory

November 17th, 2020

In this article I will lay out the foundational knowledge required to begin learning modern GTO (Game Theory Optimal) poker. At this point, the GTO vs Exploitative debate has essentially been settled and most reasonable people have acknowledged that if you want to be a profitable poker player you need to have a solid understanding of theory. Any exploitation should generally occur as a deliberate deviation from what you understand to be a theoretically correct play. But, if you do not have a foundational understanding of what theoretically correct play even is, your exploitative plays will be based on accurate hunches (at best) and complete delusion (at worst).

Advanced poker theory articles. How Equity Realization Impacts Every Hand You’ll Ever Play This article covers a concept that impacts all of your poker decisions. 3 Spots You Should Actually Consider Minimum Defense Frequency (And 3 Spots You Shouldn’t) Minimum defense frequency is a concept that you always have to consider when you face a bet. MIT 15.S50 Poker Theory and Analysis, IAP 2015View the complete course: Kevin DesmondAn overview of the course requ. This subreddit is for discussing poker theory and strategy as well as hand analysis. Do not post memes, royal flush images or other material unaffiliated with poker theory. Poker is a popular card game that combines elements of chance and strategy. There are various styles of poker, all of which share an objective of presenting the least probable or highest-scoring hand. Poker is any of a number of card games in which players wager over which hand is best according to that specific game's rules in ways similar to these rankings.Often using a standard deck, poker games vary in deck configuration, the number of cards in play, the number dealt face up or face down, and the number shared by all players, but all have rules which involve one or more rounds of betting.

Mathematical Foundations

Before diving into the foundations of GTO we will cover some very basic poker math. These concepts have been discussed ad nauseam in a variety of poker books but are still wroth being briefly mentioned.

Hand Combinatorics

There are three categories of hands:

• Suited holdings
  • 4 combos per holding
  • eg. AKs - 4 combos, one of spades, clubs, diamonds, hearts
• Unsuited Holdings
  • 12 combos per holding
  • eg. AKo - 12 combos
• Pocket Pairs
  • 6 combos per holding
  • eg. 66 - 6 combos

Pocket Pair Combinatorics

Pocket pairs have six combos. So for the holding ‘66’ there would be be:

• 6 hearts 6 diamonds
• 6 hearts 6 spades
• 6 hearts 6 clubs
• 6 diamonds 6 clubs
• 6 diamonds 6 spades
• 6 spades 6 clubs

Removing just one 6 from the deck takes the amount of possible combos to 3.Removing another 6 from the deck brings the possible combos of 6 down to one.(Check for yourself by trying to exclude a suit and see how many are left.

Suited holdings have 4 combos. Let’s say we have AKs, removing the ace of hearts would take the number of combos down to 3. Removing the king of hearts would have no effect since the AhKh combo has already been removed. Thus, the number of a suited combos in an opponent’s range is easy to calculate.

Off-suit Holdings have 12 combos. The removal factor on off-suit holdings is slightly more nuanced. Take the example of AKo

AKo - 12 combos

Each Ace is mapped to exactly three king cards (every suit of king other than it’s own suit). Likewise, each King is mapped to exactly three Ace cards.

Removing an Ace from the deck will make the possible number of AKo combos 9. Removing an King subsequently will take the number of combos down to 6.

Outs and Equity:

The amount of outs a hand has is simply the number of possible cards left in the deck that can result in a player making a draw of a pair etc.

A common mental math calculation is to multiply the number of outs by 4 on the flop for the probability of making a hand such as a flop or a straight by the river and to multiply by 2 on the turn.

Hand Type	Outs
Flush Draw	9
Open Ended Straight Draw	8
Gut-shot Straight draw	4

Pot Odds:

When an opponent bets, pot odds can be calculated as

[frac{bet}{pot + bet}]

This will give you the frequency at which you would need to call to theoretically keep your opponent indifferent to bluffing you with any two cards (more on this later). But as we will later discover this approach to defense in the game is heavily outdated, however for players who are vastly under or over-defending vs raises and bets, this concept can be helpful.

Hand Odds:

This will give you the amount of equity against your opponent’s range required to defend against a bet. Again, this model is outdated and has been disproven by solvers but is still useful for beginners.

[frac{bet}{2*bet + pot}]

Range Morphology

A range is a set of hands. Preflop, you have a range for each action, whether you open-raise, 3-bet or flat an open defines that range. You range will interact with flops in a variety of ways. There are a few words we use to describe ranges and their states.

• Linear Range: A range that contains the best hands, the worst hands, and everything in-between
• Condensed Range: A range that is condensed contains mostly medium strength hands with very few if any strong and weak hands
• Polarized: A polarized range contains the strong hands and weak hands but very few if any medium strength hands
• Capped Range: A capped range is a range that does not contain the very strongest hands on a board runout
• Uncapped: An uncapped range contains the best possible holdings

Basic Game Theory Optimal Concepts

Now that I have given you a brief summary of the foundations of poker strategy we will now dive into the basics of game theory optimal play.

History of GTO Play

The idea of an optimal strategy to poker is as old as the field of game theory itself. Von Neumann cited poker as the inspiration of his early works on game theory and wanted to formalize the strategies behind bluffing. John Nash, and Harold W. Kuhn also developed poker toy games which they solved to equilibrium in order to further develop their ideas on games of hidden information.

Why Things Are Different Now

The old way of conducting game theory research was to come up with a game, formalize the rules, and attempt to calculate an equilibrium and utility payoff structure. After deeply analyzing the game tree, mathematicians could calculate optimal solutions either by hand or with some form of computational assistance.

This was the technique used by Edward O. Thorpe when he first invented card counting for blackjack. Now, with very fast computers, we can simply run simulations of model poker games and observe their solutions.

Rather than theorizing about what might be optimal play, we can simply observe optimal play and work backwards in terms of developing heuristics to imitate software.

GTO Concept 1: Minimum Defense Frequency

In around 2013, Matthew Janda published his widely acclaimed book “Applications of No-Limit Hold’em.” This work popularized the concept of minimum defense frequencies and formed the back-bone of success behind many successful poker players such as Doug Polk, Ryan Fee and others.

Yet, the concept of MDF was not anywhere close to being representative of actual GTO play. It was however, so vastly superior to whatever techniques others were using in their calling/raising strategies at the time, that it made a great many players rich beyond their wildest dreams.

Let us suppose that you are facing a bet on the river. Your hand consists of a medium strength bluff catcher, where your opponent either holds the nuts, or total air. In other words; your opponent has a ‘perfectly polarized range’ consisting of hands with either 100% equity or 0% equity at an even distribution vs your bluff catcher.

How would be go about calculating the optimal frequency at which you need to call in order to keep villain indifferent to bluffing? In other words, how can we call in such a way that the EV of betting is effectively zero, and villain gains no additional value by bluffing?

[EV[Bluffing] = 0 = (Pot)*(1 - C) - (Bet)*C]

When villain attempts to bluff us and we fold, he wins whatever is in the pot. When villain attempts to bluff us and we call, he loses whatever it is that he has bet.By solving for the appropriate calling frequency we arrive at the conclusion that:

[C = frac{Pot}{Pot + Bet}]

The idea is that by calling at least C% of the time, we can assure that our opponent will not gain any additional utility by betting with a hand that would lose at showdown.

GTO Concept 2: Value to Bluff Ratios

• Let there be two players, an IP (in position) and an OOP (out of position) player.
• Let IP have a perfectly polarized range on the river (hands with either 100% equity or 0% equity) so that whenever OOP calls we effectively win.

Question: What proportion of our river betting range should be value hands (hands that win at showdown) vs what proportion should be bluffs to make the EV of our opponent calling zero?

• Let X be our bluffing frequency on a river bet
• Let P be the original size of the pot
• Let B be the size of our bet
• Assume EV of Calling - EV of Folding = 0 (Our opponent is indifferent to either actions)
• EV of Folding is always zero, therefore we need to only solve EV[Call]

Our opponent will win the size of the pot + our bet when he calls our bluffs and will lose the size of our bet when he calls our value bets.

[EV[Call] = 0 = (P + B)x - B(1 - x)]

Solving for x

[X = B/(P + 2B)]

Now using our equation we can determine the frequency at which we should be bluffing for any given bet size where B is a fraction of P.

• Pot Sized Bet (B = P): 33% bluffs
• 3/4 Sized Bet (B = .75P): 30% bluffs
• 1/2 Sized Bet (B = .5P): 25% bluffs
• 1/3 Sized Bet (B = .33P): 20% bluffs

Let’s of course notice that the larger our bet-size the more bluff heavy our range is! To most players this seems counter-intuitive but let’s use some very simple reasoning to think through it.

When an opponent calls our value bet, we win what’s already in the pot + whatever we risked. When an opponent calls our bluff, we only lose what we risked. Larger bet sizes cause a rational opponent to fold at a higher frequency meaning that we must accordingly add more bluffs. Additionally, we can afford to lose at showdown often when we bluff since value bets will win us much larger pots. Large bet sizes are positively correlated with a more polarized range (separation between bluffs and value bets) whereas smaller bet sizes are more useful when ranges are too heavy to want to generate folds. This is evident in many solver simulations.

Conclusion

In this article we covered the bare bones fundamentals of poker. If you are a new player I hope you enjoyed this introduction. In future posts, we will dive deeper into solver based poker strategies!

Poker is a popular card game that combines elements of chance and strategy. There are various styles of poker, all of which share an objective of presenting the least probable or highest-scoring hand. A poker hand is usually a configuration of five cards depending on the variant, either held entirely by a player or drawn partly from a number of shared, community cards. Players bet on their hands in a number of rounds as cards are drawn, employing various mathematical and intuitive strategies in an attempt to better opponents.

Given the game's many different forms and various dynamics, poker strategy becomes a complex subject. This article attempts to introduce only the basic strategy concepts.

The fundamental theorem of poker[edit]

Poker Theory Book

The fundamental theorem of poker, introduced by David Sklansky, states: Every time you play your hand the way you would if you could see your opponents' cards, you gain, and every time your opponents play their cards differently from the way they would play them if they could see your cards, you gain.^[1] This theorem is the foundation for many poker strategy topics. For example, bluffing and slow-playing (explained below) are examples of using deception to induce your opponents to play differently from how they would if they could see your cards. There are some exceptions to the fundamental theorem in certain multi-way pot situations, as described in Morton's theorem.

Pot odds, implied odds and poker probabilities[edit]

The relationship between pot odds and odds of winning is one of the most important concepts in poker strategy. Pot odds are the ratio of the size of the pot to the size of the bet required to stay in the pot.^[1] For example, if a player must call $10 for a chance to win a $40 pot (not including their $10 call), their pot odds are 4-to-1. To have a positive expectation, a player's odds of winning must be better than their pot odds. If the player's odds of winning are also 4-to-1 (20% chance of winning), their expected return is to break even (on average, losing four times and winning once for every five times they play such a pot).

Implied odds is a more complicated concept, though related to pot odds. The implied odds on a hand are based not on the money currently in the pot, but on the expected size of the pot at the end of the hand. When facing an even money situation (like the one described in the previous paragraph) and holding a strong drawing hand (say a Four flush) a skilled player will consider calling a bet or even opening based on their implied odds. This is particularly true in multi-way pots, where it is likely that one or more opponents will call all the way to showdown.

Deception[edit]

By employing deception, a poker player hopes to induce their opponent(s) to act differently from how they would if they could see their cards. David Sklansky has argued that winning at poker is often decided by how much one player can force another to change his/her style while successfully maintaining their own strategy.^[2]Bluffing is a form of deception where players bet strongly on a weak hand to induce opponents to fold superior hands. Related is the semi-bluff, in which a player who does not have a strong hand, but has a chance to improve it to a strong hand in later rounds, bets strongly on the hand in the hopes of inducing other players with weaker 'made' hands to fold. Slow-playing is deceptive play in poker that is roughly the opposite of bluffing: checking or betting weakly with a strong holding, attempting to induce other players with weaker hands to call or raise the bet instead of folding, to increase the payout.

Position[edit]

Position refers to the order in which players are seated around the table and the strategic consequences of this. Generally, players in earlier position (who have to act first) need stronger hands to bet/raise or call than players in later position. For example, if there are five opponents yet to act behind a player, there is a greater chance one of the yet to act opponents will have a better hand than if there were only one opponent yet to act. Being in late position is an advantage because a player gets to see how their opponents in earlier position act (which provides the player more information about their hands than they have about his). This information, coupled with a low bet to a late player, may allow the player to 'limp in' with a weaker hand when they would have folded the same hand if they'd had to act earlier. Position is one of the most vital elements to understand in order to be a long-term winning player. As a player's position improves, so too does the range of cards with which they can profitably enter a hand. Conversely this commonly held knowledge can be used to an intelligent poker player's advantage. If playing against observant opponents, then a raise with any two cards can 'steal the blinds,' if executed against passive players at the right time.

Reasons to raise[edit]

Unlike calling, raising has an extra way to win: opponents may fold. An opening bet may be considered a raise from a strategy perspective. David Sklansky gives seven reasons for raising, summarized below.^[1]

• To get more money in the pot when a player has the best hand: If a player has the best hand, raising for value enables them to win a bigger pot.
• To drive out opponents when a player has the best hand: If a player has a made hand, raising may protect their hand by driving out opponents with drawing hands who may otherwise improve to a better hand.
• To bluff A player raises with an inferior or 'trash' hand attempts to deceive other players about the strength of their hand, and hopefully induce a better hand to fold.
• To semi-bluff A player with a drawing hand may raise both to bluff and for value. While technically still a bluff, as the player may not end up with a made hand and is primarily trying to drive out players, the player still has the opportunity to make his or her hand and win the pot if the bluff is called.
• To block Players on drawing hands may put out a 'blocking bet' against players who are likely to bet when checked to, but unlikely to raise when bet into. This is a small bet made on a drawing hand to lessen the likelihood of having to call a larger bet from a player in late position.
• To get a free card: If a player raises with a drawing hand, their opponent may call the bet and check to them on the next betting round, giving them a chance to get a free card to improve their hand.
• To gain information: If a player raises with an uncertain hand, they gain information about the strength of their opponent's hand if they are called. Players may use an opening bet on a later betting round (probe or continuation bets) to gain information by being called or raised (or may win the pot immediately).
• To drive out worse hands when a player's own hand may be second best: A combination protection and probe raise, a player with a strong hand but not the 'nuts' (the hole cards that make the best possible hand given the current face-up cards) may raise, both to induce drawing hands that may improve to the 'nut hand' to fold, while also testing to see if another player has the 'nuts'.
• To drive out better hands when a drawing hand bets: If an opponent with an apparent drawing hand has bet before the player to act, if the player raises, opponents behind them who may have a better hand may fold rather than call two bets 'cold'. This is a form of isolation play, and has elements of blocking and protection.

Reasons to call[edit]

There are several reasons for calling a bet or raise, summarized below.

• To see more cards: With a drawing hand, a player may be receiving the correct pot odds with the call to see more cards.
• To limit loss in equity: Calling may be appropriate when a player has adequate pot odds to call but will lose equity on additional money contributed to the pot with a raise.
• To avoid a re-raise: Only calling (and not raising) denies the original bettor the option of re-raising. However, this is only completely safe in case the player is last to act (i.e. 'closing the action').
• To conceal the strength of a player's hand: If a player has a very strong hand, they might smooth call on an early betting round to avoid giving away the strength of their hand on the hope of getting more money into the pot in later betting rounds.
• To manipulate pot odds: By calling (not raising), a player offers any opponents yet to act behind them more favorable pot odds to also call. For example, if a player has a very strong hand, a smooth call may encourage opponents behind them to overcall or even raise, building the pot. Particularly in limit games, building the pot in an earlier betting round may induce opponents to call future bets in later betting rounds because of the pot odds they will be receiving.
• To set up a bluff on a later betting round: Sometimes referred to as a long-ball bluff, calling on an earlier betting round can set up a bluff (or semi-bluff) on a later betting round. For instance, a player with a strong initial hand may call instead of raise to see the flop cheaply. That flop may not benefit the player, but the player may still have many 'outs' (cards left to deal that could make a strong hand), or even if the odds are slim they can try to bluff. By raising, this scenario may appear to an opponent like a player who has 'limped in' with a weak initial hand, but after the flop now has a strong made or drawing hand. A recent online term for 'long-ball bluffing' is floating.^[3]

Gap concept[edit]

The gap concept states that a player needs a better hand to play against someone who has already opened (or raised) the betting than he would need to open himself.^[4] The gap concept reflects that players prefer to avoid confrontations with other players who have already indicated strength, and that calling only has one way to win (by having the best hand), whereas opening may also win immediately if your opponent(s) fold.

Sandwich effect[edit]

Related to the gap effect, the sandwich effect states that a player needs a stronger hand to stay in a pot when there are opponents yet to act behind him.^[3] Because the player does not know how many opponents will be involved in the pot or whether he will have to call a re-raise, he does not know what his effective pot odds actually are. Therefore, a stronger hand is desired as compensation for this uncertainty. A squeeze play exploits this principle.

Loose/tight play[edit]

Loose players play relatively more hands and tend to continue with weaker hands; hence they do not often fold. Tight players play relatively fewer hands and tend not to continue with weaker hands; hence they often fold. The following concepts are applicable in loose games (and their inverse in tight games):^[1]

Poker Theory And Practice

• Bluffs and semi-bluffs are less effective because loose opponents are less likely to fold.
• Requirements for continuing with made hands may be lower because loose players may also be playing lower value hands.
• Drawing to incomplete hands, like flushes, tends to be more valuable as draws will often get favorable pot odds and a stronger hand (rather than merely one pair) is often required to win in multi-way pots.

Aggressive/passive play[edit]

Aggressive play refers to betting and raising. Passive play refers to checking and calling. Unless passive play is being used deceptively as mentioned above, aggressive play is generally considered stronger than passive play because of the bluff value of bets and raises and because it offers more opportunities for your opponents to make mistakes.^[1]

Hand reading, tells and leveling[edit]

Hand reading is the process of making educated guesses about the possible cards an opponent may hold, based on the sequence of actions in the pot. The term 'hand reading' is actually a misnomer, as skilled players do not attempt to assign a player to an exact hand. Rather they attempt to narrow the possibilities down to a range of probable hands based on the past actions of their opponent, during both the current hand and previous hands played by this opponent.

Tells are detectable changes in opponents' behavior or demeanor which provide clues about their hands or their intentions. Educated guesses about opponents' cards and intentions can help a player avoid mistakes in his own play, induce mistakes by the opponents, or influence the opponents to take actions that they would not normally take under the circumstances. For example, a tell might suggest that an opponent has missed a draw, so a player seeing it may decide a bluff would be more effective than usual.

Leveling or multiple level thinking is accounting for what the other opponents think about the hands. This information can then be used to the player's advantage. Some players might be able to make educated guesses about opponents' hands; this could be seen as the first level. The second level could be thought of as the combination of the first level and deducing what the opponents think the player's hand may be. Skilled players can adjust their game play to be on a higher level than that of less skilled opponents.

Table image and opponent profiling[edit]

By observing the tendencies and patterns of one's opponents, one can make more educated guesses about others' potential holdings. For example, if a player has been playing extremely tightly (playing very few hands), then when he/she finally enters a pot, one may surmise that he/she has stronger than average cards. One's table image is the perception by one's opponents of one's own pattern of play. A player can leverage their table image by playing out of character, and thereby inducing his/her opponents to misjudge his/her hand and make a mistake.

In live poker, as opposed to internet, stereotypes are often used for initial 'reads'. For instance, people of retirement age are often witnessed to play tight. Players will often project this image on unknown people of retirement age. Young people wearing headphones and hoodies are often witnessed to play more aggressively and mathematically if they played a lot of winning internet poker. These stereotypes can often be good bases to start a profile.

Often, there is a rather small pool of players in a given card playing venue. People will carry their history of playing with them in these environments.

Internet poker players can use large databases of hand histories to get a more precise player profile. Statistical information about opponents is displayed on the tables in the form of a heads up display. The most commonly used software is PokerTracker and Hold'em Manager.

Equity[edit]

Players' equity in a pot is their expected share of the pot, expressed either as a percentage (probability of winning) or expected value (amount of pot * probability of winning). Negative equity, or loss in equity, occurs when contributing to a pot with a probability of winning less than 1 / (number of opponents matching the contribution+1).

Example

Alice contributes $12 to a pot and is matched by two other opponents. Alice's $12 contribution 'bought' the chance to win $36. If Alice's probability of winning is 50%, her equity in the $36 pot is $18 (a gain in equity because her $12 is now 'worth' $18). If her probability of winning is only 10%, Alice loses equity because her $12 is now only 'worth' $3.60 (amount of pot * probability of winning).

Texas hold 'em example

Alice holds J♦8♠. Bob holds K♥7♠. After the flop, the board is 5♥6♥7♦. If both hands are played to a showdown, Alice has a 45% chance to win (which she is unaware of, because she does not know what hand Bob holds), Bob has a 53% chance to win and there is a 2% chance to split the pot. The pot currently has $51. Alice goes all-in for $45 reasoning that she can take the pot immediately if Bob folds or that Bob calls with a worse hand. Bob's simple pot odds for the call are also 32%; since his equity of 53% is greater than the pot odds he has to call, Bob has a positive expected value for the call (if he knew Alice's hole cards).

Short-handed considerations[edit]

When playing short-handed (at a table with 3-6 players), players must loosen up their play (play more hands) for several reasons:^[1]

• There is less likelihood of another player having a strong hand because there are fewer players.
• Each player's share of the forced bets increases because there are fewer players contributing to the forced bets, thus waiting for premium hands becomes more expensive.

This type of situation comes up most often in tournament style play. In a cash game, the adjustments are very similar, but not quite as drastic as the table can ask for what is known as a 'rake break.' A rake break occurs when the floor-man, who represents the casino, agrees to take a smaller portion than usual for the hand. For example, a random casino might normally receive 10% of the pot up to 5 dollars for a 'rake.' In this case the table would only owe 10% up to 3 dollars until there are a sufficient number of players again. In online poker rake breaks are determined automatically.

Structure considerations[edit]

The blinds and antes and limit structure of the game have a significant influence on poker strategy. For example, it is easier to manipulate pot odds in no-limit and pot-limit games than in limit games. In tournaments, as the size of the forced bets relative to the chip stacks grows, pressure is placed on players to play pots to avoid being anted/blinded away.^[5]

Mindset considerations[edit]

In 2014, Bwin conducted a study to see what makes a professional poker player. The brain activity of poker players, of varying degrees, was monitored using EEG headsets and visualised into brain maps.^[6] Leading sports psychologist, James Hazlett, then interpreted the findings:

• More experienced players showed higher levels of focus and concentration throughout the game.
• The amateur players had less control over their emotions, and were prone to allowing negative emotions, such as frustration, distract them.
• Whilst opponents were taking their turn, the expert players opened up another table or watched replays of hands they had played poorly to improve.
• The brain maps showed that the professional players were led more by logic and intuition.

The conclusions of the study suggest that poker players can improve their strategy by considering their mindset. Mental training techniques, commonly used by athletes, could therefore help to improve performance by working on elements such as self-control and concentration.

See also[edit]

Poker Theory Mit

Poker plays[edit]

Specific games[edit]

Notes[edit]

Poker Theory Wikipedia

1. ^ ^abcdefDavid Sklansky (1987). The Theory of Poker. Two Plus Two Publications. ISBN1-880685-00-0.
2. ^Milovsky, Nicholas. 'The Basics of Game Theory and Associated Games'. Retrieved 11 February 2014.
3. ^ ^abDan Harrington and Bill Robertie (2004). Harrington on Hold'em: Expert Strategy For No-Limit Tournaments; Volume I: Strategic Play. Two Plus Two Publications. ISBN1-880685-33-7.
4. ^David Sklansky (2001). Tournament Poker for Advanced Players. Two Plus Two Publications. ISBN1-880685-28-0.
5. ^Dan Harrington and Bill Robertie (2005). Harrington on Hold'em: Expert Strategy For No-Limit Tournaments; Volume II: The Endgame. Two Plus Two Publications. ISBN1-880685-35-3.
6. ^Your Brain on Poker