After all is said and done, poker is ultimately about psychology – playing against your competitors. For example, many games often end with a very small hand, such as a pair of 7s, beating a smaller hand, such as a pair of 3s. Never forget that your hand doesn’t have to be the best hand possible. It simply has to be better than those LEFT in the hand!
Knowing what your opponents have is an art, not a science. Many successful (and unsuccessful!) poker players talk about “tells” – twitches, trembles, and other bodily signs that might give you a clue as to what your opponent has. In time, you may even learn to be able to “read” those tells. Mastering the psychology of poker, however, is much more important (and, in fact, a crucial foundation before reading tells is even possible).
Poker psychology boils down to your ability to watch how others play, and use that experience to judge how your opponents may be playing in the current hand. It is critical that you never become distracted from the game. For example, do not watch TV, even during a friendly game, for this will deprive you of the information you gain while watching your opponents. Even in a friendly game, your “friends” are trying to take your money from you!
The simplest layer of poker psychology is to watch what your opponents visibly do based on their own cards. For example, keep track of how each player bets. If you have problems doing this, start by only keeping track of those who did not fold, and don’t worry about keeping track of amounts. Simply get a feel for whether the players bet strongly or weakly. During a showdown, note the hands each player had. Were they betting heavily with a weak hand? Was the hand possibly going to “make it?” (e.g., were they drawing to a flush, and just didn’t make it? Was the flush even possible? Was it likely, or was it a long shot?)
This is not a skill learned in a day. You must play THOUSANDS of hands to master it. Gradually, you will build a feel for how players bet in response to what they have in their hands. Then focus on how they respond to other players. Did they come out betting heavily early in the game, then fade away and eventually fold to heavy raising, even if their hand looked like it improved? Did they instead re-raise or cap the betting?
The same mathematical strategies that apply to you can be of assistance here, especially in community card or stud games, which give you some information about what the other players have even before the showdown. In fact, it is during these games that poker psychology is most readily learned, because in draw games you never know what the player discarded.
Learn to classify your opponents, and adjust your strategy against how they play. For example, identify whether your opponents are loose or tight. If they are loose, they are likely to bet heavily or stay in for a long time with even a very weak hand, or on a long shot draw. Tight players, however, tend to fold at every breeze. Also categorize them in terms of passive or aggressive. When raised, do they tend to call or fold? Or do they re-raise?
Ultimately, no single strategy will ever teach you the art of poker psychology. You will either learn it over a long period of time playing many hands, or you will go broke trying!
Morton’s theorem is a poker principle articulated by Andy Morton. It states that in multiway pots, a player’s expectation may be maximised by an opponent making a correct decision.
The most common application of Morton’s theorem occurs when one player holds the best hand, but there are two or more opponents on draws. In this case, the player with the best hand might make more money in the long run when an opponent folds to a bet, even if that opponent is folding correctly and would be making a personal mistake to call the bet. This type of situation is sometimes referred to as implicit collusion.
Morton’s theorem should be contrasted with the fundamental theorem of poker, which states that you want your opponents to make decisions which minimise their own expectation. The discrepancy between the two “theorems” occurs because of the presence of more than one opponent. Whereas the fundamental theorem always applies heads-up (one opponent), it does not always apply in multiway pots. The scope of Morton’s theorem in multiway situations is a subject of controversy. For example, Morton himself expresses the belief that the fundamental theorem rarely applies to multiway situations.
An example
The following example is credited to Morton, who first posted on rec.gambling.poker. (Some numbers have been changed to allow for complete information, see below.)
Suppose in holdem you hold A♦K♣ and the flop is K♠9♥3♥, giving you top pair with best kicker. When the betting on the flop is complete, you have two opponents remaining, one of whom you know has the nut flush draw (say A♥T♥, giving him 9 outs) and one of whom you believe holds second pair with random kicker (say Q♣9♣, 4 outs), leaving you with all the remaining cards in the deck as your outs. The turn card is an apparent blank (say 6♦) and say the pot size at that point is P, expressed in big bets.
When you bet the turn player A, holding the flush draw, is sure to call and is almost certainly getting the correct pot odds to call your bet. Once player A calls, player B must decide whether to call or fold. To figure out which action player B should choose, calculate his expectation in each case. This depends on the number of cards among the remaining 42 that will give him the best hand, and the size of the pot when he is deciding. (Here, as in arguments involving the fundamental theorem, we assume that each player has complete information of their opponents’ cards.)
E( player B | folding ) = 0
Player B doesn’t win or lose anything by folding. When calling, he wins the pot 4/42 of the time, and loses one big bet the remainder of the time. Setting these two expectations equal to each other and solving for P lets us determine the pot-size at which he is indifferent to calling or folding:
E( player B | folding ) = E( player B | calling )
When the pot is larger than this, player B should chase you; otherwise, it’s in B’s best interest to fold.
To figure out which action on player B’s part you would prefer, calculate your expectation the same way
Your expectation depends in each case on the size of the pot (in other words, the pot odds B is getting when considering his call.) Setting these two equal lets us calculate the pot-size P where you are indifferent whether B calls or folds:
E( you | B calls ) = E( you | B folds )
When the pot is smaller than this, you profit when player B is chasing, but when the pot is larger than this, your expectation is higher when B folds instead of chasing.
In this case, there is a range of pot-sizes where it’s correct for B to fold, and you make more money when he does so than when he incorrectly chases. You can see this graphically below
| B SHOULD FOLD | B SHOULD CALL | v | YOU WANT B TO CALL| YOU WANT B TO FOLD | v+---+---+---+---+---+---+---+---+---> pot-size P in big bets0 1 2 3 4 5 6 7 8 XXXXXXXXXX ^ "PARADOXICAL REGION"
The range of pot sizes marked with the X’s is where you want your opponent to fold correctly, because you lose expectation when he calls incorrectly.
Analysis
In essence, in the above example, when player B calls in the “paradoxical region”, he is paying too high a price for his weak draw, but you are no longer the sole benefactor of that high price — player A is now taking B‘s money those times that A makes his flush draw. Compared to the case where you are heads up with player B, you still stand the risk of losing the whole pot, but are no longer getting 100% of the compensation from B‘s loose calls.
It is the existence of this middle region of pot sizes, where you want at least some of your opponents to fold correctly, that explains the standard poker strategy of thinning the field as much as possible when you think you hold the best hand. Even players with incorrect draws cost you money when they call your bets, because part of their calls end up in the stacks of other players drawing against you.
Because you are losing expectation from B‘s call, it follows that the aggregate of all other players (i.e., A and B) must be gaining from B’s call. In other words, if A and B were to meet in the parking lot after the game and split their profits, they would have been colluding against you. This is sometimes referred to as implicit collusion. It should be contrasted with what is sometimes called schooling. Schooling occurs when many players correctly call against a player with the best hand, whereas implicit collusion occurs when a player incorrectly calls against a player with the best hand.
One conclusion of Morton’s theorem is that, for example, in holdem, the value of suited hands goes up, because they are precisely the types of hands which will benefit from implicit collusion.
The Fundamental Theorem of Poker applies to all heads-up decisions, but it does not apply to all multi-way decisions. This is because each opponent of a player can make an incorrect decision, but the “collective decision” of all the opponents works against the player.
This type of situation occurs mostly in games with multi-way pots, when a player has a strong hand, but several opponents are chasing with draws or other weaker hands. Sometimes such a situation is referred to as implicit collusion. Experts disagree on the prevalence of implicit collusion in particular games, as well as the extent to which implicit collusion might be unethical.
The Fundamental Theorem of Poker is simply expressed and appears axiomatic, yet its proper application to the countless varieties of circumstances that a poker player may face requires a great deal of knowledge, skill, and experience.
Here is an example that illustrates how the Fundamental Theorem is applied. (This example assumes a familiarity with the basic rules and terminology of holdem.) Suppose you are playing limit holdem and are dealt 9♣ 9♠ under the gun before the flop. You call, and everyone folds to the big blind who checks. The flop comes A♣ K♦ 10♦, and the big blind bets.
You now have a decision to make based upon incomplete information. In this particular circumstance, the correct decision is almost certainly to fold. There are too many turn and river cards that could kill your hand. Even if the big blind does not have an A or a K, there are 3 cards to a straight and 2 cards to a flush on the flop, and she could easily be on a straight or flush draw. You are essentially drawing to 2 outs (another 9), and even if you catch one of these outs, your set may not hold up.
However, suppose you knew (with 100% certainty) the big blind held 8♦ 7♦. In this case, it would be correct to raise. Even though the big blind would still be getting the correct pot odds to call, the best decision is to raise. (Calling would be giving the big blind infinite pot odds, and this decision makes less money in the long run than raising.) Therefore, by folding (or even calling), you have played your hand differently from the way you would have played it if you could see your opponent’s cards, and so by the Fundamental Theorem of Poker, she has gained. You have made a “mistake”, in the sense that you have played differently from the way you would have played if you knew the big blind held 8♦ 7♦, even though this “mistake” is almost certainly the best decision given the incomplete information available to you.
This example also illustrates that one of the most important goals in poker is to induce your opponents to make mistakes. In this particular hand, the big blind has practiced deception by employing a semi-bluff — she has bet a hand, hoping you will fold, but she still has outs even if you call or raise. She has induced you to make a mistake.
The fundamental theorem of poker is a principle first articulated by David Sklansky that he believes expresses the essential nature of poker as a game of decision-making in the face of incomplete information.
Every time you play a hand differently from the way you would have played it if you could see all your opponents’ cards, they gain; and every time you play your hand the same way you would have played it if you could see all their cards, they lose. Conversely, every time opponents play their hands differently from the way they would have if they could see all your cards, you gain; and every time they play their hands the same way they would have played if they could see all your cards, you lose.
The Fundamental Theorem is stated in common language, but its formulation is based on mathematical reasoning. Each decision that is made in poker can be analyzed in terms of the concept of expected value. The expected value expresses the average payoff of a decision if the decision is made a large number of times. The correct decision to make in a given situation is the decision that has the largest expected value. (Although sometimes it is correct not to choose this decision for the larger goal of long-term deception.) If you could see all your opponents’ cards, you would always be able to calculate the correct decision with mathematical certainty. (This is certainly true heads-up, but is not always true in multi-way pots.) The less you deviate from these correct decisions, the better your expected long-term results. This is the mathematical expression of the Fundamental Theorem.
The first known discussion of game theory occurred in a letter written by James Waldegrave in 1713. In this letter, Waldegrave provides a minimax mixed strategy solution to a two-person version of the card game le Her. It was not until the publication of Antoine Augustin Cournot’s Researches into the Mathematical Principles of the Theory of Wealth in 1838 that a general game theoretic analysis was pursued. In this work Cournot considers a duopoly and presents a solution that is a restricted version of the Nash equilibrium.
Although Cournot’s analysis is more general than Waldegrave’s, game theory did not really exist as a unique field until John von Neumann published a series of papers in 1928. These results were later expanded in the 1944 book The Theory of Games and Economic Behavior by von Neumann and Oskar Morgenstern. This profound work contains the method for finding optimal solutions for two-person zero-sum games. During this time period, work on game theory was primarily focused on cooperative game theory, which analyzes optimal strategies for groups of individuals, presuming that they can enforce agreements between them about proper strategies.
In 1950, the first discussion of the Prisoner’s dilemma appeared, and an experiment was undertaken on this game at the RAND corporation. Around this same time, John Nash developed a definition of an “optimum” strategy for multiplayer games where no such optimum was previously defined, known as Nash equilibrium. This equilibrium is sufficiently general, allowing for the analysis of non-cooperative games in addition to cooperative ones.
Game theory experienced a flurry of activity in the 1950s, during which time the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value were developed. In addition, the first applications of Game theory to philosophy and political science occurred during this time.
In 1965, Reinhard Selten introduced his solution concept of subgame perfect equilibria, which further refined the Nash equilibrium (later he would introduce trembling hand perfection as well). In 1967, John Harsanyi developed the concepts of complete information and Bayesian games. He, along with John Nash and Reinhard Selten, won The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel (also known as The Nobel Prize in Economics) in 1994.
In the 1970s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionary stable strategy. In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledge[5] were introduced and analyzed.
In 2005, game theorists Thomas Schelling and Robert Aumann won the Nobel Prize in Economics. Schelling worked on dynamic models, early examples of evolutionary game theory. Aumann contributed more to the equilibrium school, developing an equilibrium coarsening correlated equilibrium and developing extensive analysis of the assumption of common knowledge.
A symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. If the identities of the players can be changed without changing the payoff to the strategies, then a game is symmetric. Many of the commonly studied 2×2 games are symmetric. The standard representations of Chicken, the Prisoner’s Dilemma, and the Stag hunt are all symmetric games. [1]
Most commonly studied asymmetric games are games where there are not identical strategy sets for both players. For instance, the Ultimatum game and similar the Dictator game have different strategies for each player. It is possible, however, for a game to have identical strategies for both players, yet be asymmetric. For example, the game pictured to the right is asymmetric despite having identical strategy sets for both players.
Zero sum and non-zero sum
A Zero-Sum Game
A
B
A
2, -2
-1, 1
B
-1, 1
3, -3
In zero-sum games the total benefit to all players in the game, for every combination of strategies, always adds to zero (or more informally put, a player benefits only at the expense of others). Poker exemplifies a zero-sum game, because one wins exactly the amount one’s opponents lose. Other zero sum games include Matching pennies and most classical board games including Go and Chess. Many games studied by game theorists (including the famous Prisoner’s Dilemma) are non-zero-sum games, because some outcomes have net results greater or less than zero. Informally, in non-zero-sum games, a gain by one player does not necessarily correspond with a loss by another.
It is possible to transform any game into a zero-sum game by adding an additional dummy player (often called “the board”), whose losses compensate the players’ net winnings.
Simultaneous and sequential
Simultaneous games are games where both players move simultaneously, or if they do not move simultaneously, the later players are unaware of the earlier players’ actions (making them effectively simultaneous). Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions. This need not be perfect knowledge about every action of earlier players; it might be very little information. For instance, a player may know that an earlier player did not perform one particular action, while she does not know which of the other available actions the first player actually performed.
The difference between simultaneous and sequential games is captured in the different representations discussed above. Normal form is used to represent simultaneous games, and extensive form is used to represent sequential ones.
Perfect information and imperfect information
A game of imperfect information (the dotted line represents ignorance on the part of player 2)
An important subset of sequential games consists of games of perfect information. A game is one of perfect information if all players know the moves previously made by all other players. Thus, only sequential games can be games of perfect information, since in simultaneous games not every player knows the actions of the others. Most games studied in game theory are imperfect information games, although some interesting games are games of perfect information, including the Ultimatum Game and Centipede Game. Many popular games are games of perfect information including Chess, Go, and Mancala.
Perfect information is often confused with complete information, which is a similar concept. Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions.
Infinitely long games
For obvious reasons, games as studied by economists and real-world game players are generally finished in a finite number of moves. Pure mathematicians are not so constrained, and set theorists in particular study games that last for infinitely many moves, with the winner (or other payoff) not known until after all those moves are completed.
The focus of attention is usually not so much on what is the best way to play such a game, but simply on whether one or the other player has a winning strategy. (It can be proved, using the axiom of choice, that there are games—even with perfect information, and where the only outcomes are “win” or “lose”—for which neither player has a winning strategy.) The existence of such strategies, for cleverly designed games, has important consequences in descriptive set theory.
Notes
^ Some scholars would consider certain asymmetric games as examples of these games as well. However, the most common payoffs for each of these games are symmetric.
The games studied by game theory are well-defined mathematical objects. A game consists of a set of players, a set of moves (or strategies) available to those players, and a specification of payoffs for each combination of strategies. There are two ways of representing games that are common in the literature.
Normal form
A normal form game
Player 2 chooses left
Player 2 chooses right
Player 1 chooses top
4, 3
-1, -1
Player 1 chooses bottom
0, 0
3, 4
The normal (or strategic form) game is a matrix which shows the players, strategies, and payoffs (see the example to the right). Here there are two players; one chooses the row and the other chooses the column. Each player has two strategies, which are specified by the number of rows and the number of columns. The payoffs are provided in the interior. The first number is the payoff received by the row player (Player 1 in our example); the second is the payoff for the column player (Player 2 in our example). Suppose that Player 1 plays top and that Player 2 plays left. Then Player 1 gets 4, and Player 2 gets 3.
When a game is presented in normal form, it is presumed that each player acts simultaneously or, at least, without knowing the actions of the other. If players have some information about the choices of other players, the game is usually presented in extensive form.
Extensive form
Extensive form games attempt to capture games with some important order. Games here are presented as trees (as pictured to the left). Here each vertex (or node) represents a point of choice for a player. The player is specified by a number listed by the vertex. The lines out of the vertex represent a possible action for that player. The payoffs are specified at the bottom of the tree.
In the game pictured here, there are two players. Player 1 moves first and chooses either F or U. Player 2 sees Player 1′s move and then chooses A or R. Suppose that Player 1 chooses U and then Player 2 chooses A, then Player 1 gets 8 and Player 2 gets 2.
Extensive form games can also capture simultaneous-move games as well. Either a dotted line or circle is drawn around two different vertices to represent them as being part of the same information set (i.e., the players do not know at which point they are).
Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns. First developed as a tool for understanding economic behavior, game theory is now used in many diverse academic fields, ranging from biology to philosophy. Game theory saw substantial growth during the Cold War because of its application to military strategy, most notably to the concept of mutually assured destruction. Beginning in the 1970s, game theory has been applied to animal behavior, including species’ development by natural selection. Because of interesting games like the Prisoner’s dilemma, where mutual self-interest hurts everyone, game theory has been used in ethics and philosophy. Finally, game theory has recently drawn attention from computer scientists because of its use in artificial intelligence and cybernetics.
In addition to its academic interest, game theory has received attention in popular culture. An important figure in game theory, John Nash was the subject of the 2001 film A Beautiful Mind. Several game shows have adopted game theoretic situations, including Friend or Foe and Deal or No Deal. [1]
Although similar to decision theory, game theory studies decisions that are made in an environment where various players interact. In other words, game theory studies choice of optimal behavior when costs and benefits of each option are not fixed, but depend upon the choices of other individuals.
Notes
^ GameTheory.net has an extensive list of references to game theory in popular culture.
References
Textbooks and general reference texts
Gibbons, Robert (1992) Game Theory for Applied Economists, Princeton University Press ISBN 0691003955 (readable; suitable for advanced undergraduates. Published in Europe by Harvester Wheatsheaf (London) with the title A primer in game theory)
Ginits, Herbert (2000) Game Theory Evolving Princeton University Press ISBN 0691009430
Osborne, Martin and Ariel Rubinstein: A Course in Game Theory, MIT Press, 1994, ISBN 0-262-65040-1 (modern introduction at the introductory graduate level)
Fudenberg, Drew and Jean Tirole: Game Theory, MIT Press, 1991, ISBN 0262061414 (the definitive reference text)
Historically important texts
Fisher, Ronald (1930) The Genetical Theory of Natural Selection Clarendon Press, Oxford.
Luce, Duncan and Howard Raiffa Games and Decisions: Introduction and Critical Survey Dover ISBN 0486659437
Maynard Smith, John Evolution and the Theory of Games, Cambridge University Press 1982
Morgenstern, Oskar and John von Neumann (1947) The Theory of Games and Economic Behavior Princeton University Press
Nash, John (1950) “Equilibrium points in n-person games” Proceedings of the National Academy of the USA 36(1):48-49.
Poundstone, William Prisoner’s Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb, ISBN 038541580X
Other print references
Camerer, Colin (2003) Behavioral Game Theory Princeton University Press ISBN 0691090394
Gauthier, David (1987) Morals by Agreement Oxford University Press ISBN 0198249926
Grim, Patrick, Trina Kokalis, Ali Alai-Tafti, Nicholas Kilb, and Paul St Denis (2004) “Making meaning happen.” Journal of Experimental & Theoretical Artificial Intelligence 16(4): 209-243.
Kavka, Gregory (1986) Hobbesian Moral and Political Theory Princeton University Press. ISBN 069102765X
Lewis, David (1969) Convention: A Philosophical Study
Maynard Smith, J. and Harper, D. (2003) Animal Signals. Oxford University Press. ISBN 0198526857
Quine, W.v.O (1967) “Truth by Convention” in Philosophica Essays for A.N. Whitehead Russel and Russel Publishers. ISBN 0846209705
Quine, W.v.O (1960) “Carnap and Logical Truth” Synthese 12(4):350-374.
Skyrms, Brian (1996) Evolution of the Social Contract Cambridge University Press. ISBN 0521555833
Skyrms, Brian (2004) The Stag Hunt and the Evolution of Social Structure Cambridge University Press. ISBN 0521533929.
Sober, Elliot and David Sloan Wilson (1999) Unto Others: The Evolution and Psychology of Unselfish Behavior Harvard University Press. ISBN 0674930479
A check-raise in poker is a common deceptive play in which a player checks early in a betting round, hoping someone else will open. The player who checked then raises in the same round.
This might be done, for example, when the first player believes that an opponent has an inferior hand and will not call a direct bet, but that he may attempt to bluff, allowing the first player to win more money than he would by betting straightforwardly.
Of course, if no other player chooses to open, the betting will be checked around and the play will fail.
While it is an important part of poker strategy, in some home games and certain small-stakes casino games, this play is not allowed. It is also frequently not allowed in the game of California lowball.
Morton’s theorem is a poker principle articulated by Andy Morton. It states that in multiway pots, a player’s expectation may be maximised by an opponent making a correct decision.
A game of imperfect information (the dotted line represents ignorance on the part of player 2)