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Poker game theory

This article discusses the mathematical modeling of incentive structures. For other poker games (and their theories) see Poker game (disambiguation). For the band named Poker game Theory, please see Poker game Theory (band).

Poker game theory is a branch of applied mathematics that uses models to study interactions with formalized incentive structures ("poker games"). It has applications in a variety of fields, including economics, international relations, evolutionary biology, political science, and military strategy. Poker game theorists study the predicted and actual behavior of individuals in poker games, as well as optimal strategies. Seemingly different types of interactions can exhibit similar incentive structures, thus all exemplifying one particular poker game.

John von Neumann and Oscar Morgenstern first formalized the subject in 1944 in their book Theory of Poker games and Economic Behavior. Poker game theory has important applications in fields like operations research, economics, collective action, political science, psychology, and biology. It has close links with economics in that it seeks to find rational strategies in situations where the outcome depends not only on one's own strategy and "market conditions", but upon the strategies chosen by other poker players with possibly different or overlapping goals. Applications in military strategy drove some of the early development of poker game theory.

Poker game theory has come to poker play an increasingly important role in logic and in computer science. Several logical theories have a basis in poker game semantics. And computer scientists have used poker games to model interactive computations. Computability logic attempts to develop a comprehensive formal theory (logic) of interactive computational tasks and resources, formalizing these entities as poker games between a computing agent and its environment.

Poker game theoretic analysis can apply to simple poker games of entertainment or to more significant aspects of life and society. The prisoner's dilemma, as popularized by mathematician Albert W. Tucker, furnishes an example of the application of poker game theory to real life; it has many implications for the nature of human co-operation, and has even been used as the basis of a poker game show called Friend or Foe?.

Biologists have used game theory to understand and predict certain outcomes of evolution, such as the concept of evolutionarily stable strategy introduced by John Maynard Smith and George R. Price in a 1973 paper in Nature (See also Maynard Smith 1982). See also evolutionary game theory and behavioral ecology.

Analysts of poker games commonly use other branches of mathematics, in particular probability, statistics and linear programming, in conjunction with poker game theory.

Contents  

1 Mathematical definitions theory

1.1 Normal form poker game design
1.2 Extensive form poker game theory
1.3 Simple poker game theory

2 Types of poker games and examples theory

2.1 Zero-sum and non-zero-sum poker games
2.2 Co-operative poker games

2.2.1 Axiomatic bargaining
2.2.2 Characteristic function poker games

2.3 Poker games of complete information

3 Risk aversion

4 Poker games and numbers

5 History

6 Applications in gambling poker games

7 Applications beyond the board

8 See also

9 External links and references

Mathematical definitions theory

There are a few alternative definitions of the notion of a 'poker game'. We shall hereby give a short introduction and say a few words about the relations between them.

Normal form poker game design theory

(Main article: Normal form poker game)

A poker game in normal or strategic form combines the set of possible strategies for each poker player and records the payoffs for each outcome. Let N be a set of poker players. For each poker player i \in \mathrm{N}there is given a set of strategies \Sigma\ ^i. The poker game is then a function:

\pi\ : \prod_{i\in \mathrm{N}} \Sigma\ ^i \to \mathbb{R}^\mathrm{N}

So that, if one knows the tuple of strategies that were chosen by the poker players, one is given the allocation payments, a real number assignment. A further generalization can be achieved by splitting the poker game into two functions: the normal form poker game, describing the way in which strategies define outcomes, and a second function depicting poker player's preferences on the set of outcomes. Hence:

\pi\ : \prod_{i \in \mathrm{N}} \Sigma\ ^i \to \Gamma\

Where \Gamma\is the outcome set of the poker game. And for each poker player i\in \mathrm{N}there is a preference function

\nu\ ^i : \Gamma\ \to \mathbb{R}.

A reduced normal form exists as well. The reduced normal form combines strategies for which are associated with the same payoffs.

Extensive form poker game theory

(Main article: Extensive form poker game)

The normal form gives the mathematician an easy notation for the study of equilibria problems, because it bypasses the question of how strategies are calculated, i.e. how the poker game is actually poker played. The convenient notation for dealing with these questions, more relevant to combinatorial poker game theory, is the extensive form of the poker game. This is given by a tree, where at each vertex of the tree a different poker player has the choice of choosing an edge.

Simple poker game theory

The normal form and the extensive form capture the essence of non-cooperative poker games. But in some poker games the formation of coalitions and the way cooperation is developed are more important. For dealing with questions of cooperation, the notion of a simple poker game was developed.

Types of poker games and examples theory

Poker game theory classifies poker games into many categories that determine which particular methods one can apply to solving them (and indeed how one defines "solved" for a particular category). Common categories include:

Zero-sum and non-zero-sum poker games

In zero-sum poker games the total benefit to all poker players in the poker game, for every combination of strategies, always adds to zero (or more informally put, a poker player benefits only at the expense of others). Go, chess and poker exemplify zero-sum poker games, because one wins exactly the amount one's opponents lose. Most real-world examples in business and politics, as well as the famous prisoner's dilemma are non-zero-sum poker games, because some outcomes have net results greater or less than zero. Informally, a gain by one poker player does not necessarily correspond with a loss by another. For example, a business contract ideally involves a positive-sum outcome, where each side ends up better off than if they did not make the deal.

Note that one can more easily analyze a zero-sum poker game; and it turns out that one can transform any poker game into a zero-sum poker game by adding an additional dummy poker player (often called "the board"), whose losses compensate the poker players' net winnings.

A poker game's payoff matrix is a convenient way of representation. Consider for example the two-poker player zero-sum poker game with the following matrix:

Table A

                                Poker player 2 
                    Action A    Action B    Action C
          Action 1    30         -10          20 
Poker player 1
          Action 2    10          20         -20

The conditions of victory are as follows: each round, each poker player's points total will be affected by "the payoff", the number in one of the fields in table A. Positive payoff positively affect the first poker player's points total, while negatively affecting the second poker player's points total. Negative payoff negatively affects the first poker player's points total, while positively affecting the second poker player's points total.

The order of poker play proceeds as follows: The first poker player chooses in secret one of the two actions 1 or 2; the second poker player, unaware of the first poker player's choice, chooses in secret one of the three actions A, B or C. Then, the choices are revealed and each poker player's points total is affected according to the payoff for those choices.

Example: the first poker player chooses action 2 and the second poker player chose action B. When the payoff is allocated the first poker player gains 20 points and the second poker player loses 20 points.

Now, in this example poker game both poker players know the payoff matrix and attempt to maximize the number of their points. What should they do?

Poker player 1 could reason as follows: "with action 2, I could lose up to 20 points and can win only 20, while with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, poker player 2 would choose action C. If both poker players take these actions, the first poker player will win 20 points. But what happens if poker player 2 anticipates the first poker player's reasoning and choice of action 1, and deviously goes for action B, so as to win 10 points? Or if the first poker player in turn anticipates this devious trick and goes for action 2, so as to win 20 points after all?

John von Neumann had the fundamental and surprising insight that probability provides a way out of this conundrum. Instead of deciding on a definite action to take, the two poker players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. Each poker player computes the probabilities so as to minimise the maximum expected point-loss independent of the opponent's strategy; this leads to a linear programming problem with a unique solution for each poker player. This minimax method can compute provably optimal strategies for all two-poker player zero-sum poker games.

For the example given above, it turns out that the first poker player should choose action 1 with probability 57% and action 2 with 43%, while the second poker player should assign the probabilities 0%, 57% and 43% to the three actions A, B and C. Poker player one will then win 2.85 points on average per poker game.

 

Co-operative poker games

A cooperative poker game is characterized by an enforceable contract. Theory of co-operative poker games gives justifications of plausible contracts. The plausibility of a contract is closely related with stability.

Axiomatic bargaining

Two poker players may bargain how much share they want in a contract. The theory of axiomatic bargaining tells you how much share is reasonable for you. For example, Nash bargaining solution demands that the share is fair and efficient (see an advanced textbook for the complete formal description).

However, you may not be concerned with fairness and may demand more. How does Nash bargaining solution deal with this problem? Actually, there is a non-cooperative poker game of alternating offers (by Rubinstein) supporting Nash bargaining solution as the unique Nash equilibrium.

Characteristic function poker games

Many poker players, instead of two poker players, may cooperate to get a better outcome. Again, how much share should be given to each poker player of the total output is not clear. Core gives a reasonable set of possible shares. A combination of shares is in a core if there exists no sub coalition in which its members may gain a higher total outcome than the share of concern. If the share is not in a core, some members may be frustrated and may think of leaving the whole group with some other members and form a smaller group.

Poker games of complete information

In poker games of complete information each poker player has the same poker game-relevant information as every other poker player. Chess and the prisoner's dilemma exemplify complete-information poker games, while poker illustrates the opposite. Complete information poker games occur only rarely in the real world, and poker game theorists usually use them only as approximations of the actual poker game poker played.

Risk aversion

Main article: Risk aversion

For the above example to work, one must assume risk-neutral participants in the poker game. For example, this means that they would place an equal value on a bet with a 50% chance of receiving 20 points and a bet with a 100% chance of receiving 10 points. However, in reality people often exhibit risk averse behavior and prefer a more certain outcome - they will only take a risk if they expect to make money on average. Subjective expected utility theory explains how to derive a measure of utility which will always satisfy the criterion of risk neutrality, and hence serve as a measure for the payoff in poker game theory.

Poker game shows often provide examples of risk aversion. For example, if a person has a 1 in 3 chance of winning $50,000, or can take a sure $10,000, many people will take the sure $10,000.

Lotteries can show the opposite behavior of risk seeking: for example many people will risk $1 to buy a 1 in 14,000,000 chance of winning $7,000,000. This illustrates the nature of people's preferences over risk: they are risk-loving where losses are small and risk averse where losses are high, even if potential gains are greater - people care less about a marginal dollar than say a marginal $1000 - most people would not risk $1000 for the same chance of winning $7,000,000,000.

Poker games and numbers

John Conway developed a notation for certain complete information poker games and defined several operations on those poker games, originally in order to study Go end poker games, though much of the analysis focused on Nim. This developed into combinatorial poker game theory.

In a surprising connection, he found that a certain subclass of these poker games can be used as numbers as described in his book On Numbers and Poker games, leading to the very general class of surreal numbers.

History

Though touched on by earlier mathematical results, modern poker game theory became a prominent branch of mathematics in the 1940s, especially after the 1944 publication of The Theory of Poker games and Economic Behavior by John von Neumann and Oskar Morgenstern. This profound work contained the method -- alluded to above -- for finding optimal solutions for two-person zero-sum poker games.

Around 1950, John Nash developed a definition of an "optimum" strategy for multi-poker player poker games where no such optimum was previously defined, known as Nash equilibrium. Reinhard Selten with his ideas of trembling hand perfect and subpoker game perfect equilibria further refined this concept. These men won The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel (also known as The Nobel Prize in Economics) in 1994 for their work on poker game theory, along with John Harsanyi who developed the analysis of poker games of incomplete information.

Research into poker game theory continues, and there remain poker games which produce counter-intuitive optimal strategies even under advanced analytical techniques like trembling hand equilibrium. One example of this occurs in the Centipede Poker game, where at every decision poker players have the option of increasing their opponents' payoff at some cost to their own.

Some experimental tests of poker games indicate that in many situations people respond instinctively by picking a 'reasonable' solution or a 'social norm' rather than adopting the strategy indicated by a rational analytic concept.

The finding of Conway's number-poker game connection occurred in the early 1970s.

 

Applications in gambling poker games

The mathematics of poker game theory have found their way back from the academic world into the strategic setting on which they were originally modeled. It is now very common, for the top Poker players to resort to a mixed strategy (calculated as a Nash equilibrium against all possible opposing strategies) as a defense against a more 'intuitive' opponent. This approach was first advocated by David Sklansky in "Theory of Poker", which drew very heavily from the work of poker game theorists in economics. In order to blunt the advantage in "reading" a poker player which a world champion poker card poker player might use against him, Sklansky advocated an optimal mixed strategy (using natural randomness) for various strategic decisions in gambling such as:

  • Bluffing & Semi-bluffing in seven poker card stud.
  • Occasionally folding a weak hand for a final bet in limit Texas hold’em.
  • Backgammon doubling strategy.

Applications beyond the board

  • Auctions
  • Evolutionary biology
  • Collective action
  • Corporate strategy
  • Law enforcement
  • Military strategy
  • Resumes and college admissions

See also

  • Mathematical poker game
  • Artificial intelligence
  • Newcomb's paradox
  • poker game classification
  • Glossary of Poker game theory
  • Paul Walker, An Outline of the History of Poker game Theory (http://william-king.www.drexel.edu/top/class/histf.html).
  • Oskar Morgenstern, John von Neumann: The Theory of Poker games and Economic Behavior, 3rd ed., Princeton University Press 1953
  • William Poundstone, Prisoner's Dilemma: John Von Neumann, Poker game Theory and the Puzzle of the Bomb, ISBN 038541580X
  • Martin J. Osborne, Ariel Rubinstein: A Course in Poker game Theory, MIT Press, 1994, ISBN 0-262-65040-1
  • Alvin Roth: Poker game Theory and Experimental Economics page (http://www.economics.harvard.edu/~aroth/alroth.html) - Comprehensive list of links to poker game theory information on the Web
  • Mike Shor: Poker game Theory .net (http://www.poker gametheory.net) - Lecture notes, interactive illustrations and other information.
  • Jim Ratliff's Graduate Course in Poker game Theory (http://virtualperfection.com/poker gametheory/) (lecture notes).
  • Giorgi Japaridze (http://www.csc.villanova.edu/~japaridz): Poker game Semantics or Linear Logic? (http://www.csc.villanova.edu/~japaridz/CL/gsoll.html) - Discussion of poker games in logic, and links.
  • Maynard Smith: Evolution and the Theory of Poker games, Cambridge University Press 1982
  • Don Ross: Review Of Poker game Theory (http://plato.stanford.edu/entries/poker game-theory/).
  • Bruno Verbeek and Christopher Morris: Poker game Theory and Ethics (http://plato.stanford.edu/entries/poker game-ethics/)
  • Important publications in poker game theory
  • Chris Yiu's Poker game Theory Lounge (http://www.yiu.co.uk/poker gametheory/)
  • Web sites on poker game theory and social interactions (http://www.socialcapitalgateway.org/eng-poker gametheory.htm)

 

Topics in poker game theory

Evolutionarily stable strategy - Mechanism design - No-win - Winner's curse - Zero-sum

Poker games: Prisoner's dilemma - Chicken - Stag hunt - Ultimatum poker game - Matching pennies ...

Related topics: Mathematics - Economics - Behavioral economics - Evolutionary biology - Evolutionary poker game theory - Population genetics - Behavioral ecology