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Encyclopedia > Payoff matrix
It has been suggested that this article or section be merged with normal form game. (Discuss)


A payoff matrix or payoff function is a concept in game theory which shows what payoff each player will receive at the outcome of the game. The payoff for each player will of course depend on the combined actions of all players. Wikipedia does not have an article with this exact name. ... In game theory, normal form is a way of describing a game. ... Game theory is a branch of applied mathematics that studies strategic situations where players choose different actions in an attempt to maximize their returns. ...


Technically, matrix is used only in the case when there are two players and the payoff function for each can be represented as a matrix. For expository purposes, we consider some examples first.

Contents


Example

We show the 2-player matrices for a version of Prisoner's dilemma. In this game there are two players and the numerical payoff for each player is the sentence (time in jail) measured in years of confinement; in this case, lower is better. Each player has two strategies: Cooperate with the other prisoner or defect, that is rat out to the police. Will the two prisoners cooperate to minimise total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? In Game Theory, the prisoners dilemma is a type of non-zero-sum game. ...


For player 1, the payoff matrix is as follows,

  Cooperate Defect
Cooperate .5 10
Defect 0 5

In this matrix, player 1's strategies are designated along the left hand column and 2's are the designated along the top row.


For player 2, the payoff matrix is

  Cooperate Defect
Cooperate .5 0
Defect 10 5

Where again, player 1's strategies are designated along the left hand column and 2's are the designated along the top row.


Often these two payoff matrices are combined into a single matrix representation. In this case one player chooses the row, another the column. The row player receives the first listed payoff the column the second. For the Prisoner's dilemma, the matrix would be:

  Cooperate Defect
Cooperate (.5, .5) (10, 0)
Defect (0, 10) (5, 5)

Note that other versions of the prisoner's dilemma game can be obtained by varying the numerical values of the payoff matrix.


Another example

2-player matrices for a version of Game of Chicken. In this game the higher the payoff, the better. Each player has two strategies: Swerve or continue. The game of chicken (also referred to as playing chicken) is a game in which two players engage in an activity that will result in serious damage unless one of them backs down. ...


For player 1, the payoff matrix is

  Swerve Continue
Swerve 0 -1
Continue +1 -20

For player 2, the matrix is:

  Swerve Continue
Swerve 0 +1
Continue -1 -20

In both these examples, the strategy sets for both players have the same cardinality; more significantly, the payoff matrices are symmetric in regard to the players. Note that the payoff matrices themselves are not symmetric matrices however. A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...


Consider a game with players referred to as 1 and 2. Each player has an assigned strategy set. Player 1 can select a strategy from {1,2, ..., m1} and player 2 can select from {1,2, ..., m2}


Definition. A payoff matrix for a two-player game is an m1 × m2 matrix of real numbers:

begin{bmatrix} a_{11} & a_{1 2} & cdots & a_{1 m_2}  a_{12} & a_{2 2} & cdots & a_{2 m_2}  vdots & vdots & vdots & vdots   a_{m_1 1} & a_{m_1 2} & cdots & a_{m_1 m_2} end{bmatrix}

Player 1's strategies are designated along the left hand column and 2's are the designated along the top row.


To specify a two-person game, we need to specify the strategy sets for each player and payoff functions for each player.


Remarks. Note that in general, games do not have to be symmetrical or in any way fair; for instance the strategy sets may have different cardinality for each player.


General formulation

A widely adopted model for non-cooperative games in general is based on the notion of finite games in normal form; this means we are given the following data In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In game theory, normal form is a way of describing a game. ...

  • There is a finite set P of players, which we label {1, 2, ..., m}
S_k = {1, 2, ldots, n_k}.

A pure strategy profile is an association of strategies to players, that is an m-tuple A pure strategy is a term used to refer to strategies in Game theory. ... In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects (an infinite sequence is a family). ...

vec{sigma} = (sigma_1, sigma_2, ldots,sigma_m)

such that

sigma_1 in S_1, sigma_2 in S_2, ldots, sigma_m in S_m

We will denote the set of strategy profiles by Σ.


A payoff function is a function

F: Sigma rightarrow mathbb{R}.

whose intended interpretation is the award given to a single player at the outcome of the game. Accordingly, to completely specify a game, the payoff function has to be specified for each player in the player set P= {1, 2, ..., m}.


Definition. A game in normal form is a structure

(P, mathbf{S}, mathbf{F})

where P = {1,2, ...,m} is a set of players,

mathbf{S}= (S_1, S_2, ldots, S_m)

is an m-tuple of pure strategy sets, one for each player, and

mathbf{F} = (F_1, F_2, ldots, F_m)

is an m-tuple of payoff functions.


Remark. There is no reason in the previous discussion to exclude games which have an infinite number of players or an infinite number of strategies per player. The study of infinite games is more difficult however, since it requires use of functional analytic techniques. Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...


Extension to mixed strategies

In game theory, one considers mixed strategies, also called randomized strategies. Each player k chooses a probability Prk for each element of Sk={1, 2, ..., nk}. We denote these probabilities as follows: A mixed strategy is used in game theory economics to describe a strategy comprising possible moves and a probability distribution which corresponds to how frequently each move is chosen. ... The word probability derives from the Latin probare (to prove, or to test). ...

operatorname{Pr}_k(1), operatorname{Pr}_k(2), ldots, operatorname{Pr}_k(n_k).

An operational interpretation of Prk for repeated plays of the game is as follows: prior to each play, player k chooses a strategy in Sk according to probability Pk. An operational definition of a quantity is a specific process whereby it is measured. ...


A mixed strategy profile is an association of mixed strategies to players, that is an m-tuple of mixed strategies In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects (an infinite sequence is a family). ...

operatorname{Pr} = (operatorname{Pr}_1, operatorname{Pr}_2, ldots,operatorname{Pr}_m)

Given a mixed strategy profile, the set Σ of pure strategy profiles becomes a probability space, where the probability of each pure strategy profile

vec{sigma} = (sigma_1, sigma_2, ldots,sigma_m)

is

operatorname{Pr}_1(sigma_1) timesoperatorname{Pr}_2(sigma_2) times cdots times operatorname{Pr}_m(sigma_m)

Any payoff function F on Σ thus becomes a random variable on (Σ Pr). The expectation of F relative to Pr is the extension of F to mixed strategies.


References

  • R. D. Luce and H. Raiffa, Games and Decisions, Dover Publications, 1989.
  • J. Weibull, Evolutionary Game Theory, MIT Press, 1996
  • J. von Neumann and O. Morgenstern, Theory of games and Economic Behavior, John Wiley Science Editions, 1964. This book was initially published by Princeton University Press in 1944.

External link


  Results from FactBites:
 
Payoff matrix - definition of Payoff matrix in Encyclopedia (756 words)
A payoff matrix or payoff function is a concept in game theory which shows what payoff each player will receive at the outcome of the game.
Technically, matrix is used only in the case when there are two players and the payoff function for each can be represented as a matrix.
Note that the payoff matrices themeselves are not symmetric matrices however.
Normal form game - Wikipedia, the free encyclopedia (824 words)
A payoff function for a player is a mapping from that the cross-product of players' strategy spaces to that player's set of payoffs (normally the set of real numbers, where the number represents a cardinal or ordinal utility - often cardinal in the normal form representation) of a player, i.e.
For example, in the game matrix on the left if a payoff is emboldened, it means that player whose payoff it is has a best response to the corresponding strategy played by the other player by playing that strategy.
In the matrix on the left, payoffs are emboldened as before to show the best response of the player whose payoff it is (note that a best response is the strategy according to that payoff, not the payoff).
  More results at FactBites »

 
 

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