Payoff matrix

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.

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.

For player 1, the payoff matrix is as follows,

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

For player 1, the payoff matrix is

Swerve Continue 0 -1 +1 -20

For player 2, the matrix is:

Swerve Continue 0 +1 -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.

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 can select from {1,2, ..., m2}

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

[itex] \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} [itex]

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

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

A pure strategy profile is an association of strategies to players, that is an m-tuple

[itex] \vec{\sigma} = (\sigma_1, \sigma_2, \ldots,\sigma_m) [itex]

such that

[itex] \sigma_1 \in S_1, \sigma_2 \in S_2, \ldots, \sigma_m \in S_m [itex]

We will denote the set of strategy profiles by Σ.

A payoff function is a function

[itex] F: \Sigma \rightarrow \mathbb{R}. [itex]

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

[itex] (P, \mathbf{S}, \mathbf{F}) [itex]

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

[itex]\mathbf{S}= (S_1, S_2, \ldots, S_m) [itex]

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

[itex] \mathbf{F} = (F_1, F_2, \ldots, F_m) [itex]

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.

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:

[itex] \operatorname{Pr}_k(1), \operatorname{Pr}_k(2), \ldots, \operatorname{Pr}_k(n_k). [itex]

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.

A mixed strategy profile is an association of mixed strategies to players, that is an m-tuple of mixed strategies

[itex] \operatorname{Pr} = (\operatorname{Pr}_1, \operatorname{Pr}_2, \ldots,\operatorname{Pr}_m) [itex]

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

[itex] \vec{\sigma} = (\sigma_1, \sigma_2, \ldots,\sigma_m) [itex]

is

[itex] \operatorname{Pr}_1(\sigma_1) \times\operatorname{Pr}_2(\sigma_2) \times \cdots \times \operatorname{Pr}_m(\sigma_m) [itex]

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.

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