Payoff matrix
From Academic Kids

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 2player 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  

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
2player 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  

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.
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, ..., m_{1}} and player can select from {1,2, ..., m_{2}}
Definition. A payoff matrix for a twoplayer game is an m_{1} × m_{2} matrix of real numbers:
 <math> \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} <math>
Player 1's strategies are designated along the left hand column and 2's are the designated along the top row.
To specify a twoperson 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 noncooperative 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}
 Each player k in P has a finite number of pure strategies
 <math> S_k = \{1, 2, \ldots, n_k\}. <math>
A pure strategy profile is an association of strategies to players, that is an mtuple
 <math> \vec{\sigma} = (\sigma_1, \sigma_2, \ldots,\sigma_m) <math>
such that
 <math> \sigma_1 \in S_1, \sigma_2 \in S_2, \ldots, \sigma_m \in S_m <math>
We will denote the set of strategy profiles by Σ.
A payoff function is a function
 <math> F: \Sigma \rightarrow \mathbb{R}. <math>
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
 <math> (P, \mathbf{S}, \mathbf{F}) <math>
where P = {1,2, ...,m} is a set of players,
 <math>\mathbf{S}= (S_1, S_2, \ldots, S_m) <math>
is an mtuple of pure strategy sets, one for each player, and
 <math> \mathbf{F} = (F_1, F_2, \ldots, F_m) <math>
is an mtuple 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 Pr_{k} for each element of S_{k}={1, 2, ..., n_{k}}. We denote these probabilities as follows:
 <math> \operatorname{Pr}_k(1), \operatorname{Pr}_k(2), \ldots, \operatorname{Pr}_k(n_k). <math>
An operational interpretation of Pr_{k} for repeated plays of the game is as follows: prior to each play, player k chooses a strategy in S_{k} according to probability P_{k}.
A mixed strategy profile is an association of mixed strategies to players, that is an mtuple of mixed strategies
 <math> \operatorname{Pr} = (\operatorname{Pr}_1, \operatorname{Pr}_2, \ldots,\operatorname{Pr}_m) <math>
Given a mixed strategy profile, the set Σ of pure strategy profiles becomes a probability space, where the probability of each pure strategy profile
 <math> \vec{\sigma} = (\sigma_1, \sigma_2, \ldots,\sigma_m) <math>
is
 <math> \operatorname{Pr}_1(\sigma_1) \times\operatorname{Pr}_2(\sigma_2) \times \cdots \times \operatorname{Pr}_m(\sigma_m) <math>
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.