Mogahed's Notes

Finite Two-Person Zero-Sum Games

A zero-sum game $\Gamma$ is expressed as follows,

where $X$ and $Y$ are the set of strategies of player 1 and player 2 respectively. The function $K:X \times Y \rightarrow \mathbb{R}^1$ is the payoff function of player 1.

The elements $x \in X$ and $y \in Y$ are called the strategies of player 1 and 2 respectively. The cartesian product $X \times Y$, i.e., the pairs of strategies $(x,y)$, are called strategy profiles. For a certain strategy profile $(x,y)$, the payoff of player 1 is $K(x,y)$, whereas the payoff of player 2 is $-K(x,y)$. Since the player 2's payoff is player 1's negated payoff, $K$ is considered the payoff of the zero-sum game.

In a game $\Gamma$, player 1 selects a strategy $x \in X$ and player 2 selects a strategy $y \in Y$. Both players perform the strategy selection simultaneously and independently.

An important representation of a game is by using a matrix. Two-player games with a finite set of strategies can be represented via a matrix. A game that is represented by a matrix is called a matrix game.

Matrix games are zero-sum two-person games with a finite set of strategies that could be represented by a payoff matrix. A matrix game $\Gamma_A$ is defined as

The strategies of player 1 and player 2 are

The strategy profile in $\Gamma_A$ is $(x_i,y_j) \in X \times Y$, and the payoff function of player 1 is $K(x_i,y_j) = a_{i,j}$, whereas the payoff function of player 2 is $-K(x_i,y_j) = -a_{i,j}$.