Bimatrix game

In game theory, a bimatrix game is a simultaneous game for two players in which each player has a finite number of possible actions. The name comes from the fact that the normal form of such a game can be described by two matrixes - matrix A describing the payoffs of player 1 and matrix B describing the payoffs of player 2.

Player 1 is often called the "row player" and player 2 the "column player". If player 1 has m possible actions and player 2 n possible actions, then each of the two matrixes has m rows by n columns. when the row player selects the $i$-th action and the column player selects the $j$-th action, the payoff to the row player is $A[i,j]$ and the payoff to the column player is $B[i,j]$.

The players can also play mixed strategies. A mixed strategy for the row player is a non-negative vector x of length m such that: $\sum_{i=1}^m x_i = 1$. Similarly, a mixed strategy for the column player is a non-negative vector y of length m such that: $\sum_{j=1}^m y_j = 1$. When the players play mixed strategies with vectors x and y, the expected payoff of the row player is: $x^T A y$ and of the column player: $x^T B y$.

Nash equilibrium in bimatrix games

Every bimatrix game has a Nash equilibrium in (possibly) mixed strategies. Finding such a Nash equilibrium is a special case of the Linear complementarity problem and can be done in finite time by the Lemke–Howson algorithm.

There is a reduction from the problem of finding a Nash equilibrium in a bimatrix game to the problem of finding a competitive equilibrium in an economy with Leontief utilities.

Related terms

A zero-sum game is a special case of a bimatrix game in which $A+B = 0$.