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 matrices - 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 has $$n$$ possible actions, then each of the two matrices 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 $$n$$ such that: $\sum_{j=1}^n y_j = 1$. When the players play mixed strategies with vectors $$x$$ and $$y$$, the expected payoff of the row player is: $$x^\mathsf{T} A y$$ and of the column player: $$x^\mathsf{T} B y$$.

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 $$A+B = 0$$.