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 -th action and the column player selects the -th action, the payoff to the row player is and the payoff to the column player is.
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:. Similarly, a mixed strategy for the column player is a non-negative vector y of length n such that:. When the players play mixed strategies with vectors x and y, the expected payoff of the row player is: and of the column player:.

Nash equilibrium in bimatrix games

Every bimatrix game has a Nash equilibrium in 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.