A payoff matrix converted from A and B where player 1 has two possible actions V and W and player 2 has actions X, Y and Z 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

describing the payoffs of player 1 and matrix

describing the payoffs of player 2.

possible actions and player 2 has

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

Similarly, a mixed strategy for the column player is a non-negative vector

When the players play mixed strategies with vectors

, the expected payoff of the row player is:

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.

[1] 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.