# 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.^{[1]}

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 -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 *m* 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 (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.^{[2]}

## Related terms

A zero-sum game is a special case of a bimatrix game in which .

## References

- 1 2 Chandrasekaran, R. "Bimatrix games" (PDF). Retrieved 17 December 2015.
- ↑ Codenotti, Bruno; Saberi, Amin; Varadarajan, Kasturi; Ye, Yinyu (2006). "Leontief economies encode nonzero sum two-player games".
*Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06*. p. 659. doi:10.1145/1109557.1109629. ISBN 0898716055.