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.

Contents

• Nash equilibrium in bimatrix games
• Related terms
• References

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: ∑ 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: ∑ 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 .