Bimatrix game
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 <math>A</math> describing the payoffs of player 1 and matrix <math>B</math> describing the payoffs of player 2.
Player 1 is often called the "row player" and player 2 the "column player". If player 1 has <math>m</math> possible actions and player 2 has <math>n</math> possible actions, then each of the two matrices has <math>m</math> rows by <math>n</math> columns. When the row player selects the <math>i</math>-th action and the column player selects the <math>j</math>-th action, the payoff to the row player is <math>A[i,j]</math> and the payoff to the column player is <math>B[i,j]</math>.
The players can also play mixed strategies. A mixed strategy for the row player is a non-negative vector <math>x</math> of length <math>m</math> such that: <math display="inline">\sum_{i=1}^m x_i = 1</math>. Similarly, a mixed strategy for the column player is a non-negative vector <math>y</math> of length <math>n</math> such that: <math display="inline">\sum_{j=1}^n y_j = 1</math>. When the players play mixed strategies with vectors <math>x</math> and <math>y</math>, the expected payoff of the row player is: <math>x^\mathsf{T} A y</math> and of the column player: <math>x^\mathsf{T} B y</math>.