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 ${\displaystyle A}$ describing the payoffs of player 1 and matrix ${\displaystyle B}$ describing the payoffs of player 2.^[1]

${\displaystyle \mathbf {A} ={\begin{bmatrix}c&d&e\\f&g&h\\\end{bmatrix}}}$	${\displaystyle \mathbf {B} ={\begin{bmatrix}p&q&r\\s&t&u\\\end{bmatrix}}}$
${\displaystyle {\begin{array}{c|c|c|c}&X&Y&Z\\\hline V&c,p&d,q&e,r\\W&f,s&g,t&h,u\\\end{array}}}$ 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

Player 1 is often called the "row player" and player 2 the "column player". If player 1 has ${\displaystyle m}$ possible actions and player 2 has ${\displaystyle n}$ possible actions, then each of the two matrices has ${\displaystyle m}$ rows by ${\displaystyle n}$ columns. When the row player selects the ${\displaystyle i}$-th action and the column player selects the ${\displaystyle j}$-th action, the payoff to the row player is ${\displaystyle A[i,j]}$ and the payoff to the column player is ${\displaystyle B[i,j]}$.

The players can also play mixed strategies. A mixed strategy for the row player is a non-negative vector ${\displaystyle x}$ of length ${\displaystyle m}$ such that: ${\textstyle \sum _{i=1}^{m}x_{i}=1}$. Similarly, a mixed strategy for the column player is a non-negative vector ${\displaystyle y}$ of length ${\displaystyle n}$ such that: ${\textstyle \sum _{j=1}^{n}y_{j}=1}$. When the players play mixed strategies with vectors ${\displaystyle x}$ and ${\displaystyle y}$, the expected payoff of the row player is: ${\displaystyle x^{\mathsf {T}}Ay}$ and of the column player: ${\displaystyle x^{\mathsf {T}}By}$.

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 ${\displaystyle A+B=0}$.