# Bimatrix game

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A finite non-cooperative game between two players. A bimatrix game is specified by two matrices $A=\|a_{ij}\|$ and $B=\|B_{ij}\|$ of the same dimension $m\times n$, which are, respectively, the payoff matrices (or gain matrices) of the players I and II. The strategy of player I is the selection of a row, that of player II the selection of a column. If player I chooses $i$ ($1\leq i\leq m$), while player II chooses $j$ ($1\leq j\leq n$), their respective payoffs (gains) will be $a_{ij}$ and $b_{ij}$; if $a_{ij}+b_{ij}=0$ for all $i,j$, the bimatrix game becomes a matrix game. The theory of bimatrix games is the simplest branch of the general theory of non-cooperative games, but even bimatrix games are not always solvable according to Nash or strongly solvable. Various algorithms are available to find equilibrium solutions in bimatrix games: a method of describing the submatrices of $A,B$ yielding all extreme points of the set of equilibrium solutions , ; and methods which reduce the problem of finding the equilibrium solutions of a bimatrix game to a problem of quadratic programming , , .