Bimatrix Games

These are two person non-zero(constant)-sum games in which each player has finitely many pure strategies. We call these non-zero(constant)sum games because the interests of the players is not required to be exactly opposed to each other. Of course, these include zero(constant)-sum games and are a true generalization of zero(constant)-sum games. But the methods used to analyze them are different. They also bring our many more difficulties as shown in the initial part. Suppose player 1 has m pure strategies and player 2 has n pure strategies. Let A be an m × n matrix representing payoffs to player 1and similarly let B be an m × n matrix representing the payoff matrix to player 2. If A+B = 0(αJ) where the right side is an m×n matrix all of whose entries are zero(J is a matrix all of whose entries are equla to 1 and α is a number), then we have the zero(constant)-sum case. Else we have the non-zero(constant)-sum case. A pair of (mixed) strategies x ∈ X = {x : x ≥ 0; ∑m i=1 xi = 1} and y ∗ ∈ Y = {y : y ≥ 0; ∑n j=1 yj = 1} which satisfy the relations: