Conversion of a Matrix Game to a Linear Programming Problem

Let us consider a *m* x *n* game with a payoff matrix

Let vector **x _{0}** = (x

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If all elements in the matrix **G ** are positive (this requirement can be fulfilled by adding a suitable positive constant to each payoff), the value of the game is also positive and we can divide each mentioned inequality by *v *without changing the sense of the inequalities. To simplify notation, we introduce new variables *t _{i}* instead of fractions

The relationship implies . Then .

Since , the last equation results in . Instead of maximizing *v*, we can minimize . Now we have the following linear programming problem:

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In a similar way, a linear programming problem can be formulated for player B with the optimal strategy **y _{0}** = (