Optimal Strategy Formulas for a 2 x 2 Game

Let us consider a game with a payoff matrix

and with optimal strategies of players x0, y0. Under the assumption that the game has no saddle point, both components of the vector y0 are positive and therefore the following equations hold: E(x0,B1)=v, E(x0,B2)=v (these equations follow from the equivalence of a matrix game to a linear programming problem and from relationships between solutions of primal and dual problems). Since E(x0,B1)=a11x01+a21x02, E(x0,B2)= a12x01+a22x02, the following equation is valid: a11x01+a21x02 = a12x01+a22x02. After the substitution x02 =1- x01 , the solution to this equation yields the following: