Consider the Two-Person,zero-Sum Game Having the Following Payoff Table

Question

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The Answer is (a)Applying the minimax criterion to player 2 and the maximin criterion to player 1 yields the following results.

Maximum 5 3 5

Minimax value Thus,strategies 1 and 2 both yield the maximin value of -2 for player 1.Strategy 2 yields the minimax value of 3 for player 2.However,the entries in the payoff table that yield the maximin and minimax values are not the same,so we do not have a saddle point.Therefore,this is not a stable game,so the players instead need to find their optimal mixed strategies for this unstable game.(b)Let x_i = probability that player 1 will use strategy i (i = 1,2,3)x₄ = value of the game for player 1.Using the formulation developed in Sec.15.5,the linear programming formulation for player 1 now is Maximize x₄,subject to 5x₁ - 2x₂ + 4x₃ - x₄ ≥ 0 -2x₁ + 3x₂ - 3x₃ - x₄ ≥ 0 3x₁ - x₂ + 5x₃ - x₄ ≥ 0 x₁ + x₂ + x₃ = 1 and x₁ ≥ 0,x₂ ≥ 0,x₃ ≥ 0.(c)The corresponding linear programming formulation for player 2 is the dual of the linear programming model (the primal)formulated for player 1 in part (b).Applying the simplex method to find the optimal solution for the primal problem also automatically identifies the optimal solution for the dual problem as well in Row 0 of the simplex tableau (as described in Sec.6.1).Therefore,yes,applying the simplex method to the linear programming problem formulated in part (b)also will identify the optimal mixed strategy for player 2.

Consider the Two-Person,zero-Sum Game Having the Following Payoff Table

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