Question 1. Consider a zero-sum game with payoff matrix Ilij where I = J = {1,..., n}. Assume IIij is antisymmetic: Ilij = -Ilji. (i) Show that if the game has an equilibrium in pure strategy, then the value of the game for both players is zero. (ii) Show that the game has an equilibrium in pure strategy if and only if there is a i* such that II;*; > 0 for all j. (iii) Show that the value to player 1 of the game in a mixed strategy equi- librium is zero. (iv) Using Gurobi if needed, compute an equilibirum in mixed strategy when 0 3 1 -2 -3 0 -2 II = -1 2 0 1 2 5 -1 0 Is there an equilibrium in pure strategy? (v) Same questions as (iv) when II = 0 -3 1 2 3 0 - 2 -5 -1 2 0 1 2 5 -1 0