(a) Construct a 4 4 matrix with no saddle entries. (b) Consider the following payoff matrix....

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(a) Construct a 4 4 matrix with no saddle entries. (b) Consider the following payoff matrix. Rose plays the mixed strategy 2 and Colin plays the mixed strategy (1, 1). 2 3 6 (i) Find the expected payoff of the matrix game in terms of x, expressing your answer in simplest form. (ii) Hence, find Colin's best mixed strategy. Which column will Colin always play? (iii) Assuming that Colin plays his best mixed strategy, find the expected payoff of the matrix game. (iv) What does the expected payoff of this matrix game tell us about the expected winnings for the players? For part (c), recall that a saddle entry is defined to be a matrix entry which is minimal in its row and maximal in its column. You may only use this definition in your proof in part (c). Do not assume any of the other properties of saddle entries that you have seen in this unit. (c) Consider the 2 2 matrix a b] c d where a, b, c and d are real numbers. Suppose that a is a saddle entry of the matrix. Prove that either row 1 dominates row 2, or column 1 dominates column 2 (or both). (Hint: Consider two possible cases: bd and bd.) (a) Construct a 4 4 matrix with no saddle entries. (b) Consider the following payoff matrix. Rose plays the mixed strategy 2 and Colin plays the mixed strategy (1, 1). 2 3 6 (i) Find the expected payoff of the matrix game in terms of x, expressing your answer in simplest form. (ii) Hence, find Colin's best mixed strategy. Which column will Colin always play? (iii) Assuming that Colin plays his best mixed strategy, find the expected payoff of the matrix game. (iv) What does the expected payoff of this matrix game tell us about the expected winnings for the players? For part (c), recall that a saddle entry is defined to be a matrix entry which is minimal in its row and maximal in its column. You may only use this definition in your proof in part (c). Do not assume any of the other properties of saddle entries that you have seen in this unit. (c) Consider the 2 2 matrix a b] c d where a, b, c and d are real numbers. Suppose that a is a saddle entry of the matrix. Prove that either row 1 dominates row 2, or column 1 dominates column 2 (or both). (Hint: Consider two possible cases: bd and bd.)