# Question

Consider the general m × n, two-person, zero-sum game. Let pij denote the payoff to player 1 if he plays his strategy i (i = 1, . . . , m) and player 2 plays her strategy j ( j = 1, . . . , n). Strategy 1 (say) for player 1 is said to be weakly dominated by strategy 2 (say) if p1j ≤ p2j for j = 1, . . . , n and p1j ≤ p2j for one or more values of j.

(a) Assume that the payoff table possesses one or more saddle points, so that the players have corresponding optimal pure strategies under the minimax criterion. Prove that eliminating weakly dominated strategies from the payoff table cannot eliminate all these saddle points and cannot produce any new ones.

(b) Assume that the payoff table does not possess any saddle points, so that the optimal strategies under the minimax criterion are mixed strategies. Prove that eliminating weakly dominated pure strategies from the payoff table cannot eliminate all optimal mixed strategies and cannot produce any new ones.

(a) Assume that the payoff table possesses one or more saddle points, so that the players have corresponding optimal pure strategies under the minimax criterion. Prove that eliminating weakly dominated strategies from the payoff table cannot eliminate all these saddle points and cannot produce any new ones.

(b) Assume that the payoff table does not possess any saddle points, so that the optimal strategies under the minimax criterion are mixed strategies. Prove that eliminating weakly dominated pure strategies from the payoff table cannot eliminate all optimal mixed strategies and cannot produce any new ones.

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