Optimization | Part IB, 2001

Explain what is meant by a two-person zero-sum game with payoff matrix A=[aij]A=\left[a_{i j}\right]. Write down a set of sufficient conditions for a pair of strategies to be optimal for such a game.

A fair coin is tossed and the result is shown to player I, who must then decide to 'pass' or 'bet'. If he passes, he must pay player II £1£ 1. If he bets, player II, who does not know the result of the coin toss, may either 'fold' or 'call the bet'. If player II folds, she pays player I £1£ 1. If she calls the bet and the toss was a head, she pays player I £2£ 2; if she calls the bet and the toss was a tail, player I must pay her £2£ 2.

Formulate this as a two-person zero-sum game and find optimal strategies for the two players. Show that the game has value 13\frac{1}{3}.

[Hint: Player I has four possible moves and player II two.]

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