Explain what is meant by a two-person zero-sum game with payoff matrix $A=\left[a_{i j}\right]$. 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 $£ 1$. 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 $£ 1$. If she calls the bet and the toss was a head, she pays player I $£ 2$; if she calls the bet and the toss was a tail, player I must pay her $£ 2$.

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

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