(a) Explain what is meant by a two-person zero-sum game with payoff matrix $A=\left(a_{i j}: 1 \leqslant i \leqslant m, 1 \leqslant j \leqslant n\right)$ and define what is an optimal strategy (also known as a maximin strategy) for each player.

(b) Suppose the payoff matrix $A$ is antisymmetric, i.e. $m=n$ and $a_{i j}=-a_{j i}$ for all $i, j$. What is the value of the game? Justify your answer.

(c) Consider the following two-person zero-sum game. Let $n \geqslant 3$. Both players simultaneously call out one of the numbers $\{1, \ldots, n\}$. If the numbers differ by one, the player with the higher number wins $£ 1$ from the other player. If the players' choices differ by 2 or more, the player with the higher number pays $£ 2$ to the other player. In the event of a tie, no money changes hands.

Write down the payoff matrix.

For the case when $n=3$ find the value of the game and an optimal strategy for each player.

Find the value of the game and an optimal strategy for each player for all $n$.

[You may use results from the course provided you state them clearly.]