Explain what is meant by a two-person zero-sum game with payoff matrix $A=\left(a_{i j}: 1 \leqslant i \leqslant m, \quad 1 \leqslant j \leqslant n\right)$ and what is meant by an optimal strategy $p=\left(p_{i}: 1 \leqslant i \leqslant m\right)$.

Consider the following betting game between two players: each player bets an amount $1,2,3$ or 4 ; if both bets are the same, then the game is void; a bet of 1 beats a bet of 4 but otherwise the larger bet wins; the winning player collects both bets. Write down the payoff matrix $A$ and explain why the optimal strategy $p=\left(p_{1}, p_{2}, p_{3}, p_{4}\right)^{T}$ must satisfy $(A p)_{i} \leqslant 0$ for all $i$. Hence find $p$.