= Solution
In a <matrix game>, entry $a_{ij}$ is the row player's payoff, and the column player's payoff is $-a_{ij}$. A <mixed strategy> is a probability vector over the available rows or columns. With strategies $p,q$, the row payoff is $p^TAq$. Its <optimal mixed strategy> maximizes its guaranteed payoff, while the column player minimizes the largest row payoff:
$$
\boxed{\max_{p\in\Delta_m}\min_{q\in\Delta_n}p^TAq
=\min_{q\in\Delta_n}\max_{p\in\Delta_m}p^TAq=v.}
$$
The equality is the finite <minimax theorem>. Equivalently the row player maximizes $\min_j(p^TA)_j$, and the column player minimizes $\max_i(Aq)_i$. Each optimal strategy guarantees its corresponding bound against every opponent strategy; for the column player's own negative payoff this is also a maximin strategy.
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