= Solution
For a real <antisymmetric matrix> $A$, $p^TAp=0$ for every real vector $p$. Thus for each row strategy $p$, the column player could choose $q=p$, giving $\min_qp^TAq\le0$. Hence the row security value is at most zero. Conversely, for each column strategy $q$, the row choice $p=q$ gives $\max_pp^TAq\ge0$, so the column security value is at least zero. The <minimax theorem> equates these values, proving
$$
\boxed{v=0.}
$$
The choices matching the opponent in this proof establish inequalities for security levels; they do not by themselves specify a single universal optimal strategy.
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