 Solution

For a real antisymmetric matrix $A$, $p^{T}Ap=0$ for every real vector $p$. Thus for each row strategy $p$, the column player could choose $q=p$, giving ${\min }_q{p}^{T}Aq\le 0$. Hence the row security value is at most zero. Conversely, for each column strategy $q$, the row choice $p=q$ gives ${\max }_p{p}^{T}Aq\ge 0$, so the column security value is at least zero. The minimax theorem equates these values, provingThe choices matching the opponent in this proof establish inequalities for security levels; they do not by themselves specify a single universal optimal strategy.