= Support certificate for an antisymmetric matrix game
{title2=$Ap\le0,\quad p\ge0,\quad\mathbf1^Tp=1$}
For an antisymmetric <payoff matrix>, such a vector guarantees that an opponent playing rows cannot win positively against it. Antisymmetry also gives $p^TA\ge0$, so the same <mixed strategy> guarantees nonnegative payoff as a row strategy and proves game value zero. Furthermore $p^TAp=0$ forces $(Ap)_i=0$ whenever $p_i>0$. To prove uniqueness, use one certified strategy $q$: any other optimal $p$ satisfies $q^TAp=-(Aq)^Tp$, so a strict negative component of $Aq$ forces the corresponding probability in $p$ to vanish.
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