Support certificate for an antisymmetric matrix game ($Ap\le 0,p\ge 0,1^{T}p=1$)

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