 Consecutive-number antisymmetric game

In this game the higher of two chosen integers wins one unit if they are consecutive, but pays two units if their difference is at least two. Ties pay zero. For any choices $1,\dots ,n$ with $n\ge 3$, both players can optimally choose $1,2,3$ with probabilities $1/4,1/2,1/4$ and assign zero probability to larger numbers. If $p$ is this vector, $Ap$ is zero in its first three entries, $-5/4$ in its fourth when present, and $-2$ in all later entries. Thus $Ap\le 0$ and $p^{T}A\ge 0$, proving the zero value and both security guarantees.