lin-0206
16.6 Relational fallback
When scalarization is rejected, retain the centered reciprocal game
\[ A=H-\tfrac 12\mathbf1\mathbf1^\top , \qquad A^\top =-A, \]
and solve
\[ \pi ^* \in \arg \min _{\pi \in \Delta (V)} \max _{q\in \Delta (V)}q^\top A\pi . \]
Proposition
16.5
Relational minimax guarantee
For every reciprocal population matrix \(H\), the game value is zero and a population minimax policy has zero pairwise exploitability. Uniform play is minimax when \(A\mathbf1=0\), but need not be minimax in asymmetric games.
Proof
Skew symmetry gives \(\pi ^\top A\pi =0\). The finite minimax theorem and exchange of the identical strategy simplices negate the value, so it equals zero.
The guarantee does not eliminate finite-sample error: an empirical minimax policy can still be exploitable under the population game.