Finite-magnet correction and tightness of the I-projection law

Derive the finite-magnet-strength quantal-response-equilibrium correction to the reach-weighted I-projection law for regularized self-play and establish formally when the approximation is tight.

Background

The paper finds that the selected Nash-equilibrium member approximately follows the reach-weighted I-projection of the reference policy onto the Nash polytope. The approximation becomes accurate under projection-based readout but is not established as an exact finite-magnet-strength identity.

The authors explicitly identify two unresolved theoretical tasks: deriving a correction for finite magnet strength, where the regularized fixed point is QRE-like rather than exactly Nash, and specifying conditions under which the I-projection approximation is tight. These questions matter for predicting selection accuracy beyond the tractable experimental settings examined in the paper.