Constant Swap Regret in General-Sum Games via Two-Scale Higher-Order Optimism

We give deterministic and uncoupled learning dynamics for finite multiplayer general-sum games under full-information feedback that achieve constant individual swap regret in self-play, independent of the horizon $T$. With $n$ players and at most $m$ actions each, every player’s individual swap regret is $O(\sqrt{n}m\log m{\log }^{5/2}(nm))$ at every finite horizon. The dynamics use the classical Blum–Mansour framework with optimism. Each player predicts the deviation gains, uses these predictions to update a row-stochastic transition matrix, and plays its stationary distribution. Our new ingredients include a tailored row normalization map and a two-scale higher-order predictor. An adversarially robust variant, obtained through a generic common-prefix switching wrapper, preserves the self-play bound up to a universal constant and guarantees individual swap regret at most $7\sqrt{mT\log m}$ in the adversarial setting.