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, independent of the horizon T. With n players and at most m actions each, the individual swap regret of every player is O(sqrt n m m 5/2(nm)) at every finite horizon. Each player predicts the deviation gains, then uses these predictions to update a row-stochastic transition matrix, and plays its stationary distribution. The proof combines a potential argument exploiting stationarity with a two-scale higher-order prediction analysis, using rooted-tree representations to handle the nonlinear dependence of deviation gains on the stationary distributions. 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 m T m in the adversarial setting.