Characterize when profile and normal-form correlated equilibria coincide or separate in polytope games

Determine necessary and sufficient conditions on two-player polytope games—where each player’s action set is a convex polytope and utilities are bilinear—for which a separation exists between profile correlated equilibria (defined as correlated strategy profiles whose players each have zero profile swap regret) and normal-form correlated equilibria (defined as correlated equilibria of the vertex normal-form game and projected into the original space). Specifically, characterize the class of polytope games G for which the projection of the set of normal-form correlated equilibria equals the set of profile correlated equilibria, and the class for which there is a strict gap, beyond the known case that the gap disappears when either player’s action set is a simplex.

Background

The paper introduces two distinct notions of correlated equilibrium for polytope games: normal-form correlated equilibria (NFCE), derived from the vertex normal-form game and implementable by a two-sided mediator, and profile correlated equilibria (PCE), defined by the outcomes of minimizing profile swap regret for both players. While these notions coincide in normal-form games, the authors prove a separation in general polytope games: there exist profile equilibria that cannot be implemented by any normal-form correlated equilibrium.

They also provide a partial characterization: the separation disappears when either player’s action set is a simplex, covering important subclasses such as Bayesian games with one-sided private information. Fully characterizing when these notions coincide or separate remains open and is important for understanding the relationship between mediator-implementable outcomes and those achievable by decentralized learning dynamics in structured games.