The Geometry of Nash Equilibria and Correlated Equilibria and a Generalization of Zero-Sum Games

A pure strategy is coherent if it is played with positive probability in at least one correlated equilibrium. A game is pre-tight if in every correlated equilibrium, all incentives constraints for non deviating to a coherent strategy are tight. We show that there exists a Nash equilibrium in the relative interior of the correlated equilibrium polytope if and only if the game is pre-tight. Furthermore, the class of pretight games is shown to include and generalize the class of two-player zero-sum games.