The pure Nash equilibrium property and the quasi-acyclic condition

This paper presents a sufficient condition for the quasi-acyclic condition. A game is quasi-acyclic if from any strategy profile, there exists a finite sequence of strict best replies that ends in a pure strategy Nash equilibrium. The best-reply dynamics must converge to a pure strategy Nash equilibrium in any quasi-acyclic game. A game has the pure Nash equilibrium property (PNEP) if there is a pure strategy Nash equilibrium in any game constructed by restricting the set of strategies to a subset of the set of strategies in the original game. Any finite, ordinal potential game and any finite, supermodular game have the PNEP. We show that any finite, two-player game with the PNEP is quasi-acyclic.