Refinements of Nash equilibrium in potential games

A strategic-form game is a potential game if the incentive of all players to change their strategy can be expressed in one global function, called the game’s potential. Potential games have many applications in Economics and other disciplines (cf. Rosenthal 1973, Monderer and Shapley 1996, Ostrovsky and Schwarz 2005, Armstrong and Vickers 2001, Myatt and Wallace 2009, inter alia). Potential games have a distinct computational advantage in that any maximizer of a potential is a pure-strategy Nash equilibrium. Therefore, the computation of an equilibrium is reduced to the solution of an optimization problem, thus obviating the need for computational fixed point theory. A Nash equilibrium need not maximize the potential function, so the set of maximizers provides a natural equilibrium selection device. Given the focus on the set of maximizers of a potential, it would be helpful to know if any maximizers are “robust” as Nash equilibria, and it is this issue that we study in this paper. Structurally, we work with games in which each player’s strategy set is a nonempty, compact metric space and for which there exists an upper semicontinuous potential. Consequently, we are working in a framework substantially more general than the case of finite strategy spaces, and the upper semicontinuity assumption adds extra flexibility in applications. Furthermore, the upper semicontinuity assumption allows us to use some basic machinery from variational analysis.1