On sustainable equilibria

Following the ideas laid out in Myerson (1996), Hofbauer (2003) defined a Nash equilibrium of a finite game as sustainable if it can be made the unique Nash equilibrium of a game obtained by deleting/adding a subset of the strategies that are inferior replies to it. This paper proves a result about sustainable equilibria and uses it to provide a refinement as well. Our result concerns the Hofbauer-Myerson conjecture about the relationship between the sustainability of an equilibrium and its index: for a generic class of games, an equilibrium is sustainable iff its index is +1. von Schemde and von Stengel (2008) proved this conjecture for bimatrix games; we show that the conjecture is true for all finite games. More precisely, we prove that an equilibrium is isolated and has index +1 if and only if it can be made unique in a larger game obtained by adding finitely many strategies that are inferior replies to that equilibrium. It follows in a straightforward way from our result that sustainable equilibria fail the Decomposition Axiom for games as formulated by Mertens (1989a). In order to rectify this problem we propose a refinement, called strongly sustainable equilibria, which is shown to exist for all regular games.