  Following the ideas laid out in Myerson (1996), Hofbauer (2000) 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 two results about
sustainable equilibria. The first 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 isolated equilibrium 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. Our second result gives an
axiomatic extension of sustainability to all games and shows that only the Nash
components with positive index can be sustainable.
