Metastability is a refinement of the Nash equilibria of a game derived from two conditions: embedding combines behavioral axioms called invariance and small–worlds, and continuity requires games with nearby best replies to have nearby equilibria. These conditions imply that a connected set of Nash equilibria is metastable if it is arbitrarily close to an equilibrium of every sufficiently small perturbation of the best–reply correspondence of every game in which the given game is embedded as an independent subgame. Metastability satisfies the same decision–theoretic properties as Mertens' stronger refinement called stability. Metastability is characterized by a strong form of homotopic essentiality of the projection map from a neighborhood in the graph of equilibria over the space of strategy perturbations. Mertens' definition differs by imposing homological essentiality, which implies a version of small–worlds satisfying a stronger decomposition property. Mertens' stability and metastability select the same outcomes of generic extensive–form games.
