Metastable Equilibria

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.