Metastable Equilibria

Metastable Equilibria

By Srihari Govindan, Robert Wilson
2007Working Paper No. 1934

We define a refinement of Nash equilibria called metastability. This refinement supposes that the given game might be embedded within any global game that leaves its local bestreply correspondence unaffected. A selected set of equilibria is metastable if it is robust against perturbations of every such global game; viz., every sufficiently small perturbation of the best-reply correspondence of each global game has an equilibrium that projects arbitrarily near the selected set. Metastability satisfies the standard decisiontheoretic axioms obtained by Mertens (1989) refinement (the strongest proposed refinement), and it satisfies the projection property in Mertens small-worlds axiom: a metastable set of a global game projects to a metastable set of a local game. But the converse is slightly weaker than Mertens decomposition property: a metastable set of a local game contains a metastable set that is the projection of a metastable set of a global game. This is inevitable given our demonstration that metastability is equivalent to a strong form of homotopic essentiality. Mertens definition invokes homological essentiality whereas we derive homotopic essentiality from primitives (robustness for every embedding). We argue that this weak version of decomposition has a natural gametheoretic interpretation.

Keywords
game theory, economic theory