Axiomatic Theory of Equilibrium Selection for Games with Two Players, Perfect Information, and Generic Payoffs

Abstract

Three axioms from decision theory are applied to refinements that select connected subsets of the Nash equilibria of games with perfect recall. The first axiom requires all equilibria in a selected subset to be admissible, i.e. each player's strategy is an admissible optimal reply to other players' strategies. The second axiom invokes backward induction by requiring a selected subset to contain a sequential equilibrium. The third axiom requires a refinement to be immune to embedding a game in a larger game with additional strategies and players, provided the original players' strategies and payoffs are preserved, viz., selected subsets must be the same as those induced by the selected subsets of any larger game in which it is embedded. These axioms are satisfied by refinements that select subsets that are stable as defined by Mertens (1989). For a game with two players, perfect information, and generic payoffs, we prove the converse that the axioms require a selected set to be stable. In the space of mixed strategies of minimal dimension, the stable set is unique and consists of the admissible equilibria with the same outcome as the unique subgame-perfect equilibrium obtained by backward induction. Each other admissible equilibrium with this outcome is the profile of players' strategies in an admissible sequential equilibrium of a larger game in which the original game is embedded, so the third axiom requires it to be included.