A general definition of perfect equilibrium

TL;DR

This paper introduces a broad, unified definition of perfect equilibrium applicable to various game types, extending classical concepts to more general settings including discontinuous payoffs and finitely additive strategies.

Contribution

It develops a comprehensive framework for perfect equilibrium using completely mixed strategy nets, unifying existing notions and extending applicability to complex game scenarios.

Findings

The new definition yields a nonempty, compact set of perfect equilibria under standard conditions.

For finite actions, it coincides with Selten's original notion.

Applicable to games with discontinuities and finitely additive strategies.

Abstract

We propose a general definition of perfect equilibrium which is applicable to a wide class of games. A key feature is the concept of completely mixed nets of strategies, based on a more detailed notion of carrier of a strategy. Under standard topological conditions, this definition yields a nonempty and compact set of perfect equilibria. For finite action sets, our notion of perfect equilibrium coincides with Selten's (1975) original notion. In the compact-continuous case, perfect equilibria are weak perfect equilibria in the sense of Simon and Stinchcombe (1995). In the finitely additive case, perfect equilibria in the sense of Marinacci (1997) are perfect. Under mild conditions, perfect equilibrium meets game-theoretic desiderata such as limit undominatedness and invariance. We provide a variety of examples to motivate and illustrate our definition. Notably, examples include applications to games with discontinuous payoffs and games played with finitely additive strategies.