Equilibrium Cycle: A "Dynamic" Equilibrium

TL;DR

This paper introduces the equilibrium cycle, a new set-valued solution concept for oscillatory game dynamics that extends classical equilibrium notions to discontinuous and complex games.

Contribution

It proposes the equilibrium cycle, capturing oscillatory outcomes and generalizing the minimal curb set to discontinuous games, applicable even without pure Nash equilibria.

Findings

Equilibrium cycle exists in games without pure Nash equilibria.

It characterizes stability and instability properties of action sets.

Relates to strongly connected sink components in finite games.

Abstract

In this paper, we introduce a novel equilibrium concept, called the equilibrium cycle, which seeks to capture the outcome of oscillatory game dynamics. Unlike the (pure) Nash equilibrium, which defines a fixed point of mutual best responses, an equilibrium cycle is a set-valued solution concept that can be demonstrated even in games where best responses do not exist (for example, in discontinuous games). The equilibrium cycle identifies a Cartesian product set of action profiles that satisfies three important properties: stability against external deviations, instability against internal deviations, and minimality. This set-valued equilibrium concept generalizes the classical notion of the minimal curb set to discontinuous games. In finite games, the equilibrium cycle is related to strongly connected sink components of the best response graph.