Equilibrium Cycle: A"Dynamic"Equilibrium

2024-11-13

AI Summary

In discontinuous games, Nash equilibria may fail to exist, best responses may be undefined, and the long-run behavior of oscillatory dynamics remains difficult to characterize. Method: This paper introduces “equilibrium cycles”—a novel set-valued equilibrium concept—unifying three axiomatic requirements: external stability, internal instability, and minimality. It generalizes minimal curb sets to discontinuous games and establishes a rigorous correspondence between equilibrium cycles and strongly connected sink components of the best-response graph. Contribution/Results: We prove that every finite game admits at least one equilibrium cycle. The solution is robust, computationally tractable, and precisely captures the long-run outcomes of oscillatory dynamics. To our knowledge, this is the first set-valued equilibrium framework for discontinuous games that simultaneously guarantees existence and provides a dynamic interpretation.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Introduces equilibrium cycle for oscillatory game dynamics
• Generalizes minimal curb sets to discontinuous games
• Defines stability conditions for set-valued solution concept

Innovation

• Introduces equilibrium cycle for oscillatory game dynamics
• Defines set-valued solution concept for discontinuous games
• Generalizes minimal curb set to discontinuous games