A general definition of perfect equilibrium

2025-11-20

AI Summary

Existing definitions of perfect equilibrium exhibit limited applicability in generalized games—such as discontinuous-payoff games and finitely additive strategy games—due to restrictive assumptions on payoff continuity or strategy-space structure. Method: This paper introduces a unified definition framework grounded in “strategy carriers” and “fully mixed strategy nets,” integrating topological analysis, limit undominatedness, and invariance theory. Contribution/Results: The framework guarantees nonemptiness and compactness of the equilibrium set under standard conditions and is the first to systematically subsume seminal definitions—including Selten’s (finite games), Simon–Stinchcombe’s (continuous games), and Marinacci’s (uncertain environments). By relaxing continuity and structural requirements, it achieves enhanced robustness and broad applicability across diverse game classes, thereby providing critical theoretical foundations for unifying and extending core concepts in game theory.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Defining perfect equilibrium for broad game classes
• Extending Selten's concept using mixed strategy nets
• Establishing equilibrium properties under topological conditions

Innovation

• General perfect equilibrium definition for diverse games
• Completely mixed nets based on strategy carriers
• Nonempty compact equilibrium sets under topological conditions