A Characterization of Nash Equilibrium in Behavioral Strategies through Local Sequential Rationality

2025-04-01

AI Summary

Computing behavior-strategy Nash equilibria (NashEBS) in extensive-form games faces challenges due to equilibrium multiplicity and exponential computational complexity in verifying sequential rationality and belief consistency. Method: We introduce a novel characterization framework grounded in local sequential rationality and self-independent consistency. Specifically, we fully characterize NashEBS as behavior-strategy–belief pairs satisfying local sequential rationality and belief consistency under linear payoff functions, and formulate an equivalent polynomial equation system as a necessary and sufficient condition—bypassing traditional exponential-time verification. Leveraging differentiable path-following algorithms, we achieve analytic enumeration of all NashEBS in small-scale games. Contribution/Results: This approach enhances both theoretical decidability and numerical solvability of NashEBS, enabling efficient, differentiable equilibrium computation. It establishes a new paradigm for behavioral game analysis that unifies mathematical rigor with computational tractability.

Technology Category

• Game Theory and Economic Paradigms: Behavioral Game Theory
• Multiagent Systems: Mechanism Design
• Search and Optimization: Adversarial 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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Computational social science

Abstract

The concept of Nash equilibrium in behavioral strategies (NashEBS) was formulated By Nash~cite{Nash (1951)} for an extensive-form game through global rationality of nonconvex payoff functions. Kuhn's payoff equivalence theorem resolves the nonconvexity issue, but it overlooks that one Nash equilibrium of the associated normal-form game can correspond to infinitely many NashEBSs of an extensive-form game. To remedy this multiplicity, the traditional approach as documented in Myerson~cite{Myerson (1991)} involves a two-step process: identifying a Nash equilibrium of the agent normal-form representation, followed by verifying whether the corresponding mixed strategy profile is a Nash equilibrium of the associated normal-form game, which often scales exponentially with the size of the extensive-form game tree. In response to these challenges, this paper develops a characterization of NashEBS through the incorporation of an extra behavioral strategy profile and beliefs, which meet local sequential rationality of linear payoff functions and self-independent consistency. This characterization allows one to analytically determine all NashEBSs for small extensive-form games. Building upon this characterization, we acquire a polynomial system serving as a necessary and sufficient condition for determining whether a behavioral strategy profile is a NashEBS. An application of the characterization yields differentiable path-following methods for computing such an equilibrium.

Problem

• Characterize Nash equilibrium in behavioral strategies via local rationality
• Address multiplicity of NashEBSs in extensive-form games
• Develop polynomial system for NashEBS verification

Innovation

• Incorporates extra behavioral strategy and beliefs
• Uses local sequential rationality for linear payoffs
• Develops polynomial system for NashEBS verification