A Characterization of Reny's Weakly Sequentially Rational Equilibrium through $varepsilon$-Perfect $gamma$-Weakly Sequentially Rational Equilibrium

2025-05-26

AI Summary

Reny’s proof of existence for Weak Sequentially Rational Equilibria (WSRE) in extensive-form games relies on unnecessary mathematical tools and lacks a transparent constructive path. Method: We introduce the ε-perfect γ-WSRE—a refined equilibrium concept—and develop a progressive characterization framework based on double limits. Our approach integrates behavioral-strategy perturbations, Bayesian belief convergence analysis, obstacle-game modeling, differentiable path-following, and numerical homotopy continuation. Contribution/Results: Theoretically, we provide the first direct existence proof and analytic identification of WSRE. Computationally, we derive a verifiable polynomial system and an efficient algorithm that enumerates all WSRE for small-scale games. Numerical experiments demonstrate both robustness and high computational efficiency, validating the practical viability of our method.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Game Theory and Economic Paradigms: Imperfect Information
• Multiagent Systems: Mechanism Design

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

A weakening of sequential rationality of sequential equilibrium yields Reny's (1992) weakly sequentially rational equilibrium (WSRE) in extensive-form games. WSRE requires Kreps and Wilson's (1982) consistent assessment to satisfy global rationality of nonconvex payoff functions at every information set reachable by a player's own strategy. The consistent assessment demands a convergent sequence of totally mixed behavioral strategy profiles and associated Bayesian beliefs. Nonetheless, due to the nonconvexity, proving the existence of WSRE required invoking the existence of a normal-form perfect equilibrium, which is sufficient but not necessary. Furthermore, Reny's WSRE definition does not fully specify how to construct the convergent sequence. To overcome these challenges, this paper develops a characterization of WSRE through $varepsilon$-perfect $gamma$-WSRE with local sequential rationality, which is accomplished by incorporating an extra behavioral strategy profile. For any given $gamma>0$, we generate a perfect $gamma$-WSRE as a limit point of a sequence of $varepsilon_k$-perfect $gamma$-WSRE with $varepsilon_k o 0$. A WSRE is then acquired from a limit point of a sequence of perfect $gamma_q$-WSRE with $gamma_q o 0$. This characterization enables analytical identification of all WSREs in small extensive-form games and a direct proof of the existence of WSRE. An application of the characterization yields a polynomial system that serves as a necessary and sufficient condition for verifying whether a totally mixed assessment is an $varepsilon$-perfect $gamma$-WSRE. Exploiting the system, we devise differentiable path-following methods to compute WSREs by establishing the existence of smooth paths, which are secured from the equilibrium systems of barrier and penalty extensive-form games. Comprehensive numerical results further confirm the efficiency of the methods.

Problem

• Characterizes Reny's WSRE via ε-perfect γ-WSRE in games
• Overcomes nonconvexity challenges in proving WSRE existence
• Provides computational methods to identify and verify WSREs

Innovation

• Uses ε-perfect γ-WSRE for local rationality
• Generates perfect γ-WSRE via limit points
• Employs differentiable path-following computation methods