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

TL;DR

This paper characterizes Reny's weakly sequentially rational equilibrium (WSRE) in extensive-form games using $oldsymbol{ ext{ extit{ extgamma}}}$-WSRE, providing a new analytical framework, existence proof, and computational methods for identifying WSREs.

Contribution

It introduces a novel characterization of WSRE through $oldsymbol{ ext{ extit{ extgamma}}}$-WSRE, enabling analytical identification, existence proof, and efficient computation of WSREs.

Findings

Provides a polynomial system for verifying $oldsymbol{ ext{ extit{ extgamma}}}$-WSRE.

Establishes the existence of smooth paths for computational methods.

Demonstrates the efficiency of path-following algorithms through numerical results.

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 $\varepsilon_k\to 0$. A WSRE is then acquired from a limit point of a sequence of perfect $\gamma_q$-WSRE with $\gamma_q\to 0$. 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.