A Characterization of Reny's Weakly Sequentially Rational Equilibrium through ε-Perfect γ-Weakly Sequentially Rational Equilibrium

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 ε-perfect γ-WSRE with local sequential rationality, which is accomplished by incorporating an extra behavioral strategy profile. For any given γ>0, we generate a perfect γ-WSRE as a limit point of a sequence of εₖ-perfect γ-WSRE with εₖ→ 0. A WSRE is then acquired from a limit point of a sequence of perfect γq-WSRE with γq→ 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 ε-perfect γ-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.