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

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