A Characterization of Sequential Equilibrium through $\epsilon$-Perfect ${\hat{I}}^3$-Sequential Equilibrium with Local Sequential Rationality and Its Computation

Abstract

Sequential equilibrium requires a consistent assessment and sequential rationality, where the consistent assessment emerges from a convergent sequence of totally mixed behavioral strategies and associated beliefs. However, the original definition lacks explicit guidance on constructing such convergent sequences. To overcome this difficulty, this paper presents a characterization of sequential equilibrium by introducing $\epsilon$-perfect ${\hat{I}}^3$-sequential equilibrium with local sequential rationality. For any ${\hat{I}}^3>0$, we establish a perfect ${\hat{I}}^3$-sequential equilibrium as a limit point of a sequence of ${\epsilon }_k$-perfect ${\hat{I}}^3$-sequential equilibrium with ${\epsilon }_k\to 0$. A sequential equilibrium is then derived from a limit point of a sequence of perfect ${\hat{I}}_q^3$-sequential equilibrium with ${\hat{I}}_q^3\to 0$. This characterization systematizes the construction of convergent sequences and enables the analytical determination of sequential equilibria and the development of a polynomial system serving as a necessary and sufficient condition for $\epsilon$-perfect ${\hat{I}}^3$-sequential equilibrium. Exploiting the characterization, we develop a differentiable path-following method to compute a sequential equilibrium.