A Characterization of Nash Equilibrium in Behavioral Strategies through Local Sequential Rationality

The concept of Nash equilibrium in behavioral strategies (NashEBS) was formulated By Nash~\cite{Nash (1951)} for an extensive-form game through global rationality of nonconvex payoff functions. Kuhn's payoff equivalence theorem resolves the nonconvexity issue, but it overlooks that one Nash equilibrium of the associated normal-form game can correspond to infinitely many NashEBSs of an extensive-form game. To remedy this multiplicity, the traditional approach as documented in Myerson~\cite{Myerson (1991)} involves a two-step process: identifying a Nash equilibrium of the agent normal-form representation, followed by verifying whether the corresponding mixed strategy profile is a Nash equilibrium of the associated normal-form game, which often scales exponentially with the size of the extensive-form game tree. In response to these challenges, this paper develops a characterization of NashEBS through the incorporation of an extra behavioral strategy profile and beliefs, which meet local sequential rationality of linear payoff functions and self-independent consistency. This characterization allows one to analytically determine all NashEBSs for small extensive-form games. Building upon this characterization, we acquire a polynomial system serving as a necessary and sufficient condition for determining whether a behavioral strategy profile is a NashEBS. An application of the characterization yields differentiable path-following methods for computing such an equilibrium.