The Algebraic Geometry of Perfect and Sequential Equilibrium

Two of the most important refinements of the Nash equilibrium concept for extensive form games with perfect recall are Selten’s (1975) perfect equilibrium and Kreps and Wilson’s (1982) more inclusive sequential equilibrium. These two equilibrium refinements are motivated in very different ways. Nonetheless, as Kreps and Wilson (1982, Section 7) point out, the two concepts lead to similar prescriptions for equilibrium play. For each particular game form, every perfect equilibrium is sequential. Moreover, for almost all assignments of payoffs to outcomes, almost all sequential equilibrium strategy profiles are perfect equilibrium profiles, and all sequential equilibrium outcomes are perfect equilibrium outcomes. We establish a stronger result: For almost all assignments of payoffs to outcomes, the sets of sequential and perfect equilibrium strategy profiles are identical. In other words, for almost all games each strategy profile which can be supported by beliefs satisfying the rationality requirement of sequential equilibrium can actually be supported by beliefs satisfying the stronger rationality requirement of perfect equilibrium. We obtain this result by exploiting the algebraic/geometric structure of these equilibrium correspondences, following from the fact that they are semi-algebraic sets; i.e., they are defined by finite systems of polynomial inequalities. That the perfect and sequential equilibrium correspondences have this semi-algebraic structure follows from a deep result from mathematical logic, the Tarski – Seidenberg Theorem; that this structure has important game-theoretic consequences follows from deep properties of semi-algebraic sets.