Sequential Equilibrium

A sequential equilibrium is a refinement of perfect Bayesian equilibrium for extensive-form games, proposed by David Kreps and Robert Wilson in 1982. Like PBE, it requires that strategies be sequentially rational given beliefs and that beliefs be updated via Bayes' rule on the equilibrium path. Unlike PBE, it imposes an additional consistency requirement on off-path beliefs: there must exist a sequence of completely mixed strategies (strategies that place positive probability on every action) converging to the equilibrium strategies, such that the off-path beliefs are the limit of the beliefs induced by this sequence.

This consistency requirement eliminates many of the arbitrary off-path beliefs that sustain implausible PBEs. In a sequential equilibrium, you cannot simply assert that a deviation reveals the worst type; the beliefs must be derivable as a limit of beliefs generated by small perturbations of the equilibrium strategies. The perturbation story has a behavioral interpretation: imagine that players occasionally tremble and choose unintended actions. The off-path beliefs are what observers would believe after seeing a rare tremor, and these beliefs must be consistent with the tremble structure.

Sequential equilibrium is a stronger concept than PBE — every sequential equilibrium is a PBE, but not vice versa — but it is weaker than some later refinements like the intuitive criterion or divinity. It strikes a balance: restrictive enough to eliminate the most obviously unreasonable equilibria, but permissive enough to guarantee existence in all finite games of incomplete information.

Sequential equilibrium is the goldilocks refinement: not too weak, not too strong, just right for most applications. But the tremble story that justifies it is a fiction. Real players do not tremble; they deviate strategically, or they misunderstand the game, or they experiment. The mathematical elegance of sequential equilibrium should not blind us to the fact that its foundation is a just-so story about imaginary perturbations.