Structural Consistency, Consistency, and Sequential Rationality
David M. Kreps
In an extensive form game, a strategy profile is a sequential equilibrium if there are consistent beliefs at all information sets which, with the strategy profile, are sequentially rational at every information set. Along the equilibrium path, consistent beliefs are computed using the strategy profile and Bayes' rule. Consistent out-of-equilibrium beliefs, on the other hand, are derived by taking the closure of belief-strategy profile pairs, for totally mixed strategies. Kreps and Wilson (1982) claim that consistent beliefs are structurally consistent; that is, out-of-equilibrium beliefs can be rationalized by some single alternative conjecture as to the strategy opponents have used. We show by example that this is incorrect. Moreover, while equilibrium beliefs are structurally consistent in some cases, the alternative conjectures that justify them are inconsistent with the hypothesis (implicit in sequential rationality) that there will be no further defections from the initially hypothesized equilibrium strategies. These difficulties disappear if one relaxes the requirement of structural consistency, so that out-of-equilibrium conjectures may be formed as convex combinations of conjectures strategies of opponents. But such convex combinations result in out-of-equilibrium conjectures in which opponents' strategies are correlated. Correlation arises "naturally" if one adopts the perspective that no action can really have zero probability. But if one admits this sort of correlation, then other forms of correlation might also seem reasonable, including correlation that is not permitted under consistency and sequential rationality.