AGM-consistency and perfect Bayesian equilibrium. Part I: definition and properties

Abstract

We provide a general notion of perfect Bayesian equilibrium which can be applied to arbitrary extensive-form games and is intermediate between subgame-perfect equilibrium and sequential equilibrium. The essential ingredient of the proposed definition is the qualitative notion of AGM-consistency, which has an epistemic justification based on the theory of belief revision introduced by Alchourrón, Gärdenfors and Makinson. AGM-consistency is a generalization of the notion of consistency introduced by Kreps and Wilson as part of the definition of sequential equilibrium.

AGM-consistency and perfect Bayesian equilibrium. Part II: from PBE to sequential equilibrium

“…In this section we recall the notion of perfect Bayesian equilibrium introduced in [6]; we employ the same notation, which makes use of the history-based de…nition of extensive-form game (see [11]). As in [6], we restrict attention to …nite extensive-form games with perfect recall.…”

“…6 Property B3 requires the system of beliefs to satisfy Bayes'rule in the sense that if h 2 E and (h) > 0 (so that E is the equivalence class of the most plausible elements of I(h)) then for every history h 0 2 I(h), (h 0 ) (the probability assigned to h 0 by ) coincides with the probability of h 0 conditional on I(h) using the probability measure E . How should one interpret the probability E (h) of De…nition 3?…”

“…How should one interpret the probability E (h) of De…nition 3? This issue was not discussed in [6]. 7 First of all, it should be noted that one cannot interpret E (h) as the "probability that history h is reached by the actual play of the game".…”

“…In [6] a solution concept for extensive-form games was introduced, called perfect Bayesian equilibrium (PBE), and shown to be a strict re…nement of subgame-perfect equilibrium ( [16]); it was also shown that, in turn, the notion of sequential equilibrium ( [10]) is a strict re…nement of PBE. Besides sequential rationality, the ingredients of the de…nition of perfect Bayesian equilibrium are (1) the qualitative notions of plausibility order and AGM-consistency and (2) the notion of Bayesian consistency relative to the plausibility order.…”

“…There are two such conditions. One is the notion of choice-measurability of the plausibility order, which was shown in [6] to be implied by sequential equilibrium. Choice measurability requires that the plausibility order -on the set of histories H that rationalizes the given assessment have a cardinal representation, in the sense that there is an integer-valued function F : H !…”