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

Abstract

In [6] a general notion of perfect Bayesian equilibrium (PBE) for extensive-form games was introduced and shown to be intermediate between subgame-perfect equilibrium and sequential equilibrium. Besides sequential rationality, the ingredients of the proposed notion are (1) the existence of a plausibility order on the set of histories that rationalizes the given assessment and (2) the notion of Bayesian consistency relative to the plausibility order. We show that a cardinal property of the plausibility order and a strengthening of the notion of Bayesian consistency provide necessary and su¢ cient conditions for a PBE to be a sequential equilibrium.

Exploring the Gap between Perfect Bayesian Equilibrium and Sequential Equilibrium

“…This definition of PBE is based on two notions (besides sequential rationality): (1) the qualitative property of AGM-consistency relative to a plausibility order 1 ; and (2) the quantitative property of Bayes consistency. This notion of PBE was further used in [8] to provide a new characterization of sequential equilibrium, in terms of a strengthening of both AGM consistency and Bayes consistency. In this paper we explore the gap between PBE and sequential equilibrium, by identifying solution concepts that lie strictly between PBE and sequential equilibrium.…”

“…The paper is organized as follows. Section 2 reviews the notation, definitions and main results of [5,8]. The new material is contained in Sections 3 and 4.…”

“…Consider the extensive-form game shown in Figure 1 7 and the assessment (σ, µ) where σ = (d, e, g) and µ is the following system of beliefs: µ(a) = 0, µ(b) = 1 3 , µ(c) = 2 3 and µ(a f ) = µ(b f ) = 1 2 . This assessment is AGM-consistent, since it is rationalized by the following plausibility order 8 : The precise definition is as follows. Let Z denote the set of terminal histories and, for every player i, let U i : Z → R be player i's von Neumann-Morgenstern utility function.…”

“…10 Note that if h, h ∈ E and h = ha 1 ...a m , then σ(a j ) > 0, for all j = 1, ..., m. In fact, since h ∼ h, every action a j is plausibility preserving and therefore, by Property P1 of Definition 2, σ(a j ) > 0. 11 For an interpretation of the probabilities ν E (h) see [8].…”

“…An assessment (σ, µ) is a sequential equilibrium if it is KW-consistent and sequentially rational. In [8] it is shown that sequential equilibrium can be characterized as a strengthening of PBE based on two properties: (1) a property of the plausibility order that constrains the supports of the belief system; and (2) a strengthening of the notion of Bayes consistency, that imposes constraints on how the probabilities can be distributed over those supports. The details are given below.…”