Independence on Relative Probability Spaces and Consistent Assessments in Game Trees

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

“…Since H is finite, the set of integer-valued representations of is non-empty. 13 Definition 8 A plausibility order on the set of histories H is choice-measurable if it has at least one integer-valued representation F that satisfies the following property: ∀h ∈ D, ∀h ′ ∈ I(h), ∀a ∈ A(h),…”

“…An assessment (σ, µ) is KW-consistent if there is an infinite sequence σ 1 , ..., σ m , ... of completely mixed strategy profiles such that, letting µ m be the unique system of beliefs obtained from σ m by applying Bayes' rule, lim m→∞ (σ m , µ m ) = (σ, µ). 1 A number of authors have tried to shed light on this topological notion by relating it to more intuitive concepts, 2 such as (convex) structural consistency ( [15]), generally reasonable extended assessment ( [10]), stochastic independence ( [4,13]). 3 .…”

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

“…For example, the perfect Bayesian equilibrium = (c; d; g) and (a) = (be) = 1 of the game of Figure 3 re ‡ects the following belief revision policy: the initial beliefs are that Player 1 will play c; conditional on learning that Player 1 did not play c, the observer would become convinced 8 The probability of reaching history h is the sum of the probabilities of the histories that have h as a pre…x and is thus interpreted as the probability that the actual play of the game is or was at h. In the above example, conditional on t = 1 or t = 2 and considering decision and terminal histories in E, the probability of reaching b is given bŷ E (d = b and t = 1) +^ E (d = bL and t = 2) +^ E (d = bR and t = 2) = : 9 The reason why we take the support of E to be E \ D + , rather than E, is that terminal histories as well as decision histories h with (h) = 0 are irrelevant for the notion of Bayesian consistency and E so de…ned is a much simpler object (compare, for instance, the simpler function E with the more extensive function^ E in the above example).…”

“…It is shown in [6] that, in turn, sequential equilibrium is a strict re…nement of PBE. 1 1 For example, [9] adopt this interpretation. For a subjective interpretation of perfect Bayesian equilibrium see [5].…”

“…A "limit free"characterization of sequential equilibrium is also provided in [9] using relative probability spaces ( ; ) (which express the notion of an event being in…nitely less likely than another) and random variables s i :…”