Perfect Regular Equilibrium



each (θi,a1, ...,at 1) Θi × A1 × ∙ ∙ ∙ × Ai 1, μt(θi, a1,..., at 1; ) is a probability measure
on
×jiβj) and 2) for every B ×i≠lβi), μt(;B) is βi) × (×t⅛×l=1β(AX mea-
surable. Here, the condition 1) requires that each
μt(θi1, ...,at 1: ) specify a probability
distribution of other players’ types over the information set Θ
-i × {(θi,a1, ...,at-1)}. The
condition 2) requires that
μt allow a well-defined expected utility functional. Let Φ be the
set of all systems of beliefs. Then, an element4 (
μ, δ) in Φ × ∏ is called an assessment.

A Von Neumann-Morgenstern utility function for player i is defined as Ui : Θ × A1 ×∙∙∙× Ar
--r R. We assume that each Ui is bounded above or bounded below and ×=0βi) ×
(×r=1×f=1 β(Ai)) measurable, which guarantees that Ui is integrable. In addition, we assume
that each
Ui can be expressed as a sum of finite-period utility functions. Formally, for each Ui,
we assume that there exist both a partition
{K} ≡ Γ of {1, 2, ...,T} and its associated finite-
period utility functions
Uκ : Θ × (×kκAk) —R R such that Ui(θ, a) = κr Uκ(θ, aκ) for

every (θ, a) Θ × A1 × ∙ ∙ ∙ × Ar where aκ = (ak)kκ ×kκAk. Here, the partition Γ is a

disjoint collection of non-empty subsets K of {1,2,..., T} such that UκrK = {1, 2,..., T}.
These two assumptions ensure the existence of a well-defined expected utility functional.

An expected utility functional for player i is implicitly defined as a unique function Ei :
∏ --r R (= R U {-∞, ∞}) satisfying the following two conditions given any arbitrary

strategy profile δ. First, if Ei(δ) is finite, then for any ε > 0, there exist both a period t' T

4 For each i and t, the measures μ((∙; ∙) and J((∙; ∙) are known as transition probabilities. For more
information on the transition probability, please refer to Neveu (1965, III), Ash (1972, 2.6), and Uglanov
(1997).



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