Perfect Regular Equilibrium

each (θ_i,a¹, ...,a^{t 1}) ∈ Θ_i × A¹ × ∙ ∙ ∙ × A^{i 1}, μt(θ_i, a¹,..., a^{t 1}; ∙) is a probability measure
on ×j≠iβ(Θj) and 2) for every B ∈ ×i≠_lβ(β_i), μt(∙;B) is β(Θ_i) × (×t⅛×l=1β(AX mea-
surable. Here, the condition 1) requires that each μ^t(θ_i,α¹, ...,a^{t 1}: ∙) specify a probability
distribution of other players’ types over the information set Θ_-i × {(θ_i,a¹, ...,a^t-1)}. The
condition 2) requires that μ^t 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 U_i : Θ × A¹ ×∙∙∙× A^r
--r R. We assume that each U_i is bounded above or bounded below and ×=₀β(Θi) ×
(×^r=₁×f=₁ β(Ai)) measurable, which guarantees that U_i is integrable. In addition, we assume
that each U_i can be expressed as a sum of finite-period utility functions. Formally, for each U_i,
we assume that there exist both a partition {K} ≡ Γ of {1, 2, ...,T} and its associated finite-
period utility functions U^κ : Θ × (×_k∈_κA^k) —R R such that U_i(θ, a) = ∑_κ∈_r U^κ(θ, a^κ) for

every (θ, a) ∈ Θ × A¹ × ∙ ∙ ∙ × A^r where a^κ = (a^k)_k∈_κ ∈ ×_k∈_κA^k. Here, the partition Γ is a

disjoint collection of non-empty subsets K of {1,2,..., T} such that U_κ∈_rK = {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 E_i :
∏ --r R (= R U {-∞, ∞}) satisfying the following two conditions given any arbitrary

strategy profile δ. First, if E_i(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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