Perfect Regular Equilibrium

and a sequence of actions (a^t'⁺¹..... a^τ) ∈ A^t'⁺¹ × ∙ ∙ ∙ × A^τ such that for any t ≥ t',

I E_i(δ)-f_θJA₁ ∙∙∙ J^ U_i(θ.a¹..... n^{! 1}. a^t. α^t+1..... a^τ)δ^t(θ. a¹..... a^t~¹; da^t)∙∙∙δ¹(d; da¹')η(dθ') ∣< ε

where for each t, δ^t denotes the product measure of {δ^..... δ^t₁} on ×{=₁₁β(A^t), that is, δ^t = δ^ ×
∙ ∙ ∙ × δ^t₁. Second, if E_i(δ} is infinite, then for any M ∈ N, there exist both a period t^l ≤ T
and a sequence of actions (a^t'⁺¹,.... a^τ) ∈ A^t'⁺¹ × ∙ ∙ ∙ × A^τ such that for any t ≥ t',

JθJa¹ ∙ ∙ ∙ JA^tU_i(θ. a¹..... a^{t 1}. a^t. a^t+1..... a^τ)δ^t(θ. a¹..... a^t~¹ da^t^) ∙ ∙ ∙ δ¹(6^l; da¹)η(dθ~)

> M when E_i(δ) = ∞ and < —M when E_i(δ) = -∞.

This definition of the expected utility functional makes sense according to Ash (1972, 2.6)⁵ .

In this definition of the expected utility functional, the necessity of the second assumption
on the utility function, which is that the utility functions U_i can be expressed as sums of
finite-period utility functions U^κ, that is, U_i = ∑_κ∈r U^K, might not be clearly seen This
assumption is necessary to well-define an expected utility functional because the definition
uses finitely iterated integrals. The following example shows that without this assumption,
we might not be able to define an expected utility functional. Consider a game with just
one player. Let a function U : {a.β}^œ  > {0.1} be a utility function for the player such
that for any a ∈ {a.β }^œ, U (a) = 0 if a contains infinitely many a, otherwise U (a) = 1.

⁵ Let Fj be a σ — field of subsets of Ω_j∙ for each j = 1,..., n. Let μ₁ be a probability measure on F₁,
and for each (ω₁,..., ω_j-) ∈ Ω₁ × ∙ ∙ ∙ × Ω_j-, let μ(ω₁,..., ω_j-; B), B ∈ Fj+₁, be a probability measure on Fj+₁
(j = 1, 2,..., n — 1). Assume that μ(ω₁,..., ω_j-; C) is measurable for each fixed C ∈ Fj+₁. Let Ω = Ω₁ × ∙ ∙ ∙
×Ω_n and F = F1 × ∙ ∙ ∙ ×F_n.

(1) There is a unique probability measure μ on F such that for each measurable rectangle A₁ × ∙ ∙ ∙ × A_n
∈ ^f, m(^a1 × ∙ ∙ ∙ × ^An⁾ = J_j4₁ J∖₂ ∙ ∙ ∙ J'^_j μ(^ω1^,... ^, wn-ɪ; ^dωn_l') ∙ ∙∙ μL⅛ ^dωt')∣∣-β^dωtf

(2) Let f : (Ω, F) —> (R ,ββUβ and f ≥ 0. Then, f_ςifdμ = J_Qi ∙∙∙ J^ f (ω1,... ,ωn)μ(ω_b... ,ω_n-1; dω_n) ∙∙∙
μ₁(dω₁).

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