and a sequence of actions (at'+1..... aτ) ∈ At'+1 × ∙ ∙ ∙ × Aτ such that for any t ≥ t',
I Ei(δ)-fθJA1 ∙∙∙ J^ Ui(θ.a1..... n! 1. at. αt+1..... aτ)δt(θ. a1..... at~1; dat)∙∙∙δ1(d; da1')η(dθ') ∣< ε
where for each t, δt denotes the product measure of {δ^..... δt1} on ×{=11β(At), that is, δt = δ^ ×
∙ ∙ ∙ × δt1. Second, if Ei(δ} is infinite, then for any M ∈ N, there exist both a period tl ≤ T
and a sequence of actions (at'+1,.... aτ) ∈ At'+1 × ∙ ∙ ∙ × Aτ such that for any t ≥ t',
JθJa1 ∙ ∙ ∙ JAtUi(θ. a1..... at 1. at. at+1..... aτ)δt(θ. a1..... at~1 dat^) ∙ ∙ ∙ δ1(6l; da1)η(dθ~)
> M when Ei(δ) = ∞ and < —M when Ei(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 Ui can be expressed as sums of
finite-period utility functions Uκ, that is, Ui = ∑κ∈r UK, 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 Ωj∙ for each j = 1,..., n. Let μ1 be a probability measure on F1,
and for each (ω1,..., ωj-) ∈ Ω1 × ∙ ∙ ∙ × Ωj-, let μ(ω1,..., ωj-; B), B ∈ Fj+1, be a probability measure on Fj+1
(j = 1, 2,..., n — 1). Assume that μ(ω1,..., ωj-; C) is measurable for each fixed C ∈ Fj+1. Let Ω = Ω1 × ∙ ∙ ∙
×Ωn and F = F1 × ∙ ∙ ∙ ×Fn.
(1) There is a unique probability measure μ on F such that for each measurable rectangle A1 × ∙ ∙ ∙ × An
∈ f, m(a1 × ∙ ∙ ∙ × An) = Jj41 J∖2 ∙ ∙ ∙ J'^j μ(ω1,... , wn-ɪ; dωnl') ∙ ∙∙ μL⅛ dωt')∣∣-βdωtf
(2) Let f : (Ω, F) —> (R ,ββUβ and f ≥ 0. Then, fςifdμ = JQi ∙∙∙ J^ f (ω1,... ,ωn)μ(ωb... ,ωn-1; dωn) ∙∙∙
μ1(dω1).
10
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