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
More intriguing information
1. The name is absent2. The name is absent
3. Philosophical Perspectives on Trustworthiness and Open-mindedness as Professional Virtues for the Practice of Nursing: Implications for he Moral Education of Nurses
4. MULTIMODAL SEMIOTICS OF SPIRITUAL EXPERIENCES: REPRESENTING BELIEFS, METAPHORS, AND ACTIONS
5. STIMULATING COOPERATION AMONG FARMERS IN A POST-SOCIALIST ECONOMY: LESSONS FROM A PUBLIC-PRIVATE MARKETING PARTNERSHIP IN POLAND
6. Tissue Tracking Imaging for Identifying the Origin of Idiopathic Ventricular Arrhythmias: A New Role of Cardiac Ultrasound in Electrophysiology
7. L'organisation en réseau comme forme « indéterminée »
8. Experimental Evidence of Risk Aversion in Consumer Markets: The Case of Beef Tenderness
9. Estimating the Economic Value of Specific Characteristics Associated with Angus Bulls Sold at Auction
10. Literary criticism as such can perhaps be called the art of rereading.