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❉❡❝✐s✐♦♥ ❚❤❡♦r② ▼❡❡ts ▲✐♥❡❛r ❖♣t✐♠✐③❛t✐♦♥ ✭❇❡②♦♥❞ ❈♦♠♣✉t❛t✐♦♥✮

❈✳ ❏❛♥s❡♥ ❚✳ ❆✉❣✉st✐♥

• ✳ ❙❝❤♦❧❧♠❡②❡r

❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s✱ ▲▼❯ ▼✉♥✐❝❤ ❊❈❙◗❆❘❯ ✷✵✶✼✱ ✶✸t❤ ♦❢ ❏✉❧②✱ ▲✉❣❛♥♦

✶ ✴ ✶✻

❙❝♦♣❡ ♦❢ t❤❡ t❛❧❦

◮ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ t❤❡♦r② ❤❛s ❜❡❡♥ s❤♦✇♥ t♦ ❜❡ ❛ ♣♦✇❡r❢✉❧ t♦♦❧ ❢♦r ✭✐♠♣r❡❝✐s❡✮ ❞❡❝✐s✐♦♥ t❤❡♦r② r❡❣❛r❞✐♥❣ ❜♦t❤

◮ ❡✣❝✐❡♥t ❝♦♠♣✉t❛t✐♦♥ ♦❢ ♦♣t✐♠❛❧ ❛❝ts ✇✳r✳t✳ ❝♦♠♣❧❡① ❝r✐t❡r✐❛

✭❝❢✳✱ ❡✳❣✳✱ ❑✐❦✉t✐ ❡t ❛❧✳ ✭✷✵✶✷✮ ♦r ❯t❦✐♥ ❛♥❞ ❆✉❣✉st✐♥ ✭✷✵✵✺✮✮

◮ ♣r♦✈✐❞✐♥❣ t❤❡♦r❡t✐❝❛❧ ✐♥s✐❣❤ts ♦♥ ♣r♦♣❡rt✐❡s ♦❢ ♦♣t✐♠❛❧ ❛❝ts

✭❝❢✳✱ ❡✳❣✳✱ ❲❡✐❝❤s❡❧❜❡r❣❡r ✭✶✾✾✻✮✮ ◮ ♦✉r ♣❛♣❡r ♣r❡s❡♥ts s♦♠❡ ♥❡✇ r❡s✉❧ts ❝♦♥❝❡r♥✐♥❣ ❜♦t❤ r❡❣❛r❞s ✐♥❝❧✉❞✐♥❣

◮ ❧✐♥❡❛r ♣r♦❣r❛♠s ❢♦r ❍♦❞❣❡s ❛♥❞ ▲❡❤♠❛♥♥ ❛♥❞ ❲❛❧❧❡②✬s ♠❛①✐♠❛❧✐t② ◮ ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ ❧❡❛st ❢❛✈♦r❛❜❧❡ ♣r✐♦rs ❛♥❞ ●❛♠♠❛✲▼❛①✐♠✐♥ ✷ ✴ ✶✻

❙❡t✉♣ ❛♥❞ ♥♦t❛t✐♦♥

❲❡ ❝♦♥s✐❞❡r t❤❡ st❛♥❞❛r❞ ♠♦❞❡❧ ♦❢ ✭✜♥✐t❡✮ ❝❛r❞✐♥❛❧ ❞❡❝✐s✐♦♥ t❤❡♦r②✿ ◮ A = {a✶, . . . , an}✿ s❡t ♦❢ ❛❝ts ◮ Θ = {θ✶, . . . , θm}✿ s❡t ♦❢ st❛t❡s ♦❢ t❤❡ ✇♦r❧❞ ◮ u : A×Θ → R✿ ✉t✐❧✐t② ❢✉♥❝t✐♦♥✱ ✇❤❡r❡ uij := u(ai, θj) ✐s t❤❡ ✉t✐❧✐t② ♦❢ ❝❤♦♦s✐♥❣ ❛❝t ai ❣✐✈❡♥ θj ✐s t❤❡ tr✉❡ st❛t❡ ♦❢ t❤❡ ✇♦r❧❞ u(ai, θj) θ✶ · · · θm a✶ u(a✶, θ✶) · · · u(a✶, θm) ✳ ✳ ✳ ✳ ✳ ✳ · · · ✳ ✳ ✳ an u(an, θ✶) · · · u(an, θm) ◮ ❢♦r ❡✈❡r② a ∈ A✱ ❞❡✜♥❡ ua : Θ → R ❜② ua(θ) := u(a, θ) ❢♦r ❛❧❧ θ ∈ Θ ◮ ❢♦r ❡✈❡r② θ ∈ Θ✱ ❞❡✜♥❡ uθ : A → R ❜② uθ(a) := u(a, θ) ❢♦r ❛❧❧ a ∈ A

✸ ✴ ✶✻

❙❡t✉♣ ❛♥❞ ♥♦t❛t✐♦♥✱ ❝♦♥t✐♥✉❡❞

❉❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❝♦♥t❡①t✱ ✇❡ ❛❧s♦ ❛❧❧♦✇ ❢♦r r❛♥❞♦♠✐③❡❞ ❛❝ts✿ ◮ ❝❛❧❧ ❡✈❡r② ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ λ ♦♥ (A, ✷A) ❛ r❛♥❞♦♠✐③❡❞ ❛❝t ❛♥❞ ❞❡♥♦t❡ ❜② G(A) t❤❡ s❡t ♦❢ ❛❧❧ r❛♥❞♦♠✐③❡❞ ❛❝ts ◮ ❝❤♦♦s✐♥❣ λ ✐s ✐♥t❡r♣r❡t❡❞ ❛s ❧❡❛✈✐♥❣ t❤❡ ✜♥❛❧ ❞❡❝✐s✐♦♥ t♦ ❛ r❛♥❞♦♠ ❡①♣❡r✐♠❡♥t ✇❤✐❝❤ ②✐❡❧❞s ❛❝t ai ✇✐t❤ ♣r♦❜❛❜✐❧✐t② λ({ai}) ◮ ❡✈❛❧✉❛t❡ ❝❤♦♦s✐♥❣ λ ❣✐✈❡♥ θ ❜② G(u)(λ, θ) := Eλ

• uθ

◮ ❢♦r λ ∈ G(A)✱ ❞❡✜♥❡ G(u)λ : Θ → R ❜② G(u)λ(θ) := G(u)(λ, θ) ◮ ✐❞❡♥t✐❢② a ∈ A ✇✐t❤ δa ∈ G(A) ❛♥❞ ♦❜s❡r✈❡ u(a, θ) = G(u)(δa, θ)

✹ ✴ ✶✻

❘❛♥❞♦♠✐③❛t✐♦♥✿ ❆ t♦② ❡①❛♠♣❧❡

◮ ❈♦♥s✐❞❡r ❛ ❣❛♠❡ ❜❡t✇❡❡♥ t✇♦ ♣❧❛②❡rs✿ P✐♥❦② ✭r♦✇s✮ ❛♥❞ ❇r❛✐♥ ✭❝♦❧✉♠♥s✮ ◮ P✐♥❦② ❝❤♦♦s❡s ♠♦✈❡s P = {p✶, p✷}✱ ❇r❛✐♥ r❡❛❝ts ❜② ♠♦✈❡s B = {b✶, b✷} ◮ P✐♥❦②✬s ✉t✐❧✐t② up : P × B → R ✐s ❣✐✈❡♥ ❜② t❤❡ ❜❡❧♦✇ t❛❜❧❡ ◮ ❇r❛✐♥✬s ✉t✐❧✐t② ub : B × P → R ✐s ❣✐✈❡♥ ❜② ub(b, p) := −up(p, b) up(·) b✶ b✷ P✐♥❦②✬s r❡✇❛r❞ p✶ ✶✵ ✷✵ ✶✵ p✷ ✸✵ ✺ ✺ ◮ P✐♥❦② t♦ss❡s ❛ ✭❢❛✐r✮ ❝♦✐♥✱ ✐✳❡✳ ❝❤♦♦s❡s r❛♥❞♦♠✐③❡❞ ❛❝t λ ≈ p✶ p✷ ✵.✺ ✵.✺

• ✳

◮ ❍❡ r❡❝❡✐✈❡s r❡✇❛r❞ ♦❢ ♠✐♥b G(up)(λ, b) = ✶✷.✺.

✺ ✴ ✶✻

❚✇♦ ✇❛②s ♦❢ ✐♥❝♦r♣♦r❛t✐♥❣ ✐♠♣❡r❢❡❝t ♣r✐♦r ❦♥♦✇❧❡❞❣❡

❈♦♥s✐❞❡r❡❞ ❤❡r❡✿ ❉❡❝✐s✐♦♥ ♣r♦❜❧❡♠s ✇✐t❤ ♣r✐♦r ✐♥❢♦r♠❛t✐♦♥ ♦♥ t❤❡ st❛t❡s Θ✳ ■❢ ♣r✐♦r ✐♥❢♦r♠❛t✐♦♥ ✐s ♣r❡❝✐s❡❧② ❣✐✈❡♥ ❜② ❛♥ ✭✉♥❞♦✉❜t❡❞✮ ♣r♦❜❛❜✐❧✐t② ♦♥ t❤❡ st❛t❡ s♣❛❝❡✱ ❛❝ts ❛r❡ ♠♦st ❝♦♠♠♦♥❧② r❛♥❦❡❞ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡✐r ❡①♣❡❝t❡❞ ✉t✐❧✐t② ✈❛❧✉❡s✳ ❖t❤❡r✇✐s❡ ✭♦❢ ✐♥t❡r❡st ❤❡r❡✮✱ ✇❡ ❞✐st✐♥❣✉✐s❤ t✇♦ ❞✐✛❡r❡♥t ❝❛s❡s✿ ✭✶✮ ❯♥❝❡rt❛✐♥t② ❛❜♦✉t ♣r❡❝✐s❡ ♣r♦❜❛❜✐❧✐t✐❡s✿ ❚❤❡r❡ ✐s ❛ ♣r❡❝✐s❡ ♣r✐♦r ♣r♦❜❛❜✐❧✐t② π ♦♥ (Θ, ✷Θ) ❛✈❛✐❧❛❜❧❡✱ ❤♦✇❡✈❡r✱ t❤❡r❡ ✐s s♦♠❡ ❞♦✉❜t ❛❜♦✉t ✐ts ❢✉❧❧ ❛♣♣r♦♣r✐❛t❡♥❡ss✳ ❊①❛♠♣❧❡✿ Pr✐♦r ❛✈❛✐❧❛❜❧❡ ❢♦r ❛♥ ❡①♣❡r✐♠❡♥t❀ s❧✐❣❤t ♠♦❞✐✜❝❛t✐♦♥ ♦❢ t❤❡ s❡t✉♣ ✭✷✮ ■♠♣r❡❝✐s❡ ♣r♦❜❛❜✐❧✐t✐❡s✿ ❆ ♣r✐♦r ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ π ♦♥ t❤❡ st❛t❡ s♣❛❝❡ Θ ❝❛♥♥♦t ❜❡ ❢✉❧❧② s♣❡❝✐✜❡❞✳ ■♥st❡❛❞ ❛ ❝r❡❞❛❧ s❡t M ♦❢ ♣r✐♦r ♣r♦❜❛❜✐❧✐t✐❡s ✐s ❝♦♠✲ ♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ❛✈❛✐❧❛❜❧❡ ✐♥❢♦r♠❛t✐♦♥ ❊①❛♠♣❧❡✿ ❊✈❡♥t E✶ ✐s ❛t ❧❡❛st ❛s ❧✐❦❡❧② ❛s E✷✱ ✐✳❡✳ M = {π|π(E✶) ≥ π(E✷)}

✻ ✴ ✶✻

✭✶✮ ❯♥❝❡rt❛✐♥t② ❛❜♦✉t ♣r❡❝✐s❡ ♣r✐♦rs✿ ❍♦❞❣❡s ✫ ▲❡❤♠❛♥♥

❖♥❡ ❝♦♠♠♦♥ ✇❛② t♦ ❞❡❛❧ ✇✐t❤ s✐t✉❛t✐♦♥ ✭✶✮ ✐s t❤❡ ❞❡❝✐s✐♦♥ ❝r✐t❡r✐♦♥ ♦❢ ❍♦❞❣❡s ✫ ▲❡❤♠❛♥♥✱ ✇❤✐❝❤ ❧✐♥❡❛r❧② tr❛❞❡s ♦❢ ❜❡t✇❡❡♥ ♠❛①✐♠✐♥ ❛♥❞ ❡①♣❡❝t❡❞ ✉t✐❧✐t②✳

❍♦❞❣❡s ✫ ▲❡❤♠❛♥♥ ♦♣t✐♠❛❧✐t②

▲❡t π ❞❡♥♦t❡ s♦♠❡ ♣r✐♦r ♦♥ (Θ, ✷Θ) ❛♥❞ ❧❡t α ∈ [✵, ✶] ❡①♣r❡ss t❤❡ ❛❣❡♥t✬s tr✉st ✐♥ ✐ts ❛♣♣r♦♣r✐❛t❡♥❡ss✳ ❚❤❡ ❢✉♥❝t✐♦♥ Φπ,α : G(A) → R ❞❡✜♥❡❞ ❜② Φπ,α(λ) = (✶ − α) · ♠✐♥

θ G(u)(λ, θ)

• Maximin utility

+ α · Eπ

• G(u)λ
• Expected utility

✐s ❝❛❧❧❡❞ ❍♦❞❣❡s ✫ ▲❡❤♠❛♥♥✲❝r✐t❡r✐♦♥ ✇✳r✳t✳ (π, α)✳ ❆♥② r❛♥❞♦♠✐③❡❞ ❛❝t λ∗ ∈ G(A) ♠❛①✐♠✐③✐♥❣ t❤❡ ❝r✐t❡r✐♦♥ ✐s t❤❡♥ ❝❛❧❧❡❞ Φπ,α✲♦♣t✐♠❛❧✳ ◆❛t✉r❛❧ q✉❡st✐♦♥✿ ❍♦✇ t♦ ❞❡t❡r♠✐♥❡✴❝♦♠♣✉t❡ Φπ,α✲♦♣t✐♠❛❧ ❛❝ts❄

✼ ✴ ✶✻

❉❡t❡r♠✐♥✐♥❣ ♦♣t✐♠❛❧ ❛❝ts ✉♥❞❡r ✭✶✮

❖♣t✐♠❛❧ r❛♥❞♦♠✐③❡❞ ❛❝ts ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝r✐t❡r✐♦♥ ♦❢ ❍♦❞❣❡s ❛♥❞ ▲❡❤♠❛♥♥ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠✿

❍♦❞❣❡s ❛♥❞ ▲❡❤♠❛♥♥ ❛❝ts

❈♦♥s✐❞❡r t❤❡ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠ (✶ − α) · (w✶ − w✷) + α ·

n

• i=✶

Eπ(uai) · λi − → ♠❛①

(w✶,w✷,λ✶,...,λn)

✇✐t❤ ❝♦♥str❛✐♥ts (w✶, w✷, λ✶, . . . , λn) ✵ ❛♥❞

• n

i=✶ λi = ✶

• w✶ − w✷ n

i=✶ uij · λi

❢♦r ❛❧❧ j = ✶, . . . , m ❚❤❡♥ ❡✈❡r② ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ (w ∗

✶ , w ∗ ✷ , λ∗ ✶, . . . , λ∗ n) ✐♥❞✉❝❡s ❛ Φπ,α✲♦♣t✐♠❛❧ r❛♥❞♦♠✐✲

③❡❞ ❛❝t λ∗ ∈ G(A) ❜② s❡tt✐♥❣ λ∗({ai}) := λ∗

i ✳

✽ ✴ ✶✻

✭✷✮ ■♠♣r❡❝✐s❡ ♣r♦❜❛❜✐❧✐st✐❝ ✐♥❢♦r♠❛t✐♦♥

❲❡ ❛ss✉♠❡ ♣r♦❜❛❜✐❧✐st✐❝ ✐♥❢♦r♠❛t✐♦♥ ✐s ❡①♣r❡ss❡❞ ❜② ❛ ♣♦❧②❤❡❞r✐❝❛❧ ❝r❡❞❛❧ s❡t M ♦❢ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s ♦♥ (Θ, ✷Θ) ♦❢ t❤❡ ❢♦r♠ M :=

• π| bs Eπ(fs) bs ∀s = ✶, ..., r
• ✇❤❡r❡✱ ❢♦r ❛❧❧ s = ✶, ..., r✱ ✇❡ ❤❛✈❡

◮ fs : Θ → R ✐s ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ♦♥ Θ ❛♥❞ ◮ (bs, bs) ∈ R✷ ✇✐t❤ bs bs ❛r❡ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ❢♦r t❤❡✐r ❡①♣❡❝t❛t✐♦♥✳

▲❡❛st ❢❛✈♦r❛❜❧❡ ♣r✐♦rs

■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ♦♥❡ ❛❞❞✐t✐♦♥❛❧ ❝♦♥❝❡♣t ✇✐❧❧ ❜❡ ♥❡❡❞❡❞✿ ◮ ❢♦r π ∈ M✱ ❧❡t B(π) := s✉♣{Eπ(ua) : a ∈ A} ❛♥❞ Aπ := ❛r❣♠❛①a∈AEπ(ua) ◮ ❝❛❧❧ π− ∈ M ❛ ❧❡❛st ❢❛✈♦r❛❜❧❡ ♣r✐♦r ✭❧❢♣✮ ✐❢ B(π−) ≤ B(π) ❢♦r ❛❧❧ π ∈ M

✾ ✴ ✶✻

❈♦♠♣✉t✐♥❣ ❧❡❛st ❢❛✈♦r❛❜❧❡ ♣r✐♦rs

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣♦s✐t✐♦♥ ❞❡s❝r✐❜❡s ❛♥ ❡❛s② ❧✐♥❡❛r ♣r♦❣r❛♠ ❢♦r ❞❡t❡r♠✐♥✐♥❣ ❛ ❧❡❛st ❢❛✈♦r❛❜❧❡ ♣r✐♦r ❞✐str✐❜✉t✐♦♥s ❢r♦♠ ❛ ❣✐✈❡♥ ❝r❡❞❛❧ s❡t✳

▲❡❛st ❢❛✈♦r❛❜❧❡ ♣r✐♦rs

▲❡t (A, Θ, u) ❛♥❞ M ❜❡ ❛s ❜❡❢♦r❡✳ ❈♦♥s✐❞❡r t❤❡ ❧✐♥❡❛r ♣r♦❣r❛♠ w✶ − w✷ − → ♠✐♥

(w✶,w✷,π✶,...,πm)

✇✐t❤ ❝♦♥str❛✐♥ts (w✶, w✷, π✶, . . . , πm) ✵ ❛♥❞

• m

j=✶ πj = ✶

• bs m

j=✶ fs(θj) · πj bs

❢♦r ❛❧❧ s = ✶, ..., r

• w✶ − w✷ m

j=✶ uij · πj ❢♦r ❛❧❧ i = ✶, . . . n

❚❤❡♥ ❡✈❡r② ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ (w ∗

✶ , . . . , π∗ m) ✐♥❞✉❝❡s ❛ ❧❡❛st ❢❛✈♦r❛❜❧❡ ♣r✐♦r π− ∈ M

❜② s❡tt✐♥❣ π−({θj}) := π∗

j ✳

✶✵ ✴ ✶✻

❉❡❝✐s✐♦♥ ♠❛❦✐♥❣ ✉♥❞❡r ✭✷✮

■❢ ✉♥❝❡rt❛✐♥t② ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❛ ❝r❡❞❛❧ s❡t M✱ ♠❛♥② ❞✐✛❡r❡♥t ❛♣♣r♦❛❝❤❡s ❢♦r ❞❡❝✐s✐♦♥ ♠❛❦✐♥❣ ❡①✐st✳ ❲❡ ❢♦❝✉s ♦♥ t❤r❡❡ ♦❢ t❤❡s❡✱ ♥❛♠❡❧② ❲❛❧❧❡②✬s ♠❛①✐♠❛❧✐t②✿ ❆♥ ❛❝t a∗ ∈ A ✐s s❛✐❞ t♦ ❜❡ M✲♠❛①✐♠❛❧✱ ✐❢ ∀ a ∈ A ∃ πa ∈ M : Eπa(ua∗) Eπa(ua) ❊✲❛❞♠✐ss✐❜✐❧✐t②✿ ❆♥ ❛❝t a∗ ∈ A ✐s s❛✐❞ t♦ ❜❡ M✲❛❞♠✐ss✐❜❧❡✱ ✐❢ ∃ π ∈ M ∀ a ∈ A : Eπ(ua∗) Eπ(ua)

• ❛♠♠❛✲▼❛①✐♠✐♥✿ ❆♥ r❛♥❞♦♠✐③❡❞ ❛❝t λ∗ ∈ G(A) ✐s s❛✐❞ t♦ ❜❡ M✲▼❛①✐♠✐♥

♦♣t✐♠❛❧ ✐✛ ❢♦r ❛❧❧ λ ∈ G(A)✿ EM

• G(u)λ∗

EM

• G(u)λ
• ✇❤❡r❡ EM(X) := ♠✐♥π∈M Eπ(X) ❢♦r r❛♥❞♦♠ ✈❛r✐❛❜❧❡s X : Θ → R✳

✶✶ ✴ ✶✻

❆ ❧✐♥❡❛r ♣r♦❣r❛♠ ❢♦r ♠❛①✐♠❛❧✐t②

❚❤❡ s❡t ♦❢ ♠❛①✐♠❛❧ ✭♥♦♥✲r❛♥❞♦♠✐③❡❞✮ ❛❝ts ❝❛♥ ❜❡ ❞❡t❡r♠✐♥❡❞ ❜② r✉♥♥✐♥❣ t❤❡ ❢♦❧✲ ❧♦✇✐♥❣ ❧✐♥❡❛r ♣r♦❣r❛♠ ❢♦r ❡✈❡r② ❛❝t a ∈ A ✉♥❞❡r ❝♦♥s✐❞❡r❛t✐♦♥✿

❈❤❡❝❦✐♥❣ ♠❛①✐♠❛❧✐t② ♦❢ ♥♦♥✲r❛♥❞♦♠✐③❡❞ ❛❝ts

▲❡t az ∈ A ❜❡ ❛♥② ❛❝t✳ ❈♦♥s✐❞❡r t❤❡ ❧✐♥❡❛r ♣r♦❣r❛♠

n

• i=✶

m

• j=✶

γij

• −

→ ♠❛①

(γ✶✶,...,γnm)

✇✐t❤ ❝♦♥str❛✐♥ts (γ✶✶, . . . , γnm) ✵ ❛♥❞

• m

j=✶ γij ✶

❢♦r ❛❧❧ i = ✶, . . . , n

• bs m

j=✶ fs(θj) · γij bs

❢♦r ❛❧❧ s = ✶, ..., r✱ i = ✶, . . . , n

• m

j=✶(uij − uzj) · γij ✵

❢♦r ❛❧❧ i = ✶, . . . , n ❚❤❡♥ az ∈ A ✐s M✲▼❛①✐♠❛❧ ✐✛ t❤❡ ♦♣t✐♠❛❧ ♦✉t❝♦♠❡ ❡q✉❛❧s n✳

✶✷ ✴ ✶✻

❆ s❧✐❣❤t ♠♦❞✐✜❝❛t✐♦♥✿ c✲❝♦♥str❛✐♥t ♠❛①✐♠❛❧✐t②

❈❤❡❝❦✐♥❣ c✲❝♦♥str❛✐♥t ♠❛①✐♠❛❧✐t② ♦❢ ♣✉r❡ ❛❝ts

▲❡t az ∈ A ❜❡ ❛♥② ❛❝t ❛♥❞ ❧❡t c ∈ [✵, ✶]✳ ❈♦♥s✐❞❡r t❤❡ ❧✐♥❡❛r ♣r♦❣r❛♠

n

• i=✶

m

• j=✶

γij

• −

→ ♠❛①

(γ✶✶,...,γnm)

✇✐t❤ ❝♦♥str❛✐♥ts (γ✶✶, . . . , γnm) ✵ ❛♥❞

• m

j=✶ γij ✶

❢♦r ❛❧❧ i = ✶, . . . , n

• bs m

j=✶ fs(θj) · γij bs

❢♦r ❛❧❧ s = ✶, ..., r✱ i = ✶, . . . , n

• m

j=✶(uij − uzj) · γij ✵

❢♦r ❛❧❧ i = ✶, . . . , n

• ✶

✷

m

j=✶ |γi✶j − γi✷j| c

❢♦r ❛❧❧ i✶, i✷ ∈ {✶, . . . , n} ❚❤❡♥ az ∈ A ✐s cM✲▼❛①✐♠❛❧ ✐✛ t❤❡ ♦♣t✐♠❛❧ ♦✉t❝♦♠❡ ❡q✉❛❧s n✳

✶✸ ✴ ✶✻

❈r♦ss✐♥❣ t❤❡ ❜♦r❞❡r ❜❡t✇❡❡♥ ✭✶✮ ❛♥❞ ✭✷✮

❋♦r t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❛♥ ε✲❝♦♥t❛♠✐♥❛t✐♦♥ ♠♦❞❡❧ ♦❢ t❤❡ ❢♦r♠ M(π✵,ε) := {(✶ − ε)π✵ + επ : π ∈ P(Θ)} ✇❤❡r❡ ◮ ε > ✵ ✐s ❛ ✜①❡❞ ❝♦♥t❛♠✐♥❛t✐♦♥ ♣❛r❛♠❡t❡r ❛♥❞ ◮ π✵ ∈ P(Θ) ✐s t❤❡ ❝❡♥tr❛❧ ❞✐str✐❜✉t✐♦♥✱

• ❛♠♠❛✲▼❛①✐♠✐♥ ✐s ♠❛t❤❡♠❛t✐❝❛❧❧② ❝❧♦s❡❧② r❡❧❛t❡❞ t♦ t❤❡ ❍♦❞❣❡s ✫ ▲❡❤♠❛♥♥✿

EM(π✵,ε)(X) = ♠✐♥

π∈P(Θ)((✶ − ε)Eπ✵(X) + εEπ(X))

= (✶ − ε)Eπ✵(X) + ε ♠✐♥

π∈P(Θ) Eπ(X)

= (✶ − ε)Eπ✵(X) + ε ♠✐♥

θ∈Θ X(θ)

M(π✵,ε)✲▼❛①✐♠✐♥ ≈ ❍♦❞❣❡s ✫ ▲❡❤♠❛♥♥ ✇✳r✳t✳ (✶ − ε) ❛♥❞ π✵✳

✶✹ ✴ ✶✻

❆ r❡s✉❧t ❜❡②♦♥❞ ❝♦♠♣✉t❛t✐♦♥

• ❛♠♠❛✲▼❛①✐♠✐♥ ❛♥❞ ❧❢♣s

▲❡t (A, Θ, u) ❛♥❞ M ❜❡ ❛s ❜❡❢♦r❡✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ ✐✮ ■❢ π− ✐s ❛ ❧❢♣ ❢r♦♠ M✱ t❤❡♥ ❢♦r ❛❧❧ ♦♣t✐♠❛❧ r❛♥❞♦♠✐③❡❞ M✲▼❛①✐♠✐♥ ❛❝ts λ∗ ∈ G(A) ✇❡ ❤❛✈❡ λ∗({a}) = ✵ ❢♦r ❛❧❧ a ∈ A \ Aπ−✳ ✐✐✮ ▲❡t π− ❞❡♥♦t❡ ❛ ❧❢♣ ❢r♦♠ M ❛♥❞ ❧❡t λ∗ ∈ G(A) ❞❡♥♦t❡ ❛ r❛♥❞♦♠✐③❡❞ M✲▼❛①✐♠✐♥ ❛❝t✳ ❚❤❡♥ ❢♦r ❛❧❧ a ∈ Aπ− ✇❡ ❤❛✈❡ Eπ− ua

• = EM
• G(u)λ∗

❈♦r♦❧❧❛r②

■❢ t❤❡r❡ ❡①✐sts ❛ ❧❢♣ π− ❢r♦♠ M s✉❝❤ t❤❛t Aπ− = {az} ❢♦r s♦♠❡ z ∈ {✶, . . . , n}✱ t❤❡♥ δaz ∈ G(A) ✐s t❤❡ ✉♥✐q✉❡ r❛♥❞♦♠✐③❡❞ M✲▼❛①✐♠✐♥ ❛❝t✳ ❙♣❡❝✐✜❝❛❧❧②✱ ❝♦♥s✐❞❡r✐♥❣ r❛♥❞♦♠✐③❡❞ ❛❝ts ✐s ✉♥♥❡❝❡ss❛r② ✐♥ s✉❝❤ s✐t✉❛t✐♦♥s✳

✶✺ ✴ ✶✻

❙✉♠♠❛r② ❛♥❞ ♦✉t❧♦♦❦

❲❡ ✐♥✈❡st✐❣❛t❡❞ ◮ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ❛♣♣r♦❛❝❤❡s ❢♦r ❞❡t❡r♠✐♥✐♥❣ ♦♣t✐♠❛❧ r❛♥❞♦♠✐③❡❞ ❛❝ts ◮ ✇❤❛t ❝❛♥ ❜❡ ❧❡❛r♥❡❞ ❜② ❞✉❛❧✐③✐♥❣ ♦✉r ♣r♦❣r❛♠s ❋✉t✉r❡ r❡s❡❛r❝❤ ✐♥❝❧✉❞❡s✿ ◮ ❝♦♥s✐❞❡r M ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡❞✱ ✐✳❡✳ π({θ}) > ✵ ❢♦r ❛❧❧ (π, θ) ∈ M × Θ ◮ t❤❡♥ ❡✈❡r② ❧❢♣ π− ❢r♦♠ M ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡❞ ❛s ✇❡❧❧ ◮ ❜② ❝♦♠♣❧❡♠❡♥t❛r② s❧❛❝❦♥❡ss ♣r♦♣❡rt②✱ t❤❡ ❝♦♥str❛✐♥ts ✐♥ t❤❡ ❧✐♥❡❛r ♣r♦❣r❛♠ ❢♦r ❞❡t❡r♠✐♥✐♥❣ ❢♦r ●❛♠♠❛✲▼❛①✐♠✐♥ ❛❝ts ❛r❡ ❜✐♥❞✐♥❣✿ ✇❤❡♥ s✉✣❝✐❡♥t❄

✶✻ ✴ ✶✻