Maxmin Expected Utility with Non-unique Prior - Google Sites

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(b) The set C in (2) is unique iff assumption A.6 is add to (1). ... Proof: This is immediate from the von Neumann-Morge
Maxmin Expected Utility with Non-unique Prior Gilboa, I. and Schmeidler, D. (1989) Journal of Mathematical Economics, 18: 141–153.

Weiye Chen Graduated School of Economics , Osaka University

Introduction

Ellsberg paradigm Gamble 1

Red

Urn

1 ball

(contains 50 red balls and 50 black balls)

(random) black

Bet 1(named π‘ΉπŸ)

The ball will be red

Bet 2(named π‘©πŸ)

$100 if it red

The ball will be black

$0 If it black

π‘ΉπŸ β‰… π‘©πŸ

$0 if it red $100 If it black

Gamble 2

red

Urn

1 ball

(contains 100 balls)

(random) black

Bet 3(named π‘ΉπŸ)

The ball will be red

Bet 4(named π‘©πŸ)

$100 if it red

The ball will be black

$0 If it black

π‘ΉπŸ β‰… 𝑩 𝟐

$0 if it red $100 If it black

Gamble 3

If you can take only one from those bets(bet 1,2,3,4) ,which one would you preference ? Why ?

π™πŸ ≃ π‘©πŸ > π™πŸ ≃ π‘©πŸ In gamble 2 β€’ We considers a set of the priors as possible.(too little information) β€’ We take into account the minimal expected utility.(uncertainty averse)

Hint: π’Žπ’Šπ’ 𝑬 𝒖 π™πŸ

= π’Žπ’Šπ’ 𝑬 𝒖 π‘©πŸ

> π’Žπ’Šπ’ 𝑬 𝒖 π™πŸ

= π’Žπ’Šπ’ 𝑬 𝒖 π‘©πŸ

Statement π‘Œ = 𝑦: 𝑋 β†’ 0,1 𝑦 π‘₯ β‰  0 π‘“π‘œπ‘Ÿ π‘œπ‘›π‘™π‘¦ 𝑓𝑖𝑛𝑖𝑑𝑒𝑙𝑦 π‘šπ‘Žπ‘¦ π‘₯′𝑠 𝑖𝑛 𝑋(𝑠𝑒𝑑, π‘›π‘œπ‘› βˆ’ π‘’π‘šπ‘π‘‘π‘¦ ) π‘Žπ‘›π‘‘

𝑦 π‘₯ =1 π‘₯βˆˆπ‘‹

For notational 𝑦 ∈ π‘Œ 𝑦 π‘₯ = 1 π‘“π‘œπ‘Ÿ π‘ π‘œπ‘šπ‘’ π‘₯ 𝑖𝑛 𝑋 βŠ‚ π‘Œ

Ζ© = 𝜬 Ζ© π’Šπ’” 𝒂𝒍𝒍 𝒂𝒏 π’‚π’π’ˆπ’†π’ƒπ’“π’‚ 𝒔𝒖𝒃𝒔𝒆𝒕𝒔 𝒐𝒇 𝑺 𝑳0 = 𝒇 ∢ 𝑆 β†’ π‘Œ 𝑓 𝑖𝑠 π‘Žπ‘› 𝛴 βˆ’ π‘šπ‘’π‘Žπ‘ π‘’π‘Ÿπ‘Žπ‘π‘™π‘’ 𝑓𝑖𝑛𝑖𝑑𝑒 𝑠𝑑𝑒𝑝 π‘“π‘’π‘›π‘π‘‘π‘–π‘œπ‘›π‘  𝑳𝑐 = 𝒇 βŠ‚ 𝑳0 𝒇 𝑖𝑠 π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ π‘“π‘’π‘›π‘π‘‘π‘–π‘œπ‘›π‘  π‘Œ 𝑆 = 𝒇 ∢ 𝑆 β†’ π‘Œ π‘Œ 𝑆 𝑖𝑠 π‘Ž π‘ π‘’π‘π‘ π‘π‘Žπ‘π‘’ π‘œπ‘“ π‘‘β„Žπ‘’ π‘™π‘–π‘›π‘’π‘Žπ‘Ÿ π‘ π‘π‘Žπ‘π‘’ 𝑳 = 𝒇 ∈ π‘Œ 𝑆 𝑳𝑐 βŠ‚ 𝑳, 𝑳 𝑖𝑠 π‘π‘œπ‘›π‘£π‘’π‘₯ βŠ‚ π‘Œ 𝑆 Example 𝑋 β‹― outcomes π‘Œ β‹― π‘Ÿπ‘Žπ‘›π‘‘π‘œπ‘š π‘œπ‘’π‘‘π‘π‘œπ‘šπ‘’π‘  π‘œπ‘Ÿ π‘Ÿπ‘œπ‘’π‘™π‘’π‘‘π‘‘π‘’ π‘™π‘œπ‘‘π‘‘π‘’π‘Ÿπ‘–π‘’π‘  𝑳 β‹― π‘Žπ‘π‘‘π‘  β„Žπ‘œπ‘Ÿπ‘ π‘’ π‘™π‘œπ‘‘π‘‘π‘’π‘Ÿπ‘–π‘’π‘  and a binary relation over 𝑳 π‘‘π‘’π‘›π‘œπ‘‘π‘’π‘‘ 𝑏𝑦 β‰₯ S β‹― π‘ π‘‘π‘Žπ‘‘π‘’π‘  π‘œπ‘“ π‘›π‘Žπ‘‘π‘’π‘Ÿπ‘’ 𝛴 β‹― 𝑒𝑣𝑒𝑛𝑑𝑠

Property (axiom)

A.1. Weak order β€’ (a) For all 𝒇 and π’ˆ in 𝑳: 𝒇 ≧ π’ˆ or π’ˆ ≧ 𝒇. (Completeness) β€’ (b) For all 𝒇, π’ˆ and 𝒉 in 𝑳: If 𝒇 ≧ π’ˆ and π’ˆ ≧ 𝒉 then 𝒇 ≧ 𝒉. (Transitivity)

A.2. Certainty-Independence (C-independence) β€’ For all 𝒇, π’ˆ in 𝑳 and 𝒉 in 𝑳𝑐 and for all 𝛼 in 0,1 : 𝒇 > π’ˆ iff 𝛼𝒇 + 1 βˆ’ 𝛼 𝒉 > π›Όπ’ˆ + 1 βˆ’ 𝛼 𝒉.

A.3. Continuity β€’ For all 𝒇, π’ˆ and 𝒉 in 𝑳: if 𝒇 > π’ˆ and π’ˆ > 𝒉 then there are 𝛼 and 𝛽 in 0,1 such that 𝛼𝒇 + (1 βˆ’ 𝛼)𝒉 > π’ˆ and π’ˆ > 𝛽𝒇 + (1 βˆ’ 𝛽)𝒉

A.4. Monotonicity β€’ For all 𝒇 and π’ˆ in 𝑳: if 𝒇 𝑠 ≧ π’ˆ(𝑠) on S then 𝒇 ≧ π’ˆ.

A.5. Uncertainty Aversion β€’ For all 𝒇, π’ˆ ∈ 𝑳 and 𝛼 ∈ 0,1 : 𝒇 ≃ π’ˆ implies 𝛼𝒇 + (1 βˆ’ 𝛼)π’ˆ ≧ 𝒇.

A.6. Non-degeneracy β€’ Not for all 𝒇 and π’ˆ in 𝑳, 𝒇 ≧ π’ˆ.

Theorem 1 Let ≧ be a binary relation on 𝐿0 . Then the following conditions are equivalent: (1) ≧ satisfies assumptions A.1-A.5 for 𝐿 = 𝐿0. (2) There exist an affine function π“Š: π‘Œ β†’ 𝑅 and a non-empty, closed and convex set π‘ͺ of finitely additive probability measures on 𝛴 such that: (*) 𝒇 ≧ π’ˆ iff π‘šπ‘–π‘›π™‹βˆˆπ‘ͺ ∫ π“Š ∘ 𝒇𝑑𝑃 ≧ π‘šπ‘–π‘›π™‹βˆˆπ‘ͺ ∫ π“Š ∘ π’ˆπ‘‘π‘ƒ (for all 𝒇, π’ˆ ∈ 𝐿0) furthermore: (a) The function π“Š in (2) is unique up to a positive linear transformation; (b) The set C in (2) is unique iff assumption A.6 is add to (1).

proof Step 1:We will proof that (1)β‡’(2) by several lemmata we suppose assumptions A.1-A.6 Define: π“Š π’š = π“Š: π‘Œ ⟢ 𝑅 π“Š π’š1 < βˆ’1, π“Š π’š2 > 1 𝑩 𝑺, Ζ© = 𝒇: 𝑺 ⟢ 𝒀 𝑩 𝑺, Ζ© 𝑖𝑠 π‘π‘œπ‘’π‘›π‘‘π‘’π‘‘ Ζ© βˆ’ π‘šπ‘’π‘Žπ‘ π‘’π‘Ÿπ‘’π‘Žπ‘π‘™π‘’ π‘Ÿπ‘’π‘Žπ‘™ π‘£π‘Žπ‘™π‘’π‘’π‘‘ π‘“π‘’π‘›π‘π‘‘π‘–π‘œπ‘›π‘  𝑩0 = 𝒇: 𝑺 ⟢ 𝒀 𝑩0 βŠ‚ 𝑩 𝑺, Ζ© π‘€β„Žπ‘–π‘β„Ž π‘Žπ‘ π‘ π‘’π‘šπ‘’ 𝑓𝑖𝑛𝑖𝑑𝑒𝑙𝑦 π‘£π‘Žπ‘™π‘’π‘’π‘  𝑲= π‘²βˆˆπ‘Ήπ‘²=π“Š 𝒀 𝑩0 𝑲 = 𝒇 ∈ 𝑩0 π“Š ∘ 𝒇 = 𝑲 π›„βˆ— = π›„βˆ— ∈ 𝑩0 π›„βˆ— = 𝛾 π‘“π‘œπ‘Ÿ 𝛾 ∈ 𝑹

Lemma 3.1. β€’ There exists an affine π“Š: π‘Œ β†’ 𝑅 such that for all π’š, 𝒛 ∈ π‘Œ: π’š ≧ 𝒛 iff π“Š π’š β‰§π“Š 𝒛 . β€’ Furthermore, π“Š is unique up to a positive linear transformation. Proof: This is immediate from the von Neumann-Morgenstern theorem.(Cindependence β‡’ Independence)

Lemma 3.2. β€’ Given a π“Š: π‘Œ β†’ 𝑅 fro Lemma 3.1, there exists a unique 𝑱: 𝑳0 β†’ 𝑅 such

that: β€’ (𝑖) 𝒇 ≧ π’ˆ iff 𝑱 𝒇 ≧ 𝑱 π’ˆ (for all 𝒇, π’ˆ ∈ 𝑳0) β€’ (𝑖𝑖) for 𝒇 = π’šβˆ— ∈ 𝑳𝐢 , 𝑱 𝒇 = π“Š π’š

Proof:π’šβˆ— = π’šβˆ— 𝑠 = π’š βˆ€π’š ∈ π‘Œ, 𝑠 ∈ 𝑆 𝒇 ∈ 𝑳0, π’š, π’š ∈ π‘Œ such thatπ’š ≀ 𝒇 ≀ π’š, there exists a unique 𝛼 ∈ 0,1 that 𝒇 = π›Όπ’š + 1 βˆ’ 𝛼 π’š, and 𝑱 𝒇 ≑ 𝑱 π›Όπ’š + 1 βˆ’ 𝛼 π’š

such

Lemma 3.3. There exists a functional 𝑰: 𝑩0 ⟢ 𝑹 such that : (𝑖) For all 𝒇 ∈ 𝑳0 , 𝑰 π“Š ∘ 𝒇 = 𝑱 𝒇 (hence 𝑰 πŸβˆ— = 1). (𝑖𝑖) 𝑰 is monotonic(i.e., for 𝒂, 𝒃 ∈ 𝑩0: 𝒂 ≧ 𝒃 β‡’ 𝑰 𝒂 ≧ 𝑰 𝒃 ). (𝑖𝑖𝑖) 𝑰 is superlinear(that is, superadditive and homogeneous of degree 1). β€’ (𝑖𝑣) 𝑰 is C-independent: for any 𝒂 ∈ 𝑩0 and 𝛾 ∈ 𝑹, 𝑰 𝒂 + π›„βˆ— = β€’ β€’ β€’ β€’

𝑰 𝒂 + 𝑰 π›„βˆ— .

Proof: Monotonicity: Lemma 3.2 Homogeneous: 𝒂, 𝒃 ∈ 𝑩0 𝑲 , 𝛼 ∈ 0,1 , 𝒂 = 𝛼𝒃 β‡’ 𝑰 𝒂 = 𝛼𝑰 𝒃 C-independent: π“Š ∘ 𝒇 = 𝒂, π“Š ∘ π’š = 𝒃, π“Š ∘ 𝒛 = 2π›„βˆ— , 𝒇 ≃ π’š β‡’ 𝑰 𝒂 + π›„βˆ— = 𝑰 𝒂 + 𝛾 Superadditive: 𝑖𝑓 𝑰 𝒂 = 𝑰 𝒃 β‡’ 𝑰 𝑰 𝒃 , 𝒄 = 𝒃 + π›„βˆ— β‡’ 𝑰

1 1 𝒂+ 𝒄 2 2

≧

1 1 𝒂+ 𝒃 ≧ 𝑰 2 2 1 1 𝑰 𝒂 + 𝑰 𝒃 2 2

1 2

1 2

𝒂 = 𝑰 𝒂 + 𝑰 𝒃 ; 𝑖𝑓 𝑰 𝒂 > 1 2

+ 𝛾

Lemma 3.4. β€’ There exists a unique continuous extension of 𝑰 to 𝑩. β€’ Furthermore, this extension is monotonic, superlinear and C-

independent.

Proof: 𝒂 ≦ 𝒃 + 𝒂 βˆ’ 𝒃

βˆ—

β‡’ 𝑰 𝒂 βˆ’π‘° 𝒃

≦ π’‚βˆ’π’ƒ

Lemma 3.5. β€’ If 𝑰 is a monotonic superlinear and C-independent functional on 𝑩 with 𝑰 1βˆ— = 1 , there exists a closed and convex set π‘ͺ of finitely additive probability measures on Ζ© such that: for all 𝒃 ∈ 𝑩, 𝑰 𝒃 = min ∫ 𝒃 𝑑𝑷 𝑷 ∈ π‘ͺ . Proof: 𝑫1 = 𝒂 ∈ 𝑩 𝑰 𝒂 > 1 , 𝑫2 = π‘π‘œπ‘›π‘£ 𝒂 ∈ 𝑩 𝒂 ≦ 1βˆ— βˆͺ 𝒂 ∈ 𝑩 𝒂 ≦ 𝒃 𝑰 𝒃 . Use separation theorem, thus 𝒑𝒃 𝒃 = 𝑰 𝒃 . Then there exists a finitely additive probability measure 𝑷𝒃 on Ζ© such that 𝒑𝒃 𝒂 = ∫ 𝒂 𝑑𝑷 for all 𝒂 ∈ 𝑩. π‘ͺ ≑ 𝑷𝒃 𝑰 𝒃 > 0 Β° β‡’ 𝑰 𝒂 ≦ min

𝒂 𝑑𝑷 𝑷 ∈ π‘ͺ .

Step 2: Then we will proof that (a) and (b). The uniqueness of π“Š up to positive linear transformation is implied is implied by Lemma 3.1. If A.6 does not hold, π‘ͺ can be any non-empty closed and convex set. If A.6 is hold, consider non-empty, closed and convex π‘ͺ1, π‘ͺ2, such that the two functions on 𝑳0 : 𝙅1 𝒇 = min ∫ π“Š 𝒇 𝑑𝑷 𝑷 ∈ π‘ͺ1 , 𝙅2 𝒇 = min ∫ π“Š 𝒇 𝑑𝑷 𝑷 ∈ π‘ͺ2 , Both represent ≧. Use separation theorem, there exists 𝒇 ∈ 𝑳0 , such that 𝙅1 𝒇 ≦ 𝙅2 𝒇 . Let π’š ∈ 𝒀 satisfy π’š ≃ 𝒀, we get 𝙅1 π’š = 𝙅1 𝒇 ≦ 𝙅2 𝒇 = 𝙅2 π’š a contradiction.

Example Consider the gamble 3.

π“Š π‘₯ = π‘₯, which π‘₯ is the quantity of money π‘ͺ=

𝓅, 1 βˆ’ 𝓅 𝓅 ∈ 0,1 , 𝓅 𝑖𝑠 π‘‘β„Žπ‘’ π‘π‘Ÿπ‘œπ‘π‘Žπ‘π‘–π‘™π‘–π‘‘π‘¦ π‘œπ‘“ π‘Ÿπ‘’π‘‘ π‘π‘Žπ‘™π‘™ π‘œπ‘π‘π‘’π‘Ÿ

Then a measure on π‘ͺ is like probability π›‘π™πŸ 𝑠 = πœ‹π™πŸ 𝑠 =

0, 𝓅 𝓅 ∈ 0,1 100,1 βˆ’ 𝓅

Then we can consider this case as: π‘šπ‘Žπ‘₯ π‘šπ‘–π‘›π™‹ ∈π‘ͺ

π“Š ∘ 𝛑𝑑𝑒𝑑𝙋

π™πŸ ≃ π‘©πŸ > π™πŸ ≃ π‘©πŸ ⇔ πŸ“πŸŽ = π‘šπ‘–π‘›π™‹ ∈π‘ͺ

π‘šπ‘–π‘›π™‹ ∈π‘ͺ

π“Š ∘ π›‘π™πŸ 𝑑𝑒𝑑𝙋 = π‘šπ‘–π‘›π™‹ ∈π‘ͺ

π“Š ∘ π›‘π™πŸ 𝑑𝑒𝑑𝙋 = π‘šπ‘–π‘›π™‹ ∈π‘ͺ

π“Š ∘ π›‘π‘©πŸ 𝑑𝑒𝑑𝙋 >

π“Š ∘ π›‘π‘©πŸ 𝑑𝑒𝑑𝙋 = 𝟎

Proposition

Proposition 4.1. β€’ Suppose that a preference relation ≧ over 𝑳0 satisfies assumptions A.1-A.5. then it has a unique extension to 𝑳 ≧ which satisfies the same assumptions π‘œπ‘£π‘’π‘Ÿ 𝑳 ≧ . Remark. In view of proposition 4.1, ≧ may be represented as in Theorem 1 on 𝑳 ∩ 𝑳 ≧ .

Independent β€’ βˆ€ 𝒇, π’ˆ ∈ 𝑳 satisfy: β€’ (1) There exists 𝑷0 ∈ π‘ͺ such that ∫ π“Š ∘ 𝒇 𝑑𝑷0 = min ∫ π“Š ∘ 𝒇 𝑑𝑷 𝑷 ∈ π‘ͺ , and ∫ π“Š ∘ π’ˆ 𝑑𝑷0 = min ∫ π“Š ∘ π’ˆ 𝑑𝑷 𝑷 ∈ π‘ͺ ; β€’ (2) π“Š ∘ 𝒇 and π“Š ∘ π’ˆ are two stochastically independent random variables with respect any extreme point of π‘ͺ [ for short: 𝐸π‘₯𝑑 π‘ͺ ].

Definition non-unique probability space: 𝑺, Ζ©, π‘ͺ is the product of π‘Ίπ’Š , Ζ©π’Š, π‘ͺπ’Š , which 𝑺 = 𝑺1 Γ— 𝑺2 , Ζ© = Ζ©1 ⨂Ʃ2 and π‘ͺ = π‘π‘œπ‘›π‘£ 𝑷1 ⨂𝑷2 𝑷1 ∈ π‘ͺ1 , 𝑷2 ∈ π‘ͺ2 . The ≧ on 𝑳 define as ≧=≧1 ⨂ ≧2 which β‰§π’Š on π‘³π’Š = π‘³π’Š β‰§π’Š , then it satisfies A.1-A.6. Give π’‡π’Š ∈ π‘³π’Š, π’‡π’Š ∈ 𝑳 is a unique trivial extension.

Proposition 4.2. β€’ Given 𝑳1 , ≧1 , 𝑳2, ≧2 and 𝑳 as above, ≧ is the unique preference

relation over 𝑳 satisfying: β€’ (1) assumptions A1-A.6; β€’ (2) for all π’‡π’Š, π’ˆπ’Š ∈ π‘³π’Š , π’‡π’Š β‰§π’Š π’ˆπ’Š iff π’‡π’Š ≧ π’ˆπ’Š π’Š = 1,2 ; β€’ (3) for all 𝒇 ∈ 𝑳1 and π’ˆ ∈ 𝑳2 , 𝒇 and π’ˆ are independent.

Note The set π‘ͺ is only depend on axiom The set π‘ͺ is not identical to the set of probabilities that can`t be ruled out based on hard evidence Compare to Choquet expected utility

More intuitive More degrees of freedom