(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: 141β153.
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 πΉπ)
The ball will be red
Bet 2(named π©π)
$100 if it red
The ball will be black
$0 If it black
πΉπ β
π©π
$0 if it red $100 If it black
Gamble 2
red
Urn
1 ball
(contains 100 balls)
(random) black
Bet 3(named πΉπ)
The ball will be red
Bet 4(named π©π)
$100 if it red
The ball will be black
$0 If it black
πΉπ β
π© π
$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 ?
ππ β π©π > ππ β π©π 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: πππ π¬ π ππ
= πππ π¬ π π©π
> πππ π¬ π ππ
= πππ π¬ π π©π
Statement π = π¦: π β 0,1 π¦ π₯ β 0 πππ ππππ¦ πππππ‘πππ¦ πππ¦ π₯β²π ππ π(π ππ‘, πππ β ππππ‘π¦ ) πππ
π¦ π₯ =1 π₯βπ
For notational π¦ β π π¦ π₯ = 1 πππ π πππ π₯ ππ π β π
Ζ© = π¬ Ζ© ππ πππ ππ πππππππ πππππππ ππ πΊ π³0 = π βΆ π β π π ππ ππ π΄ β ππππ π’πππππ πππππ‘π π π‘ππ ππ’πππ‘ππππ π³π = π β π³0 π ππ ππππ π‘πππ‘ ππ’πππ‘ππππ π π = π βΆ π β π π π ππ π π π’ππ ππππ ππ π‘βπ ππππππ π ππππ π³ = π β π π π³π β π³, π³ ππ ππππ£ππ₯ β π π Example π β― outcomes π β― ππππππ ππ’π‘πππππ ππ πππ’πππ‘π‘π πππ‘π‘πππππ π³ β― πππ‘π βπππ π πππ‘π‘πππππ and a binary relation over π³ πππππ‘ππ ππ¦ β₯ S β― π π‘ππ‘ππ ππ πππ‘π’ππ π΄ β― ππ£πππ‘π
Property (axiom)
A.1. Weak order β’ (a) For all π and π in π³: π β§ π or π β§ π. (Completeness) β’ (b) For all π, π and π in π³: If π β§ π and π β§ π then π β§ π. (Transitivity)
A.2. Certainty-Independence (C-independence) β’ For all π, π in π³ and π in π³π and for all πΌ in 0,1 : π > π iff πΌπ + 1 β πΌ π > πΌπ + 1 β πΌ π.
A.3. Continuity β’ For all π, π and π in π³: if π > π and π > π then there are πΌ and π½ in 0,1 such that πΌπ + (1 β πΌ)π > π and π > π½π + (1 β π½)π
A.4. Monotonicity β’ For all π and π in π³: if π π β§ π(π ) on S then π β§ π.
A.5. Uncertainty Aversion β’ For all π, π β π³ and πΌ β 0,1 : π β π implies πΌπ + (1 β πΌ)π β§ π.
A.6. Non-degeneracy β’ Not for all π and π in π³, π β§ π.
Theorem 1 Let β§ be a binary relation on πΏ0 . Then the following conditions are equivalent: (1) β§ satisfies assumptions A.1-A.5 for πΏ = πΏ0. (2) There exist an affine function π: π β π
and a non-empty, closed and convex set πͺ of finitely additive probability measures on π΄ such that: (*) π β§ π iff ππππβπͺ β« π β πππ β§ ππππβπͺ β« π β πππ (for all π, π β πΏ0) furthermore: (a) The function π 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 π = π: π βΆ π
π π1 < β1, π π2 > 1 π© πΊ, Ζ© = π: πΊ βΆ π π© πΊ, Ζ© ππ πππ’ππππ Ζ© β ππππ π’ππππππ ππππ π£πππ’ππ ππ’πππ‘ππππ π©0 = π: πΊ βΆ π π©0 β π© πΊ, Ζ© π€βππβ ππ π π’ππ πππππ‘πππ¦ π£πππ’ππ π²= π²βπΉπ²=π π π©0 π² = π β π©0 π β π = π² πβ = πβ β π©0 πβ = πΎ πππ πΎ β πΉ
Lemma 3.1. β’ There exists an affine π: π β π
such that for all π, π β π: π β§ π iff π π β§π π . β’ Furthermore, π 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: π β π
fro Lemma 3.1, there exists a unique π±: π³0 β π
such
that: β’ (π) π β§ π iff π± π β§ π± π (for all π, π β π³0) β’ (ππ) for π = πβ β π³πΆ , π± π = π π
Proof:πβ = πβ π = π βπ β π, π β π π β π³0, π, π β π such thatπ β€ π β€ π, there exists a unique πΌ β 0,1 that π = πΌπ + 1 β πΌ π, and π± π β‘ π± πΌπ + 1 β πΌ π
such
Lemma 3.3. There exists a functional π°: π©0 βΆ πΉ such that : (π) For all π β π³0 , π° π β π = π± π (hence π° πβ = 1). (ππ) π° is monotonic(i.e., for π, π β π©0: π β§ π β π° π β§ π° π ). (πππ) π° is superlinear(that is, superadditive and homogeneous of degree 1). β’ (ππ£) π° is C-independent: for any π β π©0 and πΎ β πΉ, π° π + πβ = β’ β’ β’ β’
π° π + π° πβ .
Proof: Monotonicity: Lemma 3.2 Homogeneous: π, π β π©0 π² , πΌ β 0,1 , π = πΌπ β π° π = πΌπ° π C-independent: π β π = π, π β π = π, π β π = 2πβ , π β π β π° π + πβ = π° π + πΎ Superadditive: ππ π° π = π° π β π° π° π , π = π + πβ β π°
1 1 π+ π 2 2
β§
1 1 π+ π β§ π° 2 2 1 1 π° π + π° π 2 2
1 2
1 2
π = π° π + π° π ; ππ π° π > 1 2
+ πΎ
Lemma 3.4. β’ There exists a unique continuous extension of π° to π©. β’ Furthermore, this extension is monotonic, superlinear and C-
independent.
Proof: π β¦ π + π β π
β
β π° π βπ° π
β¦ πβπ
Lemma 3.5. β’ If π° is a monotonic superlinear and C-independent functional on π© with π° 1β = 1 , there exists a closed and convex set πͺ of finitely additive probability measures on Ζ© such that: for all π β π©, π° π = min β« π ππ· π· β πͺ . Proof: π«1 = π β π© π° π > 1 , π«2 = ππππ£ π β π© π β¦ 1β βͺ π β π© π β¦ π π° π . Use separation theorem, thus ππ π = π° π . Then there exists a finitely additive probability measure π·π on Ζ© such that ππ π = β« π ππ· for all π β π©. πͺ β‘ π·π π° π > 0 Β° β π° π β¦ min
π ππ· π· β πͺ .
Step 2: Then we will proof that (a) and (b). The uniqueness of π up to positive linear transformation is implied is implied by Lemma 3.1. If A.6 does not hold, πͺ can be any non-empty closed and convex set. If A.6 is hold, consider non-empty, closed and convex πͺ1, πͺ2, such that the two functions on π³0 : π
1 π = min β« π π ππ· π· β πͺ1 , π
2 π = min β« π π ππ· π· β πͺ2 , Both represent β§. Use separation theorem, there exists π β π³0 , such that π
1 π β¦ π
2 π . Let π β π satisfy π β π, we get π
1 π = π
1 π β¦ π
2 π = π
2 π a contradiction.
Example Consider the gamble 3.
π π₯ = π₯, which π₯ is the quantity of money πͺ=
π
, 1 β π
π
β 0,1 , π
ππ π‘βπ ππππππππππ‘π¦ ππ πππ ππππ ππππ’π
Then a measure on πͺ is like probability πππ π = πππ π =
0, π
π
β 0,1 100,1 β π
Then we can consider this case as: πππ₯ ππππ βπͺ
π β πππ’ππ
ππ β π©π > ππ β π©π β ππ = ππππ βπͺ
ππππ βπͺ
π β πππ ππ’ππ = ππππ βπͺ
π β πππ ππ’ππ = ππππ βπͺ
π β ππ©π ππ’ππ >
π β ππ©π ππ’ππ = π
Proposition
Proposition 4.1. β’ Suppose that a preference relation β§ over π³0 satisfies assumptions A.1-A.5. then it has a unique extension to π³ β§ which satisfies the same assumptions ππ£ππ π³ β§ . Remark. In view of proposition 4.1, β§ may be represented as in Theorem 1 on π³ β© π³ β§ .
Independent β’ β π, π β π³ satisfy: β’ (1) There exists π·0 β πͺ such that β« π β π ππ·0 = min β« π β π ππ· π· β πͺ , and β« π β π ππ·0 = min β« π β π ππ· π· β πͺ ; β’ (2) π β π and π β π are two stochastically independent random variables with respect any extreme point of πͺ [ for short: πΈπ₯π‘ πͺ ].
Definition non-unique probability space: πΊ, Ζ©, πͺ is the product of πΊπ , Ζ©π, πͺπ , which πΊ = πΊ1 Γ πΊ2 , Ζ© = Ζ©1 β¨Ζ©2 and πͺ = ππππ£ π·1 β¨π·2 π·1 β πͺ1 , π·2 β πͺ2 . The β§ on π³ define as β§=β§1 β¨ β§2 which β§π on π³π = π³π β§π , then it satisfies A.1-A.6. Give ππ β π³π, ππ β π³ is a unique trivial extension.
Proposition 4.2. β’ Given π³1 , β§1 , π³2, β§2 and π³ as above, β§ is the unique preference
relation over π³ satisfying: β’ (1) assumptions A1-A.6; β’ (2) for all ππ, ππ β π³π , ππ β§π ππ iff ππ β§ ππ π = 1,2 ; β’ (3) for all π β π³1 and π β π³2 , π and π are independent.
Note The set πͺ is only depend on axiom The set πͺ 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