optimal risk sharing with optimistic and pessimistic...

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IntroductionPreliminary results

Existence of individually rational Pareto efficient allocationsOptimal risk sharing

Optimal Risk Sharing with Optimistic andPessimistic Decision Makers

A. Araujo1,2, J.M. Bonnisseau3, A. Chateauneuf3 and R.Novinski1

1Instituto Nacional de Matematica Pura e Aplicada 2Fundacao Getulio Vargas3Paris School of Economics, Université Paris 1 Panthéon-Sorbonne

Centre d’Economie de la Sorbonne, UMR CNRS 8174

Economic Theory ConferenceUniversity of Kansas

December 9 - 10, 2011

Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 1

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Outline

1 Introduction

2 Preliminary results

3 Existence of individually rational Pareto efficient allocations

4 Optimal risk sharing


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Outline

1 Introduction





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Introduction

Main motivation: we aim at proving that under mild conditionsindividually Pareto rational optima will exist even in presence ofnon-convex preferences.We essentially consider preferences on `∞+ henceaccommodating the behavior of DM in front of a countable flowx of payoffs, or else choosing among financial assets x whoseoutcomes depend on the realization of a countable set of statesof the world.Therefore, our conditions for the existence of P.O. can beinterpreted as myopia in the first context and as pessimism orreasonable optimism in the second context.


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Introduction (continued)

Replacing the uncertain context by a risky one, we then derivesome necessary properties of risk sharing. For strong riskaverters these conditions are exactly the usual ones of pairwisecomonotonicity, for strong risk lovers exclusively cornerconditions or pairwise comonotonicity can appear.A crucial result for our work is the proof that the closed unit ballB̄(0, 1) for the normed dual `∞ of the normed space `1 iscompact in the Mackey topology τ(`∞, `1).


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Outline

1 Introduction





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Preliminary results

TheoremLet B(S) be the normed space of bounded real mappings on Sendowed with the sup-norm. Then:

1) B(S) is the norm dual of the space rca(S) of allregular and bounded Borel measures for Sendowed with the discrete topology.2) The closed unit ball of B(S) is compact for theMackey topology τ(B(S), rca(S)).

Corollary

The closed unit ball B̄(0, 1) for the normed dual `∞ of thenormed space `1 is compact in the Mackey topology τ(`∞, `1).

Proof. Taking S = N, clearly `∞ = B(S).Araujo, Bonnisseau, Chateauneuf and Novinski Risk Sharing with Diverse Decision Makers 7

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Outline

1 Introduction





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Existence of individually rational Pareto efficientallocations under Mackey upper-semicontinuouspreferences

Assumption(1) For every i = 1, . . . , m, �i is a transitivereflexive preorder on `∞+ .(ii) For every i and every yi ∈ `∞+ ,{xi ∈ `∞+ | xi �i yi} is τ -closed where τ is theMackey topology τ(`∞, `1).

TheoremUnder Assumptions (1) and (2), individually rational Paretoefficient allocations exist.


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Examples of Mackey upper-semicontinuouspreferences

DefinitionLet (S,A) be a measurable space and B∞(A) be the space ofbounded A-measurable mappings from S to R.v is a capacity on A if v(∅) = 0, v(S) = 1, ∀A, B ∈ A,A ⊂ B ⇒ v(A) ≤ v(B).∀x ∈ B∞(A), the Choquet integral of x with respect to v ,denoted

∫xdv is defined by∫ 0

−∞(v(x ≥ t)− 1)dt +∫ +∞

0 v(x ≥ t)dt .


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>From Araujo et al. “General Equilibrium, Wariness andefficient bubbles", JET forthcoming, we know that preferenceson `∞ represented by a Choquet integral is Mackey usc if andonly if � is strongly myopic where:

DefinitionA preference relation � on `∞, is strongly myopic if∀(x , y) ∈ `∞ × `∞ x � y , z ∈ `∞+ , implies that there exists nlarge enough such that x � y + zEn where zEn(m) = 0 if m < nand z(m) if m ≥ n.


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Pessimistic Mackey upper-semicontinuouspreferences

DefinitionA Choquet DM will be said pessimistic if v is convex, i.e.,∀A, B ∈ A, v(A ∪ B) + v(A ∩ B) ≥ v(A) + v(B).

Recall that if v is convex, then∫xdv = min{EP(x) | P additive probability measure ≥ v}


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PropositionFor a pessimistic decision maker, the following assertions areequivalent:

(1) � is Mackey upper-semicontinuous;(2) v is outer continuous, i.e., ∀An, A ∈ A = P(N),An ↓ A ⇒ v(An) ↓ v(A).

Example I(x) = infN{x(n)} or equally I(x) =∫

xdv wherev(A) = 0 ∀A 6= N, v(N) = 1Remark that I is not Mackey lower semicontinuous.


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Optimistic Mackey upper-semicontinuous preferences

DefinitionA Choquet DM will be said optimistic if v is concave, i.e.,∀A, B ∈ A, v(A ∪ B) + v(A ∩ B) ≤ v(A) + v(B).

Recall that if v is concave,∫xdv = max{EP(x) | P additive probability measure ≥ v}

or else x , y ∈ B∞(A), x ∼ y , α ∈]0, 1[⇒ αx + (1− α)y � y .


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PropositionFor an optimistic decision maker the following assertions areequivalent:

(1) � is Mackey upper-semicontinuous;(2) � is Mackey lower-semicontinuous;(3) v is outer continuous at ∅, i.e., An ∈ P(N),An ↓ ∅ ⇒ v(An) ↓ 0.

Remark. I(x) = supN{x(n)}, i.e., I(x) =∫

xdv with v(A) = 1∀A 6= ∅, v(∅) = 0 is not Mackey upper-semicontinuous.


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Example where no existence of nontrivial individuallyrational Pareto optima if such a “too optimistic" DM ispresent

Consumption space X = `∞+ , ps = 12s .

DM1 U1(x) =∑

s∈N psxs, initial endowments ω1 = (2− 1s+1)s∈N

DM2 U2(x) = sups∈N xs, ω2 = ω1.

The unique Pareto optimal allocation is (x̄1 = ω1 + ω2, x̄2 = 0).No individually rational Pareto optimal allocation.


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Outline

1 Introduction





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Optimal risk sharing for risk lovers and risk aversedecision makers

Interpret `∞ as the set of all bounded A measurable mappingsx from N to R where A = P(N) and assume that a σ-additiveprobability P is given on (N,P(N)). So any x can now beinterpreted as a random variable.

Definitionx is less risky than y for the second order stochasticdominance denoted x �SSD y if∫ t−∞ P(x ≤ u)du ≤

∫ t−∞ P(y ≤ u)du for all t ∈ R.

x is strictly less risky than y denoted x �SSD y if one of theprevious inequality is strict.


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DefinitionLet �i be the preference relation of a DM.

�i is a strict (strong) risk averter ifx �SSD y ⇒ x �i y and x �SSD y ⇒ x �i y .�i is a strict (strong) risk lover ifx �SSD y ⇒ x �i y and x �SSD y ⇒ x ≺i y .


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Examples of Mackey upper-semicontinuous strict risklover of strict risk averter

Consider a Choquet decision maker where v = f ◦ P, i.e., v is adistortion of a probability P, i.e., f : [0, 1] → [0, 1], f (0) = 0,f (1) = 1, f strictly increasing.

If f is strictly concave and continuous, thenx ∈ `∞ → I(x) =

∫xdvf with vf = f ◦ P defines a Mackey

upper-semicontinuous strict risk lover.

If f is strictly convex and continuous, thenx ∈ `∞ → I(x) =

∫xdvf with vf = f ◦ P defines a Mackey

upper-semicontinuous strict risk averter.


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The main result

TheoremConsider m strict risk averters i = 1, . . . , m with Mackeyupper-semicontinuous preferences �i and n strict risk loversj = 1, . . . , n with Mackey upper-semicontinuous preferences �jwith initial endowments ωi ∈ `∞++, ωj ∈ `∞++.Then individual rational Pareto efficient allocations (in (`∞+ )m+n)exist.


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Theorem continued

TheoremFor such PO (xi , i = 1, . . . , m; yj , j = 1, . . . , n) we have:

1) The allocations of risk averters are pairwisecomonotonic, i.e.,(xi1(s)− xi1(t))(xi2(s)− xi2(t)) ≥ 0 ∀(s, t) ∈ N2,∀(i1, i2) ∈ {1, . . . , m}2.2) For two risk lovers j1, j2:(yj1(s)− yj1(t))(yj2(s)− yj2(t)) < 0 ⇒(yj1(s) ∧ yj1(t))(yj2(s) ∧ yj2(t)) = 0 (i.e. in at leastone state, one of these two risk lovers gets 0).3) If all allocations of risk lovers are strictlypositive, then these allocations are pairwisecomonotonic.


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