Efficient Allocations under Ambiguity Tomasz Strzalecki (Harvard - - PowerPoint PPT Presentation

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Efficient Allocations under Ambiguity Tomasz Strzalecki (Harvard - - PowerPoint PPT Presentation

Jun 27, 2023 337 likes •1.08k views

Efficient Allocations under Ambiguity Tomasz Strzalecki (Harvard University) Jan Werner (University of Minnesota) Goal Understand risk sharing among agents with ambiguity averse preferences Ambiguity 30 balls Red 60 balls Green or Blue

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Efficient Allocations under Ambiguity

Tomasz Strzalecki (Harvard University) Jan Werner (University of Minnesota)

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Goal Understand risk sharing among agents with ambiguity averse preferences

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Ambiguity

30 balls Red 60 balls Green or Blue

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Ambiguity

R G B r+ 1

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Ambiguity

R G B r+ 1 g+ 1

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Ambiguity

R G B r+ 1 g+ 1 r− 1 1

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Ambiguity

R G B r+ 1 g+ 1 r− 1 1 g− 1 1

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Goal Understand risk sharing among agents with ambiguity averse preferences

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Setup and notation

S — states of the world (finite) ∆(S) — all probabilities on S two agents exchange economy, one shot ex ante trade f : S → R+ — allocation of agent 1 g : S → R+ — allocation of agent 2

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Question 1: Full Insurance

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Full Insurance

Theorem agents have strictly risk averse EU the aggregate endowment is risk-free common beliefs = ⇒ all PO allocations are risk-free

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Question 2: Conditional Full Insurance

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Conditional Full Insurance

Theorem agents have strictly risk averse EU the aggregate endowment is G-measurable G-concordant beliefs = ⇒ all PO allocations are G-measurable

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Conditional Full Insurance

Theorem agents have strictly risk averse EU the aggregate endowment is G-measurable G-concordant beliefs = ⇒ all PO allocations are G-measurable p(· | G) = q(· | G) for all G ∈ G

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Conditional Full Insurance

Theorem agents have strictly risk averse EU the aggregate endowment is G-measurable G-concordant beliefs = ⇒ all PO allocations are G-measurable p(· | G) = q(· | G) for all G ∈ G Ep[f |G] = Eq[f |G] for all f

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Proof

u′(f (s1)) u′(f (s2)) p(s1) p(s2) = v′(g(s1)) v′(g(s2)) q(s1) q(s2)

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Proof

u′(f (s1)) u′(f (s2)) = v′(g(s1)) v′(g(s2))

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Proof

u′(f (s1)) u′(f (s2)) = v′(g(s1)) v′(g(s2)) If f (s1) > f (s2) then g(s1) > g(s2), but that can’t be since f (s1) + g(s1) = f (s2) + g(s2)

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Question 3: Comonotonicity

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Question 3: Comonotonicity [f (s1) − f (s2)][g(s1) − g(s2)] ≥ 0

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Comonotonicity

Theorem agents have strictly risk averse EU common probability beliefs = ⇒ all PO allocations are comonotone

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Proof

u′(f (s1)) u′(f (s2)) p(s1) p(s2) = v′(g(s1)) v′(g(s2)) p(s1) p(s2)

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Proof

u′(f (s1)) u′(f (s2)) p(s1) p(s2) = v′(g(s1)) v′(g(s2)) p(s1) p(s2) Concordant not enough, because I need this to hold for any two states, so boils down to p = q

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Proof

u′(f (s1)) u′(f (s2)) = v′(g(s1)) v′(g(s2))

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Proof

u′(f (s1)) u′(f (s2)) = v′(g(s1)) v′(g(s2)) If f (s1) > f (s2) then g(s1) > g(s2)

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Question:

What is the analogue of these results for ambiguity averse ?

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Main Characters

  • 1. Expected utility (EU) : U(f ) = Epu(f )
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Main Characters

  • 1. Expected utility (EU) : U(f ) = Epu(f )
  • 2. Maxmin expected utility (MEU): U(f ) = minp∈C Epu(f )
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Main Characters

  • 1. Expected utility (EU) : U(f ) = Epu(f )
  • 2. Maxmin expected utility (MEU): U(f ) = minp∈C Epu(f )

Constraint preferences: C q,ǫ = {p ∈ ∆(S) | R(p q) ≤ ǫ}

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Main Characters

  • 1. Expected utility (EU) : U(f ) = Epu(f )
  • 2. Maxmin expected utility (MEU): U(f ) = minp∈C Epu(f )

Constraint preferences: C q,ǫ = {p ∈ ∆(S) | R(p q) ≤ ǫ} Rank dependent EU: C q,γ = {p ∈ ∆(S) | p(A) ≥ γ(q(A))}

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Main Characters

  • 1. Expected utility (EU) : U(f ) = Epu(f )
  • 2. Maxmin expected utility (MEU): U(f ) = minp∈C Epu(f )

Constraint preferences: C q,ǫ = {p ∈ ∆(S) | R(p q) ≤ ǫ} Rank dependent EU: C q,γ = {p ∈ ∆(S) | p(A) ≥ γ(q(A))}

  • 3. General : strictly convex, monotone, continuous
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This gives us freedom to play with the risk-neutral probabilities without bending the utility too much

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EU

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MEU

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MEU dual space

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Variational

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Full Insurance for Ambiguity averse

What is the analogue of the common beliefs condition?

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Full Insurance for Ambiguity averse

Billot, Chateauneuf, Gilboa, and Tallon (2000) Rigotti, Shannon, and Strzalecki (2008)

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Beliefs

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Beliefs

p ∈ ∆(S) is a subjective belief at f if Ep(h) ≥ Ep(f ) for all h f

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Full Insurance

agents have strictly convex preferences the aggregate endowment is risk-free shared beliefs = ⇒ all PO allocations are risk-free

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C

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Conditions on Beliefs

EU

  • Full Insurance

same beliefs shared beliefs

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Conditions on Beliefs

EU

  • Full Insurance

same beliefs shared beliefs Conditional Full Insurance concordant beliefs ?

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Conditional Full Insurance

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Conditional Full Insurance

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Conditional Full Insurance

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Conditional Full Insurance

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Conditional Full Insurance

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Conditional Full Insurance

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Conditional Full Insurance

The problem is that MRS12 depends on what is going on in state 3 (Sure thing principle violated)

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Conditional Full Insurance

p is a subjective belief at f if Ep(h) ≥ Ep(f ) for all h f

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Conditional Full Insurance

p is a subjective belief at f if Ep(h) ≥ Ep(f ) for all h f p is a G-conditional belief at f if p is concordant with some subjective belief at f

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Conditional Full Insurance

p is a subjective belief at f if Ep(h) ≥ Ep(f ) for all h f p is a G-conditional belief at f if p is concordant with some subjective belief at f p is a consistent G-conditional belief if p is a G-conditional belief at any G-measurable f

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Conditional Full Insurance

p is a subjective belief at f if Ep(h) ≥ Ep(f ) for all h f p is a G-conditional belief at f if p is concordant with some subjective belief at f p is a consistent G-conditional belief if p is a G-conditional belief at any G-measurable f Can show: p is a consistent G-conditional belief iff Ep[h|G] h for all h Or: p is a consistent G-conditional belief iff f f + ǫ for every ǫ with Ep[ǫ|G] = 0

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When does this happen?

MEU with concave utility and set of priors C q is a consistent G-conditional belief iff pq

G ∈ C for every p ∈ C

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When does this happen?

MEU with concave utility and set of priors C q is a consistent G-conditional belief iff pq

G ∈ C for every p ∈ C

pq

G = conditionals from q, marginals from p

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When does this happen?

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Examples

Constraint preferences: C q,ǫ = {p ∈ ∆(S) | R(p q) ≤ ǫ}

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Examples

Constraint preferences: C q,ǫ = {p ∈ ∆(S) | R(p q) ≤ ǫ} Divergence preferences: C q,ǫ = {p ∈ ∆(S) | D(p q) ≤ ǫ}

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Examples

Constraint preferences: C q,ǫ = {p ∈ ∆(S) | R(p q) ≤ ǫ} Divergence preferences: C q,ǫ = {p ∈ ∆(S) | D(p q) ≤ ǫ} Rank dependent EU: C q,γ = {p ∈ ∆(S) | p(A) ≥ γ(q(A))}

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Conditional Full Insurance

Theorem agents have strictly convex preferences the aggregate endowment is G-measurable shared consistent G-conditional beliefs = ⇒ all PO allocations are G-measurable

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Conditional Full Insurance

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Comonotonicity

Theorem agents have strictly convex preferences the aggregate endowment is G-measurable shared consistent H-conditional beliefs for any H coarser than G = ⇒ all PO allocations are comonotone

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Other papers

Chateauneuf, Dana, and Tallon (2000)

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Other papers

Chateauneuf, Dana, and Tallon (2000) de Castro and Chateauneuf (2009)

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Other papers

Chateauneuf, Dana, and Tallon (2000) de Castro and Chateauneuf (2009) Kajii and Ui (2009); Martins da Rocha (forthcoming)

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Billot, A., A. Chateauneuf, I. Gilboa, and J.-M. Tallon (2000): “Sharing Beliefs: Between Agreeing and Disagreeing,” Econometrica, 68, 685–694. Chateauneuf, A., R. Dana, and J. Tallon (2000): “Risk sharing rules and Equilibria with non-additive expected utilities,” Journal of Mathematical Economics, 34, 191–215. de Castro, L. and A. Chateauneuf (2009): “Ambiguity Aversion and Trade,” mimeo. Kajii, A. and T. Ui (2009): “Interim efficient allocations under uncertainty,” Journal of Economic Theory, 144, 337–353. Martins da Rocha, V. F. (forthcoming): “Interim efficiency with MEU-preferences,” JET. Rigotti, L., C. Shannon, and T. Strzalecki (2008): “Subjective Beliefs and Ex Ante Trade,” Econometrica, 76, 1167–1190.