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
Ambiguity
r+
R
1
G
0
B
0
Ambiguity
r+
g+
R
1
0
G
0
1
B
0
0
Ambiguity
r+
g+
r−
R
1
0
0
G
0
1
1
B
0
0
1
Ambiguity
r+
g+
r−
g−
R
1
0
0
1
G
0
1
1
0
B
0
0
1
1
Goal
Understand risk sharing among agents with
ambiguity averse preferences
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
Question 1: Full Insurance
Full Insurance
Theorem
agents have strictly risk averse EU
the aggregate endowment is risk-free
common beliefs
=⇒ all PO allocations are risk-free
Question 2: Conditional Full Insurance
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
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
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
Proof
u 0 (f (s1 )) p(s1 )
v 0 (g (s1 )) q(s1 )
=
u 0 (f (s2 )) p(s2 )
v 0 (g (s2 )) q(s2 )
Proof
u 0 (f (s1 ))
v 0 (g (s1 ))
=
u 0 (f (s2 ))
v 0 (g (s2 ))
Proof
u 0 (f (s1 ))
v 0 (g (s1 ))
=
u 0 (f (s2 ))
v 0 (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 )
Question 3: Comonotonicity
Question 3: Comonotonicity
[f (s1 ) − f (s2 )][g (s1 ) − g (s2 )] ≥ 0
Comonotonicity
Theorem
agents have strictly risk averse EU
common probability beliefs
=⇒ all PO allocations are comonotone
Proof
u 0 (f (s1 )) p(s1 )
v 0 (g (s1 )) p(s1 )
=
u 0 (f (s2 )) p(s2 )
v 0 (g (s2 )) p(s2 )
Proof
u 0 (f (s1 )) p(s1 )
v 0 (g (s1 )) p(s1 )
=
u 0 (f (s2 )) p(s2 )
v 0 (g (s2 )) p(s2 )
Concordant not enough, because I need this to hold for any two
states, so boils down to p = q
Proof
u 0 (f (s1 ))
v 0 (g (s1 ))
=
u 0 (f (s2 ))
v 0 (g (s2 ))
Proof
u 0 (f (s1 ))
v 0 (g (s1 ))
=
u 0 (f (s2 ))
v 0 (g (s2 ))
If f (s1 ) > f (s2 ) then g (s1 ) > g (s2 )
Question:
What is the analogue of these results for ambiguity averse %?
Main Characters
1. Expected utility (EU) : U(f ) = Ep u(f )
Main Characters
1. Expected utility (EU) : U(f ) = Ep u(f )
2. Maxmin expected utility (MEU): U(f ) = minp∈C Ep u(f )
Main Characters
1. Expected utility (EU) : U(f ) = Ep u(f )
2. Maxmin expected utility (MEU): U(f ) = minp∈C Ep u(f )
Constraint preferences: C q, = {p ∈ ∆(S) | R(p k q) ≤ }
Main Characters
1. Expected utility (EU) : U(f ) = Ep u(f )
2. Maxmin expected utility (MEU): U(f ) = minp∈C Ep u(f )
Constraint preferences: C q, = {p ∈ ∆(S) | R(p k q) ≤ }
Rank dependent EU: C q,γ = {p ∈ ∆(S) | p(A) ≥ γ(q(A))}
Main Characters
1. Expected utility (EU) : U(f ) = Ep u(f )
2. Maxmin expected utility (MEU): U(f ) = minp∈C Ep u(f )
Constraint preferences: C q, = {p ∈ ∆(S) | R(p k q) ≤ }
Rank dependent EU: C q,γ = {p ∈ ∆(S) | p(A) ≥ γ(q(A))}
3. General % : strictly convex, monotone, continuous
This gives us freedom to play with the risk-neutral probabilities
without bending the utility too much
EU
MEU
MEU dual space
Variational
Full Insurance for Ambiguity averse %
What is the analogue of the common beliefs condition?
Full Insurance for Ambiguity averse %
Billot, Chateauneuf, Gilboa, and Tallon (2000)
Rigotti, Shannon, and Strzalecki (2008)
Beliefs
Beliefs
p ∈ ∆(S) is a subjective belief at f if Ep (h) ≥ Ep (f ) for all h % f
Full Insurance
agents have strictly convex preferences
the aggregate endowment is risk-free
shared beliefs
=⇒ all PO allocations are risk-free
C
Conditions on Beliefs
Full Insurance
EU
%
same beliefs
shared beliefs
Conditions on Beliefs
EU
%
Full Insurance
same beliefs
shared beliefs
Conditional Full Insurance
concordant beliefs
?
Conditional Full Insurance
Conditional Full Insurance
Conditional Full Insurance
Conditional Full Insurance
Conditional Full Insurance
Conditional Full Insurance
Conditional Full Insurance
The problem is that MRS12 depends on what is going on in state 3
(Sure thing principle violated)
Conditional Full Insurance
p is a subjective belief at f if Ep (h) ≥ Ep (f ) for all h % f
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
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
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 + for every with Ep [|G] = 0
When does this happen?
MEU with concave utility and set of priors C
q is a consistent G-conditional belief iff
pGq ∈ C for every p ∈ C
When does this happen?
MEU with concave utility and set of priors C
q is a consistent G-conditional belief iff
pGq ∈ C for every p ∈ C
pGq = conditionals from q, marginals from p
When does this happen?
Examples
Constraint preferences: C q, = {p ∈ ∆(S) | R(p k q) ≤ }
Examples
Constraint preferences: C q, = {p ∈ ∆(S) | R(p k q) ≤ }
Divergence preferences: C q, = {p ∈ ∆(S) | D(p k q) ≤ }
Examples
Constraint preferences: C q, = {p ∈ ∆(S) | R(p k q) ≤ }
Divergence preferences: C q, = {p ∈ ∆(S) | D(p k q) ≤ }
Rank dependent EU: C q,γ = {p ∈ ∆(S) | p(A) ≥ γ(q(A))}
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
Conditional Full Insurance
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
Other papers
Chateauneuf, Dana, and Tallon (2000)
Other papers
Chateauneuf, Dana, and Tallon (2000)
de Castro and Chateauneuf (2009)
Other papers
Chateauneuf, Dana, and Tallon (2000)
de Castro and Chateauneuf (2009)
Kajii and Ui (2009); Martins da Rocha (forthcoming)
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.