Choice under Aggregate Uncertainty - Kellogg School of ......Choice under Aggregate Uncertainty...
Transcript of Choice under Aggregate Uncertainty - Kellogg School of ......Choice under Aggregate Uncertainty...
Choice under Aggregate Uncertainty
Nabil I. Al-Najjar & Luciano Pomatto
May 4, 2015
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Aggregation
The utility of a decision maker depends on outcomeprofiles:
s = (x1, . . . , xn)
S = X1 × · · · × Xn
An attractive representation is via linear aggregators:
V (x1, . . . , xn) =n∑
i=1
vi(xi)
Incorporate uncertainty (??): given P ∈ ∆(S), EU is:∫S
n∑i=1
v(xi) dP(s) ≡∫
SV (s) dP(s)
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Example 1: Dixit-Stiglitz CES Aggregator
Representative consumer derives utility from consuming avariety of products
The Dixit-Stiglitz aggregator is:(n∑
i=1
q ρi
) 1ρ
, 0 < ρ < 1
qi ≥ 0: units of variety i
ρ measures taste for variety
Is ρ cardinally meaningful? Risk neutrality to incomegambles?
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Example 2: Utilitarian Social Welfare Functions
A social planner’s utility is an additive function of theutilities of members of society
n∑i=1
vi(xi)
vi : utility function of individual i
xi : his consumption bundle
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Example 3 : Games Against Multiple Opponents
Player’s payoff depends on actions of n opponents
Linear dependence expressed as:
n∑i=1
vi(ai)
vi (ai ): payoff impact of individual i ’s action on utility
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Aggregation and Aggregate Uncertainty
Aggregate uncertainty is irrelevant with linear aggregators:
∫S
V (s) dP(s) ≡∫
S
(n∑
i=1
v(xi)
)dP(s)
≡n∑
i=1
(∫S
v(xi) dP(s)
)
≡n∑
i=1
Epi (xi )v(xi)
pi(xi): marginal distribution of P on the i ’s coordinate
Only the marginals of P matter; correlation is irrelevant
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
.. but aggregate uncertainty does (should) matter!
Society responds differently to aggregate vs. idiosyncraticrisks with the same marginals
Terrorism vs. idiosyncratic traffic deaths
Idiosyncratic vs. aggregate strategic uncertainty in games
Robson (1996): Evolutionary reasons why Nature maydesign utility to distinguish between the two
Halevy and Feltkamp (2005) model of uncertainty aversion
Motivation: Many paradoxes motivating non-Bayesian decisioncriteria are consequences of using additive aggregators andinsensitivity to aggregate uncertainty
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
1 Background
2 Model & Representation
3 Agg. Uncertainty
4 Aggregation of Idiosyncratic Risk
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Mathematical Structure
S = X1 × · · · × Xn
n ≥ 3
Xi is connected, complete, separable metric space
∆(S) Borel probability measures on profiles
pi is the marginal of P ∈ ∆(S) on Xi
P is independent if
P = p1 × · · · × pn
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Expected Utility
Preference relation % on the set of social lotteries ∆(S)
Expected Utility (EU)% has a representation:∫
SU(s) dP(s),
for a cardinally unique and continuous U.
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Aggregation
New medical procedure with outcomes: recovery x , or death y
Three profiles:x = (x , x , . . . , x)
y = (y , y , . . . , y)
s = (x , . . . , x︸ ︷︷ ︸n/2
, y , . . . , y︸ ︷︷ ︸n/2
)
ObviouslyU(x) > U(y)
U(s)??
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Aggregation
Fix a non-empty subset I ⊂ {1, . . . ,n}
sI is profile s restricted to I
Conditional preference: h is preferred to h′ given s means
(hI , sIc ) % (h′I , sIc )
Sure Thing Principle (STP)
For all profiles s, s′,h,h′,
(hI , sIc ) % (h′I , sIc ) ⇐⇒ (hI , s′Ic ) % (h′I , s′Ic )
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Theorem
% satisfies EU and STP
⇐⇒% has an aggregative utility representation:
U(P) =
∫S
u
(1n
n∑i=1
vi(si)
)dP(s)
vi ’s are continuous and non-constant functionsvi : Xi → R, i = 1, . . . ,n
u is increasing, continuous functionu : range
(1n∑n
i=1 vi)→ R.
The vi ’s are unique up to common positive affinetransformation, and u is cardinally unique given the vi ’s.
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Sketch of the Proof
Debreu (1960)’s aggregation theorem says that there existcontinuous, cardinally unique v1, . . . , vn such that
V (s) =n∑
i=1
vi(xi)
represents % restricted to S
By identifying s with the dirac measure δs that puts unitmass on s, we have
S ⊂ ∆(S)
U,V : S → R represent identical ordinal ranking
There exists a strictly increasing function u
U(s) = u (V (s))
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
1 Background
2 Model & Representation
3 Agg. Uncertainty
4 Aggregation of Idiosyncratic Risk
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Sensitivity to Aggregate Uncertainty
Indifference to Aggregate Uncertainty
% is indifferent to aggregate uncertainty if
P ∼ Q
whenever P,Q have equal marginals:
pi = qi , ∀i .
Theorem% is indifferent to aggregate uncertainty if and only if u is affine.
Motivation interms of riskaversion:
α:lottery overprizes 1$, 0$
α:average oftwo dollaramounts1$, 0$
α
1
U(0)
U(1)
0$ 1$
α
Averaging Monetary Values
0 1
Pro
bab
ilit
yM
ixtu
res
Riskaversion:
The twomixtureoperationsare notequivalent
u
α
1
U(0)
U(1)
0$ 1$
u−1(α) α
Averaging Monetary Values
0 1
Pro
bab
ilit
yM
ixtu
res
Probabilitymixture:α% chanceeveryone getsx1−α% chanceeveryone getsy
Populationmixture:
α% get x1− α% get y
α
1
U(y)
U(x)
V (y) V (x)
α
Population Mixtures
0 1
Pro
bab
ilit
yM
ixtu
res
Hedging viafractionalprofiles:
Aggregativeutility withconcave uimplieswillingness tosubstituteprobabilitymixtures bypopulationmixtures
u
α
1
U(y)
U(x)
V (y) V (x)
u−1(α) α
Population Mixtures
0 1
Pro
bab
ilit
yM
ixtu
res
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Dixit-Stiglitz Revisited
The standard Dixit-Stiglitz CES aggregator is often written(1n
∑qρi
) 1ρ
Not clear whether the exponent 1ρ has cardinal meaning
Using our representation and the assumption that vi = vjand u are all CES, we obtain
U(P) =
∫ (1n
∑qρi
)κρ
dP
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Dixit-Stiglitz Revisited
(1n
∑qρi
)κρ
ρ is the familiar elasticity of substitution
Consider bundles: αq = (αq1, . . . , αqn), α > 0 then
κ =αU ′′(αq)
U ′(αq)
κ is the induced relative risk aversion wrt to changes inconsumption levels
κρ has cardinal meaning of risk aversion relative to taste forvariety
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Portfolio Choice
In portfolio theory returns are perfectly fungible
U(P) =
∫u(
1n
∑xi
)dP.
This is aggregative utility where the vi = the identity.
More general form is
U(P) =
∫u(
1n
∑vi(xi)
)dP.
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Mental Accounting
Consider the general form:
u(
1n
∑vi(xi)
)
The concavity of the vi ’s may be interpreted as capturing:“utility unrelated to consumption. [...] An investor may interpret abig loss on a stock as a sign that he is a second-rate investor, thusdealing his ego a painful blow, and he may feel humiliation in frontof friends and family when word about the failed investment leaksout.” (Barberis and Huang)
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Halevy-Feltkamp 2005
Halevy-Feltkamp (RES 2005):“Bayesian Model of Uncertainty Aversion”
Urn with known composition and an “ambiguous urn”
Agent’s payoff is a concave function of two draws from thesame urn
u(x1 + x2)
They show thatAgent prefers risky urn (with known composition)Agents have a strict preference to randomize when facingthe ambiguous urn
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
1 Background
2 Model & Representation
3 Agg. Uncertainty
4 Aggregation of Idiosyncratic Risk
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Aggregation of idiosyncratic risk
Regularity Assumptions
For every n, the aggregative utility Un satisfies:
(i) The range of vi is contained in [0,1] for every i ;
(ii) The range of u is contained in [0,1];
(iii) u is Lipschitz continuous, with Lipschitz constant K .
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Aggregation of idiosyncratic risk
Compare aggregative utility of the nth problem:
Un(P) =
∫S
u
(1n
n∑i=1
vi(si)
)dP(s) (1)
Next, imagine “moving P inside u(·)”:
Un(P) = u
(1n
n∑i=1
Epi vi(si)
)(2)
As n increases, u(1
n∑
vi(si))
“concentrates” around Un(P)
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Aggregation of idiosyncratic risk
Use concentration inequalities to prove:
Theorem ∣∣Un(P)− Un(P)∣∣ < ε+ 2e−2n ( ε
K )2
For everyIndependent P
vi ’s and u satisfying the regularity assumption
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
The Conditionally i.i.d. Case
Si = Sj for all i , j
Regularity Assumption holds
Pµ is conditionally i.i.d.1 Draw θ using µ
2 Use θ to draw profile s i.i.d.
Background Model & Representation Agg. Uncertainty Aggregation of Idiosyncratic Risk
Theorem
∣∣∣∣∣∫
Θ
∫S
u
(1n
n∑i=1
vi(si)
)dPθ(s) dµ(θ)
−∫Θ
u
(1n
n∑i=1
Epθivi(si)
)dµ(θ)
∣∣∣∣∣.−→ 0
uniformly in µ
Compound lotteries reduce, of course, but idiosyncratic riskmakes
∑vi(si) concentrate around its mean while aggregate
risk does not