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)


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?


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


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


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


.. 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


1 Background

2 Model & Representation

3 Agg. Uncertainty

4 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


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.


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)??


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 )


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.


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))


1 Background


3 Agg. Uncertainty



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


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


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


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.


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)


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


1 Background


3 Agg. Uncertainty



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 .



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)



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


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.


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

choice under aggregate uncertainty - kellogg school of ......choice under aggregate uncertainty...

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