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Axiomatic Foundations of Multiplier Preferences
Tomasz Strzalecki
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Multiplier preferences
Expected Utility inconsistent with observed behavior
We (economists) may not want to fully trust any probabilisticmodel.
Hansen and Sargent: “robustness against model misspecification”
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Multiplier preferences
Unlike many other departures from EU, this is very tractable:
Monetary policy – Woodford (2006)Ramsey taxation – Karantounias, Hansen, and Sargent (2007)Asset pricing: – Barillas, Hansen, and Sargent (2009)
– Kleshchelski and Vincent (2007)
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Multiplier preferences
But open questions:
→ Where is this coming from?What are we assuming about behavior (axioms)?
→ Relation to ambiguity aversion (Ellsberg’s paradox)?
→ What do the parameters mean (how to measure them)?
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Multiplier preferences
But open questions:
→ Where is this coming from?What are we assuming about behavior (axioms)?
→ Relation to ambiguity aversion (Ellsberg’s paradox)?
→ What do the parameters mean (how to measure them)?
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Multiplier preferences
But open questions:
→ Where is this coming from?What are we assuming about behavior (axioms)?
→ Relation to ambiguity aversion (Ellsberg’s paradox)?
→ What do the parameters mean (how to measure them)?
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Sources of Uncertainty
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Ellsberg Paradox
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Ellsberg Paradox
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Ellsberg Paradox
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Ellsberg Paradox
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Ellsberg Paradox
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Ellsberg Paradox
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Ellsberg Paradox
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Ellsberg Paradox
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Sources of Uncertainty
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Sources of Uncertainty
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Sources of Uncertainty
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Sources of Uncertainty
Small Worlds (Savage, 1970; Chew and Sagi, 2008)
Issue Preferences (Ergin and Gul, 2004; Nau 2001)
Source-Dependent Risk Aversion (Skiadas)
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Main Result
Within each source (urn) multiplier preferences are EU
But they are a good model of what happens between the sources
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Main Result
Within each source (urn) multiplier preferences are EU
But they are a good model of what happens between the sources
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Criterion
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Savage Setting
S – states of the world
Z – consequences
f : S → Z – act
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Expected Utility
u : Z → R – utility function
q ∈ ∆(S) – subjective probability measure
V (f ) =∫S u(fs) dq(s) – Subjective Expected Utility
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Expected Utility
u : Z → R – utility function
q ∈ ∆(S) – subjective probability measure
V (f ) =
∫S u(fs) dq(s)
– Subjective Expected Utility
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Expected Utility
u : Z → R – utility function
q ∈ ∆(S) – subjective probability measure
V (f ) =
∫S u(
fs
) dq(s)
– Subjective Expected Utility
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Expected Utility
u : Z → R – utility function
q ∈ ∆(S) – subjective probability measure
V (f ) =
∫S
u(fs)
dq(s)
– Subjective Expected Utility
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Expected Utility
u : Z → R – utility function
q ∈ ∆(S) – subjective probability measure
V (f ) =∫S u(fs) dq(s) – Subjective Expected Utility
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Multiplier preferences
V (f ) =
minp∈∆(S)
∫u(fs) dp(s)
+ θ R (p‖q)
Kullback-Leibler divergencerelative entropy:R(p‖q) =
∫log( dp
dq
)dp
θ ∈ (0,∞]θ ↑⇒ model uncertainty ↓θ =∞⇒ no model uncertainty
q – reference measure (best guess)
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Multiplier preferences
V (f ) =
minp∈∆(S)
∫u(fs) dp(s)
+ θ R (p‖q)
Kullback-Leibler divergencerelative entropy:R(p‖q) =
∫log( dp
dq
)dp
θ ∈ (0,∞]θ ↑⇒ model uncertainty ↓θ =∞⇒ no model uncertainty
q – reference measure (best guess)
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Multiplier preferences
V (f ) = minp∈∆(S)
∫u(fs) dp(s)
+ θ R (p‖q)
Kullback-Leibler divergencerelative entropy:R(p‖q) =
∫log( dp
dq
)dp
θ ∈ (0,∞]θ ↑⇒ model uncertainty ↓θ =∞⇒ no model uncertainty
q – reference measure (best guess)
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Multiplier preferences
V (f ) = minp∈∆(S)
∫u(fs) dp(s) + θ R (p‖q)
Kullback-Leibler divergencerelative entropy:R(p‖q) =
∫log( dp
dq
)dp
θ ∈ (0,∞]θ ↑⇒ model uncertainty ↓θ =∞⇒ no model uncertainty
q – reference measure (best guess)
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Multiplier preferences
V (f ) = minp∈∆(S)
∫u(fs) dp(s) + θ R (p‖q)
Kullback-Leibler divergencerelative entropy:R(p‖q) =
∫log( dp
dq
)dp
θ ∈ (0,∞]θ ↑⇒ model uncertainty ↓θ =∞⇒ no model uncertainty
q – reference measure (best guess)
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Multiplier preferences
V (f ) = minp∈∆(S)
∫u(fs) dp(s) + θ R (p‖q)
Kullback-Leibler divergencerelative entropy:R(p‖q) =
∫log( dp
dq
)dp
θ ∈ (0,∞]θ ↑⇒ model uncertainty ↓θ =∞⇒ no model uncertainty
q – reference measure (best guess)
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Multiplier preferences
V (f ) = minp∈∆(S)
∫u(fs) dp(s) + θ R (p‖q)
Kullback-Leibler divergencerelative entropy:R(p‖q) =
∫log( dp
dq
)dp
θ ∈ (0,∞]θ ↑⇒ model uncertainty ↓θ =∞⇒ no model uncertainty
q – reference measure (best guess)
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Observational Equivalence
When only one source of uncertainty
Link between model uncertainty and risk sensitivity:
Jacobson (1973); Whittle (1981); Skiadas (2003)
dynamic multiplier preferences = (subjective) Kreps-Porteus-Epstein-Zin
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Observational Equivalence
Multiplier Criterion EU Criterion
%
→ u and θ not identified
→ Ellsberg’s paradox cannot be explained
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Observational Equivalence
φθ(u) =
{− exp
(− u
θ
)for θ <∞,
u for θ =∞.
φθ ◦ u is more concave than umore risk averse
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Observational Equivalence
φθ(u) =
{− exp
(− u
θ
)for θ <∞,
u for θ =∞.
φθ ◦ u is more concave than umore risk averse
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Observational Equivalence
Dupuis and Ellis (1997)
minp∈∆S
∫S
u(fs) dp(s) + θR(p‖q) = φ−1θ
(∫Sφθ ◦ u(fs) dq(s)
)
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Observational Equivalence
Observation (a) If % hasa multiplier representationwith (θ, u, q), then it has aEU representation with(φθ ◦ u, q).
Observation (b) If % hasa EU representation with(u, q), where u is boundedfrom above, then it hasa multiplier representationwith (θ, φ−1
θ ◦ u, q) for anyθ ∈ (0,∞].
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Observational Equivalence
Observation (a) If % hasa multiplier representationwith (θ, u, q), then it has aEU representation with(φθ ◦ u, q).
Observation (b) If % hasa EU representation with(u, q), where u is boundedfrom above, then it hasa multiplier representationwith (θ, φ−1
θ ◦ u, q) for anyθ ∈ (0,∞].
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Boundedness Axiom
Axiom There exist z ≺ z ′ in Z and a non-null event E , such thatwEz ≺ z ′ for all w ∈ Z
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Enriching the Domain: TwoSources
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Enriching Domain
f : S → Z – Savage act (subjective uncertainty)
∆(Z ) – lottery (objective uncertainty)
f : S → ∆(Z ) – Anscombe-Aumann act
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Anscombe-Aumann Expected Utility
fs ∈ ∆(Z )
u(fs) =∑
z u(z)fs(z)
V (f ) =
∫Su(fs) dq(s)
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Anscombe-Aumann Expected Utility
fs ∈ ∆(Z )
u(fs) =∑
z u(z)fs(z)
V (f ) =
∫Su(fs) dq(s)
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Axiomatization
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Variational Preferences
Multiplier preferences are a special case of variational preferences
V (f ) = minp∈∆(S)
∫u(fs) dp(s) + c(p)
axiomatized by Maccheroni, Marinacci, and Rustichini (2006)
Multiplier preferences:
V (f ) = minp∈∆(S)
∫u(fs) dp(s) + θR(p‖q)
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MMR Axioms
A1 (Weak Order) The relation % is transitive and complete
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MMR Axioms
A2 (Weak Certainty Independence) For all acts f , g and lotteriesπ, π′ and for any α ∈ (0, 1)
αf + (1− α)π % αg + (1− α)π
mαf + (1− α)π′ % αg + (1− α)π′
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MMR Axioms
A3 (Continuity) For any f , g , h the sets{α ∈ [0, 1] | αf + (1− α)g % h} and{α ∈ [0, 1] | h % αf + (1− α)g} are closed
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MMR Axioms
A4 (Monotonicity) If f (s) % g(s) for all s ∈ S , then f % g
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MMR Axioms
A5 (Uncertainty Aversion) For any α ∈ (0, 1)
f ∼ g ⇒ αf + (1− α)g % f
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MMR Axioms
A6 (Nondegeneracy) f � g for some f and g
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MMR Axioms
Axioms A1-A6
mVariational Preferences
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MMR Axioms
A7 (Unboundedness) There exist lotteries π′ � π such that, for allα ∈ (0, 1), there exists a lottery ρ that satisfies eitherπ � αρ+ (1− α)π′ or αρ+ (1− α)π � π′.
A8 (Weak Monotone Continuity) Given acts f , g , lottery π,sequence of events {En}n≥1 with En ↓ ∅
f � g ⇒ πEnf � g for large n
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MMR Axioms
Axioms A1-A6
mVariational Preferences
Axiom A7 ⇒ uniqueness of the cost function c(p)
Axiom A8 ⇒ countable additivity of p’s.
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Axioms for MultiplierPreferences
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P2 (Savage’s Sure-Thing Principle)
For all events E and acts f , g , h, h′ : S → Z
fEh % gEh =⇒ fEh′ % gEh′
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P2 (Savage’s Sure-Thing Principle)
For all events E and acts f , g , h, h′ : S → Z
fEh % gEh =⇒ fEh′ % gEh′
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P4 (Savage’s Weak Comparative Probability)
For all events E and F and lotteries π � ρ and π′ � ρ′
πEρ % πFρ =⇒ π′Eρ′ % π′Fρ′
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P4 (Savage’s Weak Comparative Probability)
For all events E and F and lotteries π � ρ and π′ � ρ′
πEρ % πFρ =⇒ π′Eρ′ % π′Fρ′
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P6 (Savage’s Small Event Continuity)
For all Savage acts f � g and π ∈ ∆(Z ), there exists a finitepartition {E1, . . . ,En} of S such that for all i ∈ {1, . . . , n}
f � πEig and πEi f � g .
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Main Theorem
Axioms A1-A8, together with P2, P4, and P6, are necessary andsufficient for % to have a multiplier representation (θ, u, q).
Moreover, two triples (θ′, u′, q′) and (θ′′, u′′, q′′) represent thesame multiplier preference % if and only if q′ = q′′ and there existα > 0 and β ∈ R such that u′ = αu′′ + β and θ′ = αθ′′.
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Proof Idea
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Proof: Step 1
% on lotteries → identify u (uniquely)
MMR axioms → V (f ) = I(u(f )
)I defines a preference on utility acts x , y : S → R
x %∗ y iff I (x) ≥ I (y)
Where I (x + k) = I (x) + k for x : S → R and k ∈ R
(Like CARA, but utility effects, rather than wealth effects)
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Proof: Step 2
P2, P4, and P6, together with MMR axioms imply other Savageaxioms, so
f % g iff∫ψ(fs) dq(s) ≥
∫ψ(gs) dq(s)
π′ % π iff ψ(π′) ≥ ψ(π) iff u(π′) ≥ u(π).
ψ and u are ordinally equivalent, so there exists a strictlyincreasing function φ, such that ψ = φ ◦u.
f % g iff∫φ(u(fs)
)dq(s) ≥
∫φ(u(gs)
)dq(s)
Because of Schmeidler’s axiom, φ has to be concave.
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Proof: Step 3
x %∗ y iff∫φ(x) dq ≥
∫φ(y) dq
iff (Step 1) x + k %∗ y + k iff∫φ(x + k) dq ≥
∫φ(y + k) dq
So %∗ represented by φk(x) := φ(x + k) for all k
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Proof: Step 4
So functions φk are affine transformations of each other
Thus, φ(x + k) = α(k)φ(x) + β(k) for all x , k .
This is Pexider equation. Only solutions are φθ for θ ∈ (0,∞]
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Proof: Step 5
f % g ⇐⇒∫φθ
(u(fs)
)dq(s) ≥
∫φθ
(u(gs)
)dq(s)
From Dupuis and Ellis (1997)
φ−1θ
(∫Sφθ ◦u(fs) dq(s)
)= min
p∈∆S
∫Su(fs) dp(s) + θR(p‖q)
So
minp∈∆S
∫u(fs) dp(s) + θR(p‖q) ≥ min
p∈∆S
∫u(gs) dp(s) + θR(p‖q)
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Proof: Step 5
f % g ⇐⇒∫φθ
(u(fs)
)dq(s) ≥
∫φθ
(u(gs)
)dq(s)
From Dupuis and Ellis (1997)
φ−1θ
(∫Sφθ ◦u(fs) dq(s)
)= min
p∈∆S
∫Su(fs) dp(s) + θR(p‖q)
So
minp∈∆S
∫u(fs) dp(s) + θR(p‖q) ≥ min
p∈∆S
∫u(gs) dp(s) + θR(p‖q)
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Proof: Step 5
f % g ⇐⇒∫φθ
(u(fs)
)dq(s) ≥
∫φθ
(u(gs)
)dq(s)
From Dupuis and Ellis (1997)
φ−1θ
(∫Sφθ ◦u(fs) dq(s)
)= min
p∈∆S
∫Su(fs) dp(s) + θR(p‖q)
So
minp∈∆S
∫u(fs) dp(s) + θR(p‖q) ≥ min
p∈∆S
∫u(gs) dp(s) + θR(p‖q)
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Interpretation
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V (f ) =∫
φθ(u(fs)
)dq(s)
(1) u(z) = z θ = 1
(2) u(z) = − exp(−z) θ =∞
Anscombe-Aumann
u is identified
(1) 6= (2)
Savage
Only φθ ◦ u is identified
(1) = (2)
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Ellsberg Paradox
V (f ) =
∫φθ
(∑z
u(z)fs(z)
)dq(s)
Objective gamble: 12 · 10 + 1
2 · 0 → φθ
(12 · u(10) + 1
2 · u(0))
Subjective gamble: → 12φθ
(u(10)
)+ 1
2φθ
(u(0)
)
For θ =∞ objective ∼ subjective
For θ <∞ objective � subjective
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Measurement of Parameters
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Ellsberg Paradox – Measuring Parameters
V (f ) =
∫φθ
(∑z
u(z)fs(z)
)dq(s)
Certainty equivalent for the objective gamble:φθ
(u(x)
)= φθ
(12 · u(10) + 1
2 · u(0))
Certainty equivalent for the subjective gamble:φθ
(u(y)
)= 1
2φθ
(u(10)
)+ 1
2φθ
(u(0)
)
x → curvature of u
(x − y) → value of θ
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Sources of Uncertainty
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Multiplier Preferences
V (f ) =
∫φθ
(∑z
u(z)fs(z)
)dq(s)
Anscombe-Aumann Expected Utility
V (f ) =
∫ (∑z
u(z)fs(z)
)dq(s)
V (f ) =
∫ (∑z
φθ
(u(z)
)fs(z)
)dq(s)
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Second Order Expected Utility
Neilson (1993)
V (f ) =
∫φ
(∑z
u(z)fs(z)
)dq(s)
Ergin and Gul (2009)
V (f ) =
∫Sb
φ
(∫Sa
u(f (sa, sb)) dqa(sa)
)dqb(sb)
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Conclusion
Axiomatization of multiplier preferences
Multiplier preferences measure the difference of attitudes towarddifferent sources of uncertainty
Measurement of parameters of multiplier preferences
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Thank you
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