Range-dependent utility
Kontek K., Lewandowski M.
Two elements:
I
range-dependent utility: a general framework for decisions
under risk
I
decision utility model: its operational special case used for
prediction
Plan:
1. Motivation
2. Idea
3. Axiomatization
4. Paradoxes
5. Monotonicity/continuity
6. Extensions
Eye-adaptation process
What we see in a dark room
Just after entering
After 15 minutes
What an eye adapts to:
I Mean luminance level
I Or luminance range?
Eye-adaptation process
What we see in a dark room
Just after entering
After 15 minutes
What an eye adapts to:
I Mean luminance level
I Or luminance range?
Two psychophysical theories:
I Adaptation-level theory (Helson 1963) ⇒ reference point
⇒ Prospect Theory
I Range-frequency theory (Parducci 1964) ⇒ range
⇒ Our model
TK 1992
experimental data
I
How to explain this data?
Reference point:
I
I
I
EU - no probability weighting (poor fit)
CPT with probability weighting (good fit)
Range-dependence
I
Our model with NO probability weighting (good
fit)
No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
xl
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
50
50
50
50
50
50
50
50
100
100
100
100
100
xu
50
50
50
100
100
100
100
100
200
200
200
200
200
400
400
100
100
100
150
150
150
150
150
200
200
200
200
200
p
0.10
0.50
0.90
0.05
0.25
0.50
0.75
0.95
0.01
0.10
0.50
0.90
0.99
0.01
0.99
0.10
0.50
0.90
0.05
0.25
0.50
0.75
0.95
0.05
0.25
0.50
0.75
0.95
CE
9.0
21.0
37.0
14.0
25.0
36.0
52.0
78.0
10.0
20.0
76.0
131.0
188.0
12.0
377.0
59.0
71.0
83.0
64.0
72.5
86.0
102.0
128.0
118.0
130.0
141.0
162.0
178.0
xl
0
0
0
0
0
xu
50
50
50
50
50
p
0.00
0.10
0.50
0.90
1.00
CE
0
9.0
21.0
37.0
50.0
.
.
.
xl
100
100
100
100
100
100
100
xu
200
200
200
200
200
200
200
p
0.00
0.05
0.25
0.50
0.75
0.95
1.00
CE
100.0
118.0
130.0
141.0
162.0
178.0
200.0
I
Fix the lottery range [xl , xu ]
I
Assign u[xl ,xu ] (xl ) = 0 and
u[xl ,xu ] (xu ) = 1
I
Following the vNM idea:
u[xl ,xu ] (CE ) = p
I
We fit a nonlinear function
u[xl ,xu ] with two restrictions
Fitting range-dependent utility functions
Fitting range-dependent utility functions
I
I
Conceptually
interesting
Operationally
demanding:
I
Eliciting
different
utility
funtions for
different
lottery
ranges
Fitting the decision utility function
I
Observation
I
Range-dependent utilities
differ mostly in stretch
and shift of lottery
consequences
I
Normalize all lottery ranges
into a common interval
[0, 1]
I
Define a single function,
called the decision utility
function
Fitting the decision utility function
I
Observation
I
Range-dependent utilities
differ mostly in stretch
and shift of lottery
consequences
I
Normalize all lottery ranges
into a common interval
[0, 1]
I
Define a single function,
called the decision utility
function
Range-dependent utility axiomatization
I
To compare lotteries:
I
I
I
I
Therefore axioms in terms of CEs rather than a preference
relation
EU representation in terms of CEs
I
I
Expected Utility values are meaningless if lottery ranges
differ
We need to use CE values
Hardy, Littlewood, Polya (1934) result on quasilinear mean
We relax their axioms to make it range-dependent
Range-dependent utility axiomatization
Formally
I
Lotteries are real-valued random variables defined on a
probability space: x : Ω → R
I
The set of all simple lotteries with range [xl , xu ] denoted by
L([xl , xu ])
Axiom (1) (Certainty)
For a degenerate lottery x = x ∗ :
CE (x) = x ∗
Axiom (2) (Within-Range Monotonicity)
Fix range [xl , xu ]. For every x, y ∈ L([xl , xu ]):
Fx FOSD Fy
⇒ CE (x) > CE (y)
Range-dependent utility - main result
Axiom (3) (Within-Range Independence)
Fix range [xl , xu ]. For every x, y, z ∈ L([xl , xu ]) and α ∈ (0, 1)
CE (x) = CE (y) ⇐⇒ CE [(x, α; z, 1 − α)] = CE [(y, α; z, 1 − α)]
Theorem (1) (Range-dependent utility)
For every lottery x ∈ L with some range [xl , xu ] there exists a
unique real number CE (x) satisfying axioms 1-3 if and only if there
exists a continuous and strictly increasing1 function u[xl ,xu ] (·), such
that:
−1
CE (x) = u[x
Eu[xl ,xu ] (x)
(1)
l ,xu ]
1
On the closed interval [A, B], such that [xl , xu ] ⊂ [A, B].
Decision utility
Axiom (4) (Shift and scale invariance)
For every lottery x ∈ L, the certainty equivalent is unique up to
positive affine transformation of lottery payoffs:
CE (αx + β) = αCE (x) + β, ∀α > 0, β ∈ R
Theorem (2) (Decision utility)
For every lottery x ∈ L with some range [xl , xu ] there exists a
unique real number CE (x) satisfying axioms 1-4 if and only if there
exists a continuous and strictly increasing function D(·), called
decision utility, defined on the interval [0, 1], such that:
CE (x) = xl + (xu − xl )D
−1
ED
x − xl
xu − xl
(2)
Two-outcome lotteries: probability weighting and decision
utility
For a binary lottery (xl , 1 − p; xu , p)
I
the decision utility model:
CE (x) = xl + (xu − xl )D −1 (p)
I
a probability weighting model
CE (x) = xl + (xu − xl )w (p)
I
I
For D −1 (p) = w (p) we have the same predictions
Both models are observationally equivalent in this case
I
yet interpretations differ entirely
Plan
1. Motivation
2. Idea
3. Axiomatization
4. Paradoxes
5. Monotonicity/Continuity
6. Extensions
Fair full insurance (House H damaged with prob. p)
p
0
0
No insurance
H − pH
Insurance
1−p H
1
D(r )
1
D(1 − p)
1−p
r
0
1-p
1
1−p
Fair gambling (Prize P won with probability p)
p
P − pP
0
Lottery
No lottery
1
1 − p −pP
1
0
D(r )
p
D(p)
0
r
p
1
p
Common ratio effect
0.8
4000
A
B
0.2
0
1
0.2
3000
0.25 3000
4000
C
D
0.8
0
0.75 0
Discontinuity in the legs in the Machina triangle
p1
1 best prize x1
p3
medium prize x2
worst prize x3
Monotonicity
Proposition
00
00
(η)
(1−η)D (η)
If − ηD
is non-increasing for
D 0 (η) is non-decreasing and
D 0 (η)
all η ∈ [0, 1], then the following holds:
x FOSD y ⇒ CE (x) > CE (y), ∀x, y ∈ L.
Range-dependent utility extensions
Range-dependent utility concept may be extended in several
different ways:
I
Adding wealth/income effects (changing inflection point)
I
Extension to uncertainty aversion (changing slope)
I
Risk and time extension
I
Distribution dependent utility
Relaxing axiom 4: Gonzales, Wu (1999) data
I
Old Axiom 4: Shift and scale invariance
CE (αx + β) = αCE (x) + β, ∀α > 0, β ∈ R
I
New Axiom 4’: Scale invariance
∃W ∈ R : CE (α(W + x)) = αCE (W + x), ∀α > 0
0-25
a=0.34
:
0-50
a=0.17
0-75
a=0.14
1.0
1.0
1.0
0.8
0.8
0.8
0.6
0.6
,
0.4
,
0.4
0.2
0.4
0.6
0.8
1.0
0.2
0.2
0-100
a=0.13
0.4
0.6
0.8
1.0
0.2
0-150
a=0.11
1.0
1.0
0.8
0.8
0.8
0.6
0.6
,
0.4
,
0.4
0.2
0.6
0.8
1.0
0.6
0.8
1.0
0.6
,
0.4
0.2
0.4
0.4
0-200
a=0.06
1.0
0.2
,
0.4
0.2
0.2
0.6
0.2
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
Ellsberg paradox in decision utility model
Decision utility and time preferences
I
For a given time period t, we face the following lottery
(ct , S(t))
I
The relative certainty equivalent for this lottery is r (t)
I
Apply the decision utility model: D(r (t)) = S(t)
I
And get r (t) = D −1 (S(t))
Conclusions
I
A new range-dependent framework for decisions under risk
I
I
I
The decision utility model used for prediction
I
I
I
I
I
a simple and well defined modification of EUT
requires no probability weighting
assumes shift and scale invariance of CEs
fits data well
explains many paradoxes
is consistent with FOSD
Possible extensions (current and future work):
I
I
I
Adding wealth effect
Decisions under uncertainty
Decisions in time
Expected Utility Theory
[vNM - Expected Utility]
[de Finetti - quasilinear mean]
The observable choices are modeled by
a binary relation on L, written
%⊂ L × L.
The observable certainty equivalence for a
lottery is modelled by a unique real
number CE (P) for any P ∈ L.
Weak order: % is complete and
transitive
Certainty: For a degenerate lottery
P ∈ L : {P(x ∗ ) = 1, x ∗ ∈ X }:
CE (P) = x ∗
Continuity: For every P, Q, R ∈ L,
P Q R ⇒ ∃α, β ∈ (0, 1) :
Monotonicity: For every P, Q ∈ L:
P FOSD Q
⇒ CE (P) > CE (Q)
αP + (1 − α)R Q βP + (1 − β)R
Independence 1: For every
P, Q, R ∈ L, and every α ∈ (0, 1)
Independence 2: For every P, Q, R ∈ L
and every α ∈ (0, 1)
P % Q ⇐⇒
CE (P) = CE (Q) ⇐⇒
αP + (1 − α)R % αQ + (1 − α)R
CE (αP + (1 − α)R) = CE (αQ + (1 − α)R)
Expected Utility Theory
Theorem (vNM)
Theorem (de Finetti)
%⊂ L × L satisfies Weak Order,
Continuity and Independence 1
For every lottery P ∈ L with finite
support there exists a unique number
CE (P) which satisfies Certainty,
Monotonicity and Independence 2
if and only if
if and only if
there exists u : X → R, strictly
increasing and continuous, such that, for
every P, Q ∈ L
there exists u : X → R, strictly
increasing and continuous, such that for
every P ∈ L:
P % Q ⇐⇒
X
X
P(x)u(x) ≥
Q(x)u(x)
x∈X
u[CE (P)] =
X
P(x)u(x)
x∈X
x∈X
Moreover, in this case u is unique up to
a positive affine transformation.
Moreover, in this case u is unique up to
a positive affine transformation.
Decision utility vs Expected Utility
Within Expected Utility Theory the following is true:
Theorem
CE is shift-invariant iff vNM utility function is CARA
CE is scale-invariant iff vNM utility function is CRRA
CE is shift- and scale-invariant iff vNM utility function is affine
So within Expected Utility Theory we can get maximum two of the
following elements:
I
risk aversion
I
CE shift-invariance
I
CE scale-invariance
And within Decision Utility Model we can get all three at the same
time.