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Combining Bayesian Beliefs and Willingness to Bet to Analyze Attitudes
towards Uncertainty
by Peter P. Wakker, Econ. Dept.,Erasmus Univ. Rotterdam
(joint with Mohammed Abdellaoui & Aurélien Baillon) RUD, Tel Aviv, June 24 '07
Topic: Uncertainty/Ambiguity.
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Making uncertainty/ambiguity more operational:
measuring, predicting,quantifying completely,in tractable manner.
No (new) maths; but "new" (mix of) concepts: uniform sources; source-dependent probability transformation.
1. Introduction
Good starting point for uncertainty: Risk.Many nonEU theories exist, virtually all amounting to:
x y 0; xpy w(p)U(x) + (1–w(p))U(y);
Relative to EU: one more graph …
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inverse-S, (likelihood
insensitivity)
p
w
expected utility
mot
ivat
iona
l
cognitive
pessimism
extreme inverse-S ("fifty-fifty")
prevailing finding
pessimistic "fifty-fifty"
Abdellaoui (2000); Bleichrodt & Pinto (2000); Gonzalez & Wu 1999; Tversky & Fox, 1997.
Now to Uncertainty (unknown probabilities);
In general, on the x-axis we have events. So, no nice graphs …
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Many advanced theories;mostly ambiguity-averse
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CEU (Gilboa 1987;
Schmeidler 1989)
PT
(Tversky &
Kahnem
an 1992)Multiple priors (Gilboa
& Schmeidler 1989)
Endogeneous definitions
(Epstein, Zhang, Kopylov,
Ghirardato, Marinacci)
Smooth (KMM; Nau)Variational
model (Maccheroni,
Marinacci,
Rustichini)
Many tractable empirical studies;also inverse-S
Curley & Yates 1985
Fox & Tve
rsky
1995
Biseparable
(Ghirardato &
Marinacci 2001)
Choice-based
Kilka & We-ber 2001
Cabantous
2005di Mauro & Maffioletti 2005
Nice graphs, but x-axis-problem: choice-less probability-inputs there
We connect
Einhorn & Hogarth 1985
next p.p. 9 (theory)
Einhorn & Hogarth 1985 (+ 1986 + 1990).Over 400 citations after '88.
For ambiguous event A, take "anchor probability" pA (c.f. Hansen & Sargent). Weight S(pA):
S(pA) = (1 – )pA + (1 – pA);
: index of inverse-S (regression to mean); ½.: index of elevation (pessimism/ambiguity aversion);
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Einhorn & Hogarth 1985
Graphs: go to pdf file of Hogarth & Einhorn (1990, Management Science 36, p. 785/786).
Problem of the x-axis …
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p. 6 (butter fly-theories)
2. Theory
Only binary acts with gains.
All popular static nonEU theories (except variational):
x y 0; xEy W(E)U(x) + (1–W(E))U(y).
(Ghirardato & Marinacci 2001).
For rich S, such as continuum, general W is too complex.
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Machina & Schmeidler (1992), probabilistic sophistication:
x y; xEy w(P(E))U(x) + (1–w(P(E)))U(y).
Then still can get nice x-axis for uncertainty!
However,
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Common preferences between gambles for $100:(Rk: $100) (Ru: $100)(Bk: $100) (Bu: $100)
>
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Ellsberg paradox. Two urns with 20 balls.
Ball drawn randomly from each. Events:Rk: Ball from known urn is red. Bk, Ru, Bu are similar.
Known urnk
10 R10 B
20 R&B in unknown proportion
Unknown urnu
? 20–?
P(Rk) > P(Ru) P(Bk) > P(Bk)
+1
+1 ><Under probabilistic sophistication
with a (non)expected utility model:
Ellsberg: There cannot exist probabilities in any sense. No "x-axis" and no nice graphs …
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(Or so it seems?)
>
Common preferences between gambles for $100:(Rk: $100) (Ru: $100) (Bk: $100) (Bu: $100)
20 R&B in unknown proportion
Ellsberg paradox. Two urns with 20 balls.
Ball drawn randomly from each. Events:Rk: Ball from known urn is red. Bk, Ru, Bu are similar.
10 R10 B
Known urnk Unknown urnu
? 20–?
P(Rk) > P(Ru) P(Bk) > P(Bk)
+ +1 1 ><Under probabilistic sophistication
with a (non)expected utility model:
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twomodels, depending on source
reconsidered.
Step 1 of our approach (to operationalize uncertainty/ambiguity):
Distinguish between different sources of uncertainty.
Step 2 of our approach:Define sources within which probabilistic sophistication holds. We call them
Uniform sources.
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Step 3 of our approach:
Develop a method for (theory-free)* eliciting probabilities within uniform sources; empirical elaboration of Chew & Sagi's exchangeability.
* Important because we will use different decision theories for different sources
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Step 4 of our approach:Decision theory for uniform sources S, source-dependent. E denotes event w.r.t. S.
x y; xEy wS(P(E))U(x) + (1–wS(P(E)))U(y).
wS: source-dependent probability transformation. (Einhorn & Hogarth 1985; Kilka & Weber 2001)
Ellsberg: wu(0.5) < wk(0.5) u: k: unknown known
(Choice-based) probabilities can be maintained.We get back our x-axis, and those nice graphs!
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We have reconciled Ellsberg 2-color with Bayesian beliefs! (Also KMM/Nau did partly.)
We cannot do so always; Ellsberg 3-color(2 sources!?).
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`c =0.08
w(p)
Fig.a. Insensitivity index a: 0;pessimism index b: 0.
Figure 5.2. Quantitative indexes of pessimism and likelihood insensitivity
00.11= c
10.89
d =0.11
Fig.b. Insensitivity index a: 0;pessimism index b: 0.22.
c =0.11
d =0.11
Fig.c. Insensitivity index a: 0.22;pessimism index b: 0.
0
d =0.14
Fig.d. Insensitivity index a: 0.22; pessimism index b: 0.06.
d =01
p
c = 0 0
Theory continued:
(Chateauneuf, Eichberger, & Grant 2005 ; Kilka & Weber 2001; Tversky & Fox 1995)
3. Let the Rubber Meet the Road: An Experiment
Data:
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4 sources:1. CAC40;2. Paris temperature;3. Foreign temperature;4. Risk.
Method for measuring choice-based probabilities20
EE EEE E
Figure 6.1. Decomposition of the universal event
a3/4
E
a1/2a1/4a1/8a3/8
E
b1a5/8 a7/8b0
a3/4a1/2a1/4
E E
b1b0
E E
a1/2
E
b1b0
E
E = S
b1b0
The italicized numbers and events in the bottom row were not elicited.
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Median choice-based probabilities (real incentives)
Real data over 19002006
0.035201510
0.8
0.6
0.4
0.2
1.0
Figure 7.2. Probability distributions for Paris temperature
Median choice-based probabilities (hypothetical choice)
0.0
Median choice-based probabilities (real incentives)
Real data over the year 2006
0 1 2 3123
0.8
0.6
0.4
0.2
1.0
Figure 7.1. Probability distributions for CAC40
Median choice-based probabilities (hypothetical choice)
Results for choice-based probabilities
Uniformity confirmed 5 out of 6 cases.
Certainty-equivalents of 50-50 prospects.Fit power utility with w(0.5) as extra unknown.
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0
HypotheticalReal
1 2 30
1
0.5
Figure 7.3. Cumulative distribution of powers
Method for measuring utility
Results for utility
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0.125
00
Figure 8.3. Probability transformations for participant 2
Fig. a. Raw data and linear interpolation.
0.25
0.875
0.75
1
0.50
0.125 0.8750.25 0.50 0.75 1
Paris temperature; a = 0.78, b = 0.12
foreign temperature; a = 0.75, b = 0.55
risk:a = 0.42, b = 0.13
Within-person comparisons
Many economists, erroneously, take this symmetric weighting fuction as unambiguous or ambiguity-neutral.
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participant 2;a = 0.78, b = 0.69
0 *
Fig. a. Raw data and linear interpolation.
*
Figure 8.4. Probability transformations for Paris temperature
0.25
0.125
0.875
0.75
1
0.50
0.125 0.8750.250 0.50 0.75 1
participant 48;a = 0.21, b = 0.25
Between-person comparisons
Example of predictions [Homebias; Within-Person Comparison; subject lives in Paris]. Consider investments.Foreign-option: favorable foreign temperature: $40000unfavorable foreign temperature: $0Paris-option:favorable Paris temperature: $40000unfavorable Paris temperature: $0
Assume in both cases: favorable and unfavo-rable equally likely for subject 2; U(x) = x0.88. Under Bayesian EU we’d know all now.NonEU: need some more graphs; we have them!
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Paris temperature Foreign temperature
decision weightexpectationcertainty equivalentuncertainty premiumrisk premiumambiguity premium
0.49 0.2020000 2000017783 6424
1357622175879 5879
7697–3662
Within-person comparisons (to me big novelty of Ellsberg):
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Subject 2,p = 0.125
decision weightexpectationcertainty equivalentuncertainty premiumrisk premiumambiguity premium
0.35 0.675000 35000
12133159742732
571710257–3099
Subject 48,p = 0.125
Subject 2,p = 0.875
Subject 48,p = 0.875
5000 350000.08 0.52
2268
654 9663–39–4034
–71332078
962419026 25376
Between-person comparisons; Paris temperature
Conclusion:
By (1) recognizing importance of uniform sources and source-dependent probability transformations;(2) carrying out quantitative measurements of (a) probabilities (subjective), (b) utilities, (c) uncertainty attitudes (the graphs), all in empirically realistic and tractable manner,
we make ambiguity completely operational at a quantitative level.
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