Behavior in the loss domain

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Behavior in the loss domain  an experiment using
the probability trade-off consistency
conditionOlivier LHaridonGRID, ESTP-ENSAM
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Introduction

• Kahneman and Tverskys Prospect Theory a popular
  and convincing way to study and describe choices
  under risk

But.
Which version of Prospect Theory should we use ?
1979 Original Prospect Theory (OPT)?
With direct transformation of the initial
probabilities, or
1992 Cumulative Prospect Theory (CPT)?
With a rank dependent specification.
On a theoretical ground CPT must be chosen
- more general
- respects First Order Stochastic Dominance
- extends from risk to uncertainty
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But from a descriptive point of view???
Results are mixed
1. Some axioms underlying CPT could be violated
Wu (1994) violations of ordinal independence
Birnbaum and McIntosh (1996) violations of
branch independence
Starmer (1999) OPT can predict some
violations of transitivity
2. As regards the predicting power
Camerer and Ho, 1994 Wu and Gonzales, 1996
? OPT fits the data better than CPT
Fennema and Wakker (1997)
? CPT fits the data better than OPT
? CPT fits better in simple gambles ? OPT fits
better in complex gambles
Wu, Zhang and Abdellaoui (2005)
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Most of the previous studies investigate the gain
domain
Losses are an important part of prospect theory
? Behavior could be very different in the gain
and the loss domain
- Different attitudes toward consequences
- diminishing sensitivity
- loss aversion
- Different attitudes toward probabilities
greater probability weighting in the loss
domain (Lattimore, Baker and Witte,
1992Abdellaoui 2000)
This paper investigates the loss domain
- Different composition rules??
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This paper presents an experiment built on the
test constructed by Wu, Zhang and Abdellaoui
(2005)
Starting point OPT and CPT combine differently
consequences and probabililities
? Composition rules are different
? Probability tradeoff consistency conditions are
different
Method focusing on the probability trade-off
consistency gives a simple way to test the
composition rules used by individuals
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• 1. Probability tradeoff consistency conditions
  under OPT and CPT

Just consider a 3 outcomes gambles p1,L p2 ,l 
p3,0 with L l 0
What is the valuation of this gamble?
Under OPT VOPT p1,L p2 ,l  p3,0 )
w(p1)u(L)
w(p2)u(l)

w(p1 p2) - w(p1)u(l)
Under CPT VCPT(p1,L p2 ,l  p3,0 )
w(p1)u(L)

The difference between the 2 models lies in the
way probabilities are processed
For example, if sub-additivity is satisfied then
w(p1p2) w(p1) w(p2)
? OPT assigns a higher decision weight to the
intermediary outcome.
? whereas CPT valuation focuses on extreme
outcomes.
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Under OPT VOPT p1,L p2 ,l  p3,0 )
w(p1)u(L) w(p2)u(l)
Under CPT VCPT(p1,L p2 ,l  p3,0 )
w(p1)u(L) w(p1 p2) - w(p1)u(l)
? we need to filter out utility
In order to discriminate
? and compare probability weighting
? probability tradeoffs (PTO) can do this!
PTO comparisons of pairs of probabilities
representing probability replacement
3 outcomes ? we can represent the PTO condition
in the Marshak-Machina simplex
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Example of binary choices in the Marshak-Machina
simplex
Binary choices between
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- a safe lottery  S 
- a risky lottery  R 
larger probability of receiving the worst and
zero outcomes
Upper Consequence Probability (p3)
The difference in p1, probability of receiving
the worst outcome, serves as a measuring rod
 Risky 
 Safe 
0
1
1
Lower Consequence Probability (p1)
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The PTO in the Marshak-Machina simplex (under CPT)
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We construct 4 gambles
- by translating the initial gamble on axis p3
- by translating these gambles A on axis p1
Upper Consequence Probability (p3)
R2B
R2A
S2B
S2A
R1A
R1B
S1B
S1A
1
0
1
1
Lower Consequence Probability (p1)
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The PTO in the Marshak-Machina simplex (under CPT)
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We construct 4 gambles
- by translating the initial gamble on axis p3
- by translating these gambles A on axis p1
Upper Consequence Probability (p3)
The PTO condition restricts the set of choices
If the DM chooses R1A and S2A
R2B
R2A
? She cannot choose S1B and R2B
S2B
S2A
Impossible !
R1A
R1B
S1B
S1A
1
0
1
1
Lower Consequence Probability (p1)
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The PTO in the Marshak-Machina simplex (under CPT)
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We construct 4 gambles
- by translating the initial gamble on axis p3
- by translating these gambles A on axis p1
Upper Consequence Probability (p3)
The PTO condition restricts the set of choices
If the DM chooses R1A and S2A
R2B
R2A
? She cannot choose S1B and R2B
S2B
S2A
If the DM chooses S1A and R2A
Impossible !
? She cannot choose R1B and S2B
R1A
R1B
S1B
S1A
1
0
1
1
Lower Consequence Probability (p1)
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The PTO in the Marshak-Machina simplex (under CPT)
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An example with indifference curves
- the DM chooses the safe S2A option
- the DM chooses the risky R1A option
? Indifference curves fan-out among these gambles
Upper Consequence Probability (p3)
The PTO condition restricts the set of choices
If the DM chooses R1A and S2A
R2B
R2A
? She cant choose S1B and R2B
S2B
S2A
R1A
R1B
S1B
S1A
1
0
1
1
Lower Consequence Probability (p1)
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The PTO in the Marshak-Machina simplex (under CPT)
1
An example with indifference curves
- the DM chooses the safe S2A option
- the DM chooses the risky R1A option
? Indifference curves fan-out among these gambles
Upper Consequence Probability (p3)
The PTO condition restricts the set of choices
If the DM chooses R1A, S2A and R2B
R2B
R2A
? She cannot choose S1B
S2B
S2A
? She must choose R1B
R1A
R1B
S1B
S1A
1
0
1
1
Lower Consequence Probability (p1)
Consistency requires that fanning-in is
impossible among gambles B
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1
R2C
S2C
R1C 
S1C
Upper Consequence Probability (p3)
R2B
R2A
S2B
S2A
PTO consistency condition, CPT
R1A
R1B
S1B
S1A
1
0
Lower Consequence Probability (p1)
Under CPT, the PTO condition requires a
consistency in the fanning of indifference
curves among gambles A and B
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Under OPT, the PTO condition is different
R2C
OPT requires a consistency in the fanning of
indifference curves among gambles B and C
S2C
The focus is on the intermediary outcome
R1C 
(the hypothenuse)
S1C
Upper Consequence Probability (p3)
PTO consistency condition, OPT
R2B
R2A
S2B
S2A
PTO consistency condition, CPT
R1A
R1B
S1B
S1A
1
0
Lower Consequence Probability (p1)
Under CPT, the PTO condition requires a
consistency in the fanning of indifference
curves among gambles A and B
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1
R2C
S2C
R1C 
S1C
Upper Consequence Probability (p3)
PTO consistency condition, OPT
R2B
R2A
S2B
S2A
PTO consistency condition, CPT
R1A
R1B
S1B
S1A
1
0
Lower Consequence Probability (p1)
If one observes a different fanning of
indifference curves between gambles A and
gambles C ? the observed fanning for gambles B
discriminates between OPT and CPT
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Example suppose we observe
R2C
- some fanning-out in Gambles A
S2C
- some fanning-in in Gambles C
R1C 
S1C
Upper Consequence Probability (p3)
OPT
R2B
R2A
S2B
S2A
CPT
R1A
R1B
S1B
S1A
1
0
Lower Consequence Probability (p1)
If indifference curves fan out among gambles B
- CPT probability trade-off consistency condition
satisfied
- OPT probability trade-off consistency condition
violated
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1
Example suppose we observe
R2C
- some fanning-out in Gambles A
S2C
- some fanning-in in Gambles C
R1C 
S1C
Upper Consequence Probability (p3)
OPT
R2B
R2A
S2B
S2A
CPT
R1A
R1B
S1B
S1A
1
0
Lower Consequence Probability (p1)
If indifference curves fan in among gambles B
- CPT probability trade-off consistency condition
violated
- OPT probability trade-off consistency condition
satisfied
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2. Experiment
The experiment is based on 4 sets of gambles in
the fashion of Wu, Zhang and Abdellaoui, 2005.
Pilot sessions revealed that a different
measuring rod was necessary in the loss domain
? 30 binary choices between gambles with 3
outcomes in the loss domain
? 34 individual sessions using a computer-based
questionnaire
? Random ordering of tasks and displays
? a training session with four tasks
Gambles were visualized as decision trees
containing probabilities and outcomes pies
charts representing probabilities
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Typical display used in the experiment
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3. Results
3.1 Paired choice analysis and fanning of
indifference curves
We used the Z-test constructed by Conslisk (1989)
- under the null hypothesis expected utility holds

• under the alternative hypothesis violations of
  expected utility are systematic
• rather than random

Fanning-in among gambles C but with low
significance
Mixed results among gambles B
Fanning out significant among gambles A
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Consistency between fanning among gambles B and
the two other sets of gambles? ? MLE estimation
3.2 Maximum likelihood estimation
2 types of subjects ? type 1 fanning-in ?
type 2 fanning-out
If the proportion is different between gambles A
et B ? CPT rejected
If the proportion is different between gambles B
et C ? OPT rejected
Comparison of 2 models
- model 1 same proportion between gambles ? MLE1
- model 2 different proportions between gambles
? MLE2
Likelihood ratio test statistic
2lnMLE1-MLE2?2(1)
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Tableau 2 results of the likelihood test for the
four simplexes
CPT fits the data in simplex I? OPT seems to be
more appropriate in simplex II?
? The likelihood test is not significant, both
versions of PT explain the data
Wu and al. (2005) found that OPT is better in
such gambles for gains we dont.
Preferences are consistent with CPT in simplexes
III and IV
As Wu, Zhand and Abdellaoui (2005) CPT is better
in such gambles
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An abstract in one sentence?
CPT is never rejected by the data in the loss
domain