Behavior in the loss domain : an experiment using the probability trade-off consistency condition...
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Behavior in the loss domain : an experiment using the probability trade-off consistency condition
Olivier L’HaridonGRID, ESTP-ENSAM
Introduction
Kahneman and Tversky’s 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)?
1992: Cumulative Prospect Theory (CPT)?
With direct transformation of the initial probabilities, or
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
Wu, Zhang and Abdellaoui (2005)
But from a descriptive point of view???
Camerer and Ho, 1994Wu and Gonzales, 1996
OPT fits the data better than CPT
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:
Fennema and Wakker (1997) CPT fits the data better than OPT
CPT fits better in simple gambles OPT fits better in complex gambles
Results are mixed:
This paper investigates the loss domain
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:
- Different attitudes toward probabilities: greater probability weighting in the loss domain (Lattimore, Baker and Witte, 1992;Abdellaoui 2000)
- Different composition rules??
- loss aversion
- diminishing sensitivity
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
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?
The difference between the 2 models lies in the way probabilities are processed
For example, if sub-additivity is satisfied then: w(p1+p2) ≤ w(p1) + w(p2)
OPT assigns a higher decision weight to the intermediary outcome.
whereas CPT valuation focuses on extreme outcomes.
Under OPT: VOPT {p1,L; p2 ,l ; p3,0} ) =
Under CPT: VCPT({p1,L; p2 ,l ; p3,0} ) =
w(p1)u(L) w(p2)u(l)
w(p1)u(L) [w(p1 +p2) - w(p1)]u(l)
+
+
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
probability tradeoffs (PTO) can do this!
PTO= comparisons of pairs of probabilities representing probability replacement
In order to discriminate
3 outcomes we can represent the PTO condition in the Marshak-Machina simplex
and compare probability weighting
Lower Consequence Probability (p1)
Upper Consequence Probability
(p3)
1
1
Example of binary choices in the Marshak-Machina simplex
Binary choices between:
- a safe lottery « S »
- a risky lottery « R » : larger probability of receivingthe worst and zero outcomes
0« Safe »
« Risky »
The difference in p1, probability of receiving the worst outcome, serves as a measuring rod
1
R2A
S2A
R1A
S1A
Lower Consequence Probability (p1)
Upper Consequence Probability
(p3)
10
1
1
The PTO in the Marshak-Machina simplex (under CPT)
- by translating the initial gamble on axis p3
We construct 4 gambles
- by translating these gambles A on axis p1
R2B
S2B
S1B
1
R1B
R2A
S2A
R1A
S1A
Lower Consequence Probability (p1)
Upper Consequence Probability
(p3)
10
1
1
The PTO in the Marshak-Machina simplex (under CPT)
- by translating the initial gamble on axis p3
We construct 4 gambles
- by translating these gambles A on axis p1
R2B
S2B
S1B
1
The PTO condition restricts the set of choices:
If the DM chooses R1A and S2A
She cannot choose S1B and R2B
R1B
Impossible !
R2A
S2A
R1A
S1A
Lower Consequence Probability (p1)
Upper Consequence Probability
(p3)
10
1
1
The PTO in the Marshak-Machina simplex (under CPT)
- by translating the initial gamble on axis p3
We construct 4 gambles
- by translating these gambles A on axis p1
R2B
S2B
S1B
1
The PTO condition restricts the set of choices:
If the DM chooses R1A and S2A
She cannot choose S1B and R2B
R1B
Impossible !
If the DM chooses S1A and R2A
She cannot choose R1B and S2B
R2A
S2A
R1A
S1A
Lower Consequence Probability (p1)
Upper Consequence Probability
(p3)
10
1
1
The PTO in the Marshak-Machina simplex (under CPT)
An example with indifference curves
R2B
S2B
S1B
1
The PTO condition restricts the set of choices
If the DM chooses R1A and S2A
She can’t choose S1B and R2B
R1B
- the DM chooses the safe S2A option
- the DM chooses the risky R1A option
Indifference curves fan-out among these gambles
R2A
S2A
R1A
S1A
Lower Consequence Probability (p1)
Upper Consequence Probability
(p3)
10
1
1
The PTO in the Marshak-Machina simplex (under CPT)
An example with indifference curves
R2B
S2B
S1B
1
The PTO condition restricts the set of choices
If the DM chooses R1A, S2A and R2B
She cannot choose S1B
R1B
- the DM chooses the safe S2A option
- the DM chooses the risky R1A option
Indifference curves fan-out among these gambles
Consistency requires that fanning-in is impossible among gambles B
She must choose R1B
R2A
S2A
R1A
S1A
S2B
R2B
S1BR1B
R2C
S2C
R1C
S1C
PTO consistency condition, CPT
10
1
Lower Consequence Probability (p1)
Upper Consequence
Probability (p3)
Under CPT, the PTO condition requires a consistency in the fanning of indifference curves among gambles A and B
R2A
S2A
R1A
S1A
S2B
R2B
S1BR1B
R2C
S2C
R1C
S1C
PTO consistency condition, CPT
PTO consistency condition, OPT
10
1
Lower Consequence Probability (p1)
Upper Consequence
Probability (p3)
Under CPT, the PTO condition requires a consistency in the fanning of indifference curves among gambles A and B
Under OPT, the PTO condition is different:
OPT requires a consistency in the fanning of indifference curves among gambles B and C
The focus is on the intermediary outcome(the hypothenuse)
R2A
S2A
R1A
S1A
S2B
R2B
S1BR1B
R2C
S2C
R1C
S1C
PTO consistency condition, CPT
PTO consistency condition, OPT
10
1
Lower Consequence Probability (p1)
Upper Consequence
Probability (p3)
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
R2A
S2A
R1A
S1A
S2B
R2B
S1BR1B
R2C
S2C
R1C
S1C
CPT
OPT
10
1
Lower Consequence Probability (p1)
Upper Consequence
Probability (p3)
Example: suppose we observe
- some fanning-out in Gambles A
- some fanning-in in Gambles C
If indifference curves fan out among gambles B
- CPT probability trade-off consistency condition satisfied
- OPT probability trade-off consistency condition violated
R2A
S2A
R1A
S1A
S2B
R2B
S1BR1B
R2C
S2C
R1C
S1C
CPT
OPT
10
1
Lower Consequence Probability (p1)
Upper Consequence
Probability (p3)
Example: suppose we observe
- some fanning-out in Gambles A
- some fanning-in in Gambles C
If indifference curves fan in among gambles B
- CPT probability trade-off consistency condition violated
- OPT probability trade-off consistency condition satisfied
2. Experiment
34 individual sessions using a computer-based questionnaire
30 binary choices between gambles with 3 outcomes in the loss domain
Random ordering of tasks and displays
The experiment is based on 4 sets of gambles in the fashion of Wu, Zhang andAbdellaoui, 2005.
Pilot sessions revealed that a different measuring rod was necessary in the loss domain
Gambles were visualized as decision trees containing probabilities and outcomes+ pies charts representing probabilities
a training session with four tasks
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 amonggambles C but
with low significance
Fanning out significant among gambles A
Mixed results among gambles B
Gambles A Gambles B Gambles C
Type z p Type z p Type z p
Simplex I Fanning out -1,68 0,05 Fanning in 1,16 0,12 Fanning in 0,30 0,38
Simplex II Parallel 0,00 0,50 Parallel 0,00 0,50 Fanning in 3,78 0,00
Simplex III Fanning out -2,23 0,01 Fanning out -2,18 0,01 Fanning in 1,14 0,13
Simplex IV Fanning out -1,87 0,03 Fanning out -2,96 0,00 Fanning in 0,63 0,27
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
- model 1: same proportion between gambles MLE1
- model 2: different proportions between gambles MLE2
Likelihood ratio test statistic: 2ln[MLE1-MLE2]~2(1)
Comparison of 2 models
Consistency between fanning among gambles B and the two other sets of gambles? MLE estimation
CPT OPT
2(1) p-value 2(1) p-value
Simplex I 1.88 0.1694 0.11 0.7311
Simplex II 0.03 0.8637 2.40 0.1211No certainty
effectPooled (I+ II) 0.72 0.3955 0.72 0.3948
Simplex III 0.27 0.5978 2.40 0.1218
Simplex IV 1.09 0.2955 3.62 0.0573Certainty effect
Pooled (III+IV) 0.13 0.7105 5.93 0.0148
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 don’t.
Preferences are consistent with CPT in simplexes III and IV
As Wu, Zhand and Abdellaoui (2005): CPT is better in such gambles