Introduction

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



(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



(p3)

10

1

1





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



(p3)

10

1

1





R2B

S2B

S1B

1

The PTO condition restricts the set of choices:


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



(p3)

10

1

1


An example with indifference curves

R2B

S2B

S1B

1

The PTO condition restricts the set of choices


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



(p3)

10

1

1


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


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, OPT

10

1


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, OPT

10

1


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


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


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

Typical display used in the experiment:

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

CPT is never rejected by the data in the loss domain

An abstract in one sentence?

Introduction

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