Behavior in the loss domain : an experiment using the probability trade-off consistency condition Olivier L’Haridon GRID, 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, 1994 Wu 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 {p 1,L; p 2,l ; p 3,0} 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(p 1 +p 2 ) ≤ w(p 1 ) + w(p 2 )  OPT assigns a higher decision weight to the intermediary outcome.  whereas CPT valuation focuses on extreme outcomes. Under OPT: V OPT {p 1,L; p 2,l ; p 3,0} ) = Under CPT: V CPT ({p 1,L; p 2,l ; p 3,0} ) = w(p 1 )u(L)w(p 2 )u(l) w(p 1 )u(L) [w(p 1 +p 2 ) - w(p 1 )]u(l) + +

Under OPT: V OPT {p 1,L; p 2,l ; p 3,0} ) = w(p 1 )u(L) + w(p 2 )u(l) Under CPT: V CPT ({p 1,L; p 2,l ; p 3,0} ) = w(p 1 )u(L) + [w(p 1 +p 2 ) - w(p 1 )]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 receiving the 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) 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) 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) 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) 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) 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 S1B R1B R2C S2C R1C S1C PTO consistency condition, CPT 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 S1B R1B R2C S2C R1C S1C PTO consistency condition, CPT PTO consistency condition, OPT 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 S1B R1B R2C S2C R1C S1C PTO consistency condition, CPT PTO consistency condition, OPT 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 S1B R1B R2C S2C R1C S1C CPT OPT 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 S1B R1B R2C S2C R1C S1C CPT OPT 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 and Abdellaoui, 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 among gambles C but with low significance Fanning out significant among gambles A Mixed results among gambles B

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

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?