2. Behavior in the loss domain : an experiment using the probability trade-off consistency condition Olivier L’Haridon GRID, ESTP-ENSAM

3. 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

4. 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:

5. 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

6. 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

7. 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) + +

8. 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

9. 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

10. R2A S2A R1A S1A Lower Consequence Probability (p1) Upper Consequence Probability (p3) 1 0 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

11. R2A S2A R1A S1A Lower Consequence Probability (p1) Upper Consequence Probability (p3) 1 0 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 !

12. R2A S2A R1A S1A Lower Consequence Probability (p1) Upper Consequence Probability (p3) 1 0 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

13. R2A S2A R1A S1A Lower Consequence Probability (p1) Upper Consequence Probability (p3) 1 0 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

14. R2A S2A R1A S1A Lower Consequence Probability (p1) Upper Consequence Probability (p3) 1 0 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

15. R2A S2A R1A S1A S2B R2B S1B R1B R2C S2C R1C S1C PTO consistency condition, CPT 1 0 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

16. R2A S2A R1A S1A S2B R2B S1B R1B R2C S2C R1C S1C PTO consistency condition, CPT PTO consistency condition, OPT 1 0 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)

17. R2A S2A R1A S1A S2B R2B S1B R1B R2C S2C R1C S1C PTO consistency condition, CPT PTO consistency condition, OPT 1 0 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

18. R2A S2A R1A S1A S2B R2B S1B R1B R2C S2C R1C S1C CPT OPT 1 0 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

19. R2A S2A R1A S1A S2B R2B S1B R1B R2C S2C R1C S1C CPT OPT 1 0 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

20. 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, 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

21. Typical display used in the experiment:

22. 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

23. 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

24. 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

25. CPT is never rejected by the data in the loss domain An abstract in one sentence?