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Valuation Behavior in the Loss Domain: An Experiment on Probability Trade-off Consistency

This study compares Prospect Theory versions OPT and CPT in the loss domain, investigating how individuals value risky prospects under different models. The experiment explores probability trade-off consistency conditions.

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Valuation Behavior in the Loss Domain: An Experiment on Probability Trade-off Consistency

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  1. Behavior in the loss domain : an experiment using the probability trade-off consistency conditionOlivier L’HaridonGRID, ESTP-ENSAM

  2. Introduction But…. Which version of Prospect Theory should we use ? Kahneman and Tversky’s Prospect Theory: a popular and convincing way to study and describe choices under risk 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

  3. 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)

  4. 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, 1992;Abdellaoui 2000) - Different composition rules?? This paper investigates the loss domain

  5. 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

  6. 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} ) = 1. Probability tradeoff consistency conditions under OPT and CPT 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(p1+p2) ≤ w(p1) + w(p2) OPT assigns a higher decision weight to the intermediary outcome.  whereas CPT valuation focuses on extreme outcomes.

  7. 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) In order to discriminate we need to filter out utility  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

  8. Example of binary choices in the Marshak-Machina simplex Binary choices between: 1 - 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)

  9. The PTO in the Marshak-Machina simplex (under CPT) 1 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)

  10. The PTO in the Marshak-Machina simplex (under CPT) 1 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)

  11. The PTO in the Marshak-Machina simplex (under CPT) 1 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)

  12. 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 and S2A R2B R2A  She can’t choose S1B and R2B S2B S2A R1A R1B S1B S1A 1 0 1 1 Lower Consequence Probability (p1)

  13. 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

  14. 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

  15. 1 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

  16. 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

  17. 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 out among gambles B - CPT probability trade-off consistency condition satisfied - OPT probability trade-off consistency condition violated

  18. 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

  19. 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

  20. Typical display used in the experiment:

  21. 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

  22. 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: 2ln[MLE1-MLE2]~2(1)

  23. 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

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

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