1. New Paradoxes of Risky Decision Making that Refute Prospect Theories Michael H. Birnbaum Fullerton, California, USA

2. Outline I will review tests between Cumulative Prospect Theory (CPT) and Transfer of Attention eXchange (TAX) model. Emphasis will be on critical properties that test between these two non- nested theories.

3. Cumulative Prospect Theory/ Rank-Dependent Utility (RDU)

4. “Prior” TAX Model Assumptions:

5. TAX Parameters For 0 < x < $150 u(x) = x Gives a decent approximation. Risk aversion produced by  

6. TAX CE with delta = 0

7. TAX: Effect of Delta

8. TAX Model

9. TAX and CPT nearly identical for binary (two-branch) gambles CE (x, p; y) is an inverse-S function of p according to both TAX and CPT, given their typical parameters. Therefore, there is no point trying to distinguish these models with binary gambles.

10. Non-nested Models

11. CPT and TAX nearly identical inside the prob. simplex

12. Testing CPT Coalescing Stochastic Dominance Lower Cum. Independence Upper Cumulative Independence Upper Tail Independence Gain-Loss Separability TAX:Violations of:

13. Testing TAX Model 4-Distribution Independence (RS’) 3-Lower Distribution Independence 3-2 Lower Distribution Independence 3-Upper Distribution Independence (RS’) Res. Branch Indep (RS’) CPT: Violations of:

14. Stochastic Dominance A test between CPT and TAX: G = (x, p; y, q; z) vs. F = (x, p – s; y’, s; z) Note that this recipe uses 4 distinct consequences: x > y’ > y > z > 0; outside the probability simplex defined on three consequences. CPT  choose G, TAX  choose F Test if violations due to “error.”

15. Violations of Stochastic Dominance 122 Undergrads: 59% two violations (BB) 28% Pref Reversals (AB or BA) Estimates: e = 0.19; p = 0.85 170 Experts: 35% repeat violations 31% Reversals Estimates: e = 0.20; p = 0.50 Chi-Squared test reject H0: p violations < 0.4

16. Pie Charts

17. Aligned Table: Coalesced

18. Summary: 23 Studies of SD, 8653 participants Large effects of splitting vs. coalescing of branches Small effects of education, gender, study of decision science Very small effects of probability format, request to justify choice. Miniscule effects of event framing (framed vs unframed)

19. Lower Cumulative Independence R: 39% S: 61%.90 to win $3.90 to win $3.05 to win $12.05 to win $48.05 to win $96.05 to win $52 R'': 69% S'': 31%.95 to win $12.90 to win $12.05 to win $96.10 to win $52

20. Upper Cumulative Independence R': 72% S': 28%.10 to win $10.10 to win $40.10 to win $98.10 to win $44.80 to win $110.80 to win $110 R''': 34% S''': 66%.10 to win $10.20 to win $40.90 to win $98.80 to win $98

21. Summary: UCI & LCI 22 studies with 33 Variations of the Choices, 6543 Participants, & a variety of display formats and procedures. Significant Violations found in all studies.

22. Restricted Branch Indep. S ’:.1 to win $40.1 to win $44.8 to win $100 S:.8 to win $2.1 to win $40.1 to win $44 R ’ :.1 to win $10.1 to win $98.8 to win $100 R:.8 to win $2.1 to win $10.1 to win $98

23. 3-Upper Distribution Ind. S ’ :.10 to win $40.10 to win $44.80 to win $100 S2 ’ :.45 to win $40.45 to win $44.10 to win $100 R ’ :.10 to win $4.10 to win $96.80 to win $100 R2 ’ :.45 to win $4.45 to win $96.10 to win $100

24. 3-Lower Distribution Ind. S ’ :.80 to win $2.10 to win $40.10 to win $44 S2 ’ :.10 to win $2.45 to win $40.45 to win $44 R ’ :.80 to win $2.10 to win $4.10 to win $96 R2 ’ :. 10 to win $2.45 to win $4.45 to win $96

25. Gain-Loss Separability

26. Notation

27. Wu and Markle Result

28. Birnbaum & Bahra--% F

29. Allais Paradox Dissection Restricted Branch Independence CoalescingSatisfiedViolated SatisfiedEU, PT*,CPT*CPT ViolatedPTTAX

30. Summary: Prospect Theories not Descriptive Violations of Coalescing Violations of Stochastic Dominance Violations of Gain-Loss Separability Dissection of Allais Paradoxes: viols of coalescing and restricted branch independence; RBI violations opposite of Allais paradox.

31. Summary-2 PropertyCPTRAMTAX LCINo ViolsViols UCINo ViolsViols UTINo ViolsR ’ S1Viols LDIRS2 ViolsNo Viols 3-2 LDIRS2 ViolsNo Viols

32. Summary-3 PropertyCPTRAMTAX 4-DI RS ’ Viols No Viols SR ’ Viols UDIS ’ R2 ’ Viols No ViolsR ’ S2 ’ Viols RBI RS ’ ViolsSR ’ Viols

33. Results: CPT makes wrong predictions for all 12 tests Can CPT be saved by using different formats for presentation? More than a dozen formats have been tested. Violations of coalescing, stochastic dominance, lower and upper cumulative independence replicated with 14 different formats and thousands of participants.

34. Implications Results indicate that neither PT nor CPT are descriptive of risky decision making. Editing rules of combination, cancellation, & dominance detection refuted. TAX correctly predicts the violations of CPT. CPT implies violations of TAX that either fail or show the opposite pattern from predicted by CPT.