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. Non-nested Models

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

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

9. 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:

10. Stochastic Dominance A test between CPT and TAX: G = (x, p; y, q; z) vs. F = (x, p – q; y’, q; z) Note that this recipe uses 4 distinct values of consequences. It falls outside the probability simplex defined on three consequences. CPT  G, TAX  F We can test if violations due to “error”

11. Violations of Stochastic Dominance 122 Undergrads: 59% repeat the violation (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 < 0.4

12. Pie Charts

13. Aligned Table: Coalesced

14. 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)

15. 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

16. 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

17. 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.

18. 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

19. 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

20. 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

21. Gain-Loss Separability

22. Notation

23. Wu and Markle Result

24. Birnbaum & Bahra--% F

25. 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.

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

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

28. 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.

29. Implications Results indicate that neither PT nor CPT are descriptive of risky decision making. TAX correctly predicts the violations of CPT. CPT implies violations of TAX that either fail or show the opposite pattern from predicted by CPT.