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 d.
d = 1 .

6. Non-nested Models

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

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

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

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 $ to win $3
.05 to win $ to win $48
.05 to win $ to win $52
R'': 69% S'': 31%
.95 to win $ to win $12
.05 to win $ to win $52
Results shown here are for Birnbaum (1999b) student sample tested in lab x 2 reps.

16. Upper Cumulative Independence
R': 72% S': 28%
.10 to win $ to win $40
.10 to win $ to win $44
.80 to win $ to win $110
R''': 34% S''': 66%
.10 to win $ to win $40
.90 to win $ to win $98
These results are for the student sample of Birnbaum (1999b), lab sample.

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: to win $2
.1 to win $40
R’: .1 to win $10
.1 to win $98
.8 to win $100
R: to win $2
.1 to win $10
Significantly more have the preference pattern SR’ than the reverse RS’. “Median heuristic”--
The median describes this choice, but not details of the study.
Choice depends on the values of the consequences.
Inverse-S implies RS’ pattern.

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
Here, TAX predicts a reversal: R’S2’. Indeed, of 1075 participants, more show this switch
Than the opposite switch, predicted by CPT. Here RAM allows no switch.
56% (sig more than half) prefer R’ and 67% (sig more than half) prefer S2’.

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’: to win $2
.10 to win $4
.10 to win $96
R2’: .10 to win $2
.45 to win $4
.45 to win $96
Here, TAX predicts no reversal. Indeed, of 1075 participants, more show this switch
Than the opposite switch, predicted by CPT. Here RAM allows no switch.
56% (sig more than half) prefer R’ and 67% (sig more than half) prefer S2’.

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 Property CPT RAM TAX LCI No Viols Viols UCI UTI R’S1Viols
LDI
RS2 Viols
3-2 LDI

27. Summary-3 Property CPT RAM TAX 4-DI RS’Viols No Viols SR’ Viols UDI
RBI
RS’ 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.