Lower Distribution Independence

1. Lower Distribution Independence Michael H. Birnbaum California State University, Fullerton

2. Testing Among Models of Risky Decision Making • Previous studies tested properties implied by CPT. • Violations of Stochastic Dominance, Coalescing, Upper and Lower Cumulative Independence, Upper Tail Independence refute CPT. • “Unfair” to test CPT this way?

3. Test Predicted Effects • Instead of testing implied invariance of CPT • Test predicted violations of EU • Put RAM and TAX in position of defending the null hypothesis against violations predicted by CPT

4. LDI is Violated by CPT • LDI is implied by EU. • CPT violates LDI but RAM and Special TAX models satisfy it. • In this test, RAM and TAX defend the null hypothesis against predictions of specific violations made by CPT.

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

6. RAM Model

7. RAM Model Parameters

8. RAM implies inverse-S

9. Special TAX Model

10. TAX Model • The weight of a branch depends on the branch’s probability. • Each branch gains weight from branches with higher consequences. • Each branch gives up weight to branches with lower consequences. • Predictions nearly identical to those of CPT and RAM for binary gambles.

11. The lower branch, z, has different probabilities in the two choices.

12. Lower Distribution Independence (3-LDI)

13. Example Test

14. Generic Configural Model The generic model includes RDU, CPT, RAM, TAX, GDU, & others as special cases.

15. Violation of 3-LDI A violation will occur if SfR and

16. 2 Types of Violations: SR2: RS2:

17. EU allows no violations • In EU, the weights are the probabilities; therefore

18. CPT implies violations • If W(P) = P, CPT reduces to EU; however, when W(P) is nonlinear, CPT violates LDI systematically. • From previous data, we can calculate where to expect violations and predict which type of violation should be observed.

19. CPT Model of Tversky and Kahneman (1992)

21. CPT implies RS2 Violations • When g = 1, CPT reduces to EU. • Given the inverse-S weighting function, the fitted CPT model implies RS2 pattern. • If g > 1, however, the model can handle the opposite pattern. • A series of tests can be devised to provide overlapping combinations of parameters.

22. RAM allows no Violations • RAM model with any parameters satisfies 3-LDI.

23. Special TAX: No Violations • The Special TAX model, with one configural parameter, allows no violations of 3-LDI. • The middle branch gains as much weight as it gives up for any p.

24. Summary of Predictions • RAM, TAX, & EU satisfy 3-LDI • CPT violates 3-LDI • Fitted CPT implies RS2 pattern of violation • Here CPT is the most flexible model, RAM and TAX defend the null hypothesis.

25. Web-Based Studies • Two from a Series of Studies tests: classical and new paradoxes in decision making. • People come on-line via WWW (some tested in lab for comparison). • Choose between gambles; 1 person per month (about 1% of participants) wins the prize of one of their chosen gambles. 20 or 22 choices. • Data arrive 24-7; sample sizes are large; results are clear.

26. Results n = 503

27. Results: n = 1075

28. 3-2 Lower Distribution Independence • In this property, the probability of the branch with the lowest consequence goes to zero and the branch is removed. • CPT again predicts violations • Special TAX and RAM again satisfy the property

29. Test of 3-2 LDI; n = 1075 Fitted CPT predicts RS2 Pattern

30. Summary: Predicted Violations of CPT failed to Materialize • TAX model, fit to previous data correctly predicted the modal choices. • RAM makes the same predictions in this case. • Fitted CPT was correct when it agreed with TAX, wrong otherwise except 1 case in 12.

31. To Rescue CPT: • CPT can handle the result of any single test, by choosing suitable parameters. • For CPT to handle these data, the values of b must be much smaller or g much larger than those reported in the literature.

33. RAM and TAX have been found more accurate than CPT in other tests • Are they simply more flexible? No. • In the tests of 3-LDI and 3-2-LDI, CPT is the most flexible model. • Why then has there been a growing consensus for CPT? I suspect lack of familiarity with the results of studies like these.

34. Case against CPT/RDU • Violations of Stochastic Dominance • Violations of Coalescing (Event-Splitting) • Violations of 3-Upper Tail Independence • Violations of Lower Cumulative Independence • Violations of Upper Cumulative Independence

35. More Evidence against CPT/RDU/RSDU • Violations of Gain-Loss separability. • Violations of Restricted Branch Independence are opposite predictions of fitted CPT. • Violations of 4-distribution independence, 3-UDI favor TAX over RAM and opposite predictions of CPT. • Failure of predicted violations of 3-LDI and 3-2 LDI to materialize.

36. Preview of Next Program • The next programs reviews tests of Upper Distribution Independence, assuming the viewer has seen this program. • EU and RAM predict no violations, CPT and TAX predict opposite patterns. Data agree with TAX.

37. For More Information: mbirnbaum@fullerton.edu http://psych.fullerton.edu/mbirnbaum/ Download recent papers from this site. Follow links to “brief vita” and then to “in press” for recent papers.