1. 1 Upper Cumulative Independence Michael H. Birnbaum California State University, Fullerton

2. 2 UCI is implied by CPT CPT, RSDU, RDU, EU satisfy UCI. RAM and TAX violate UCI. Violations are direct internal contradiction in RDU, RSDU, CPT, EU.

3. 3 In this test, we reduce z ’ in both gambles and coalesce it with x ’ (in R ’ ), and we decrease x and coalesce it with y (in S ’ only).

4. 4 Lower Cumulative Independence (3-LCI)

5. 5 UCI implied by any model that satisfies: Comonotonic restricted branch independence Consequence monotonicity Transitivity Coalescing (Proof on next page.)

6. 6 Comonotonic RBI Consequence monotonicity Transitivity Coalescing

7. 7 Example Test

8. 8 Generic Configural Model

9. 9 3-2-LCI in CPT Suppose CPT satisfies coalescing;

10. 10 2 Types of Reversals: R ’ S ’’’ : This is a violation of UCI. It refutes CPT. S ’ R ’’’ : This reversal is consistent with LCI. (S ’ made worse relative to R ’.)

11. 11 RAM Weights

12. 12 RAM Violations RAM violates 3-2-UCI. If t(p) is negatively accelerated, RAM violates coalescing: coalescing branches with better consequences makes the gamble worse and coalescing the branches leading to lower consequences makes the gamble better. Even though we made S relatively worse, the coalescings made it relatively better.

13. 13 TAX Model

14. 14 TAX: Violates UCI Special TAX model violates 3-2-UCI. Like RAM, the model violates coalescing. Predictions were calculated in advance of the studies, which were designed to investigate those specific predictions.

15. 15 Summary of Predictions EU, CPT, RSDU, RDU satisfy UCI TAX & RAM violate UCI CPT defends the null hypothesis against specific predictions made by both RAM and TAX.

16. 16 Birnbaum (‘99): n = 124

17. 17 Lab Studies of UCI Birnbaum & Navarrete (1998): 27 tests; n = 100; (p, q) = (.25,.25), (.1,.1), (.3,.1), (.1,.3). Birnbaum, Patton, & Lott (1999): n = 110; (p, q) = (.2,.2). Birnbaum (1999): n = 124; (p, q) = (.1,.1), (.05,.05).

18. 18 Web Studies of UCI Birnbaum (1999): n = 1224; (p, q) = (.1,.1), (.05,.05). Birnbaum (2004b): 12 studies with total of n = 3440 participants; different formats for presenting gambles probabilities; (p, q) = (.1,.1), (.05,.05).

19. 19 Additional Replications A number of as unpublished studies (as of Jan, 2005) have replicated the basic findings with a variety of different procedures in choice.

20. 20

21. 21 Error Analysis “True and Error” Model implies violations are “real” and cannot be attributed to error.

22. 22 Violations predicted by RAM & TAX, not CPT EU, CPT, RSDU, RDU are refuted by systematic violations of UCI. TAX & RAM, as fit to previous data correctly predicted the violations. Predictions published in advance of the studies. Violations are to CPT as the Allais paradoxes are to EU.

23. 23 To Rescue CPT: For CPT to handle these data, make it configural. Let  1 for three-branch gambles.

24. 24 Add to the case against CPT/RDU/RSDU Violations of Upper Cumulative Independence are a strong refutation of CPT model as proposed.

25. 25 Next Program: UTI The next programs reviews tests of Upper Tail Independence (UTI). Violations of 3-UTI contradict any form of CPT, RSDU, RDU, including EU. Violations contradict Lower GDU. They are consistent with RAM and TAX.

26. 26 For More Information: http://psych.fullerton.edu/mbirnbaum/ Download recent papers from this site. Follow links to “brief vita” and then to “in press” for recent papers. mbirnbaum@fullerton.edu