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

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

3. 3 In this test, we increase z in both gambles and coalesce it with y ’ (in R), and we increase y and coalesce it with x (in S only). So we have improved S relative to R.

4. 4 Lower Cumulative Independence (3-2-LCI)

5. 5 LCI implied by any model that satisfies: Consequence monotonicity Transitivity Coalescing Comonotonic restricted branch independence

6. 6 Example Test of 3-2-LCI

7. 7 Generic Configural Model

8. 8 CPT Cannot Handle Violation of 3-2-LCI Suppose CPT satisfies coalescing;

9. 9 2 Types of Reversals: SR ” : This is a violation of LCI. If systematic, it refutes CPT. RS ” : This reversal is perfectly consistent with LCI. (We Improved S ” relative to R ”.)

10. 10 EU allows no violations EU implies LCI:

11. 11 RAM Weights

12. 12 RAM Violations RAM violates 3-LCI. Even when the a(i,n) are all equal, 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 improved S, we made R seem better.

13. 13 TAX Model

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

15. 15 TAX Model Violates LCI According to parameters estimated from previous data, TAX implies a violation of LCI in the example used. The prediction is fairly robust with respect to the parameters, and holds for  >.36 and  < 1.06. As shown later, violations are observed here, showing that at least some participants have parameters in this region. By adjusting the consequences, we can test other combinations of parameters.

16. 16 Summary of Predictions EU, CPT satisfy LCI TAX & RAM violate LCI Here CPT defends the null hypothesis against a specific prediction made by both RAM and TAX. Predictions were made in advance and studies designed to test them.

17. 17 Birnbaum (‘99), n = 124

18. 18 Lab Studies of LCI 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).

19. 19 Web Studies of LCI 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).

20. 20 Additional Replications A number of as yet unpublished studies have replicated the basic findings with a variety of different procedures in choice.

21. 21

22. 22 Error Analysis We can fit “true and error” model to data with replications to separate “real” violations from those attributable to “error”. Model implies violations are “real” and cannot be attributed to error.

23. 23 Violations predicted by RAM & TAX, not CPT EU and CPT are refuted by systematic violations of LCI. TAX & RAM, as fit to previous data correctly predicted the modal choices. Predictions published in advance of the studies. Violations of LCI are to CPT as the Allais paradoxes are to EU.

24. 24 To Rescue CPT: CPT cannot handle the results unless it becomes a configural model. For CPT to handle these data, allow different W(P) functions depending on the number of branches in the gambles. Let  1 for three-branch gambles.

25. 25 Add to the case against CPT/RDU/RSDU Violations of Lower Cumulative Independence are a strong refutation of CPT model as proposed.

26. 26 Next Program: UCI The next programs reviews tests of Upper Cumulative Independence (UCI). Violations of 3-2-UCI contradict any form of RDU, CPT, including EU. They are consistent with (and were predicted by) RAM and TAX.

27. 27 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