Upper Cumulative Independence

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

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. 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. Lower Cumulative Independence (3-LCI)

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

6. Comonotonic RBI Consequence monotonicity Transitivity Coalescing

7. Example Test

8. Generic Configural Model

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

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. RAM Weights

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. TAX Model

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. 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. Birnbaum (‘99): n = 124

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 Additional Replications A number of as unpublished studies (as of Jan, 2005) have replicated the basic findings with a variety of different procedures in choice.

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

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. To Rescue CPT: For CPT to handle these data, make it configural. Let g < 1 for two-branch gambles and g > 1 for three-branch gambles.

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

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. 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.