Upper Tail Independence

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

2. UTI is implied by CPT Wu (1994) reported violations of UTI which he noted violate CPT, RSDU, RDU, and EU. • UTI is not equivalent to Ordinal Independence, which is defined on (at least) four-branch gambles. • RAM and TAX violate UTI. • Violations are direct internal contradiction in RDU, RSDU, CPT, and EU. They also violate lower GDU.

3. We decrease z’ in both gambles and coalesce it with x’ (in R’). This results in a choice between a two-branch and a three-branch gamble. This definition more general than that used by Wu, where y = y’ = 0.

4. Upper Tail Independence (UTI)

5. UTI implied by any model that satisfies: • Transitivity • Upper Coalescing: • Comonotonic restricted branch independence

6. Proof: From first to second lines, comonotonic restricted branch independence allows reduction of z’ on both sides. Next, we used upper coalescing on the right. The third line follows from transitivity.

7. Example Test of UTI

8. Generic Configural Model

9. CPT satisfies UTI Suppose CPT satisfies coalescing;

10. 2 Types of Violations: R’S1: This violates UTI. It refutes CPT, but this type predicted by TAX and RAM S’R1: This also violates UTI, not predicted by TAX/RAM

11. RAM/TAX Violations • RAM/TAX violate UTI. Coalescing branches with better consequences makes a gamble worse. By coalescing the two upper branches of R’, we made R1 relatively worse. Hence, RAM/TAX imply R’S1 violation.

12. Summary of Predictions • EU, CPT, lower GDU satisfy UTI • TAX & RAM violate UTI • Here CPT and GDU defend the null hypothesis against a specific prediction made by both RAM and TAX.

13. Birnbaum (‘05): n = 503 R’S1 signicantly more frequent than S’R1, predicted by TAX.

14. Studies of UTI • Wu, G. (1994). An empirical test of ordinal independence. Journal of Risk and Uncertainty, 9, 39-60. • Birnbaum, M. H. (2001). A Web-based program of research on decision making. In U.-D. Reips & M. Bosnjak (Eds.), Dimensions of Internet science (pp. 23-55). Lengerich, Germany: Pabst Science Publishers. • Birnbaum, M. H. (2005). Three new tests that differentiate models of risky decision making.

15. To Rescue CPT: • CPT cannot handle the results unless it becomes a configural model. • Wu (1994) tried to explain the results in terms of configural editing rules, in particular, cancellation.

16. Violations predicted by RAM & TAX, not CPT • EU, CPT, lower GDU are refuted by systematic violations of UTI. • TAX & RAM, as fit to previous data correctly predicted the modal choices. • Violations of UTI are to CPT as the Allais paradoxes are to EU.

17. Next Program: 4-DI • The next programs reviews tests of 4-Distribution Independence. • Violations of 4-DI contradict RAM and EU but are consistent with RDU, CPT. • Violations are opposite predictions of the inverse-S weighting function. • They are consistent with (and were predicted by) TAX.

18. 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.