1. 1 Restricted Branch Independence Michael H. Birnbaum California State University, Fullerton

2. 2 RBI is Violated by CPT EU satisfies RBI as does SWU and PT, extended to 3-branch gambles. Cancellation CPT violates RBI (it MUST to explain the Allais Paradoxes) RAM and TAX violate RBI in the opposite direction as CPT.

3. 3 In this test, we move the common branch from lowest, z, to highest, z ’ consequence.

4. 4 Restricted Branch Independence (3-RBI)

5. 5 Types of Branch Independence The term “restricted” is used to indicate that the number of branches and probability distribution is the same in all four gambles. When we further constrain z and z ’ to keep the same ranks in all four gambles, it is termed “comonotonic” (restricted) branch independence.

6. 6 A Special Case We can make a still more restricted case of restricted branch independence, in order to test the predictions of any weakly inverse-S weighting function. Let p = q. This distribution has been used in most, but not all of the studies.

7. 7 Example Test S:.80 to win $2.10 to win $40.10 to win $44 R:.80 to win $2.10 to win $4.10 to win $96 S’:.10 to win $40.10 to win $44.80 to win $100 R’:.10 to win $4.10 to win $96.80 to win $100

8. 8 Generic Configural Model The generic model includes RDU, CPT, EU, RAM, TAX, GDU, as special cases.

9. 9 Violation of 3-RBI A violation will occur if S  R and

10. 10 2 Types of Violations: SR ’ : RS ’ :

11. 11 EU allows no violations In EU, the weights equal the probabilities; therefore

12. 12 RAM Weights

13. 13 RAM Violations RAM model violates 3-RBI.

14. 14 CPT/ RDU

15. 15

16. 16 CPT implies RS ’ violation If W(P) = P, CPT reduces to EU. However, if W(P) is any weakly inverse- S function, CPT implies the RS ’ pattern. (A strongly inverse- S function is weakly inverse- S plus it crosses the identity line. If we reject weak, then we reject the strong as well.)

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18. 18 Transfer of Attention Exchange (TAX) Each branch (p, x) gets weight that is a function of branch probability Utility is a weighted average of the utilities of the consequences on branches. Attention (weight) is drawn from one branch to others. In a risk-averse person, weight is transferred to branches with lower consequences.

19. 19 “Special” TAX Model Assumptions:

20. 20 “Prior” TAX Model Parameters were chosen to give a rough approximation to Tversky & Kahneman (1992) data. They are used to make new predictions.

21. 21 TAX Model Weights Each term has the same denominator; middle branch gives up what it receives when p = q.

22. 22 Special TAX: SR ’ Violations Special TAX model violates 3-RBI when delta is not zero.

23. 23 Summary of Predictions EU, SWU, OPT satisfy RBI CPT violates RBI: RS ’ TAX & RAM violate RBI: SR ’ Here CPT is the most flexible model, RAM and TAX make opposite prediction from that of CPT.

24. 24 Results: n = 1075 SR ’ (CPT predicted RS ’ )

25. 25 Lab Studies of RBI Birnbaum & McIntosh (1996): 2 studies, n = 106; n = 48, p = 1/3 Birnbaum & Chavez (1997): n = 100; 3-RBI and 4-RBI, p =.25 Birnbaum & Navarrete (1998): 27 tests; n = 100; p =.25, p =.1. Birnbaum, Patton, & Lott (1999): n = 110; p =.2. Birnbaum (1999): n = 124; p =.1, p =.05.

26. 26 Web Studies of RBI Birnbaum (1999): n = 1224; p =.1, p =.05 Birnbaum (2004b): 12 studies with total of n = 3440 participants; different formats for presenting gambles probabilities; p =.1,.05. Birnbaum (2004a): 3 conditions with n = 350; p =.1. Tests combined with Allais paradox.

27. 27 Additional Replications SR ’ pattern is significantly more frequent than RS ’ pattern in judgment studies as well. (Birnbaum & Beeghley, 1997; Birnbaum & Veira, 1998; Birnbaum & Zimmermann, 1999). A number of as yet unpublished studies have also replicated the basic findings with a variety of different procedures in choice.

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29. 29 Error Analysis We can fit “true and error” model to data with replications to separate “real” violations from those attributable to “error”. Model estimates that SR ’ violations are “real” and probability of RS ’ is equal to zero.

30. 30 Violations predicted by RAM & TAX, not CPT EU, SWU, OPT are refuted in this case by systematic violations. Editing “cancellation” refuted. TAX & RAM, as fit to previous data correctly predicted the modal choices. Violations opposite those implied by CPT with its inverse- S W(P) function. Fitted CPT correct when it agrees with TAX, wrong otherwise.

31. 31 To Rescue CPT: CPT can handle the result of any single test, by choosing suitable parameters. For CPT to handle these data, let  > 1; i.e., an S -shaped W(P) function, contrary to previous inverse- S.

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33. 33 Adds to the case against CPT/RDU/RSDU Violations of RBI as predicted by TAX and RAM but are opposite predictions of CPT. Maybe CPT is right but its parameters are just wrong. As we see in the next program, we can generate internal contradiction in CPT.

34. 34 Next Program: LCI The next programs reviews tests of Lower Cumulative Independence (LCI). Violations of 3-LCI contradict any form of RDU, CPT. They also refute EU but are consistent with RAM and TAX.

35. 35 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