1. Among those who cycle most have no regrets Michael H. Birnbaum Decision Research Center, Fullerton

2. Outline Family of Integrative Contrast Models Special Cases: Regret Theory, Majority Rule (aka Most Probable Winner) Predicted Intransitivity: Forward and Reverse Cycles Pilot Experiment & Planned Work with Enrico Diecidue Results: Pilot tests. Comments welcome

3. Integrative, Interactive Contrast Models

4. Assumptions

5. Special Cases Majority Rule (aka Most Probable Winner) Regret Theory These can be represented with different functions. I will illustrate with different functions, f.

6. Majority Rule Model

7. Regret Model

8. Predicted Intransitivity These models violate transitivity of preference Regret and MR cycle in opposite directions However, both REVERSE cycle under permutation over events; i.e., “juxtaposition.”

9. Concrete Example Urn: 33 Red, 33White, 33 Blue One marble drawn randomly Prize depends on color drawn. A = ($4, $5, $6) means win $4 if Red, win $5 if White, $6 if Blue.

10. Majority Rule Prediction A = ($4, $5, $6) B = ($5, $7, $3) C = ($9, $1, $5) AB: choose B BC: choose C CA: choose A Notation: 222 A’ = ($6, $4, $5) B’ = ($5, $7, $3) C’ = ($1, $5, $9) A’B’: choose A’ B’C’: choose B’ C’A’: choose C’ Notation: 111

11. Regret Prediction A = ($4, $5, $6) B = ($5, $7, $3) C = ($9, $1, $5) AB: choose A BC: choose B CA: choose C Notation: 111 A’ = ($6, $4, $5) B’ = ($5, $7, $3) C’ = ($1, $5, $9) A’B’: choose B’ B’C’: choose C’ C’A’: choose A’ Notation: 222

12. Pilot Test 240 Undergraduates Tested via computers (browser) Clicked button to choose 30 choices (includes counterbalanced choices) 10 min. task, 30 choices repeated.

14. ABC Design Results

15. True and Error Model Assumptions Each choice in an experiment has a true choice probability, p, and an error rate, e. The error rate is estimated from inconsistency of response to the same choice by same person over repetitions

16. One Choice, Two Repetitions AB A B

17. Solution for e The proportion of preference reversals between repetitions allows an estimate of e. Both off-diagonal entries should be equal, and are equal to:

18. Estimating e

19. Estimating p

20. Testing if p = 0

21. A’B’C’ Results

22. ABC X A’B’C’ Analysis

23. ABC-A’B’C’ Analysis

24. Results Most people are transitive. Most common pattern is 112, pattern predicted by TAX with prior parameters. However, 2 people were perfectly consistent with MR on 24 choices. No one fit Regret theory perfectly.

25. Results: Continued Among those few (est. ~10%) who cycle (intransitive), most have no regrets (i.e., they appear to satisfy MR). Suppose 5-10% of participants are intransitive. Do we think that they indeed use a different process? Is there an artifact in the experiment? If not, can we increase the rate of intransitivity?

26. Advice Welcome: Our Plans We plan to test participants from the same pool was used to elicit regret function. Assignment: Devise a theorem of integrative interactive contrast model that will lead to self-contradiction (“paradox” of regret theory). These contrast models also imply RBI, which is refuted by our data.

27. Summary Regret and MR imply intransitivity whose direction can be reversed by permutation of the consequences. Very few people are intransitive but a few do indeed appear to be consistent with MR and 2 actually show the pattern in 24 choices.