Among those who cycle most have no regrets

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. 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 Majority Rule Prediction

11. 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 Regret Prediction

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

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.