Among those who cycle most have no regrets
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Among those who cycle most have no regrets. Michael H. Birnbaum Decision Research Center, Fullerton. Outline. Family of Integrative Contrast Models Special Cases: Regret Theory, Majority Rule (aka Most Probable Winner) Predicted Intransitivity: Forward and Reverse Cycles

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Among those who cycle most have no regrets

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Among those who cycle most have no regrets

Among those who cycle most have no regrets

Michael H. Birnbaum

Decision Research Center, Fullerton


Outline

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


Integrative interactive contrast models

Integrative, Interactive Contrast Models


Assumptions

Assumptions


Special cases

Special Cases

  • Majority Rule (aka Most Probable Winner)

  • Regret Theory

  • These can be represented with different functions. I will illustrate with different functions, f.


Majority rule model

Majority Rule Model


Regret model

Regret Model


Predicted intransitivity

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.”


Concrete example

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.


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

Majority Rule Prediction


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

Regret Prediction


Pilot test

Pilot Test

  • 240 Undergraduates

  • Tested via computers (browser)

  • Clicked button to choose

  • 30 choices (includes counterbalanced choices)

  • 10 min. task, 30 choices repeated.


Abc design results

ABC Design Results


True and error model assumptions

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


One choice two repetitions

One Choice, Two Repetitions


Solution for e

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:


Estimating e

Estimating e


Estimating p

Estimating p


Testing if p 0

Testing if p = 0


A b c results

A’B’C’ Results


Abc x a b c analysis

ABC X A’B’C’ Analysis


Abc a b c analysis

ABC-A’B’C’ Analysis


Results

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.


Results continued

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?


Advice welcome our plans

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.


Summary

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.


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