Among those who cycle most have no regrets

Michael H. BirnbaumDecision 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• Pilot Experiment & Planned Work with

Enrico Diecidue• Results: Pilot tests. Comments welcome

Integrative, Interactive Contrast Models

€

AfB⇔ φ(Ei)ψ(ai,bi)i=1

n∑A=(a1,E1;a2,E2;K;an,En)B=(b1,E1;b2,E2;K;bn,En)

Assumptions

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ψ(ai,bi)=−ψ(bi,ai)ψ(ai,bi)=0⇔ai=biDifferenceModel:ψ(ai,bi)=f[u(ai)−u(bi)]

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

€

f[u(a)−u(b)]=1 ifu(a)>u(b)0 ifu(a)=u(b)−1 ifu(a)<u(b)

⎡

⎣ ⎢ ⎢ ⎢

Regret Model

€

f [u(a) − u(b)] = u(a) − u(b)β, u(a) > u(b)

β >1

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

• 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

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

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 ResultsDATA PREDICTIONS

PATTERN One Rep not 2Two Reps One not 2 two reps true probs111 9.25 1 14.5 1.3 0.00112 31.75 50.5 38.9 37.4 0.55121 10.25 2.5 11.3 1.9 0.00122 14.25 4.5 19.0 3.4 0.02211 14.75 1.5 12.2 1.3 0.01212 27.75 16 30.6 13.3 0.13221 15.25 16 16.0 14.0 0.21222 15.75 9 17.7 7.1 0.09

TOTAL 139 101 160.2 79.8 1.00

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

A B

A

B€

pe2

+ ( 1 − p )( 1 − e )2

p ( 1 − e ) e + ( 1 − p )( 1 − e ) e

p ( 1 − e ) e + ( 1 − p )( 1 − e ) e

€

p ( 1 − e )2

+ ( 1 − p ) e2

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:

( 1 − e ) e

Estimating eProbability of Reversals in Repeated Choice

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

Error Rate (e)

Estimating p

Observed = P(1 - e)(1 - e)+(1 - P)ee

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

True Choice Probabiity, P

Error Rate = 0

Error Rate = .02

Error Rate = .04

Error Rate = .06

Error Rate = .08

Error Rate = .10

Error Rate = .12

Error Rate = .14

Error Rate = .16

Error Rate = .18

Error Rate = .20

Error Rate = .22

Error Rate = .24

Error Rate = .26

Error Rate = .28

Error Rate = .30

Error Rate = .32

Error Rate = .34

Error Rate = .36

Error Rate = .38

Error Rate = .40

Error Rate = .42

Error Rate = .44

Error Rate = .46

Error Rate = .48

Error Rate = .50

Testing if p = 0

Test if P = 0

0

0.1

0.2

0 0.1 0.2 0.3 0.4 0.5

Probability of Reversals 2e(1 - e)

A’B’C’ ResultsDATA PREDICTIONS

PATTERN One Rep not 2 Two Reps One not 2 two reps true probs

111 12.75 7 19.5 5.8 0.05

112 31.75 71.5 43.8 55.6 0.70

121 13.5 6 11.3 5.9 0.06

122 16.25 2 19.4 2.3 0.00

211 11.5 2 10.1 1.9 0.01

212 25.25 8 25.7 7.4 0.04

221 10.5 8 10.2 8.8 0.10

222 11.5 2.5 9.7 2.5 0.03

TOTAL 133 107 149.8 90.2 1

ABC X A’B’C’ Analysis111 112 121 122 211 212 221 222

111 1.0 3.0 0.5 1.3 0.5 1.0 0.8 2.3112 2.0 59.5 0.8 3.0 1.5 12.8 0.8 2.0121 2.0 2.8 2.3 2.0 2.0 0.3 1.3 0.3122 1.8 5.0 3.5 1.8 1.3 1.5 2.8 1.3211 1.3 3.3 1.5 2.5 0.5 3.3 1.8 2.3212 3.3 22.3 2.0 1.8 1.5 10.0 0.8 2.3221 1.8 3.0 6.3 3.8 3.8 2.0 8.8 2.0222 6.8 4.5 2.8 2.3 2.5 2.5 1.8 1.8

ABC-A’B’C’ AnalysisABC-A'B'C'PATTERN Est. true probs

111111 0.00112112 0.59 TAX121121 0.04122122 0.01211211 0.00212212 0.08221221 0.16222222 0.02222111 0.09 MR111222 0.02 Regret

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

• 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

• 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

• 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.