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
• Pilot Experiment & Planned Work with
Enrico Diecidue
• Results: Pilot tests. Comments welcome
Integrative, Interactive
Contrast Models
n
A
B    (E i ) (ai ,bi )
i1
A  (a1, E1;a2 , E 2 ; ;an , E n )
B  (b1, E1;b2, E 2 ; ;bn , E n )
Assumptions
 (ai, bi )   (bi , ai )
 (ai, bi )  0  ai  bi
Difference Model :
 (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
1 if u(a)  u(b)

f [u(a)  u(b)]  0 if u(a)  u(b)

1 if u(a)  u(b)
Regret Model

f [u(a)  u(b)]  u(a)  u(b) , u(a)  u(b)
 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 Results
PATTERN
111
112
121
122
211
212
221
222
TOTAL
DATA
PREDICTIONS
One Rep not 2 Two Reps One not 2 two reps true probs
9.25
1
14.5
1.3
0.00
31.75
50.5
38.9
37.4
0.55
10.25
2.5
11.3
1.9
0.00
14.25
4.5
19.0
3.4
0.02
14.75
1.5
12.2
1.3
0.01
27.75
16
30.6
13.3
0.13
15.25
16
16.0
14.0
0.21
15.75
9
17.7
7.1
0.09
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
A
B

B
pe 2  (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)e 2

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 e
Estimating p
Testing if p = 0
A’B’C’ Results
PATTERN
111
112
121
122
211
212
221
222
TOTAL
DATA
PREDICTIONS
One Rep not 2 Two Reps One not 2 two reps
true probs
12.75
7
19.5
5.8
0.05
31.75
71.5
43.8
55.6
0.70
13.5
6
11.3
5.9
0.06
16.25
2
19.4
2.3
0.00
11.5
2
10.1
1.9
0.01
25.25
8
25.7
7.4
0.04
10.5
8
10.2
8.8
0.10
11.5
2.5
9.7
2.5
0.03
133
107
149.8
90.2
1
ABC X A’B’C’ Analysis
111
112
121
122
211
212
221
222
111
1.0
2.0
2.0
1.8
1.3
3.3
1.8
6.8
112
3.0
59.5
2.8
5.0
3.3
22.3
3.0
4.5
121
0.5
0.8
2.3
3.5
1.5
2.0
6.3
2.8
122
1.3
3.0
2.0
1.8
2.5
1.8
3.8
2.3
211
0.5
1.5
2.0
1.3
0.5
1.5
3.8
2.5
212
1.0
12.8
0.3
1.5
3.3
10.0
2.0
2.5
221
0.8
0.8
1.3
2.8
1.8
0.8
8.8
1.8
222
2.3
2.0
0.3
1.3
2.3
2.3
2.0
1.8
ABC-A’B’C’ Analysis
ABC-A'B'C'
PATTERN Est. true probs
111111
0.00
112112
0.59 TAX
121121
0.04
122122
0.01
211211
0.00
212212
0.08
221221
0.16
222222
0.02
222111
0.09 MR
111222
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.