Some New Approaches to Old
Problems: Behavioral Models
of Preference
Michael H. Birnbaum
California State University,
Fullerton
Testing Algebraic Models with
Error-Filled Data
• Algebraic models assume or imply
formal properties such as stochastic
dominance, coalescing, transitivity,
gain-loss separability, etc.
• But these properties will not hold if data
contain “error.”
Some Proposed Solutions
• Neo-Bayesian approach (Myung,
Karabatsos, & Iverson.
• Cognitive process approach
(Busemeyer)
• “Error” Theory (“Error Story”) approach
(Thurstone, Luce) combined with
algebraic models.
Variations of Error Models
• Thurstone, Luce: errors related to separation
between subjective values. Case V: SST
(scalability).
• Harless & Camerer: errors assumed to be
equal for certain choices.
• Today: Allow each choice to have a different
rate of error.
• Advantage: we desire error theory that is both
descriptive and neutral.
Basic Assumptions
• Each choice in an experiment has a true
choice probability, p, and an error rate,
e.
• The error rate is estimated from (and is
the “reason” given for) 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
Ex: Stochastic Dominance
: 05 tickets to win $12
B: 10 tickets to win $12
05 tickets to win $14
05 tickets to win $90
90 tickets to win $96
85 tickets to win $96
122 Undergrads: 59% repeated viols (BB)
28% Preference Reversals (AB or BA)
Estimates: e = 0.19; p = 0.85
170 Experts: 35% repeated violations
31% Reversals
Estimates: e = 0.196; p = 0.50
Chi-Squared test reject H0: p < 0.4
Testing 2, 3, 4-Choice
Properties
• Extending this model to properties using
2, 3, or 4 choices is straightforward.
• Allow a different error rate on each
choice.
• Allow a true probability for each choice
pattern.
Response Combinations
Notation
000
001
010
011
100
101
110
111
(A, B)
A
A
A
A
B
B
B
B
(B, C)
B
B
C
C
B
B
C
C
(C, A)
C
A
C
A
C
A
C
A
*
*
Weak Stochastic Transitivity
P(A
B)  1 2 & P(B
C)  1 2  P(A
C) 
P(A
B)  P(000)  P(001)  P(010)  P(011)
P(B
P(C
C)  P(000)  P(001)  P(100)  P(101)
A)  P(000)  P(010)  P(100)  P(110)
1
2
WST Can be Violated even
when Everyone is Perfectly
Transitive
P(001)  P(010)  P(100) 
P(A
B)  P(B
C)  P(C
1
3
A)  2 3
Model for Transitivity
P(000)  p000 (1 e1 )(1 e2 )(1 e3 )  p001 (1 e1 )(1 e2 )e3 
 p010 (1 e1 )e2 (1 e3 )  p011 (1 e1 )e2e3 
 p100e1 (1 e2 )(1 e3 )  p101e1 (1 e2 )e3 
 p110e1e2 (1 e3 )  p111e1e2e3
A similar expression is written for the
other seven probabilities. These can in turn be
expanded to predict the probabilities of showing
each pattern repeatedly.
Expand and Simplify
• There are 8 X 8 data patterns in an
experiment with 2 repetitions.
• However, most of these have very small
probabilities.
• Examine probabilities of each of 8 repeated
patterns.
• Probability of showing each of 8 patterns in
one replicate OR the other, but NOT both.
Mutually exclusive, exhaustive partition.
New Studies of Transitivity
• Work currently under way testing
transitivity under same conditions as
used in tests of other decision
properties.
• Participants view choices via the WWW,
click button beside the gamble they
would prefer to play.
Some Recipes being Tested
•
•
•
•
•
•
Tversky’s (1969) 5 gambles.
LS: Preds of Priority Heuristic
Starmer’s recipe
Additive Difference Model
Birnbaum, Patton, & Lott (1999) recipe.
New tests: Recipes based on Schmidt
changing utility models.
Priority Heuristic
• Brandstaetter, Gigerenzer, & Hertwig (in
press) model assumes people do NOT
weight or integrate information.
• Each decision based on one reason
only. Reasons tested one at a time in
fixed order.
Choices between 2-branch
gambles
• First, consider minimal gains. If the difference
exceeds 1/10 the maximal gain, choose best
minimal gain.
• If minimal gains not decisive, consider
probability; if difference exceeds 1/10, choose
best probability.
• Otherwise, choose gamble with the best
highest consequence.
Priority Heuristic Preds.
A: .5 to win $100
.5 to win $0
B: $40 for sure
Reason: lowest
consequence.
C: .02 to win $100
.98 to win $0
Reason: highest
consequence.
D: $4 for sure
Priority Heuristic Implies
• Violations of Transitivity
• Satisfies New Property: Priority Dominance.
Decision based on dimension with priority
cannot be overrulled by changes on other
dimensions.
• Satisfies Independence Properties: Decision
cannot be altered by any dimension that is
the same in both gambles.
Fit of PH to Data
• Brandstaetter, et al argue that PH fits
the data of Kahneman and Tversky
(1979) and Tversky and Kahneman
(1992) and other data better than does
CPT or TAX.
• It also fits Tversky’s (1969) violations of
transitivity.
Tversky Gambles
• Some Sample Data, using Tversky’s 5
gambles, but formatted with tickets instead of
pie charts.
• Data as of May 17, 2005, n = 251.
• No pre-selection of participants.
• Participants served in other studies, prior to
testing (~1 hr).
Three of Tversky’s (1969)
Gambles
• A = ($5.00, 0.29; $0, 0.79)
• C = ($4.50, 0.38; $0, 0.62)
• E = ($4.00, 0.46; $0, 0.54)
Priority Heurisitc Predicts:
A preferred to C; C preferred to E,
And E preferred to A. Intransitive.
Results-ACE
pattern
000
001
010
011
100
101
110
111
sum
Rep 1
10
11
14
7
16
4
176
13
251
Rep 2
21
13
23
1
19
3
154
17
251
Both
5
9
1
0
4
1
133
3
156
Test of WST
A
A
B
C
D
E
0.712
0.762
0.771
0.852
0.696
0.798
0.786
0.696
0.770
B
0.339
C
0.174
0.287
D
0.101
0.194
0.244
E
0.148
0.182
0.171
0.593
0.349
Comments
• Results are surprisingly transitive.
• Differences: no pre-test, selection;
• Probability represented by # of tickets (100
per urn);
• Participants have practice with variety of
gambles, & choices;
• Tested via Computer.
Response Patterns
Choice ( 0 = first;
1 = second)
($26,.1;$0)
($25,.1;$20)
L
P
H
1
L
H
P
1
P
L
H
1
P
H
L
1
H
L
P
1
H
P
L
1
T
A
X
1
($100,.1;$0)
($25,.1;$20)
1
1 1 0 0 0
1
($26,.99;$0)
($25,.99;$20)
1
1 1 1 1 1
1
($100,.99;$0)
($25,.99;$20)
1
1 1 0 0 0
0
Data Patterns (n = 260)
Frequency Both
Rep 1 or 2 not both
Est. Prob
0000
1
2.5
0.03
0001
0
4.5
0.02
0010
0
3.5
0.01
0011
0
1
0
0100
0
8.5
0
0101
4
16
0.02
0110
6
22
0.04
0111
98
42.5
0.80
1000
1
2.5
0.01
1001
0
0
0
1010
0
1
0
1011
0
.5
0
1100
0
.5
0
1101
0
6.5
0
1110
0
5
0
1111
9
24.5
0.06
Summary
• True & Error model with different error rates seems
a reasonable “null” hypothesis for testing transitivity
and other properties.
• Requires data with replications so that we can use
each person’s self-agreement or reversals to
estimate whether response patterns are “real” or
due to “error.”
• Priority Heuristic model’s predicted violations of
transitivity do not occur and its prediction of priority
dominance is violated.