Testing Lexicographic SemiOrder Models: Generalizing
the Priority Heuristic
Michael H. Birnbaum
California State University,
Fullerton
Outline
• Priority Heuristic for risky decisions
• New Critical Tests: Allow each person
to have a different LS with different
parameters
• Results for three tests: Interaction,
Integration, and Transitivity.
• Discussion and questions
Priority Heuristic
• Brandstätter, Gigerenzer, & Hertwig
• Consider one dimension at a time.
• If that “reason” is decisive, other
reasons not considered. No
integration; no interactions.
• Very different from utility models.
• Very similar to CPT.
Priority Heuristic
• Brandstätter, et al (2006) model
assumes people do NOT weight or
integrate information.
• Each decision based on one dimension
only.
• Only 4 dimensions considered.
• Order fixed: L, P(L), H, P(H).
PH for 2-branch gambles
• First: minimal gains. If the difference
exceeds 1/10 the (rounded) maximal gain,
choose by 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 examples
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
Example: Allais Paradox
C: .2 to win $50
.8 to win $2
D: .11 to win $100
.89 to win $2
E: $50 for sure
F: .11 to win $100
.8 to win $50
.09 to win $2
PH Reproduces Some Data…
Predicts 100% of modal choices in
Kahneman & Tversky, 1979.
Predicts 85% of choices in Erev, et al.
(1992)
Predicts 73% of Mellers, et al. (1992)
data
…But not all Data
•
•
•
•
Birnbaum & Navarrete (1998): 43%
Birnbaum (1999): 25%
Birnbaum (2004): 23%
Birnbaum & Gutierrez (in press): 30%
Problems
• No attention to middle branch, contrary to
results in Birnbaum (1999)
• Fails to predict stochastic dominance in
cases where people satisfy it in Birnbaum
(1999). Fails to predict violations when
70% violate stochastic dominance.
• Not accurate when EVs differ.
• No individual differences and no free
parameters. Different data sets have
different parameters. Delta > .12 & Delta <
.04.
Modifications:
• People act as if they compute ratio of EV
and choose higher EV when ratio > 2.
• People act as if they can detect stochastic
dominance.
• Although these help, they do not improve
model to more than 50% accuracy.
• Today: Suppose different people have
different LS with different parameters.
Family of LS
• In two-branch gambles, G = (x, p; y), there
are three dimensions: L = lowest outcome
(y), P = probability (p), and H = highest
outcome (x).
• There are 6 orders in which one might
consider the dimensions: LPH, LHP, PLH,
PHL, HPL, HLP.
• In addition, there are two threshold
parameters (for the first two dimensions).
New Tests of Independence
• Dimension Interaction: Decision should
be independent of any dimension that has
the same value in both alternatives.
• Dimension Integration: indecisive
differences cannot add up to be decisive.
• Priority Dominance: if a difference is
decisive, no effect of other dimensions.
Taxonomy of choice models
Transitive Intransitive
Interactive &
EU, CPT,
TAX
Integrative
Non-interactive & Additive,
Integrative
CWA
Not interactive or 1-dim.
integrative
Regret,
Majority Rule
Additive
Diffs, SD
LS, PH*
Testing Algebraic Models
with Error-Filled Data
• Models assume or imply formal properties
such as interactive independence.
• But these properties may not hold if data
contain “error.”
• Different people might have different
“true” preference orders, and different
items might produce different amounts of
error.
Error Model Assumptions
• Each choice pattern in an experiment
has a true probability, p, and each
choice has an error rate, e.
• The error rate is estimated from
inconsistency of response to the same
choice by same person over
repetitions.
Priority Heuristic Implies
• Violations of Transitivity
• Satisfies Interactive Independence:
Decision cannot be altered by any
dimension that is the same in both gambles.
• No Dimension Integration: 4-choice
property.
• Priority Dominance. Decision based on
dimension with priority cannot be overruled
by changes on other dimensions. 6-choice.
Dimension Interaction
Risky
Safe
TAX LPH HPL
($95,.1;$5)
($55,.1;$20)
S
S
R
R
S
R
($95,.99;$5) ($55,.99;$20)
Family of LS
• 6 Orders: LPH, LHP, PLH, PHL, HPL, HLP.
• There are 3 ranges for each of two
parameters, making 9 combinations of
parameter ranges.
• There are 6 X 9 = 54 LS models.
• But all models predict SS, RR, or ??.
Results: Interaction n = 153
Risky
Safe
%
Safe
Est. p
($95,.1;$5)
($55,.1;$20)
71%
.76
($95,.99;$5)
($55,.99;$20) 17%
.04
Analysis of Interaction
•
•
•
•
•
•
Estimated probabilities:
P(SS) = 0 (prior PH)
P(SR) = 0.75 (prior TAX)
P(RS) = 0
P(RR) = 0.25
Priority Heuristic: Predicts SS
Probability Mixture Model
• Suppose each person uses a LS on any
trial, but randomly switches from one
order to another and one set of
parameters to another.
• But any mixture of LS is a mix of SS,
RR, and ??. So no LS mixture model
explains SR or RS.
Dimension Integration Study
with Adam LaCroix
• Difference produced by one dimension
cannot be overcome by integrating
nondecisive differences on 2 dimensions.
• We can examine all six LS Rules for each
experiment X 9 parameter combinations.
• Each experiment manipulates 2 factors.
• A 2 x 2 test yields a 4-choice property.
Integration Resp. Patterns
Choice
Risky= 0
Safe = 1
($51,.5;$0)
($50,.5;$50)
L
P
H
1
L
P
H
1
L
P
H
0
H
P
L
1
H
P
L
1
H
P
L
0
T
A
X
1
($51,.5;$40)
($50,.5;$50)
1
0 0 1
1
0 1
($80,.5;$0)
($50,.5;$50)
1
1
0 0 1
0 1
($80,.5;$40)
($50,.5;$50)
1
0 0 0 1
0 0
54 LS Models
• Predict SSSS, SRSR, SSRR, or RRRR.
• TAX predicts SSSR—two
improvements to R can combine to
shift preference.
• Mixture model of LS does not predict
SSSR pattern.
Choice Percentages
Risky
Safe
($51,.5;$0)
($50,.5;$50)
93
($51,.5;$40) ($50,.5;$50)
82
($80,.5;$0)
($50,.5;$50)
79
($80,.5;$40) ($50,.5;$50)
19
% safe
Test of Dim. Integration
• Data form a 16 X 16 array of
response patterns to four choice
problems with 2 replicates.
• Data are partitioned into 16 patterns
that are repeated in both replicates
and frequency of each pattern in one
or the other replicate but not both.
Data Patterns (n = 260)
Pattern
Frequency Both
Rep 1
Rep 2
Est. Prob
0000
1
1
6
0.03
0001
1
1
6
0.01
0010
0
6
3
0.02
0011
0
0
0
0
0100
0
3
4
0.01
0101
0
1
1
0
0110 *
0
2
0
0
0111
0
1
0
0
1000
0
13
4
0
1001
0
0
1
0
1010 *
4
26
14
0.02
1011
0
7
6
0
1100 PHL, HLP,HPL *
6
20
36
0.04
1101
0
6
4
0
98
149
132
0.80
9
24
43
0.06
1110 TAX
1111 LPH, LHP, PLH *
Results: Dimension
Integration
• Data strongly violate independence
property of LS family
• Data are consistent instead with
dimension integration. Two small,
indecisive effects can combine to
reverse preferences.
• Replicated with all pairs of 2 dims.
New Studies of Transitivity
• LS models violate transitivity: A > B and B >
C implies A > C.
• Birnbaum & Gutierrez tested transitivity
using Tversky’s gambles, but using typical
methods for display of choices.
• Also used pie displays with and without
numerical information about probability.
Similar results with both procedures.
Replication of Tversky (‘69)
with Roman Gutierez
• Two studies used Tversky’s 5 gambles,
formatted with tickets instead of pie
charts. Two conditions used pies.
• Exp 1, 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,
But E preferred to A. Intransitive.
TAX (prior): E > C > A
Tests of WST (Exp 1)
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
Response Combinations
Notation
000
001
010
011
100
101
110
111
(A, C)
A
A
A
A
C
C
C
C
(C, E)
C
C
E
E
C
C
E
E
(E, A)
E
A
E
A
E
A
E
A
* PH
TAX
*
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
Results-ACE
pattern
000 (PH)
001
010
011
100
101
110 (TAX)
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
Comments
• Results were surprisingly transitive, unlike
Tversky’s data.
• Differences: no pre-test, selection;
• Probability represented by # of tickets
(100 per urn); similar results with pies.
• Participants have practice with variety of
gambles, & choices;
• Tested via Computer.
Summary
• Priority Heuristic model’s predicted
violations of transitivity are rare.
• Dimension Interaction violates any member
of LS models including PH.
• Dimension Integration violates any LS
model including PH.
• Data violate mixture model of LS.
• Evidence of Interaction and Integration
compatible with models like EU, CPT, TAX.