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Testing Lexicographic Semi-Order 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 - PowerPoint PPT Presentation
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Testing Lexicographic Semi-Order Models:
Generalizing the Priority Heuristic
Michael H. BirnbaumCalifornia 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 sureReason: lowest consequence.
C: .02 to win $100 .98 to win $0Reason: 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 & Integrative
EU, CPT, TAX
Regret, Majority Rule
Non-interactive & Integrative
Additive,CWA
Additive Diffs, SD
Not interactive or integrative
1-dim. 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
($95,.99;$5) ($55,.99;$20) R S R
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
LPH
LPH
LPH
HPL
HPL
HPL
TAX
($51,.5;$0) ($50,.5;$50) 1 1 0 1 1 0 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 % 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
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
BothRep 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
1110 TAX 98 149 132 0.80
1111 LPH, LHP, PLH * 9 24 43 0.06
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 B C D E
A 0.712 0.762 0.771 0.852
B 0.339 0.696 0.798 0.786
C 0.174 0.287 0.696 0.770
D 0.101 0.194 0.244 0.593
E 0.148 0.182 0.171 0.349
Response CombinationsNotation (A, C) (C, E) (E, A)
000 A C E * PH
001 A C A
010 A E E
011 A E A
100 C C E
101 C C A
110 C E E TAX
111 C E A *
WST Can be Violated even when Everyone is Perfectly
Transitive
P(001) P(010) P(100) 13
P(AB) P(BC) P(C A) 23
Results-ACEpattern Rep 1 Rep 2 Both
000 (PH) 10 21 5
001 11 13 9
010 14 23 1
011 7 1 0
100 16 19 4
101 4 3 1
110 (TAX) 176 154 133
111 13 17 3
sum 251 251 156
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.