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.