Testing Heuristic Models of Risky Decision Making

Michael H. BirnbaumCalifornia State University,

Fullerton

Outline• Priority Heuristic• New Critical Tests: Allow each

person to have a different LS with different parameters

• Four tests: Interaction, Integration, Transitivity, & Priority Dominance.

Priority Heuristic• Brandstätter, et al (2006) model

assumes people do NOT weight or integrate information.

• Examine dimensions in order• 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 better highest consequence.

Priority Heuristic examplesA: .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

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.

Modification:• 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*

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 InteractionRisky Safe TAX LP

HHPL

($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 0.030001 1 1 6 0.010010 0 6 3 0.020011 0 0 0 00100 0 3 4 0.010101 0 1 1 00110 * 0 2 0 00111 0 1 0 01000 0 13 4 01001 0 0 1 01010 * 4 26 14 0.021011 0 7 6 01100 PHL, HLP,HPL * 6 20 36 0.041101 0 6 4 01110 TAX 98 149 132 0.801111 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.

Three of Tversky’s (1969) Gambles

• A = ($5.00, 0.29; $0, 0.71)• C = ($4.50, 0.38; $0, 0.62)• E = ($4.00, 0.46; $0, 0.54)Priority Heurisitc Predicts: A > C; C > E, but E > 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

Results-ACEpattern Rep 1 Rep 2 Both000 (PH) 10 21 5001 11 13 9010 14 23 1011 7 1 0100 16 19 4101 4 3 1110 (TAX) 176 154 133111 13 17 3sum 251 251 156

Summary• Priority Heuristic’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.