Testing Critical Properties of Models of Risky Decision Making Michael H. Birnbaum Fullerton,...

Testing Critical Properties of Models of Risky Decision

Making

Michael H. BirnbaumFullerton, California, USA

Sept. 13, 2007 Luxembourg

Outline

• I will discuss critical properties that test between nonnested theories such as CPT and TAX.

• I will also discuss properties that distinguish Lexicographic Semiorders and family of transitive integrative models (including CPT and TAX).

Cumulative Prospect Theory/ Rank-Dependent Utility

(RDU)

Probability Weighting Function, W(P)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Decumulative Probability

Decu

mu

lati

ve W

eig

ht

CPT Value (Utility) Function

0

20

40

60

80

100

120

140

0 20 40 60 80 100 120 140

Objective Cash Value

Su

bje

ctiv

e V

alu

e

CPU(G ) [W ( pj ) W ( pj )j1

i 1

j1

i

i1

n

]u(xi )

“Prior” TAX Model

Assumptions:

U(G) Au(x) Bu(y) Cu(z)

A B C

A t( p) t(p) /4 t(p) /4

B t(q) t(q) /4 t(p) /4

C t(1 p q) t(p) /4 t(q) /4

G (x, p;y,q;z,1 p q)

TAX Parameters

Probability transformation, t(p)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Probability

Tra

nsf

orm

ed P

rob

abil

ity

For 0 < x < $150u(x) = x Gives a decentapproximation.Risk aversion produced by

TAX Model

TAX and CPT nearly identical for binary (two-branch) gambles

• CE (x, p; y) is an inverse-S function of p according to both TAX and CPT, given their typical parameters.

• Therefore, there is no point trying to distinguish these models with binary gambles.

Non-nested Models

CPT and TAX nearly identical inside the prob. simplex

Testing CPT

• Coalescing• Stochastic

Dominance• Lower Cum.

Independence• Upper

Cumulative Independence

• Upper Tail Independence

• Gain-Loss Separability

TAX:Violations of:

Testing TAX Model

• 4-Distribution Independence (RS’)

• 3-Lower Distribution Independence

• 3-2 Lower Distribution Independence

• 3-Upper Distribution Independence (RS’)

• Res. Branch Indep (RS’)

CPT: Violations of:

Stochastic Dominance

• A test between CPT and TAX:G = (x, p; y, q; z) vs. F = (x, p – s; y’, s;

z)Note that this recipe uses 4 distinct

consequences: x > y’ > y > z > 0; outside the probability simplex defined on three consequences.

CPT choose G, TAX choose FTest if violations due to “error.”

Violations of Stochastic Dominance

: 05 tickets to win $12

05 tickets to win $14

90 tickets to win $96

B: 10 tickets to win $12

05 tickets to win $90

85 tickets to win $96

122 Undergrads: 59% two violations (BB) 28% Pref Reversals (AB or BA) Estimates: e = 0.19; p = 0.85170 Experts: 35% repeat violations 31% Reversals Estimates: e = 0.20; p = 0.50 Chi-Squared test reject H0: p violations < 0.4

Pie Charts

Aligned Table: Coalesced

Summary: 23 Studies of SD, 8653 participants

• Large effects of splitting vs. coalescing of branches

• Small effects of education, gender, study of decision science

• Very small effects of probability format, request to justify choice.

• Miniscule effects of event framing (framed vs unframed)

Lower Cumulative Independence

R: 39% S: 61% .90 to win $3 .90 to win $3 .05 to win $12 .05 to win $48 .05 to win $96 .05 to win $52

R'': 69% S'': 31%.95 to win $12 .90 to win $12.05 to win $96 .10 to win $52

R S R S

Upper Cumulative Independence

R': 72% S': 28% .10 to win $10 .10 to win $40 .10 to win $98 .10 to win $44 .80 to win $110 .80 to win $110

R''': 34% S''': 66% .10 to win $10 .20 to win $40 .90 to win $98 .80 to win $98

R S R S

Summary: UCI & LCI

22 studies with 33 Variations of the Choices, 6543 Participants, & a variety of display formats and procedures. Significant Violations found in all studies.

Restricted Branch Indep.

S’: .1 to win $40

.1 to win $44 .8 to win $100

S: .8 to win $2 .1 to win $40 .1 to win $44

R’: .1 to win $10

.1 to win $98 .8 to win $100

R: .8 to win $2 .1 to win $10 .1 to win $98

3-Upper Distribution Ind.

S’: .10 to win $40

.10 to win $44 .80 to win $100

S2’: .45 to win $40

.45 to win $44 .10 to win $100

R’: .10 to win $4

.10 to win $96 .80 to win $100

R2’: .45 to win $4

.45 to win $96 .10 to win $100

3-Lower Distribution Ind.

S’: .80 to win $2 .10 to win $40 .10 to win $44

S2’: .10 to win $2 .45 to win $40 .45 to win $44

R’: .80 to win $2 .10 to win $4 .10 to win $96

R2’: .10 to win $2 .45 to win $4 .45 to win $96

Gain-Loss Separability

G F

G F

G F

Notation

x1 x2 xn 0 ym y2 y1

G (x1,p1;x2 ,p2;;xn ,pn;ym ,qm;;y2 ,q2;y1,q1)

G (0, pii1

n ;ym ,pm;;y2 ,q2;y1,q1)

G (x1,p1;x2 ,p2;;xn ,pn;0, qi1

mi)

Wu and Markle ResultG F % G TAX CPT

G+: .25 chance at $1600

.25 chance at $1200

.50 chance at $0

F+: .25 chance at $2000

.25 chance at $800

.50 chance at $0

72 551.8 >

496.6

551.3 <

601.4

G-: .50 chance at $0

.25 chance at $-200

.25 chance at $-1600

F-: .50 chance at $0

.25 chance at $-800

.25 chance at $-1000

60 -275.9>

-358.7

-437 <

-378.6

G: .25 chance at $1600

.25 chance at $1200

.25 chance at $-200

.25 chance at $-1600

F: .25 chance at $2000

.25 chance at $800

.25 chance at $-800

.25 chance at $-1000

38 -300 <

-280

-178.6 <

-107.2

Birnbaum & Bahra--% FChoice % G Prior TAX Prior CPT

G F G F G F

25 black to win $100

25 white to win $0

50 white to win $0

25 blue to win $50

25 blue to win $50

50 white to win $0

0.71 14 21 25 19

50 white to lose $0

25 pink to lose $50

25 pink to lose $50

50 white to lose $0

25 white to lose $0

25 red to lose $100

0.65 -21 -14 -20 -25

25 black to win $100

25 white to win $0

25 pink to lose $50

25 pink to lose $50

25 blue to win $50

25 blue to win $50

25 white to lose $0

25 red to lose $100

0.52 -25 -25 -9 -15

25 black to win $100

25 white to win $0

50 pink to lose $50

50 blue to win $50

25 white to lose $0

25 red to lose $100

0.24 -15 -34 -9 -15

Allais Paradox Dissection

Restricted Branch Independence

Coalescing Satisfied Violated

Satisfied EU, PT*,CPT* CPT

Violated PT TAX

Summary: Prospect Theories not Descriptive

• Violations of Coalescing • Violations of Stochastic Dominance• Violations of Gain-Loss Separability• Dissection of Allais Paradoxes: viols

of coalescing and restricted branch independence; RBI violations opposite of Allais paradox.

Summary-2

Property CPT RAM TAX

LCI No Viols Viols Viols

UCI No Viols Viols Viols

UTI No Viols R’S1Viols R’S1Viols

LDI RS2 Viols No Viols No Viols

3-2 LDI RS2 Viols No Viols No Viols

Summary-3

Property CPT RAM TAX

4-DI RS’Viols No Viols SR’ Viols

UDI S’R2’

ViolsNo Viols R’S2’

Viols

RBI RS’ Viols SR’ Viols SR’ Viols

Results: CPT makes wrong predictions for all 12 tests

• Can CPT be saved by using different formats for presentation? More than a dozen formats have been tested.

• Violations of coalescing, stochastic dominance, lower and upper cumulative independence replicated with 14 different formats and thousands of participants.

Lexicographic Semiorders

• Renewed interest in issue of transitivity.• Priority heuristic of Brandstaetter,

Gigerenzer & Hertwig is a variant of LS, plus some additional features.

• In this class of models, people do not integrate information or have interactions such as the probability x prize interaction in family of integrative, transitive models (EU, CPT, TAX, and others)

LPH LS: G = (x, p; y) F = (x’, q; y’)

• If (y –y’ > ) choose G• Else if (y ’- y > ) choose F• Else if (p – q > ) choose G• Else if (q – p > ) choose F• Else if (x – x’ > 0) choose G• Else if (x’ – x > 0) choose F• Else choose randomly

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).

Non-Nested Models

TAX, CPTTAX, CPTLexicographicLexicographic SemiordersSemiorders

Intransitivity

Allais Paradoxes

Priority Dominance

Integration

Interaction

Transitive

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 (2007) 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 Gutierrez

• 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(A B) P(B C) 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.