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.