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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.
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