Testing Critical Properties of
Models of Risky Decision
Making
Michael H. Birnbaum
Fullerton, 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)
n
i
i 1
i 1
j 1
j 1
CPU(G )  [W ( pj ) W ( pj )]u(xi )
1
140
Probability Weighting
Function, W(P)
120
Subjective Value
Decumulative Weight
0.8
CPT Value (Utility) Function
0.6
0.4
100
80
60
40
0.2
20
0
0
0
0.2
0.4
0.6
0.8
Decumulative Probability
1
0
20
40
60
80
100
120
Objective Cash Value
140
“Prior” TAX Model
Assumptions:
G  (x, p;y,q;z,1 p  q)
Au(x)  Bu(y)  Cu(z)
U(G) 
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
TAX Parameters
1
For 0 < x < $150
u(x) = x
Gives a decent
approximation.
Risk aversion
produced by .
Transformed Probability
Probability transformation, t(p)
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
Probability
0.8
1
1.
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
TAX:Violations of:
• Coalescing
• Stochastic
Dominance
• Lower Cum.
Independence
• Upper
Cumulative
Independence
• Upper Tail
Independence
• Gain-Loss
Separability
Testing TAX Model
CPT: Violations of:
• 4-Distribution
Independence (RS’)
• 3-Lower Distribution
Independence
• 3-2 Lower Distribution
Independence
• 3-Upper Distribution
Independence (RS’)
• Res. Branch Indep (RS’)
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 F
Test if violations due to “error.”
Violations of Stochastic Dominance
: 05 tickets to win $12
B: 10 tickets to win $12
05 tickets to win $14
05 tickets to win $90
90 tickets to win $96
85 tickets to win $96
122 Undergrads: 59% two violations (BB)
28% Pref Reversals (AB or BA)
Estimates: e = 0.19; p = 0.85
170 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
S  R
S 
R:
39%
.90 to win $3
.05 to win $12
.05 to win $96
S: 61%
.90 to win $3
.05 to win $48
.05 to win $52
R'': 69%
.95 to win $12
.05 to win $96
S'': 31%
.90 to win $12
.10 to win $52
Upper Cumulative
Independence
R S  R
S 
R': 72%
.10 to win $10
.10 to win $98
.80 to win $110
S':
28%
.10 to win $40
.10 to win $44
.80 to win $110
R''': 34%
.10 to win $10
.90 to win $98
S''': 66%
.20 to win $40
.80 to win $98
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
R’: .1 to win $10
.1 to win $44
.8 to win $100
S:
.8 to win $2
.1 to win $40
.1 to win $44
.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
R’: .80 to win $2
.10 to win $4
.10 to win $96
S2’: .10 to win $2
.45 to win $40
.45 to win $44
R2’: .10 to win $2
.45 to win $4
.45 to win $96
Gain-Loss Separability
G

G

F

F


G
F
Notation
x1  x 2 
 xn  0  ym 
y 2  y1
G  (x1 ,p1;x2 ,p2; ;xn ,pn;ym ,qm ; ;y2 ,q2;y1 ,q1)
G  (0, pi ;ym ,pm ; ;y2 ,q2;y1 ,q1 )

n
i 1
G  (x1 ,p1;x2 ,p2; ;xn ,pn ;0, q )

m
i 1 i
G
+
G :
-
G :
G:
Wu and Markle Result
.25 chance at $1600
F
+
F : .25 chance at $2000
.25 chance at $1200
.25 chance at $800
.50 chance at $0
.50 chance at $0
.50 chance at $0
F : .50 chance at $0
-
.25 chance at $-200
.25 chance at $-800
.25 chance at $-1600
.25 chance at $-1000
.25 chance at $1600
F:
.25 chance at $2000
.25 chance at $1200
.25 chance at $800
.25 chance at $-200
.25 chance at $-800
.25 chance at $-1600
.25 chance at $-1000
%G
TAX
CPT
72
551.8 >
551.3 <
496.6
601.4
-275.9>
-437 <
-358.7
-378.6
-300 <
-178.6 <
-280
-107.2
60
38
Birnbaum & Bahra--% F
Cho ice
G
%G
F
25 black to win $100
25 blue to win $50
25 white to win $0
25 blue to win $50
50 white to win $0
50 white to win $0
50 white to lose $0
50 white to lose $0
25 pink to lose $50
25 white to lose $0
25 pink to lose $50
25 red to lose $100
25 black to win $100
25 blue to win $50
25 white to win $0
25 blue to win $50
25 pink to lose $50
25 white to lose $0
25 pink to lose $50
25 red to lose $100
25 black to win $100
50 blue to win $50
25 white to win $0
25 white to lose $0
50 pink to lose $50
25 red to lose $100
Prior TAX
Prior CPT
G
G
F
F
0. 7 1
14
21
25
19
0. 6 5
-2 1
-1 4
-2 0
-2 5
0. 5 2
-2 5
-2 5
-9
-1 5
0. 2 4
-1 5
-3 4
-9
-1 5
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’
Viols
No 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’ > D) choose G
Else if (y ’- y > D) 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, CPT
Integration
Allais Paradoxes Lexicographic
Semiorders
Intransitivity
Interaction
Transitive
Priority Dominance
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 &
EU, CPT,
TAX
Integrative
Non-interactive & Additive,
Integrative
CWA
Not interactive or 1-dim.
integrative
Regret,
Majority Rule
Additive
Diffs, SD
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
R
S
R
($95,.99;$5) ($55,.99;$20)
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
($51,.5;$0)
($50,.5;$50)
L
P
H
1
L
P
H
1
L
P
H
0
H
P
L
1
H
P
L
1
H
P
L
0
T
A
X
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
($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
% safe
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 Both
Rep 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
98
149
132
0.80
9
24
43
0.06
1110 TAX
1111 LPH, LHP, PLH *
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
A
B
C
D
E
0.712
0.762
0.771
0.852
0.696
0.798
0.786
0.696
0.770
B
0.339
C
0.174
0.287
D
0.101
0.194
0.244
E
0.148
0.182
0.171
0.593
0.349
Response Combinations
Notation
000
001
010
011
100
101
110
111
(A, C)
A
A
A
A
C
C
C
C
(C, E)
C
C
E
E
C
C
E
E
(E, A)
E
A
E
A
E
A
E
A
* PH
TAX
*
WST Can be Violated even
when Everyone is Perfectly
Transitive
P(001)  P(010)  P(100) 
P(A
B)  P(B
C)  P(C
1
3
A)  2 3
Results-ACE
pattern
000 (PH)
001
010
011
100
101
110 (TAX)
111
sum
Rep 1
10
11
14
7
16
4
176
13
251
Rep 2
21
13
23
1
19
3
154
17
251
Both
5
9
1
0
4
1
133
3
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.