New Paradoxes of Risky Decision Making that Refute Prospect Theories Michael H. Birnbaum Fullerton,...
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Transcript of New Paradoxes of Risky Decision Making that Refute Prospect Theories Michael H. Birnbaum Fullerton,...
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New Paradoxes of Risky Decision Making that Refute
Prospect Theories
Michael H. BirnbaumFullerton, California, USA
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Outline
• I will review tests between Cumulative Prospect Theory (CPT) and Transfer of Attention eXchange (TAX) model.
• Emphasis will be on critical properties that test between these two non-nested theories.
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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 )
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“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)
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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
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TAX CE with delta = 0
0
20
40
60
80
100
0.00 0.20 0.40 0.60 0.80 1.00
Probability to Win $100 0
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TAX: Effect of Delta
= 1; 1
0
20
40
60
80
100
0.0 0.2 0.4 0.6 0.8 1.0
Probability to win $100 0
delta = 1.0
delta = 0.5
delta = 0
delta = -0.5
delta = -1.0
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TAX Model
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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.
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Non-nested Models
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CPT and TAX nearly identical inside the prob. simplex
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Testing CPT
• Coalescing• Stochastic
Dominance• Lower Cum.
Independence• Upper
Cumulative Independence
• Upper Tail Independence
• Gain-Loss Separability
TAX:Violations of:
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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:
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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.”
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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
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Pie Charts
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Aligned Table: Coalesced
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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)
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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
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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
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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.
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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
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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
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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
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Gain-Loss Separability
G F
G F
G F
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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)
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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
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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
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Allais Paradox Dissection
Restricted Branch Independence
Coalescing Satisfied Violated
Satisfied EU, PT*,CPT* CPT
Violated PT TAX
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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.
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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
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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
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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.
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Implications
• Results indicate that neither PT nor CPT are descriptive of risky decision making. Editing rules of combination, cancellation, & dominance detection refuted.
• TAX correctly predicts the violations of CPT. CPT implies violations of TAX that either fail or show the opposite pattern from predicted by CPT.