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,...
New Paradoxes of Risky Decision Making that Refute
Prospect Theories
Michael H. BirnbaumFullerton, California, USA
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.
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 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
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
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.
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.