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
Decumulative Weight
CPT Value (Utility) Function
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120 140
Objective Cash Value
Subjective Value
“Prior” TAX Model
Assumptions:
€
U(G) =Au(x) + Bu(y) + Cu(z)
A + B + C
€
A = t( p) −δt(p) /4 −δt(p) /4
B = t(q) −δt(q) /4 + δt(p) /4
C = t(1− p − q) + δt(p) /4 + δt(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
Transformed Probability
For 0 < x < $150u(x) = x Gives a decentapproximation.Risk aversion produced by
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 – q; y’, q;
z)Note that this recipe uses 4 distinct
values of consequences. It falls outside the probability simplex defined on three consequences.
CPT G, TAX FWe can test 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% repeat the violation (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 < 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 p S ⇒ ′ ′ R pp ′ ′ 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 f ′ S ⇒ ′ ′ ′ R ff ′ ′ ′ 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 F
+
G
−
f F
−
⇒
G f F
Notation
€
x1
< x2
< K < xn
< 0 ≤ ym
< K y2
< y1
€
G = ( x1
, p1
; x2
, p2
; K ; xn
, pn
; ym
, qm
; K ; y2
, q2
; y1
, q1
)
€
G
+
= ( 0 , pi
i = 1
n
∑ ; ym
, pm
; K ; y2
, q2
; y1
, q1
)
€
G
−
= ( x1
, p1
; x2
, p2
; K ; xn
, pn
; 0 , q
i = 1
m
∑i
)
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
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.
• TAX correctly predicts the violations of CPT. CPT implies violations of TAX that either fail or show the opposite pattern from predicted by CPT.