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.