New Paradoxes of Risky Decision Making that Refute Prospect Theories

1
New Paradoxes of Risky Decision Making that
Refute Prospect Theories

• Michael H. Birnbaum
• Fullerton, California, USA

2
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.

3
Cumulative Prospect Theory/ Rank-Dependent
Utility (RDU)
4
Prior TAX Model

• Assumptions

5
TAX Parameters
For 0 lt x lt 150 u(x) x Gives a
decent approximation. Risk aversion produced by
d. d 1 .
6
Non-nested Models
7
CPT and TAX nearly identical inside the prob.
simplex
8
Testing CPT
TAXViolations of

• Coalescing
• Stochastic Dominance
• Lower Cum. Independence
• Upper Cumulative Independence
• Upper Tail Independence
• Gain-Loss Separability

9
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)

10
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 ? F
• We can test if violations due to error

11
Violations of Stochastic Dominance
122 Undergrads 59 repeat the violation (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
lt 0.4
12
Pie Charts
13
Aligned Table Coalesced
14
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)

15
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
16
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
17
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.
18
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

19
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

20
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

21
Gain-Loss Separability
22
Notation
23
Wu and Markle Result
24
Birnbaum Bahra-- F
25
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.

26
Summary-2
Property CPT RAM TAX
LCI No Viols Viols Viols
UCI No Viols Viols Viols
UTI No Viols RS1Viols RS1Viols
LDI RS2 Viols No Viols No Viols
3-2 LDI RS2 Viols No Viols No Viols
27
Summary-3
Property CPT RAM TAX
4-DI RSViols No Viols SR Viols
UDI SR2 Viols No Viols RS2 Viols
RBI RS Viols SR Viols SR Viols
28
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.

29
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.