The Case Against Prospect
Theories of Risky Decision
Making
Michael H. Birnbaum
California State University,
Fullerton
1

My last time at UCSD: 1972-
73--photo by NHA

From Bernoulli (1738)
Exposition of a new theory on the measurement of risk
Bernoulli (1738) quotes from a 1728 letter from Gabriel Cramer to Nicolas
Bernoulli, addressing a problem ( St.
Petersburg paradox) Nicolas had posed in 1713 to Montmort:

In Exposition of a new theory on the measurement of risk, Daniel Bernoulli (1738)
Quotes Cramer (1728):
"You ask for an explanation of the discrepancy between the mathematical calculation and the vulgar evaluation... in their theory , mathematicians evaluate money in proportion to its quantity while, in practice , people with common sense evaluate money in proportion to the utility they can obtain from it”

Bernoulli (1738)
If a poor man had a lottery ticket that would pay 20,000 ducats or nothing with equal probability, he would NOT be ill-advised to sell it for 9,000 ducats. A rich man would be ill-advised to refuse to buy it for that price.

Expected Utility Theory
• Could explain why people would buy and sell gambles
• Explain sales and purchase of insurance
• Explain the St. Petersburg Paradox
• Explain risk aversion

Allais (1953) “Constant
Consequence” Paradox
Called “paradox” because preferences contradict Expected Utility.
A : $1M for sure f B : .10 to win $2M
.89 to win $1M
.01 to win $0
C : .11 to win $1M p D: .10 to win $2M
.89 to win $0 .90 to win $0


Expected Utility (EU) Theory
EU ( G )
 n  p i u ( x i i

1
)
A f B  u ($1M) > .10
u ($2M) + .89
u ($1M) +.01
u ($0)
Subtr. .89
u ($1M): .11
u ($1M) > .10
u ($2M)+.01
u ($0)
Add .89
u ($0): .11
u ($1M)+.89
u ($0) > .10
u ($2M)+.90
u ($0)
 C f D . So, Allais Paradox refutes EU.


0.8
0.6
0.4
0.2
0
0
1
Cumulative Prospect Theory/
Rank-Dependent Utility (RDU)
CPU ( G )
 n  i

1
[ W ( i  j

1 p j
)

W ( i

1  j

1 p j
) ] u ( x i
)
Probability Weighting
Function, W(P)
0.2
0.4
0.6
0.8
Decumulative Probability
1
140
120
100
80
60
40
20
0
0
CPT Value (Utility) Function
20 40 60 80 100 120 140
Objective Cash Value

Cumulative Prospect Theory/
RDU
• Tversky & Kahneman (1992) CPT is more general than EU or (1979) PT, accounts for risk-seeking, risk aversion, sales and purchase of gambles & insurance.
• Accounts for Allais Paradoxes, chief evidence against EU theory.
• Accounts for certain violations of restricted branch independence.
• Nobel Prize in Economics (2002)


RAM/TAX Models x
1
 x
2
  x i
  x n

0
RAMU ( G )
 n  i

1 a ( i , n ) t ( p i
) u ( x i n  i

1 a ( i , n ) t ( p i
)
)



RAM Model Parameters a (1, n )

1; a (2, n )

2; ; a ( i , n )
 i ; ; a ( n , n )
 n
1
0.8
Probability Weighting
Function, t(p)
0.6
0.4
0.2
0
0 0.2
0.4
0.6
0.8
Objective Probability, p
1
140
120
100
80
60
40
Utility (Value) Function
20
0
0 20 40 60 80 100 120 140
Objective Cash Value ($)

RAM implies inverseS
100
Certainty Equivalents of
($100, p; $0)
80
60
40
20
0
0 0.2
0.4
0.6
0.8
Probability to Win $100
1

Allais “Constant Consequence”
Paradox
Can be analyzed to compare CPT vs
RAM/TAX
A : $1M for sure f B : .10 to win $2M
.89 to win $1M
.01 to win $0
C : .11 to win $1M p D: .10 to win $2M
.89 to win $0 .90 to win $0




Allais Paradox Analysis
• Transitivity:
A f B and B f C

A f C
• Coalescing:
GS = (x, p; x, q; z, r) ~ G = (x, p + q; z, r)
• Restricted Branch Independence:
 S

( x , p ; y , q ; z , r ) R

( x

, p ; y

, q ; z , r )
S
 
( x , p ; y , q ; z

, r ) R
 
( x

, p ; y

, q ; z

, r )

A : $1M for sure f B : .10 to win $2M
.89 to win $1M
.01 to win $0
A ’ : .10 to win $1M f B : .10 to win $2M
.89 to win $1M .89 to win $1M
.01 to win $1M .01 to win $0
A ” : .10 to win $1M f B ’ : .10 to win $2M
.89 to win $0 .89 to win $0
.01 to win $1M .01 to win $0
C : .11 to win $1M f D : .10 to win $2M
.89 to win $0 .90 to win $0

Decision Theories and Allais
Paradox
Branch Independence
Coalescing Satisfied Violated
Satisfied
Violated
EU, CPT*
OPT*
RDU, CPT*
SWU, OPT* RAM, TAX

Kahneman (2003)
“…Our model implied that ($100, .01; $100, .01)
— two mutually exclusive .01 chances to gain $100 — is more valuable than the prospect ($100, .02)… most decision makers will spontaneously transform the former prospect into the latter and treat them as equivalent in subsequent operations of evaluation and choice. To eliminate the problem, we proposed that decision makers, prior to evaluating the prospects, perform an editing operation that collects similar outcomes and adds their probabilities. ”

Web-Based Research
• Series of Studies tests: classical and new paradoxes in decision making.
• People come on-line via WWW (some in lab).
• Choose between gambles; 1 person per month (about 1% of participants) wins the prize of one of their chosen gambles.
• Data arrive 24-7; sample sizes are large; results are clear.

Choice data TAX CE
S R
% R S R
15 red $50
85 black $7
10 red $50
05 blue $50
85 white $7
10 blue $100
90 white $7
10 black $100
05 purple $7
85 green $7
80*
49
14 < 18
16 > 15
85 red $100
10 white $50
05 blue $50
85 black $100
15 yellow $50
85 black $100
10 yellow $100
05 purple $7
63* 68 < 70
95 red $100
05 white $7
20* 76 > 62

Allais Paradoxes
• Do not require large, hypothetical prizes.
• Do not depend on consequence of $0.
• Do not require choice between “sure thing” and 3-branch gamble.
• Largely independent of event-framing
• Best explained as violation of coalescing
(violations of BI run in opposition).
• See JMP 2004, 48 , 87-106.


Stochastic Dominance
If the probability to win x or more given
A is greater than or equal to the corresponding probability given gamble B , and is strictly
Higher for at least one x , we say that A
Dominates B by First Order Stochastic Dominance.
P ( x
 t | A )

P ( x
 t | B )
 t

A B


Preferences Satisfy Stochastic
Dominance
Liberal Standard:
If A stochastically dominates B ,
P ( A B )

1
2
Reject only if Prob of choosing
B is signficantly greater than 1/2.

RAM/TAX

Violations of Stochastic
Dominance

Which gamble would you prefer to play?
Gamble A
90 reds to win $96
05 blues to win $14
05 whites to win $12
Gamble B
85 reds to win $96
05 blues to win $90
10 whites to win $12
70% of undergrads choose B

Which of these gambles would you prefer to play?
Gamble C
85 reds to win $96
05 greens to win $96
05 blues to win $14
05 whites to win $12
Gamble D
85 reds to win $96
05 greens to win $90
05 blues to win $12
05 whites to win $12
90% choose C over D

RAM/TAX

Violations of
Stochastic Dominance

Violations of Stochastic Dominance
Refute CPT/RDU, predicted by
RAM/TAX
Both RAM and TAX models predicted this violation of stochastic dominance prior to the experiment, using parameters fit to other data.
These models do not violate Consequence monotonicity).

Questions
• How “often” do RAM/TAX models predict violations of Stochastic
Dominance?
• Are these models able to predict anything?
• Is there some format in which CPT works?

Do RAM/TAX models imply that people should violate stochastic dominance?
Rarely. Only in special cases.
Consider “random” 3-branch gambles:
*Probabilities ~ uniform from 0 to 1.
*Consequences ~ uniform from $1 to $100.
Consider pairs of random gambles. 1/3 of choices involve Stochastic Dominance, but only 1.8 per 10,000 are predicted violations by TAX. Random study of
1,000 trials would unlikely have found such violations by chance. (Odds: 7:1 against)

Can RAM/TAX account for anything?
• No. These models are forced to predict violations of stochastic dominance in the special recipe, , given the facts that people are (a) risk-seeking for small p and (b) risk-averse for medium to large p in two-branch gambles .

Analysis: SD in TAX model

Formats:
Birnbaum & Navarrete (1998)
.05 .05 .90
$12 $14 $96
.10 .05 .85
$12 $90 $96

Birnbaum & Martin (2003)
I: .05 to win $12 J: .10 to win $12
.05 to win $14 .05 to win $90
.90 to win $96 .85 to win $96

Web Format (1999b)
5. Which do you choose?
I: .05 probability to win $12
.05 probability to win $14
.90 probability to win $96
OR
J: .10 probability to win $12
.05 probability to win $90
.85 probability to win $96

Reversed Order
5. Which do you choose?
 I: .90 probability to win $96
.05 probability to win $14
.05 probability to win $12
OR
 J: .85 probability to win $96
.05 probability to win $90
.10 probability to win $12

Pie Charts

Tickets Format
 I : 90 tickets to win $96
05 tickets to win $14
05 tickets to win $12
OR
 J : 85 tickets to win $96
05 tickets to win $90
10 tickets to win $12

List Format
I: $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96
$14
$12
OR
J: $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96, $96
$90
$12, $12

Semi-Split List
I: $96, $96, $96, $96, $96, $96, $96, $96, $96, $96,
$96, $96, $96, $96, $96, $96, $96, $96
$14
$12
OR
J: $96, $96, $96, $96, $96, $96, $96, $96, $96, $96,
$96, $96, $96, $96, $96, $96, $96
$90
$12, $12

Marbles: Event-Framing
5. Which do you choose?
 I: 90 red marbles to win $96
05 blue marbles to win $14
05 white marbles to win $12
OR
 J: 85 red marbles to win $96
05 blue marbles to win $90
10 white marbles to win $12

Decumulative Probability
Format
5. Which do you choose?
 I: .90 probability to win $96 or more
.95 probability to win $14 or more
1.00 probability to win $12 or more
OR
 J: .85 probability to win $96 or more
.90 probability to win $90 or more
1.00 probability to win $12 or more

Another Test of Coalescing
Gamble A
90 reds to win $96
05 blues to win $12
05 blues to win $12
Gamble B
85 reds to win $96
05 reds to win $96
10 blues to win $12
Here coalescing

A = B, but 67% of 503 Judges chose B.

G – = ($96,.85 – r; $90,.05; $12,.1 + r)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.2
0.3
Observed
Pred-TAX
Pred_RAM
0.4
0.5
0.6
0.7
Probability to win $96 in G–
0.8
0.9

G – = ($96, .85 – r; $90, .05 + r; $12, .1)
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.1
0.2
Observed
Pred_TAX
Pred_RAM
0.3
0.4
0.5
0.6
Probability to win $96 in G–
0.7
0.8
0.9

Summary: 23 Studies of SD,
8653 participants
• Huge effects of splitting vs. coalescing of branches
• Small effects of education, gender, study of decision science
• Very small effects of probability format
• Miniscule effects of event framing
(framed vs unframed)

Case against CPT/RDU
• Violations of Stochastic Dominance
• Violations of Coalescing (Event-Splitting)
• Violations of 3-Upper Tail Independence
• Violations of Lower Cumulative Independence
• Violations of Upper Cumulative Independence


Upper Cumulative
Independence
R

S
  
S

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%
.10 to win $10
.90 to win $98
S''': 66%
.20 to win $40
.80 to win $98


R
Lower Cumulative
Independence
S
 
S

R: 39%
.90 to win $3
.05 to win $12
.05 to win $96
S: 61%
.90 to win $3
.05 to win $48
.05 to win $52
R'': 69%
.95 to win $12
.05 to win $96
S'': 31%
.90 to win $12
.10 to win $52

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.

More Evidence against
CPT/RDU/RSDU
• Violations of Restricted Branch Independence are opposite predictions of inverseS weighting function used to explain Allais
Paradoxes.
• Violations of 4-distribution independence, 3-
LDI, 3-UDI favor RAM over TAX --also opposite of predictions of CPT with inverse-
S .

Restricted Branch Indep.
S
’
: .1 to win $40
.1 to win $44
.8 to win $100
R
’
: .1 to win $10
.1 to win $98
.8 to win $100
S: .8 to win $2
.1 to win $40
.1 to win $44
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
R
’
: .10 to win $4
.10 to win $96
.80 to win $100
S2
’
: .45 to win $40
.45 to win $44
.10 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

For More Information: mbirnbaum@fullerton.edu
http://psych.fullerton.edu/mbirnbaum/
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