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The Case Against Prospect Theories of Risky Decision
Making
Michael H. BirnbaumCalifornia State University,
Fullerton
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 ) = pii =1
n
∑ u(xi )
A B u($1M) > .10u($2M) + .89u($1M) +.01u($0) Subtr. .89u($1M): .11u($1M) > .10u($2M)+.01u($0)
Add .89u($0): .11u($1M)+.89u($0) > .10u($2M)+.90u($0)
C D. So, Allais Paradox refutes EU.
Cumulative Prospect Theory/ Rank-Dependent
Utility (RDU)
€
CPU(G ) = [W ( pj )− W ( pj )j =1
i −1
∑j =1
i
∑i =1
n
∑ ]u(xi )
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
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
€
x1 > x2 > K > xi > K > xn > 0
€
RAMU(G ) =
a( i,n)t( pi )u(xi )i =1
n
∑
a( i,n)t( pi )i =1
n
∑
RAM Model Parameters
Probability Weighting Function, t(p)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1Objective Probability, p
€
a(1,n) = 1; a(2,n) = 2;K ; a( i,n) = i;K ; a(n ,n) = n
RAM implies inverse-SCertainty Equivalents of
($100, p; $0)
0
20
40
60
80
100
0 0.2 0.4 0.6 0.8 1
Probability to Win $100
Certainty Equivalent
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 B and B C A 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) f R = ( ′ x , p; ′ y ,q;z,r)
€
′ S = (x, p;y,q; ′ z ,r) f ′ R = ( ′ x , p; ′ y ,q; ′ z ,r)
€
⇔
A: $1M for sure B: .10 to win $2M.89 to win $1M
.01 to win $0
A’: .10 to win $1M 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 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 D: .10 to win $2M .89 to win $0 .90 to win $0
Decision Theories and Allais Paradox
Branch Independence
Coalescing Satisfied Violated
Satisfied EU, CPT*OPT*
RDU, CPT*
Violated 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 blue $100
90 white $7
80* 14 < 18
10 red $50
05 blue $50
85 white $7
10 black $100
05 purple $7
85 green $7
49 16 > 15
85 red $100
10 white $50
05 blue $50
85 black $100
10 yellow $100
05 purple $7
63* 68 < 70
85 black $100
15 yellow $50
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 f B
Preferences Satisfy Stochastic Dominance
Liberal Standard: If A stochastically dominates B,
€
P(A f B) ≥ 12
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 Gamble B
90 reds to win $9605 blues to win $1405 whites to win $12
85 reds to win $9605 blues to win $9010 whites to win $12
70% of undergrads choose B
Which of these gambles would you prefer to play?
Gamble C Gamble D
85 reds to win $9605 greens to win $9605 blues to win $1405 whites to win $12
85 reds to win $9605 greens to win $9005 blues to win $1205 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
TAX Model
-20
0
20
-1 0 1
Value of δ
γ = 2
γ = 1
γ = .85
γ = .7
γ = .6
γ = .5
Formats:Birnbaum & Navarrete
(1998)
.05 .05 .90 .10 .05 .85$12 $14 $96 $12 $90 $96
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
Birnbaum & Martin (2003)
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 ORJ: $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 Gamble B
90 reds to win $9605 blues to win $1205 blues to win $12
85 reds to win $9605 reds to win $9610 blues to win $12
Here coalescing A = B, but 67% of 503 Judges chose B.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Probability to win $96 in G–
Proportion of Violations of SD
Observed
Pred-TAX
Pred_RAM
G– = ($96,.85 – r; $90,.05; $12,.1 + r)
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Probability to win $96 in G–
Proportion of Violations of SD
Observed
Pred_TAX
Pred_RAM
G– = ($96, .85 – r; $90, .05 + r; $12, .1)
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': 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
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
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 inverse-S 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
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
For More Information:
http://psych.fullerton.edu/mbirnbaum/
Download recent papers from this site. Follow links to “brief vita” and then to “in press” for recent papers.
