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Stochastic Dominance
Michael H. BirnbaumDecision Research CenterCalifornia State University,
Fullerton
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SD is not only normative, but is assumed or implied
by many descriptive theories.
We can test the property to test between the class of models that satisfies and the class that violates this property.
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CPT satisfies SD
• CPT, RDU, RSDU, EU and many other models satisfy first order stochastic dominance.
• RAM, TAX, GDU, OPT, and others violate the property.
• We can test between two classes of theories by testing SD.
• Design studies to test specific predictions by RAM/TAX models.
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SD is an acceptable normative principle
• It is hard to construct a convincing argument that anyone should violate SD.
• Understanding when and why violations occur has both practical and theoretical value.
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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
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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.
• Implies certain violations of restricted branch independence.
• Shared Nobel Prize in Econ. (2002)
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RAM Model
€
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
∑
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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
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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
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TAX Model
• TAX, like RAM, assumes that weight is affected by probability by a power function, t(p) = p.
• Weight is also transferred from branches leading to higher consequences to branches leading to lower consequences.
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Special TAX Model
€
G = (x, p;y,q;z,1− p − q)
U(G) =Au(x) + Bu(y) + Cu(z)
A + B + CA = 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
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“Prior” TAX Model
€
u(x) = x; 0 < x < $150
t( p) = pγ ; γ = 0.7
δ =1 (Model rewritten so that = 1
here is the same as = –1 from previous version).
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TAX also implies inverse-S
TAX model Certainty Equivalents of ($100, p; $0)
0
10
20
30
40
50
60
70
80
90
100
0 0.2 0.4 0.6 0.8 1
Probability to Win $100
Calculated Certainty Equivalent
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Recipe for Violations
• In 1996, I was asked to show that the “configural weight models” are different from other rank-dependent models.
• Derived some tests, including UCI and LCI, published them in 1997.
• Juan Navarrete and I then set out to test these predictions.
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Analysis of Stochastic Dominance
• 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)• Consequence Monotonicity:
€
′ G = (x, p;y,q; ′ z ,r) f G = (x, p;y,q;z,r); ′ z f z
′ ′ G = (x, p; ′ y ,q;z,r) f G; ′ y f y
′ ′ ′ G = ( ′ x , p;y,q;z,r) f G; ′ x f x
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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
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Preferences Satisfy Stochastic Dominance
Liberal Standard: If A stochastically dominates B,
€
P(A f B) ≥ 12
Reject only if Prob to choose B is signficantly greater than 1/2.
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Recipe for Violation of SD according to RAM/TAX
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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 chose B
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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
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RAM/TAX Violations of Stochastic Dominance
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Violations of Stochastic Dominance Refute CPT/RDU, predicted by RAM/TAX
Both RAM and TAX models predicted this violation of stochastic dominance in advance of experiments, using parameters fit to TK 92 data. These models do not violate Consequence monotonicity).
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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?
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Do RAM/TAX models imply that people always 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)
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Can RAM/TAX account for anything?
• No. These models are forced to predict violations of stochastic dominance in the special recipe, given these properties:
• (a) risk-seeking for small p and • (b) risk-averse for medium to large
p in two-branch gambles.
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Analysis: SD in TAX model
TAX Model
-20
0
20
-1 0 1
Value of
= 2
= 1
= .85
= .7
= .6
= .5
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Coalescing and SD
• Birnbaum (1999): 62% of sample of 124 undergraduates violated SD in the coalesced choice AND satisfied it in the split version of the same choice.
• It seems that coalescing is the principle that fails, causing violations.
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Transparent Coalescing
Gamble A Gamble B
90 red to win $9605 white to win $1205 blue to win $12
85 green to win $9605 yellow to win $9610 orange to win $12
Here coalescing A = B, but 67% of 503 Judges chose B.
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Comment
• It is sometimes argued that EU theory is as good as CPT, if not better, for 3-branch gambles.
• However, this conclusion stems from research inside the Marshak-Machina triangle, where there are only 3 possible consequences.
• This recipe for violations of SD requires 4 distinct consequences. This test is outside the triangle.
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Summary
Violations of First Order Stochastic Dominance refute the CPT model, as well as many other models propsed as descriptive of DM.
Violations were predicted by RAM/TAX models and confirmed by experiment.
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Papers on SD• Birnbaum, M. H. (1997). Violations of monotonicity in
judgment and decision making. In A. A. J. Marley (Eds.), Choice, decision, and measurement: Essays in honor of R. Duncan Luce (pp. 73-100). Mahwah, NJ: Erlbaum.
• Birnbaum, M. H., & Navarrete, J. B. (1998). Testing descriptive utility theories: Violations of stochastic dominance and cumulative independence. Journal of Risk and Uncertainty, 17, 49-78.
• Birnbaum, M. H., Patton, J. N., & Lott, M. K. (1999). Evidence against rank-dependent utility theories: Violations of cumulative independence, interval independence, stochastic dominance, and transitivity. Organizational Behavior and Human Decision Processes, 77, 44-83.
• Birnbaum, M. H. (1999b). Testing critical properties of decision making on the Internet. Psychological Science, 10, 399-407.
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Next Program: Formats
• The next program asks whether there is some format for presenting choices that strongly reduces violations of CPT.
• It will turn out that violations are substantial in all formats, and that coalescing/splitting has a big effect in all of the formats studied, contrary to CPT.
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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.
mbirnbaum@fullerton.edu
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