# SD by wulinqing

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• pg 1
```									Stochastic Dominance

Michael H. Birnbaum
Decision Research Center
California State University,
Fullerton
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.

2
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.
3
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.

4
Cumulative Prospect Theory/
Rank-Dependent Utility (RDU)
n         i                                               i 1
CPU(G )  [W ( pj ) W ( pj )]u(xi )
i 1       j 1                                             j 1
1

Probability Weighting                                               140
CPT Value (Utility) Function
Function, W(P)
0.8                                                                           120
Decumulative Weight

100

Subjective Value
0.6
80

0.4                                                                            60

40
0.2
20

0                                                                              0
0     0.2      0.4      0.6    0.8        1                                   0    20    40    60   80    100    120   140
Decumulative Probability                                                         Objective Cash Value                5
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      xi           xn  0
n

 a(i,n)t( p )u(x )
i       i

RAMU(G)      i 1
n

 a(i,n)t( p )i
i 1

7
RAM Model Parameters
a(1,n)  1; a(2,n)  2;                                               ; a(i,n)  i; ; a(n,n)  n
1
Probability Weighting                                  140
Utility (Value) Function
Probability Weight, t(p)

Function, t(p)
0.8                                                              120

Subjective Utility
100
0.6
80

0.4                                                              60

40
0.2
20

0                                                                0
0     0.2   0.4   0.6   0.8     1                                0   20   40   60   80   100 120 140

Objective Probability, p                                         Objective Cash Value (\$)

8
RAM implies inverse-S
100

Certainty Equivalents of
(\$100, p; \$0)
80
Certainty Equivalent

60

40

20

0
0      0.2     0.4      0.6        0.8   1
Probability to Win \$100             9
TAX Model
• TAX, like RAM, assumes that weight is
affected by probability by a power
function, t(p) = pg.
• Weight is also transferred from
lower consequences.

10
Special TAX Model
G  (x, p;y,q;z,1 p  q)
Au(x)  Bu(y)  Cu(z)
U(G) 
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
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“Prior” TAX Model

u(x)  x; 0  x  \$150
g
t( p)  p ; g  0.7
  1 (Model rewritten so that = 1 here
is the same as  = –1 from previous
version).
12
TAX also implies inverse-S
100
TAX model Certainty Equivalents of (\$100, p; \$0)
90
Calculated Certainty Equivalent

80

70

60

50

40

30

20

10

0
0       0.2         0.4      0.6        0.8        1
Probability to Win \$100                  13
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) G  (x, p;y,q;z,r); z  z

G  (x, p; y ,q;z,r) G; y  y


G   ( x  p;y,q;z,r) G; x  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               B

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Preferences Satisfy Stochastic
Dominance
Liberal Standard:
If A stochastically dominates B,

P(A         B)     1
2
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 \$96     85 reds to win \$96
05 blues to win \$14    05 blues to win \$90
05 whites to win \$12   10 whites to win \$12

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Which of these gambles would
you prefer to play?
Gamble C               Gamble D

85 reds to win \$96     85 reds to win \$96
05 greens to win \$96   05 greens to win \$90
05 blues to win \$14    05 blues to win \$12
05 whites to win \$12   05 whites to win \$12

90% choose C over D                      20
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
fit to TK 92 data. These models do not
violate Consequence monotonicity).

22
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?

23
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

26
Coalescing and SD
• Birnbaum (1999): 62% of sample of 124
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 \$96     85 green to win \$96
05 white to win \$12   05 yellow to win \$96
05 blue to win \$12    10 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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