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					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)

                                                6
            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
  branches leading to higher
  consequences to branches leading to
  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
                                         11
       “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.


                                          14
       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
             ,
                                                       15
        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

                                                     16
 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.
                                     17
Recipe for Violation of SD according to
              RAM/TAX




                                          18
       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


 70% of undergrads chose B
                                              19
  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




                          21
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).



                                       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)
                                                    24
  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.

                                         25
Analysis: SD in TAX model




                            26
        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.



                                          27
       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.
                                             28
                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.
                                               29
             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.


                                         30
                      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.




                                                                      31
       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.



                                                      32
        For More Information:
mbirnbaum@fullerton.edu
http://psych.fullerton.edu/mbirnbaum/


 Download recent papers from this site.
 Follow links to “brief vita” and then to
 “in press” for recent papers.


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