Post on 23-Jun-2020
VIOLATIONS OF STOCHASTIC DOMINANCE IN MEDICAL DECISION MAKING
A Thesis Proposal
Presented to the
Faculty of
California State University, Fullerton
In Partial Fulfillment
of the Requirements for the Degree
Master of Arts
in
Psychology
By
Jeff Bahra
Approved by:
Michael Birnbaum, Committee Chair DateDepartment of Psychology Melinda Blackman DateDepartment of Psychology June Havlena DateDepartment of Psychology
Introduction
Violations of Stochastic Dominance in medical Decision Making
How do people make decisions in a world in which the consequences of one’s
actions cannot be known for certain? For example, a person might be asked to decide
between accepting $45 in cash or to draw a ticket from an urn, knowing that the prize
would depend on the ticket drawn. Suppose the urn has 50 tickets that win a prize of
$100 and 50 tickets result in no gain ($0). Most people choose the sure cash ($45), rather
than the gamble, even though the gamble has a higher expected value ($50). This study
looks at different decision making theories and why individuals will make one choice
over another.
Before 1738, it was considered that a rational person should choose between
gambles according to their expected values. The expected value (EV) of a gamble with n
mutually exclusive and exhaustive possible outcomes can be written:
(1)
where and are the probability and prize for outcome i. (Birnbaum, 2001)
For many years, Expected Utility (EU) theory was regarded as both a normative
theory of how people should choose between risky prospects and a descriptive theory of
how people actually do make decisions. For gambles consisting of chances
(probabilities) have a recovery of cash prizes, expected utility can be written:
(1)
where represents a gamble have a recovery of consequence
with probability , and is the utility of winning a cash prize of x dollars.
“Under expected utility, risk aversion (preferring expected value of a prospect to the
prospect itself) holds if and only if [the utility function, u] is “concave.” Concave means
that it is increasing at a progressively slower pace so curved downward, as in the figure”
(Wakker, 2004). This means that as an individual acquires more and more money the
acquisition of each additional dollar adds less utility. (Bernoulli, 1738). Bernoulli
suggested u(x) = log(x), but also mentioned Cramer’s idea that u(x) = x^.5.
In the 1950s, a number of “paradoxes” were created that show that EU theory is
not a descriptive theory of how people actually make decisions. Many of these
paradoxes, including those by Allais and by Ellsberg, were reviewed by Kahneman and
Tversky (1979) in a paper that became the most widely cited paper in the journal,
Econometrika. “The Allais Paradox is perhaps the most widely cited piece of evidence
on decision making under uncertainty, not because it is a large piece of evidence, but
rather because it is a contrary piece of evidence. It is the most famous of a number of
examples that many economists would like to embrace.” (Conlisk, 1989)
“Formal models of decision making under risk can be found in three disciplinary
guises. Until quite recently, almost all economists believed that decision makers
both should and do select risks that maximize expected utility. In contrast,
investment professionals have seen investors as selecting portfolios that achieve
an optimal balance between risk and return. Psychologists, too, have explored
expected utility theory and portfolio theory as possible descriptive models, and
they have also developed original information processing models focused on how
people choose rather than what people choose” (Lopes & Oden, 1999).
For the most part all three disciplines psychology, economics and investment
professionals have not really worked together until the last 30 years (Lopes & Oden,
1999). “Both psychologists and economists have been exploring a non-linear
modification of the expected utility model that we term the “decumulatively weighted
utility” model” (Lopes & Oden, 1999). RDU ….Quiggin, Luce & Narens….
One of the most widely cited articles in the economics literature is the one by
Kahneman and Tversky (1979) on prospect theory (Laury & Holt, 2000), the newer form
of which is known as cumulative prospect theory (CPT), which was recognized in the
2002 Nobel Prize in Economics. As noted by Laury and Holt (2000), “This theory is
motivated by laboratory experiments and can be thought of as a model of the decision
process. A key observation is that decision making begins by identifying a reference
point, often the current wealth position, from which people tend to be risk averse for
gains and risk loving for losses.” (Markowitz, 1952? Edwards, 1954) used the reference
point of status quo.
Cumulative Prospect theory can be written as follows:
(2)
where and are the probability of a loss being equal to or worse (lower) than
and strictly lower than , respectively; and are the probabilities of winning a
prize of or more and strictly more than , respectively. Is the utility (called
“value”) of the gamble in CPT. CPT Theory accounts for the Allais paradoxes and other
evidence cited against EU by Tversky and Kahneman (1979) and Tversky and Kahneman
(1992).
Convergence of ideas: Luce and Narens, Luce & Fishburn, 1991; 1995; Starmer
& Sugden, Quiggin, Schmeidler, Yaari, Wakker… Birnbaum and Stegner had a
configural weight model, rank-affected model 1979, with similarities to the RDU later
introduced. Success. Wakker, Camerer, Starmer (JEL, 2000), Wu and Gonzalez (2004?)
….summary papers concluding that CPT ….Nobel prize here.
Despite its success, CPT has come under fire. Birnbaum (1999; 2004a; 2004b;
2005) has summarized eleven “new paradoxes” that refute CPT in the same way that the
Allais paradoxes refuted EU. This paper will look at violations of stochastic dominance
and compare CPT theory against transfer of attention exchange (TAX) theory. In The
special TAX model (Birnbaum & Chavez, 1997), all weight transfers are the same
proportion of probability weight as follows:
(3)
where the configural weight parameter is . Different values of are permitted for
gambles configured of all negative or mixed consequences. If = 0, and t (p) = p, this
model reduces to EU. The effect of is to transfer weight from one branch to another,
representing the attention directed to different branches (Birnbaum & Bahra). “To explain
risk aversion in the TAX model, it is assumed that weight is taken from branches with
higher valued consequence and given to branches with lower valued consequences”
(Birnbaum, 2004). In this paper, we use previously estimated parameters from Birnbaum
(1999a), , = 1, u(x) = x, for .
The TAX model describes how the relative weight of each probability-
consequence branch depends on probability and rank of its consequence… In this model,
splitting the branch with the highest consequence can make a gamble better and splitting
the branch with the lowest consequence makes a gamble worse (Birnbaum, 2004). This
study will compare predictions of the TAX model against the CPT model. This will also
test a property known as stochastic dominance (SD).
“The TAX model can account for a variety of phenomena in risky decision
making, and including the original and extended forms of Allais common consequence
paradoxes, common ratio paradox, violations of restricted branch independence,
violations of lower and upper cumulative independence, violations of coalescing and
violations of first order stochastic dominance” (Birnbaum, 1999a; 1999b; 2004a; 2004b;
2005). “ These findings have been called “new paradoxes” because they violate CPT in
the same sense that Allais violates EU. Furthermore, TAX fits majority choices in
studies of different phenomena, obtained under uniform conditions, using the same
parameters”. (Birnbaum, Bahra, 2005).
“Constant consequence paradoxes can be decomposed into three simpler
properties: transitivity, coalescing, and restricted branch independence (Birnbaum,
1999a). If people satisfied these three properties, they would not show Allais paradoxes.
These properties are defined as follows:
Transitivity, assumed in all of the models discussed here, holds that A B and B
C A C, where denotes the preference relation.
Coalescing is the assumption that if a gamble has two or more (probability-
consequence) branches yielding identical consequences, those branches can be combined
by adding their probabilities. For example, if , a gamble
with two branches to win $100 and otherwise win zero, then ,
where ~ denotes indifference and is the coalesced form of G. Violations of coalescing
combined with transitivity are termed event-splitting effects (Humphrey, 1995; Starmer &
Sugden, 1993; Birnbaum, 1999a; 1999b). For example, if and , we say there is
an event-splitting effect. Assuming transitivity, event-splitting effects (branch-splitting
effects) are violations of coalescing.
Upper and lower coalescing are defined as follows. Let ,
where .
Upper coalescing assumes:
Lower coalescing assumes:
.
When the term “coalescing” is used without qualification, it refers to the assumption of
all forms of coalescing. Several of these experiments will examine upper and lower
coalescing properties separately.
Restricted Branch independence is the assumption that if two gambles have a
common probability-consequence (or event-consequence) branch, one can change the
common consequence without affecting the preference induced by the other components.
For three-branch gambles with p + q + r = 1, , restricted branch
independence can be written as follows for all consequences and
all consequences, , with no other restrictions on their order
(1)
The term “restricted” refers to the constraint that the numbers of branches and probability
distributions are the same in all four gambles of Expression 1. Restricted branch
independence is weaker than Savage’s (1954) “sure thing” axiom. When the rank orders
of all consequences are also restricted to be the same in all four gambles, the property is
called comonotonic restricted branch independence.
Analysis of Allais Paradox
If choices satisfy transitivity, coalescing, and restricted branch independence then
they would not show the constant consequence paradoxes of Allais. For example,
consider the following series of choices:
A: $1M for sure B: .10 to win $2M.89 to win $1M.01 to win $0
(coalescing & transitivity)
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
(restricted branch independence)
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
(coalescing & transitivity)
C: .11 to win $1M D: .10 to win $2M.89 to win $0 .90 to win $0
From the first to second step, A is converted to a split form, ; should be indifferent
to A by coalescing, and by transitivity, should be preferred to B. From the second to
third steps, the consequence on the common branch (.89 to win $1M) has been changed
to $0 on both sides, so by restricted branch independence, should be preferred to .
By coalescing branches with the same consequences on both sides, we see that C should
be preferred to D. However, many people choose A over B and D over C; indeed, this
behavior is the Allais paradox, so we know that at least one of these three assumptions
must be false.
All of the models considered here satisfy transitivity. By dissecting coalescing
and branch independence, we can test among alternative descriptive models (Birnbaum,
1999a; 2004a). CPT implies coalescing and assumes that Allais paradoxes are the result
of violations of restricted branch independence” (Birnbaum, 2005).
Stochastic dominance is the relation between gambles A and B such that the
probability of getting a prize of x or more in Gamble A is at least as high as and
sometimes higher than the probability of getting as high a prize in gamble B. Few people
would think it rational to violate stochastic dominance. However, Birnbaum’s TAX
model implies that people should in special occasions be observed to violate it.
According to Birnbaum: “People should violate stochastic dominance in choices such as
the following: From which urn would you prefer to draw a ticket at random, if the ticket
drawn determines your prize?
A: 90 tickets to win $96
05 tickets to win $14
05 tickets to win $12
B: 85 tickets to win $96
05 tickets to win $90
10 tickets to win $12
When the probability of winning prize x or greater given gamble A is greater than
or equal to the probability have a recovery of x or more in gamble B, for all x,
and if this probability is strictly higher for at least one value of x, we say that
gamble A stochastically dominates gamble B. In this example, A dominates B.
The probability have a recovery of $96 or more is .90 in A, and only .85 in B; the
probability have a recovery of $90 or more is the same, the probability have a
recovery of $14 or more is higher in A than B; and the probability have a recovery
of $12 or more is the same in both gambles” (Birnbaum, (2005).
Hypothesis 1: TAX will predict violations of stochastic dominance better than CPT:
Because CPT satisfies outcome transitivity, and coalescing, this class of theories must
satisfy stochastic dominance in this paradigm. Therefore, these models imply that
stochastic dominance must be satisfied at least 50% of the time. In this study it is
predicted that the TAX model will predict choices made by participants better than the
cumulative prospect theory CPT.
Hypothesis 2: TAX will predict coalescing better than CPT:
To test coalescing more directly, we split consequences in choice 9 and 13 to create four-
outcome gambles. The split versions of these examples are choice 9 = (12%, .05;
14%, .05; 96%, .90) versus choice 13 = (12%, .05; 12%, .05; 90%, .05; 96%, .85). The
choice, S in row 9 of table 1 versus R in row 9 is really the same choice as S versus R, in
row 13 of table 1 except for coalescing. The TAX model of Birnbaum and Chavez
(1997) with parameters estimated in previous research, predicts that participants should
violate stochastic dominance by preferring choice R in row 9 to choice, S. and by
preferring choice S in row 13 to choice, R.
Method
Each participant served in two sessions of 1.5 hours each, separated by one week.
They made choices between gambles, knowing that 10 of them would play one of their
chosen gambles for real cash prizes. They were told that any of their choices might be
the one selected for play, so they should choose carefully.
Each choice was displayed as in the following example:
1. First Gamble: 50 tickets to win $100 50 tickets to win $0OR Second Gamble: 50 tickets to win $35 50 tickets to win $25
Choose First Gamble Choose Second Gamble
Trials were blocked in groups of 25 to 48 choices each, and were randomized
within blocks. Trials from each subdesign were randomly ordered within blocks, and
would not be repeated until after at least 107 intervening trials, from at least four
subdesigns, had been presented.
Experimental Subdesigns
Tables 1, 2, and 3 show the choices used in the LH, LP, and PH designs,
respectively. These were the designs testing transitivity.
Insert Tables 1 and 2 about here.
The complete instructions and materials can be viewed at the following URL:
HTTP://psych.fullerton.edu/mbirnbaum/decisions/thanks.htm
Participants
Participants were undergraduates enrolled in lower division psychology at
California State University, Fullerton. All of the participants were tested in the lab. Of
these, ( ) indicated they were female, and ( ) were males. Ages ranged from ( ) to ( ).
Participants viewed instructions and materials via the WWW, and worked at their
own paces to fill the time allotted. This meant that some participants completed more
replications than others. A few additional participants were tested who failed to follow
instructions or complete enough trials for meaningful within-subjects analysis. There
were () whose data were internally inconsistent. There were ( ) who completed 19 or 20
replications of each choice in the three main subdesigns, LH, LP, and PH. In addition,
there were ________who completed at least ______ of each.
Table 1. Gambles used in the LH design testing transitivity.
Series I: LH Design
A = ($84, .50; $24)
B = ($88, .50; $20)
C = ($92, .50; $16)
D = ($96, .50; $12)
E = ($100, .50; $8)
Table 2. Gambles used in the LP design, testing transitivity.
Series II: LP Design
A = ($100, .50; $24)
B = ($100, .54; $20)
C = ($100, .58; $16)
D = ($100, .62; $12)
E = ($100, .66; $8)
Table 3. Gambles used in the PH design testing transitivity.
Series III: PH Design
A = ($100, .50; $0)
B = ($96, .54; $0)
C = ($92, .58; $0)
D = ($88, .62; $0)
E = ($84, .66; $0)
References
Birnbaum, M. H. (2005). A comparison of five models that predict violations of first-
order stochastic dominance in risky decision making. Journal of Risk and Uncertainty,
31, 263-287, in press.
Birnbaum, M. H. (in press). Tests of branch splitting and branch-splitting independence
in Allais paradoxes with positive and mixed consequences. Organizational Behavior and
Human Decision Processes, in press.
Birnbaum, M. H. & Bahra, J. P. (2005) Gain-loss separability and coalescing in risky
decision making. In press.
Conlisk, J. (1989). Three variants on the Allais example. The American Economic
Review, 79,
392-407.
Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under
risk.
Econometrica, 47, 263-291.
Laury, S. K. & Holt, C. A. (2000). Further reflections on prospect theory.
http://www.people.virginia.edu/~cah2k/reflect.pdf
Leshno, M., & Levy, H. (2004). Stochastic dominance and medical decision making.
Health Care Management Science, 7, 207-215.
Lopes, L. L. & Oden, G. C. (1999). The role of aspiration level in risky choice: A
comparison of cumulative prospect theory and SP/A theory. Journal of
Mathematical Psychology, 43, 286-313.
Pliskin, J. S., Shepard, D. S., & Weinstein, M. C. (1980). Utility functions for life years
and health status. Operations Research, 28, (1) 206-224.
Tversky, A. & Kahneman, D. (1992). Advances in prospect theory: Cumulative
representation
of uncertainty. Journal of Risk and Uncertainty, 5, 297-323.
Wakker, P.P. (2004). Measuring utility; introductory comments. www1.fee.uva.nl/creed/wakker/
Wu, G., & Markle, A. B. (2004). An empirical test of gain-loss separability in prospect
theory. Unpublished paper. gsbwww.uchicago.edu/fac/george.wu/research/
Figure Caption
Figure 1: Concave means that is increasing at progressively slower pace, so curved
downward.
Example of Concave Utility Function
0
2
4
6
8
10
0 20 40 60 80 100Objective Consequence, x
Subj
ectiv
e U
tility
of x
Table . Tests of Stochastic Dominance (), Coalescing (), Allais Paradoxes (and), RBI ()TAX CPT
S R S R S R
Table . Tests of Lower ( ) and Upper Cumulative Independence ( ), Restricted Branch Independence ( ) Independence, 3-Lower Distribution Independence ( ), 3-Upper Distribution Independence ( ),
TAX CPTS R S R S R
LH TRIALSLH_Trials S R TAX CPT EV
A 50 tickets to win $8450 tickets to win $24
44.0 48.1 54
B 50 tickets to win $8850 tickets to win $20
42.7 47.1 54
C 50 tickets to win $9250 tickets to win $16
41.3 46.1 54
D 50 tickets to win $9650 tickets to win $12
40.0 45.0 54
E 50 tickets to win $10050 tickets to win $8
36.7 43.7 54
VAR. Name33 90 tickets to win $98
10 tickets to win $290 tickets to win $5010 tickets to win $45
34 50 tickets to win $9850 tickets to win $2
50 tickets to win $5050 tickets to win $45
35 50 tickets to win $9850 tickets to win $2
50 tickets to win $3550 tickets to win $30
36 50 tickets to win $9850 tickets to win $2
50 tickets to win $3050 tickets to win $25
37 10 tickets to win $9890 tickets to win $2
10 tickets to win $3090 tickets to win $25
LP TRIALS
LP_Trials S R TAX CPT EV
A 50 tickets to win $10050 tickets to win $24
49.3 54.4 62
B 54 tickets to win $10046 tickets to win $20
48.2 53.7 63.2
C 58 tickets to win $10042 tickets to win $16
47.1 53.3 64.7
D 62 tickets to win $10038 tickets to win $12
46.3 53.0 66.6
E 66 tickets to win $10034 tickets to win $8
45.7 53.0 68.7
VAR. Name33 90 tickets to win $99
10 tickets to win $190 tickets to win $4010 tickets to win $35
34 50 tickets to win $9950 tickets to win $1
50 tickets to win $4050 tickets to win $35
35 10 tickets to win $9990 tickets to win $1
10 tickets to win $4090 tickets to win $35
36 90 tickets to win $4010 tickets to win $35
90 tickets to win $9910 tickets to win $1
37 50 tickets to win $4050 tickets to win $35
50 tickets to win $9950 tickets to win $1
38 10 tickets to win $4090 tickets to win $35
10 tickets to win $9990 tickets to win $1
PH TRIALSPH_Trials S R TAX CPT EV
A 50 tickets to win $10050 tickets to win $0
33.3 37.3 50
B 54 tickets to win $9646 tickets to win $0
33.8 38.2 51.8
C 58 tickets to win $9242 tickets to win $0
34.1 38.9 53.4
D 62 tickets to win $8838 tickets to win $0
34.3 39.5 54.6
E 66 tickets to win $8434 tickets to win $0
34.4 40.0 55.4
VAR. Name33 50 tickets to win $100
50 tickets to win $050 tickets to win $4550 tickets to win $40
34 30 tickets to win $10020 tickets to win $9850 tickets to win $0
50 tickets to win $4530 tickets to win $4120 tickets to win $40
35 50 tickets to win $10030 tickets to win $220 tickets to win $0
30 tickets to win $4520 tickets to win $4450 tickets to win $40
36 30 tickets to win $10020 tickets to win $9850 tickets to win $0
50 tickets to win $3030 tickets to win $2620 tickets to win $25
37 50 tickets to win $10030 tickets to win $220 tickets to win $0
30 tickets to win $3020 tickets to win $2950 tickets to win $25