VIOLATIONS OF STOCHASTIC DOMINANCE IN …psych.fullerton.edu/mbirnbaum/collaborations/bahra/... · Web viewViolations of Stochastic Dominance in medical Decision Making How do people

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


B: 85 tickets to win $96



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

http://psych.fullerton.edu/mbirnbaum/decisions/thanks.htm

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







LP TRIALS

LP_Trials S R TAX CPT EV


49.3 54.4 62


48.2 53.7 63.2


47.1 53.3 64.7


46.3 53.0 66.6


45.7 53.0 68.7













PH TRIALSPH_Trials S R TAX CPT EV


33.3 37.3 50


33.8 38.2 51.8


34.1 38.9 53.4


34.3 39.5 54.6


34.4 40.0 55.4



34 30 tickets to win $10020 tickets to win $9850 tickets to win $0








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