A Prescriptive Model of Averse-Prone Risk Attitudes

NOT FOR QUOTATION WITHOUT THE PERMISSION OF THE AUTHOR

A Prescriptive Model for Averse-Prone Risk Attitudes

C h a r l e s

M.

H a r v e y

October

1986 WP-86-70

Working P a p e r s are interim r e p o r t s on work of t h e International Institute for Applied Systems Analysis and have received only limited review. Views o r opinions expressed herein do not necessarily r e p r e s e n t those of t h e Institute or of its National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

Foreword

This paper presents

a

prescriptive model f o r

a

decision maker's risk attitude toward financial outcomes that have important non-monetary effects, f o r example, effects on how the decision maker is judged by himself and by others. The model represents the risk attitude of

a

decision maker who is risk averse in the absence of such psychological effects, but who is risk prone in their presence f o r actions leading

to

net losses o r the status quo. The model is examined f o r its adherence

to

normative principles. In particular, it is argued that the principle of dominance should be specified without any assumptions on preferences between conjunctions of lotteries; such assumptions a r e shown

to

imply the apparently stronger princi- ple of risk neutrality.

Alexander B. Kurzhanski Chairman

System and Decision Sciences Program

A Prescriptive Model for Averse-Prone Risk Attitudes Charles

M.

Harvey

Introduction

Behavioral studies of people's r i s k attitudes h a v e found t h a t in a v a r i e t y of c o n t e x t s t h e majority of people are r i s k a v e r s e in t h e i r p r e f e r e n c e s among

ac-

tions leading

to

^gains

or

t h e s t a t u s quo but are r i s k p r o n e in t h e i r p r e f e r e n c e s among actions leading

to

losses

or

t h e s t a t u s quo. Such p r e f e r e n c e s will b e

re-

f e r r e d

to

in t h i s p a p e r

as

an

averse-prone r i s k a t t i t u d e .

During t h e e a r l y development of e x p e c t e d utility t h e o r y , Friedman and Savage (1948) and Markowitz (1952) discussed averse-prone r i s k a t t i t u d e s (and even t h r e e - and four-piece r i s k attitudes). Relevant empirical work includes five stu- dies, Barnes a n d Reinmuth (1976), Grayson (1960). Green (1963). H a l t e r a n d Dean (1971), and Swalm (1966), t h a t

were

examined by Fishburn and Kochenberger (1979), a n d were reexamined t o g e t h e r with o t h e r empirical studies by Hershey, Kunreuther, and Schoemaker (1982). Other empirical work and analyses

are

in Dickson (1981), Fuchs (1976), Hershey and Schoemaker (198Oa), Langhhunn, Payne, and Crum (1980), Schoemaker and Kunreuther (1979), Wehrung

,

Bassler, MacCrimmon, and S t a n b u r g (1978), Wehrung

,

MacCrimmon, and B r o t h e r s (1984), and Williams (1966).

This p a p e r is

a

modeling a n d a subsequent analysis of averse-prone r i s k atti- tudes. In t h e c u r r e n t milieu of r e s e a r c h on p r e f e r e n c e models, i t i s important

to

emphasize t h a t t h e models p r e s e n t e d h e r e

are

p r e s c r i p t i v e r a t h e r t h a n normative

or

descriptive. They

are

intended

to

help a decision maker whose a c t u a l p r e f e r - ences, i.e., t h e p r e f e r e n c e s lying behind his cognitive limitations,

are

such t h a t h e h a s a n averse-prone r i s k attitude. These models may o r may not conform to vari- o u s assumptions of rationality (a m a t t e r t h a t is examined in t h e second p a r t of t h i s p a p e r ) . Moreover, t h e y may

or

may not provide a n a c c u r a t e

or

predictive model- ing of t h e behavior of most people ( a m a t t e r t h a t i s commented on b u t it is not

test-

e d in t h i s p a p e r ) . Bell, Raiffa, a n d Tversky (1984) and Schoemaker (1982) contain r e c e n t , g e n e r a l discussions of t h e distinctions between t h e normative, descriptive, and p r e s c r i p t i v e modeling of p r e f e r e n c e s .

Section 1 discusses t h e types of psychological responses

to

financial gains and losses t h a t may induce

an

averse-prone risk attitude, and p r e s e n t s a r a t h e r general model of such r i s k attitudes. Section 2 then discusses a specialization of t h i s model in which t h e psychological effects of gains and losses

are

priced-out

as a

"reward" t h a t i s independent of t h e magnitude of t h e gain and a "penalty" t h a t is independent of t h e magnitude of t h e loss. In t h i s model, anyone who i s r i s k a v e r s e when t h e psychological e f f e c t s

are

absent will have

an

averse-prone r i s k attitude o v e r s o m e m g e of monetary changes when t h e psychological e f f e c t s

are

present.

Sections 3 and 4 examine which normative principles

are or

are not violated by a n averse-prone r i s k attitude. The salient conclusion is not a list of yes/no

answers

but t h a t c e r t a i n conditions implicit in t h e common normative principles need

to

b e made explicit and then examined in isolation. In particular, i t is shown t h a t c e r t a i n conditions on p r e f e r e n c e s between conjunctions of lotteries t h a t

are

often implicit in t h e dominance principle imply t h e apparently s t r o n g e r principle of r i s k neutrality. I t is argued t h a t these conditions on conjunctions (and a condi- tion implicit in t h e framing principle) should not b e included as normative princi- ples.

Section 5 then r e t u r n s

to

a prescriptive viewpoint and'discusses t h e potential uses and abuses of a n averse-prone r i s k model f o r a decision analysis application in which r i s k aversion f o r gains but r i s k proneness f o r losses is

an

important p r e f e r e n c e issue.

1.

P s y c h o l o g i c a l - E f f e d s

Y

odela

S e v e r a l r e s e a r c h e r s have suggested t h a t

a

decision maker's r i s k attitude may depend on various psychological effects of financial outcomes t h a t might b e includ- e d as p a r t of the description of t h e decision maker's possible consequences. For example, P e t e r Fishburn, Ralph Keeney, and Richard Meyer (in t h e discussion fol- lowing a p a p e r presented by Tversky, 1977) suggest t h e use of

an

additional

attri-

bute to explain r i s k p r o n e p r e f e r e n c e s f o r potential losses in financial choices. A similar idea is suggested in Keeney (1984) by his argument t h a t t h e ethics of

a

deontological moralist need not violate t h e expected utility conditions provided t h a t t h e social consequences of policy decisions

are

adequately defined (p. 122).

Moreover, Raiffa (1984) discusses psychological effects by imagining t h a t t h e deci- sion maker has a n e x t e r n a l or internal kibitzer whose r e m a r k s could b e used

to

provide a

m o r e

sophisticated description of t h e consequences

to

t h e decision maker.

In t h i s section, a p r e f e r e n c e model is developed t h a t is consonant with t h e above ideas. Suppose t h a t financial outcomes t o t h e decision maker

are

measured by a variable

z

such t h a t

z >

⁰ r e p r e s e n t s n e t gains,

z <

⁰ r e p r e s e n t s n e t losses, and

z =

0 r e p r e s e n t s t h e s t a t u s quo. An individual who i s making

a

decision t h a t a p p e a r s

to

depend only on t h e financial amounts

z

may be influenced by t h e antici- pation of his psychological responses

to

t h e possible consequences of

a

chosen ac- tion. To d e s c r i b e t h e s e effects, it will b e helpful t o distinguish between

a

decision maker who is acting on his own behalf and

a

decision maker who acting as a n agent f o r a n organization.

A person who is acting on his own behalf, e.g., a private e n t r e p r e n e u r o r a n individual investor, may feel embarrassment

or a

loss of self-esteem if

a

financial loss should occur. This person may also feel p r i d e

or

increased self-esteem as

a

r e s u l t of

a

financial gain. Such a t y p e of psychological response i s h e r e dis- tinguished from a change in financial position. t h a t is, from t h e event t h a t t h e per- son will have

a

c e r t a i n l e s s e r or g r e a t e r bmount

to

spend o v e r his lifetime.

For a person who i s acting

as

a n agent f o r

an

organization. e.g., a business ex- ecutive

or

a government administrator. t h e r e

are

e x t e r n a l p r e s s u r e s in addition

to

t h e type of ego involvement described above. The person wishes

to

b e favorably judged by those

to

whom h e i s accountable; f o r example, t h e person's primary con- c e r n may b e

to

maintain

or to

enhance his reputation.

A s

a

primary assumption in t h i s p a p e r , suppose t h a t t h e psychological effects

to

b e considered can b e measured as components of t h e consequences

to

t h e deci- sion maker. For modeling purposes, t h e s e types of psychological effects will b e called

m c t s

attributes. Each effects a t t r i b u t e will b e measured by a variable zf

.

ⁱ

⁼

^{1 ,}

^{..., n .}

Consequences will b e described by both t h e variable

z

f o r t h e monetary a t t r i b u t e and t h e variables zl,

..., z,

f o r t h e effects attributes; thus.

each consequence will b e denoted by

a

v e c t o r ( z

,z I,...

,z, ).

By contrast, Bell (1982). (1983). (1985) has modeled psychological responses such

as

"regret" and "disappointment" t h a t depend upon t h e decision maker's per- ception of a l l of t h e available l o t t e r i e s

or

of t h e e n t i r e lottery t h a t i s chosen. A s s t a t e d in Bell (1982) "regret i s measured

. . . as

t h e difference in value between t h e

assets

actually received and t h e highest level of

assets

produced by other al- ternatives [italics Bell's].

"

In

a

specific decision context, t h e amounts z l ,

..., z,

may b e highly c o r r e l a t e d

to

t h e financial amount

z

and may even b e functionally dependent on

z .

However, i t will b e assumed t h a t t h e decision maker i s familiar with

a

sufficiently wide v a r i e t y of c o n t e x t s such t h a t his p r e f e r e n c e s

can

b e considered on

a

p r o d u c t

set

of potential consequences ( z ,z l,... ,z, ) where t h e variables

z

, z l,... ,z,

are

defined on specific intervals.

(A) Suppose t h a t t h e decision maker's t r a d e o f f s satisfy t h e willingness-to-pay conditions (see, e.g., Keeney and Raiffa, 1976, pp. 125-127, and Harvey, 1985).

Then, p r e f e r e n c e s among consequences in t h e product

set can

b e r e p r e s e n t e d by

a

value function of t h e form

Here, t h e amounts gi (zi ),i

=

1,

...,

n , c a n b e assessed as p r i c i n g - o u t a m o u n t s f o r t h e psychological responses; t h a t is, f o r some specified r e s p o n s e

zi *

of t h e i-th e f f e c t s a t t r i b u t e , gi (zi) i s t h a t amount such t h a t t h e decision maker would just b e willing t o pay gi (zi) in o r d e r

to

obtain

zi

r a t h e r t h a n

zi * .

The value V of a conse- quence ( z , z l ,

...,

^z,) c a n b e i n t e r p r e t e d as t h e financial amount such t h a t t h e consequence ( z ,

z

l,.

.. , z,

) i s indifferent

to

t h e consequence (V,z l*

,...

^,z,

*

).

(B) Suppose t h a t t h e decision maker's r i s k a t t i t u d e satisfies t h e conditions of expected utility. Then, p r e f e r e n c e s among l o t t e r i e s c a n b e r e p r e s e n t e d by

a

utili- t y function of t h e form

H e r e , t h e function

u

c a n b e i n t e r p r e t e d as a conditional utility function on finan- cial amounts

z

given t h a t t h e psychological e f f e c t s

are

t h e specified amounts zl*

,...,

^Zn

*.

Consider n e x t t h e causal relations between receiving a n e t gain or a n e t loss

z

i s a specific c o n t e x t and t h e resulting psychological effects. Suppose t h a t in any o n e decision context t h e psychological e f f e c t s described by

zl. ..., z, are

func- tionally dependent on t h e financial outcome

z .

Then, conditional on a given con- text:

zi = ri

( z ) , i

=

^{1 ,}

...,

n , f o r some functions

r*.

The functions

ri

w i l l b e re- f e r r e d

to as

r e s p o n s e j b n c t i o n s . Note t h a t

a

decision context does not a f f e c t t r a d e o f f s between t h e a t t r i b u t e s b u t r a t h e r r e s t r i c t s t h e domain of potential consequences.

For prescriptive purposes, i t will be useful

to

compare t h e following two types of decision contexts.

(i)

The a c t s contezt:

Here t h e decision maker anticipates t h a t his psycho- logical responses are important

to

him and should b e included in describing t h e consequences of his actions.

(C) Suppose t h a t t h e response functions zi

= ri

( z ), i

= 1 , ... ,n , as

discussed above denote t h e decision maker's psychological responses in t h e effects context.

Suppose t h a t l a r g e r monetary amounts z lead

to

responses zi

= ri

( z ), i

= 1 , ... ,n ,

that are

at

least

as

preferred. The responses

to

maintaining t h e status quo, i.e.,

to

z

= 0 ,

will be denoted by zi

* = ri (0)'

i

= 1 ,..., n .

The amounts zi

*,

i

= 1 ,..., n ,

will b e called

standard a e c t s .

(ii) The

no-effects contezt:

Here t h e psychological responses in t h e effects context are e i t h e r absent

or

are omitted f r o m consideration.

( D ) Suppose t h a t in t h e no-effects context the psychological responses

to

any financial change z

are

t h e standard effects zi

*

^{, i}

=

1,

..., ⁿ .

Thus, t h e effects of any change z in t h e no-effects context

are

t h e

same as

t h e effects of no change, z

= 0 ,

in t h e effects context.

Definition

1.

A preference model

as

described in conditions ( A ) - ( D ) will b e called

a p ~ a j c h o l o g i c a Z ~ e c t s model.

Any utility function

u ( z )

defined on monetary amounts z in t h e no-effects context will be called a

ho-q#kcts u t i l i t y function;

any utility function

w

( z ) defined on monetary amounts z in t h e effects context will be called a n

sects u t i l i t y &nction.

For

a

net gain

or

loss z , consider t h e associated

a c t s amount e

( z ) defined by

where

ri

( z ), i

=

^1,

... ^{,n ,}

are t h e psychological effects of z in t h e effects context.

The monetary amount e ( z ) can be interpreted

as

t h a t amount such t h a t t h e conse- quence

( 0 , z l,...

,zn ) i s indifferent

to

t h e consequence

( e

( z ) , z

....

^,zn

*

), t h a t is,

e ( z )

i s t h e total pricing-out amount of t h e psychological effects zi

= r i ( z ) ,

i

= 1 ,..., n .

Theorem

1.

For a psychological-effects model, if

u

( z ) i s any utility function for t h e decision maker's risk attitude in t h e no-effects context and w ( z ) i s any utility function f o r t h e decision maker's risk attitude in t h e effects context, then

f o r

s o m e

normalization constants a

>

0 and 6 .

The psychological-effects model discussed above is f a r

too

r e s t r i c t i v e

to

describe t h e heuristic biases t h a t might b e responsible for

at

least p a r t of a n ob- served averse-prone r i s k attitude. The model is intended as a possible formulation of t h a t p a r t of a n averse-prone r i s k attitude which t h e decision maker r e g a r d s

as

due

to

his underlying preferenoes. For such

a

prescriptive purpose, i t i s useful

to

f u r t h e r specialize t h e model; t h e following section presents one means of doing so.

2. The Rermrd-Penalty Podel

This section discusses

a

special type of psychological-effects model f o r

com-

paring a decision maker's p r e f e r e n c e s in t h e effects context and in t h e no-effects context. The model is intended

to

b e sufficiently specific

to

b e t r a c t a b l e for deci- sion analysis applications.

As a strong causal assumption, suppose t h a t in the effects context t h e psycho- logical responses are constant in t h e sense t h a t

f o r some constant amounts zi+ and zi-, i

=

1,

..., n.

Thus, t h e effects

zi

in t h e ef- f e c t s context depend only on whether t h e financial change is a gain or a loss, and not on t h e magnitude of t h e gain

or

loss. This type of dependence may by a p p m p r i -

ate as

a modeling simplification f o r a variety of decision situations. For example, a decision maker acting

as an

agent may believe t h a t h e is being judged in p a r t in

a

superficial manner by whether h e succeeds ( z

>

0), maintains t h e status quo ( z

=

^0),

or

fails ( z

<

^0).

I t follows directly from equations (3) and

(5)

t h a t r i ( z ) = '

I

^p f o r z > O e ( z ) = O f o r z = O

-n f o r

z <

0

zi+

f o r z > O

2; f o r z = O

zi-

f o r

z <

0

b

f o r some constant financial amounts p 2 0 _and n 2 0. The tradeoffs amounts p and

n

can b e r e g a r d e d

as

t h e e x t r a "reward" and "penalty" in t h e effects context of

a

gain and a loss respectively.

Definition 2.

A psychological effects model t h a t satisfies t h e condition of con- s t a n t psychological e f f e c t s summarized in (5), (6) will b e called

a

reward-penalty model.

Figure 1 illustrates

a

reward-penalty model in which t h e utility function w (z) f o r t h e effects context corresponds

to

a utility function u (z) for t h e no-effects context t h a t r e p r e s e n t s r i s k aversion. In t h e effects context, t h e penalty

n >

⁰

leads

to

r i s k proneness among l o t t e r i e s whose possible consequences

are

n e t losses and t h e s t a t u s quo; t h e reward p

>

⁰ leads

to

g r e a t e r risk aversion among l o t t e r i e s whose possible consequences are net gains and t h e s t a t u s quo. These p r o p e r t i e s a r e illustrated by t h e dotted lines in Figure 1. Thus, w ( z ) r e p r e s e n t s a n averse-prone r i s k attitude o v e r a r a n g e of amounts z including both n e t gains and n e t losses.

Figure 1. A Utlllty Functlon for the EfYeats Context

A numerical illustration of

a

reward-penalty model can b e calculated by con- sidering t h e following well-known choice problem presented in Tversky and Kahne- man (1981). The p e r c e n t a g e s noted by t h e a l t e r n a t i v e actions are based on t h e responses of 150 students

at

Stanford University and t h e University of British Columbia.

Problem 1. Imagine t h a t you f a c e t h e following p a i r of c o n c u r r e n t decisions.

F i r s t examine both decisions, then indicate t h e options you p r e f e r .

Decision (i). Choose between:

A. a sure gain of $240 [84 p e r c e n t ]

B. 25% chance

to

gain 8 l 0 0 0 , and 75% chance

to

gain nothing

116

p e r c e n t ]

Decision (ii). Choose between:

C.

a

s u r e

loss

of S750 [13 p e r c e n t ]

D.

75% c h a n c e

to

lose 8 l 0 0 0 , and 25% chance

to

lose nothing [87 percent].

A majority [73 p e r c e n t ] of t h e respondents chose actions A and D. The ex- pected monetary values of t h e f o u r actions are: E ( A )

=

⁸²⁴⁰

<

^{E @ )}

=

^{8250, and}

E ( C )

=

-S750

=

E(D). Thus, t h e students were r i s k a v e r s e in decision (i) but r i s k p r o n e in decision (ii).

To model t h e s e p r e f e r e n c e s with

a

reward-penalty model, suppose t h a t in t h e no-effects context a p e r s o n h a s constant r i s k aversion ( a t least o v e r t h e r a n g e of gains and losses considered) and

assesses a

certainty equivalent of $245 f o r t h e l o t t e r y B. Then, in t h e no-effects context, t h e person i s mildly r i s k a v e r s e ; h e p r e f e r s B

to

A and p r e f e r s C

to

D ( t h e opposite of t h e p r e f e r e n c e s observed by Tversky and Kahneman).

Now, consider t h e person's p r e f e r e n c e s in t h e e f f e c t s context. F o r

a

reward amount of p

= S7 or

more, t h e p e r s o n will become sufficiently r i s k a v e r s e among gains l o t t e r i e s so t h a t h e p r e f e r s A

to

B. For

a

penalty amount of rr

=

$21 or more, t h e person will become sufficiently r i s k p r o n e among losses l o t t e r i e s so t h a t h e p r e f e r s D

to

C. Therefore, psychological responses

to

gains and

losses

t h a t lead

to

only modest t r a d e o f f s amounts p and

n _are

sufficient

to

induce t h e choice behavior observed by Tversky and Kahneman.

The assumptions of a reward-penalty model imply in general a number of pro- perties f o r t h e decision maker's preferences in t h e effects context. These pro- perties are listed in t h e result below. Here, a lottery having a net gain

or

loss of

z

with probability p and

a

net gain

or

loss of

z'

with probability q

=

1 - p i s denoted by

<

p $2, q Sz '

>.

Theorem

2. Suppose t h a t

a

decision maker's preferences satisfy t h e conditions of a reward-penalty model.

Part

L If in t h e no-effects context t h e decision maker is risk a v e r s e , then in the effects context his preferences will have t h e following properties:

(a) If t h e penalty K is positive, then preferences

are

risk prone f o r moderate losses, t h a t is, f o r any probability 0

<

p

<

1 ,

<

p Sz

,

q 80

>

^{i s}

p r e f e r r e d

to

t h e amount Spz f o r sufficiently small negative

z.

(b) Preferences

are

risk a v e r s e f o r moderate

or

l a r g e gains, t h a t is, for any probability 0

<

p

<

1 , t h e amount Spz i s p r e f e r r e d

to <

p $2, q 80

>

^{f o r}

any positive

z .

(c) If K i s g r e a t e r than p, then preferences

are

r i s k a v e r s e f o r symmetric lotteries, t h a t is, 80 is p r e f e r r e d

to <

$2,

+

^{S ( z )}

^>

f o r any non-zero amount

z .

Part II.

If in t h e no-effects context t h e decision maker has decreasing risk aver- sion and has unbounded utility from below (i.e., f o r any amounts

z < z'

t h e r e is an amount

z "< z

such t h a t

z

i s p r e f e r r e d

to <

Sz ",

+

^Sz'

^>

^), then in t h e effects context his preferences have t h e following additional properties:

(d) P r e f e r e n c e s

are

risk a v e r s e f o r l a r g e losses, t h a t is, f o r any probability 0

<

p

<

1, Spz is p r e f e r r e d

to <

p $2, q SO

>

f o r sufficiently l a r g e negative

z .

(e) If K

>

0, then preferences

are

more r i s k a v e r s e in t h e effects context then in t h e no-effects context for s t r i c t losses, t h a t is, f o r any

t w o

s t r i c t losses

z

, z '

<

0, t h e certainty equivalent of

<

p $2, q $2'

>

i s less in the effects context than in t h e no-effects context.

(f) If p

>

0, then preferences

are less

risk a v e r s e in t h e effects context than in t h e no-effects context f o r s t r i c t gains, t h a t is, for any

t w o strict

gains

z

, z '

>

0, t h e certainty equivalent of

<

p $2, q $2'

>

is

more

in t h e effects context than in t h e no-effects context.

The purpose of Theorem 2 i s not

to

provide a means of testing t h e descriptive a c c u r a c y of a reward-penalty model. The intent i s r a t h e r

to

make explicit

to a

de- cision maker some of t h e implications of adopting such a p r e s c r i p t i v e model.

3. Conformity w i t h Normative Principlw

This section examines whether

a

r i s k attitude in a reward-penalty model, or a averse-prone r i s k a t t i t u d e in general, confirms

to

c e r t a i n conditions on p r e f e r - ences t h a t have been r e g a r d e d from a normative viewpoint as principals f o r ra- tional decision making. The violation of t h e s e principles by people's choice behavior in non-transparent decision problems h a s been w e l l documented, f o r ex- ample, in t h e work of D. Kahneman and A. Tversky (1979), (1981), and (1986).

3.1.

T r a n s i t i v i t y , Dominance, a n d Independence..

F o r any reward-penalty model in which t h e r e i s r i s k aversion in t h e no-effects context and

a

non-zero penalty

n,

t h e r e will b e in t h e e f f e c t s context a n averse-prone r i s k a t t i t u d e as described in p a r t s (a), (b) of Theorem 2. Such p r e f e r e n c e s

are

r e p r e s e n t e d by

a

utility func- tion w ( z )

as

in (4), and hence

are

consistent with t h e normative principles of ex- pected utility. I t follows t h a t , in p a r t i c u l a r , t h e s e averse-prone r i s k attitudes satisfy t h e principles of transitivity, stochastic dominance, and independence.

3.2. f i a m i n g Consistency.

Consider any averse-prone r i s k attitude with r e s p e c t

to

a variable

z.

In t h i s subsection, t h e r i s k attitude may

or

may not confirm

to

^{t h e}

principles of e x p e c t e d utility. Let I

= <

^{pi ,zi}

>

denote a

lottery

having possible consequences

zi

with probabilities pi where i

=

1 ,

...

^,m

.

The v a r i a b l e

z

is intended

to

measure changes in monetary position. i.e., ei- t h e r changes in t h e decision maker's personal finances

or

changes in t h e finances of

an

organization f o r which t h e decision maker is acting as a n agent. The conse- quences

to

t h e decision maker can

also

b e described

as

final

asset

positions meas- u r e d by a v a r i a b l e y . Assume that: (1)

a

c u r r e n t

asset

position

c

c a n b e defined (but not necessarily evaluated), (2) t h e amounts

z are

n e t monetary gains and losses, e.g., n e t p r e s e n t values

or

n e t after-tax profits, and (3) t h e final

asset

po- sition y implied by

c

and

z

is specified by t h e formula y

=

c

+ z.

For a situation in which

z

measures

rates

of r e t u r n o r net pre-tax profits. t h e formula relating y t o

z

f o r a fixed c u r r e n t asset position c will differ from y

=

c

+ z

, but t h e r e will still b e a one-to-one relationship y

= f (z I

^c ) between financial changes

z

and final

asset

positions y . The r e s u l t s in t h i s subsection easily generalize

to

such situations.

The p r e f e r e n c e relation concerning financial changes

z

conditional on a c u r r e n t asset position c will b e denoted by & ~ c . An associated preference rela- tion

&

concerning final

asset

positions y is defined by

where y i

=

c

+ ^zt .

^{y j}

^{= c + zj}

^{f o r i}

⁼

¹

^{,..., m}

^{and j}

⁼

¹

^{,..., m '.}

The equivalence (7) between p r e f e r e n c e s when outcomes

are

described by net gains

or losses

and p r e f e r e n c e s when outcomes are described by asset positions will be called a

fram-

i n g

tranqfomation.

Definition 3. A decision maker's p r e f e r e n c e s such t h a t t h e framing transforma- tion (7) is satisfied f o r every c u r r e n t

asset

position

c

will b e called

framing con- sistent.

The condition of framing consistency can also b e regarded

as

a consistency .

condition on p r e f e r e n c e s concerning financial changes f o r different c u r r e n t asset positions c and c ' . More precisely, framing consistency implies t h e condition t h a t

where c , c f a r e any two c u r r e n t

asset

positions and d

=

c

-

^{c ' .} Conversely, t h e condition (8) implies t h a t if a p r e f e r e n c e relation

&,

on final asset positions i s de- fined by (7) with a specific amount c , then (7) is also satisfied f o r any o t h e r m o u n t

c'.

Kahneman and Tversky (1979, p. 273) have demonstrated t h a t t h e manifest behavior of students who are presented with hypothetical choice problems is not in accord with t h e condition of framing consistency; often t h e r e a r e r e v e r s a l s in preference depending on whether t h e problem is framed in

terms

of

a

c u r r e n t as- s e t position c o r in terms of a n o t h e r c u r r e n t

asset

position c ' .

Theorem 3. The condition t h a t t h e preference relations & ~ c are averse-prone risk attitudes f o r more than one c u r r e n t

asset

position c i s inconsistent with t h e principle of framing consistency.

The r e s u l t identifies a conflict between

a

p r e f e r e n c e condition having

a

strong normative appeal and t h e choice behavior r e p o r t e d in many a r t i c l e s , e.g., Fuchs (1976), Green (1963). G r e t h e r and Plott (1979). Halter and Dean (1971), Hershey and Schoemaker (1980a), Hershey and Schoemaker (1980b), Kahneman and Tversky (1979), Langhhunn, Payne, and Crum (1980), Payne, Langhhunn, and Crum (1980), Payne, Langhhunn, and Crum (1981), Slovic, Fischhoff, Lichtenstein, Corrigan, and Combs (1977), Swalm (1966), and Williams (1966). Theorem 3 and t h e remarks in subsection 3.1 imply t h a t t h e conflict often alluded

to

in t h e l i t e r a t u r e between averse-prone choice behavior and t h e r u l e s of rationality is not

a

conflict with t h e expected utility conditions, e.g., independence, but i s a conflict with a consistency requirement. I t i s possible, f o r example, f o r

a

person's p r e f e r e n c e s

Lk to

b e a n averse-prone r i s k attitude and

to

satisfy expected utility conditions f o r each c u r r e n t

asset

position

c

and y e t not

to

b e framing consistent.

This identification of

a

specific normative principle t h a t is in conflict with averse-prone choice behavior suggests a question f o r f u t u r e empirical work. For which persons and in which contexts is

a

person's averse-prone choice behavior a r e s u l t of

a

p r e f e r e n c e issue t h a t i s of genuine importance

to

t h e person r a t h e r than a r e s u l t of his information processing limitations and of his limited experi- ence in risk taking. Dickson (1981) found t h a t a risk manager, i.e.,

a

person "con- fronted continually with problems holding out t h e chance of loss" i s likely

to

^{b e}

considerably

m o r e

r i s k a v e r s e f o r potential loss-producing choices than i s a non- r i s k manager, i.e.,

a

person who "will r a r e l y be concerned with decisions where only a loss

or

b r e a k even point i s in prospect". A survey by Freifelder and Smith (1984) h a s similar results. Hershey and Schoemaker (1980a) and Schoemaker and Kunreuther (1979) found t h a t risk aversion is f a r g r e a t e r when decisions

are

presented in a n insurance context than when t h e

s a m e

decisions

are

presented as standard gambles. Experimental studies in Hershey and Schoemaker (1980b) t h a t examine t h e reflection hypothesis of Kahneman and Tversky (1979)

at

both t h e across-sub ject and within-subject levels "seriously question t h e generality of prospect theory's reflection hypothesis."

One approach

to

t h e question posed above would b e

to

investigate t h e e x t e n t

to

which t h e

same

person in different contexts and different persons in t h e same context would modify t h e i r averse-prone choice behavior a f t e r being informed as

to

i t s conflict with t h e normative principle of framing consistency. Slovic and Tversky (1974) examine similar questions concerning a person's modification of his p r e f e r e n c e s when informed as t o i t s conflict with Savage's independence principle for expected utility.

4. C o m b i n a t i o n s o f L o t t e r i e s

Tversky and Kahnemm (1981), (1986) discuss a type of dominance principle t h a t i s concerned with t h e sum of two lotteries. In t h i s section, a g e n e r a l defini- tion of t h i s principle i s specified, and t h e principle i s shown

to

b e violated by any averse-prone r i s k attitude. Dominance concerning sums and multiples of l o t t e r i e s a p p e a r s t o b e a p p r o p r i a t e as

a

normative principle; however, i t i s shown

to

have implications t h a t r e n d e r i t i t f a r too r e s t r i c t i v e . Indeed, in a n expected utility model, t h i s t y p e of dominance i s shown

to

imply t h a t t h e decision maker i s r i s k neu- tral.

Consider t h e specific choice problem due t o Tversky and Kahneman (1981) t h a t i s described in Section 2. While most of t h e students in t h e sample (73 per- c e n t ) p r e f e r r e d A t o B and also p r e f e r r e d D

to C ,

none of t h e s e students p r e - f e r r e d t h e sum of A and D

to

t h e sum of B and C:

Problem 2. Choose between:

A and D. 25% c h m c e t o win 8240, and

75% c h a n c e

to

lose 8760. [0 p e r c e n t ] B and

C.

25% c h a n c e t o win 8250, and

75% chance

to

lose 8750. [ l O O p e r c e n t ]

Tversky and Kahneman a r g u e t h a t t h e students' decisions

are

"violations of t h e r u l e s of rational choice" and, more specifically, "violations of dominance" f o r t h o s e students who p r e f e r r e d A

to B

and p r e f e r r e d D

to

C even though t h e sum of B and

C

stochastically dominates t h e sum of A and D.

I t will b e useful t o define precisely t h e principle t h a t i s being violated by t h e students' decisions in t h i s example. Definition 4 below i s intended

to

d o so; t h e r e are o t h e r definitions t h a t have

a

g r e a t e r

or

lesser a p p a r e n t generality but

are

in f a c t equivalent.

D e f i n i t i o n 4. A p r e f e r e n c e relation on l o t t e r i e s will b e called

sum-dominance c o n s i s t e n t

provided t h a t f o r m y probability 0

<

p

<

1, both of t h e p r e f e r e n c e s

d o not o c c u r whenever t h e l o t t e r y

<

p S(z

+

z; ), Q S(z

+

z j )

>

i s dominated by t h e l o t t e r y < p S ( z 2

+

z; ), qS(z3 + Z i )

>

in t h a t

In t h e above Problems 1 and 2, p

=

.25 and zl

=

8240,

z2 =

$1000, 2 3

=

80, Zi

=

-$750, 2;

=

SO, a n d z j = -8l000

.

Before discussing t h e implications of sum-dominance consistency, w e will describe a second, analogous type of dominance. To focus t h e discussion, consider

a

person who wishes t o invest

a

c e r t a i n amount of money, which will be r e f e r r e d

to

as t h e person's

current

f i n d . I t is meaningful t o measure t h e consequences of t h e available investment l o t t e r i e s not only by net gains and losses but also by percent increases and d e c r e a s e s in t h e person's c u r r e n t fund. Consider, f o r example, t h e following choice problem.

Problem 3. Imagine t h a t you f a c e t h e following p a i r of c o n c u r r e n t decisions.

First examine both decisions, then indicate t h e options you p r e f e r .

Decision (i). Choose between:

A.

a

s u r e gain of 6X

B.

a

one-fourth chance t o gain 25X. and

a

three-fourths chance t o gain nothing

Decision (ii). Choose between:

C. a s u r e loss of 15X

D.

a

three-fourths chance

to

lose 20%. and

a

one-fourth chance

to

lose nothing.

This problem h a s not been empirically tested. However, i t i s similar t o t h e Problem 1 t e s t e d by Tversky and Kahneman in t h a t mild risk aversion f o r gains leads t o a p r e f e r e n c e of A o v e r B and any d e g r e e of r i s k proneness f o r losses leads

to

a p r e f e r e n c e of D o v e r C.

Now, consider t h e sequential o c c u r r e n c e of t h e investment lotteries in t h e de- cisions (i) and (ii). (The resulting p e r c e n t changes

are

t h e

same

r e g a r d l e s s of whether decision (i) o r decision (ii) o c c u r s first.)

Problem 4. Choose between:

A and D. a one-fourth chance

to

gain 6X a three-fourths chance

to

lose 15.2X B and C. a one-fourth chance

to

gain 6.25X

a three-fourths chance

to

lose 15X

Note t h a t t h e combination of t h e lotteries B and C dominates t h e combination of t h e l o t t e r i e s A and D in this problem just

as

t h e combination of B and C dom- inates t h e combination of A and D in Problem 2.

In general,

let z

denote a p e r c e n t increase ( z >OX), a p e r c e n t d e c r e a s e (-100%

< z <

^{OX), or} no change ( z

=

OX) in

a

person's c u r r e n t fund. Let

<

^p (zX), q (z'X)

>

denote

a

lottery having

a

p e r c e n t gain or loss of

z

with proba-

bility p and a p e r c e n t gain or loss of

z

^' with probability q

=

1

-

^p

^.

Note t h a t if f i r s t a p e r c e n t change

z l

o c c u r s and second

a

p e r c e n t change z 2 occurs, then t h e overall p e r c e n t change is

z + z z +

( z l z 2/ 100). Let

zl0z2

denote this resulting p e r c e n t change.

Definition

5.

A p r e f e r e n c e relation on l o t t e r i e s will be called percent- dominance c o n s i s t e n t provided t h a t f o r any probability 0

<

p

<

1 , both of t h e p r e f e r e n c e s

d o not o c c u r whenever t h e l o t t e r y <p(z1oz; X), q (zlOz j X)

>

is dominated by t h e lottery <p ( z 2 0 z i X), q(z30zi X)

>

in t h a t

Z ~ O Z ~

<

Z ~ O Z ~ and

zlozj <

^2302;

.

⁽¹²⁾

In t h e above Problems 3 and 4, p

=

.25 and

z l =

6X,

z 2 =

25X,

z 3 =

OX,

Z i

=

-15%. Z i

=

OX, and

z j =

-20%.

Definitions 4 and 5 formalize c e r t a i n requirements on p r e f e r e n c e s

as

pro- posed normative principles. The implications of t h e s e proposed principles

are as

follows.

Theorem 4. Consider a decision problem in which t h e possible consequences are financial changes measured e i t h e r as absolute gains and losses

z

o r as p e r c e n t gains and losses

z .

Assume t h a t t h e decision maker's p r e f e r e n c e relation on lot- t e r i e s satisfies (i) t h e conditions of expected utility and (ii) t h e condition t h a t l a r g e r amounts

are

p r e f e r r e d . Then:

(a) The p r e f e r e n c e relation is sum-dominance consistent if and only if t h e de- cision maker h a s

a

constant risk attitude f o r possible net gains and losses.

(b) The p r e f e r e n c e relation is percent-dominance consistent if and only if t h e decision maker h a s a constant proportional risk attitude f o r possible amounts of his c u r r e n t fund.

(c) Therefore, t h e p r e f e r e n c e relation i s both sum-dominance consistent and percent-dominance consistent if and only if t h e decision maker is risk neutral.

Sum-dominance consistency and percent-dominance consistency a p p e a r

to

b e reasonable

as

normative principles. They exclude t h e type of choice behavior t h a t

w a s

observed by Tversky and Kahneman (1981) f o r Problems 1 , 2, and t h a t is con- jectured

to

o c c u r f o r Problems 3, 4. A s Theorem 4 demonstrates, however, these principles exclude not only a n averse-prone risk attitude but any attitude toward risk o t h e r than r i s k neutrality.

The assumptions (i), (ii) in Theorem 4 r e q u i r e t h a t t h e r e exists

a

s t r i c t l y in- creasing utility function f o r t h e p r e f e r e n c e relation. They d o not require, howev- e r , t h a t the utility function h a s derivatives of any o r d e r

or

even

that

i t is continu- ous. Thus, Theorem 4 applies in p a r t i c u l a r

to

t h e p r e f e r e n c e relation f o r t h e ef- f e c t s context in a reward-penalty model.

5. Uses of the Reward-Penalty Yodel

This section discusses f i r s t t h e possible testing of t h e reward-penalty model

as a

descriptive model, and second i t s possible usefulness as a prescriptive model in a decision analysis study.

The reward-penalty model is sufficiently r e s t r i c t i v e

to

b e testable

as a

descriptive model of p r e f e r e n c e s in choice behavior. In this sense, i t is not sub- ject

to

t h e criticism of Tversky (1977) t h a t introducing additional variables

as

descriptors of psychological consequences i s a d hoc. However, any empirical study of the model t h a t compares

a

person's p r e f e r e n c e s in

t w o

different contexts (what

w e

have oalled t h e e f f e c t s context and t h e no-effects context) will need

to

b e carefully designed e i t h e r

to

convey

to

t h e persons being interviewed a

clear

understanding of t h e e f f e c t s context and of t h e no-effects context

or to

elicit by indirect questioning t h e i r p r e f e r e n c e s in t h e no-effects context. The design prob- lems will b e simpler f o r a n empirical study t h a t considers only t h e effects context and

tests

t h e implications (a)-(f) in Theorem 2 of a reward-penalty model. Such

a

study could o f f e r indirect evidence

as to

t h e descriptive accuracy of a reward- penalty model f o r different types of persons and under different circumstances.

The evidence is strong t h a t

w e

humans have systematic and stubborn weaknesses in o u r ability

to

process information on

our

p r e f e r e n c e s and o u r pro- bibalistic beliefs (see, f o r example, Kahneman, Slovic, and Tversky, 1982 and t h e review by Schoemaker, 1982). For this very reason, t h e distinction between descriptive models and prescriptive models can b e useful; descriptive models have t h e purpose of aiding

our

understanding

as

scientists of how people make decisions and prescriptive models have t h e purpose of aiding o u r ability as decision makers

to

process information f o r making decisions (see, f o r example, Bell, Raiffa, and Tversky, 1984, Howard, 1980, Keeney, 1982, and Raiffa, 1961.) Leaving aside t h e operational difficulties of assessing simple preferences, this argument depends on t h e premise t h a t a p r e s c r i p t i v e model can c a p t u r e t h e major p r e f e r e n c e concerns of t h e decision maker. Thus,

it

is idso important

to

distinguish between those behavioral deviations from a p r e s c r i p t i v e model t h a t

are

due

to

suboptimal infor- mation processing, i.e., heuristic biases, on t h e p a r t of t h e decision maker and those deviations t h a t are due

to

p r e f e r e n c e concerns of t h e decision maker t h a t

are

not included in t h e model.

The reward-penalty model i s prescriptive r a t h e r t h a n descriptive in t h a t i t is not intended

to

model

a

person's heuristic biases. R a t h e r , i t i s intended

to

identify those p s y c h o l o g i d e f f e c t s that

are

r e g a r d e d by the decision maker

as

a n impor- t a n t a s p e c t of his consequences (and t h a t cause his p r e f e r e n c e s

to

deviate from t h e p r e f e r e n c e condition of framing consistency). The recognition of t h e s e effects in t h e model is

to

b e r e g a r d e d

as

t h e inclusion of

a

p r e f e r e n c e issue of importance to t h e decision maker r a t h e r than

as

t h e inclusion of systematic imperfections in t h e reasoning of t h e decision maker. Thus, t h e purpose of t h e reward-penalty model is

m o r e

r e s t r i c t e d than t h e purposes of such general models as p r o s p e c t theory (Kahneman and Tversky, 1979), weighted utility theory (Chew and MacCrim- mon, 1984), and SSB theory (Fishburn, 1982).

In addition

to

t h e question of whether averse-prone choice behavior may ac- curately r e f l e c t a decision maker's p r e f e r e n c e s , and in t h a t sense is rational, t h e r e is also t h e question of whether such choice behavior i s in t h e b e s t i n t e r e s t of a n organization

or

of society, and in t h a t sense is ethical. The psychological qualities t h a t

are

hypothesized in Section 1

to

b e associated with averse-prone choice behavior are not flattering. In t h e case of a n individual who is acting on his own behalf, t h e s e qualities can b e negatively described

as

ego-centric and inflexi- ble; t h e person would r a t h e r r i s k losing a l a r g e r amount than admit even

to

himself t h a t h e h a s suffered a loss. In t h e case of a n individual who is acting on behalf of

a n organization, t h e r e is t h e additional quality t h a t his concern f o r his own repu- tation may

at

times b e

at

variance with concerns f o r t h e effectiveness of t h e

or-

ganization; in common parlance, t h e person may be primarily concerned with "cov- ering his ass."

Thus, f o r decision analysis applications in which a n averse-prone r i s k attitude is a n important f e a t u r e of t h e decision maker's p r e f e r e n c e s , t h e r e may b e cogent reasons f o r formulating

t w o

versions of t h e decision analysis model, one t h a t in- cludes

n

and p

as

additional parameters. and one t h a t does not (and thus assumes t h a t p r e f e r e n c e s are framing consistent). Both t h e effects context and t h e no- effects context would t h e r e b y by considered.

This caution

as

r e g a r d s t h e modeling of a decision maker's complete p r e f e r - e n c e s is similar

to

t h e point made in Bell (1985)

"that

what i s c u r r e n t l y omitted from expected utility analysis deserves

to

b e omitted and t h a t a formal analysis may b e exactly what is needed

to

prevent a decision maker's intuition from forcing economically inefficient decisions [italics Bell's].

"

This point is made

also

in Raiffa (1984). In a decision analysis application f o r a n organization, t h e most important p r e f e r e n c e s

to

b e examined may b e those of a n exemplifying individual r e p r e s e n t - ing t h e e n t i r e organization r a t h e r than those of any single individual within t h e

or-

ganization.

Appendix: Proofs of resulta

Proof of Theorem 1. If u ( z ) is any utility function f o r t h e no-effects context, then

u

( z )

=

U(z , z l*

,..., zn

* ) f o r some utility function U

as

described in (2). If w ( z ) is any utility function f o r t h e e f f e c t s context, then w ( z )

=

f i ( z , r l(z),... , r n ( 2 ) ) f o r some (possibly different) utility function

fi as

in (2).

hforeover,

fi =

a U

+

b f o r some constants a

>

^{0 and b .} Therefore,

Proof of

Theorem

2. Assume t h a t t h e utility function

u

f o r t h e no-effects context is normalized so t h a t

u

( 0 )

=

^0. Let L denote t h e

lottery to

b e compared.

To prove (a), observe t h a t f o r

z <

0 , w ( p z )

=

u ( p z

-

n)

=

u(-n)

+

o ( 1 ) while w ( 1 )

=

p w ( z )

+

q w ( 0 )

=

p u ( z - n )

=

p u ( - n )

+

o ( 1 ) . Thus p u ( ~ r )

> u(--rr)

implies w ( 1 )

>

w ( p z ) f o r

z <

⁰ sufficiently n e a r

to

0.

To p r o v e (b), o b s e r v e t h a t f o r

z >

0 , w ( p z )

=

u ( p z

+

^p) while w ( 1 )

=

PU ( 2 + P )

+

q u ( 0 ) . Since

u

is strictly concave and p r 0 ,

+ p) + q u ( O )

<

u ( p ( z + p ) ) S u ( p z + p ) , a n d t h u s w ( 1 )

<

w ( p z ) . To prove (c), i t suffices

to

consider

z >

^{0 .} Then,

since p

<

n and u is s t r i c t l y increasing.

To p r o v e (d),

w e

must show t h a t w ( 1 ) = p u ( z

-

ⁿ⁾ is less than w ( p z )

= u

^{( p z} -n) f o r sufficiently l a r g e negative

z .

A s

a

f i r s t observation, t h e condition of unbounded utility from below is well-known

to

imply t h a t t h e strictly increasing function ^{U ( Z )} is unbounded from below (hence t h e name). There are

t w o cases

t o consider: f i r s t , t h a t in which t h e amounts

z

have

a

finite lower bound a (with a

<

-n so t h a t w

( z )

i s defined f o r some

z <

^{0 )} and lim

u ( z ) = -,

and

= + a +

second, t h a t in which t h e amounts

z

have no lower bound and lirn u

( z ) = -.

r -.-

In t h e f i r s t case, r -(a +n)+ l i m p u

( z

-7r)

= -

^while r +(a +n)+ ^{lim u ( p z -n)}

⁼

u

( a +(p -l)(a + n ) ) is finite since (p -l)(a +n)

>

0. Therefore, p u

( z

--T)

< u

^(pz -n) f o r all

z

less than some amount

zp

between a

+

n and 0.

In t h e second c a s e , t h e condition of decreasing risk aversion implies t h a t lirn

u ' ( z ) = +-.

To show this, note t h a t

u

" ( 2 ) 6 0 f o r all

z

implies

that u ' ( z )

is r -.-

decreasing f o r all

z .

Thus, lirn

u ' ( z )

is

a

finite number b o r is ^+a. If

r

--

lirn

u ' ( z ) =

b

,

then lim s u p ~ " ( 2 )

=

0 and hence lirn inf -u

" ( z ) / u

' ( 2 )

=

0.

r

--

^{r +- r--}

However, decreasing risk aversion implies t h a t t h e local risk aversion function -u " ( 2 )/

u

^{' ( 2} ) i s positive and decreasing ( P r a t t , 1964). and this p r o p e r t y contra- dicts t h e previous statement.

Now, assume t h a t 0

<

p

<

1 i s fixed. If n

=

0 , then f o r any

z <

0 t h e line from t h e point

( z , u

( 2 ) )

to

t h e point (0,O) lies below the graph of

u ,

and t h e r e f o r e p u ( z )

+

q u ( 0 ) = p u ( z )

<

u ( p z

+

9.0)

=

u ( p z ) .

If n

>

0 , then t h e r e exists a number

z o < -

n such t h a t t h e line from

(zo,u ( z o ) )

t o (-n.0) lies above t h e g r a p h of

u .

Since lirn r +- u ' ( 2 )

= -

^and

^{u ( z )}

^{i s}

unbounded from below, t h e r e exists a unique

z l < z o

such t h a t this line intersects

t h e g r a p h of u

also

at ( z l , u (xi)). Moreover,

for

any zp < z l , t h e line

f r o m

( z p , u ( z p ) ) to (-n,O)

w i l l also

i n t e r s e c t t h e graph of u at a unique point (z; , u (z;

))

with z0 <

^Z;

< -

^TT.

Choosing zp so t h a t pzp -

^TT

< z; ,

it

follows t h a t f o r any amount z < zp , pu ( z - TT) +

q

- 0 < u

(p

(Z -TT) +

q

(-TT)) and hence

pu

(Z -TT) < u (pz -TT)

as w a s

to

be

shown.

To p r o v e (e), note t h a t t h e certainty equivalent

of t is

zo = u

-'(pa

( z ) +

qu

( z

'))

in t h e no-effects context and zl = w -l(pw ( z ) +

^qw

( z

'))

where w ( t

)

= u ( t -TT)

is

t h e

effects

context. Since u r e p r e s e n t s a n attitude of de- creasing r i s k aversion,

it follows

by

a

r e s u l t in P r a t t (1964, Theorem

2)

t h a t

Z1

<

^Zo.

Finally, t h e

proof of

u )

is similar

to t h a t

of (e)

above e x c e p t t h a t w ( t ) = u ( t + p), p > 0, and hence zl > zo.

Proof of

Theorem

3.

Suppose t h a t f o r t w o different c u r r e n t asset positions

c

<

^c',

t h e p r e f e r e n c e relations kk and satisfy t h e condition

of

a n averse-

p r o n e risk attitude. Then, framing consistency implies t h a t t h e p r e f e r e n c e

rela-

tion &

is

r i s k a v e r s e on t h e r a n g e

y

>

c

and r i s k prone on t h e r a n g e

y

<

^{c ' .}

For t h e intersection range, c <

^y

< c

',

t h e s e r i s k attitudes are contradictory.

Proof of

Theorem

4. To show part

(a), w e

w i l l relate

t h e condition of sum- dominance consistency to t h e condition of a constant r i s k attitude. Let I denote t h e interval

of possible

monetary changes z. One definition H m e y (1981, 1986)

of a

constant risk attitude

is

t h a t

for

any amounts

h l

<

h 2

<

h 3

and any probability 0

< p

< I , if

f o r some z such t h a t z + h i , z + h o , and z +

h 3

are in I, then

f o r a n y z ' s u c h t h a t z ' +

hl.

z' + h 2 , a n d z ' + h 3 a r e i n I .

For

a

p r e f e r e n c e relation 2 t h a t satisfies t h e conditions

of

expected utility

and such t h a t l a r g e r amounts z

are

p r e f e r r e d , t h i s condition holds if and only if 2

c a n

be

r e p r e s e n t e d by a utility function of t h e form

f o r some parameter value

r

(Harvey, 1986). Note t h a t no assumptions of differen- tiability o r even of continuity of t h e utility function

u

( z ) are required.

Assume t h a t t h e r e is a constant risk attitude and, in particular, t h a t ;G is represented by a utility function u ( z )

=

e x p ( r z ) f o r some

r >

0. Consider amounts

zi

, Z i i

=

1,2,3, and a probability 0

<

p

<

1 as in Definition 4. Then,

Therefore, t h e preferences (9) imply t h a t

which implies t h a t (10) i s false. Thus,

2

is sum-dominance consistent. Similar ar- guments can b e given f o r t h e

cases u

( 2 )

= z

and

u

( z )

=

--exp(rz ),

r <

0.

To show t h e converse implication, f i r s t assume t h a t t h e r e i s not a constant risk attitude but t h a t t h e utility function

u

( z ) i s continuous. Then, t h e r e exists amounts

z +

h i ,

z ' +

h i , i

=

1,2,3, in I and a probability 0

<

p

<

1 such t h a t (Al) is t r u e but (A2) i s false. By

a

slight change in p , i t follows t h a t t h e indifferences in (Al), (AZ) can b e replaced by opposite preference. Without loss of generality, assume t h a t t h e r e i s t h e p r e f e r e n c e s

+

^{in (Al)} and t h e preference 4 in (A2).

Then, by t h e continuity of u ( z ) , these preferences will remain t r u e when

z +

h l in (Al) is changed

to a

slightly smaller amount

z +

h i

-

^{d , d}

^>

0. I t follows t h a t f o r

~ ~ = ~ + h ~ - d , ~ 2 = ~ + h ~ , z 3 = z + h 3 and Z i = Z ' + h l , Z i = z f + h 2 ,

z; = z' +

h 3 t h e conditions (9) and (10) are satisfied, i.e., t h e r e is

a

violation of sum-dominance consistency.

Second, assume t h a t t h e utility function u ( z ) is not

a

continuous function on t h e interval I. Since

u

( z ) i s strictly increasing, i t has only a finite

or

countably infinite number of points of discontinuity. Thus,

w e

may choose a point

z l at

which

u

( z ) is discontinuous and

a

point

Z i at

which

u

( z ) is continuous. Suppose, f o r ex- ample, t h a t sup

{u

( 2 ) :

z < z l j < u

(zl). Choose h

>

h '

>

0 such t h a t

z 2 = z 1 +

h a n d z i

= z i +

h ' a r e i n 1 . There e x i s t s a p r o b a b i l i t y 0 ^{< p} < l s u c h t h a t

f o r all

z < z

Since

u

( z ) is continuous

at z i

, w e can choose an

z j < z i

suff i- ciently n e a r

to z i so

t h a t

Now, choose

z3

sufficiently n e a r t o

zl

s o t h a t

z l -

^{2 3 is}

^less

^than

Z i - z$ .

T h e n , z l + ( z i + h f ) < ( z l + h ) + z i a n d z l + z j < z 3 + z i . H e n c e , t h e r e i s a violation of sum-dominance consistency. A similar argument can be given f o r t h e case i n f i u ( z ) :

z

> z l ]

>

u ( z l ) .

W e will derive p a r t (b) of Theorem 4 from p a r t (a) by means of a change of variable argument. Suppose that t h e decision maker's c u r r e n t fund is denoted by a variable y

=

c

+ z

where c is his initial fund and

z

is t h e subsequent net change in t h e fund. W e assume t h a t y

>

0 f o r all net gains o r losses

z

in t h e interval I.

Consider a new variable, w

=

log y

.

The preference relation associated with y , which w e will denote h e r e by

h e

induces a preference relation associated with w , which

w e

will denote by

&,.

I t is well-known that t h e preference relation

;L21

satisfies t h e condition of a constant proportional risk attitude if and only if t h e preference relation

&

satis- fies the condition of a aonstant risk attitude (see, e.g., Harvey, 1986 f o r a detailed discussion). Thus, t o prove p a r t (b), i t suffices

to

show t h a t is percent- dominance consistent if and only if &, is sum-dominance consistent.

A lottery

<

^p (z X), q (z 'X)

>

expressed in terms of percent changes z

,

z ' is equivalent t o t h e lottery

<

p S b

,

^q Sk 'y

>

with k

=

1

+

(z / l o o ) , k '

=

1

+

(z '/ 100) expressed in terms of asset positions. Therefore. Definition 5 can be restated as t h e condition t h a t f o r any probability 0

<

p

<

1 both of t h e preferences

do not occur whenever

klk;

<

k 2 k i and k l k j < k 3 k i

.

^(A41

However. a lottery

<

p *ky

,

^q *k 'y

>

f o r t h e preference relation

&

corresponds t o a lottery

<

p (Log k

+

Log y ), q (Log k '

+

Log y )

>

f o r t h e preference relation

&,.

Thus, t h e r e exist amounts ki , k i

,

i

=

1.2.3. such t h a t (A3), (A4)

are

satisfied if and only if t h e r e exist amounts z i ,

Zi

^{, i}

=

1.2,3, such that (9), (10)

are satis-

fied.

P a r t (c) follows immediately from p a r t s (a) and (b) since a utility function

u

( y ) is both linear-exponential and logarithmic-power if and only if it is linear.