Journal of Risk and Uncertainty, 20:5᎐44 Ž2000.
䊚 1999 Kluwer Academic Publishers. Manufactured in The Netherlands.
Utility Functions for Wealth
DAVID E. BELL
Har¨ ard Business School, Boston, MA 02163
PETER C. FISHBURN
AT & T Labs-Research, Florham Park, NJ 07932
Abstract
We specify all utility functions on wealth implied by four special conditions on preferences between
risky prospects in four theories of utility, under the presumption that preference increases in wealth.
The theories are von Neumann-Morgenstern expected utility ŽEU., rank dependent utility ŽRDU.,
weighted linear utility ŽWLU., and skew-symmetric bilinear utility ŽSSBU.. The special conditions are a
weak version of risk neutrality, Pfanzagl’s consistency axiom, Bell’s one-switch condition, and a
contextual uncertainty condition. Previous research has identified the functional forms for utility of
wealth for all four conditions under EU, and for risk neutrality and Pfanzagl’s consistency axiom under
WLU and SSBU. The functional forms for the other condition-theory combinations are derived in this
paper.
Key words: expected utility, rank dependent utility, weighted utility, skew-symmetric bilinear utility,
consistency axiom, one-switch condition, contextual uncertainty
JEL Classification: D81
There are two predominant interpretations for utility of wealth in theories of
decision among risky prospects with monetary outcomes. The first, which dates
from Bernoulli Ž1738., treats utility of wealth uŽ w . as a measure of a person’s
intensity of preference for wealth w that is to be assessed in a strength-of-preference manner without reference to risk or outcome probabilities. The second, which
originated in von Neumann and Morgenstern Ž1944., eschews the earlier intensity
interpretation and views uŽ w . as a consequence of simple preference comparisons
between risky prospects that is assessed with the aid of outcome probabilities,
largely through indifference judgments. In-depth analyses of the two interpretations are presented in Ellsberg Ž1954. and Fishburn Ž1989., but see also Allais
Ž1953, 1979. for a dissenting opinion which maintains that von Neumann and
Morgenstern intended a Bernoullian interpretation for their notion of utility.
The present paper lies squarely in the tradition of von Neumann and Morgenstern. Its aim is to specify all functional forms for uŽ w . and related functions that
occur in four types of utility representations when special conditions on prefer-
6
BELL AND FISHBURN
ences are imposed on their structures. The four utility theories are von NeumannMorgenstern expected utility and three generalizations: rank dependent expected
utility, weighted linear utility, and skew-symmetric bilinear utility. Our special
conditions, also four in number, range from a restrictive condition of risk neutrality
to Bell’s Ž1988. more accommodating one-switch rule.
We assume throughout that more wealth is preferred to less. Although psychologically innocuous, this is important mathematically because it rules out nonmonotonic solutions to functional equations that arise in our derivations.
Section 2 outlines the numerical representations of the four utility theories we
consider and describes the four special conditions we apply to those theories. The
most flexible special condition is the one-switch rule which says that if preference
between two lotteries on increments to wealth reverses as wealth increases, then no
further reversals occur beyond the initial switch point. Section 3 presents our
results in theorems which show how the special conditions affect the numerical
representations. The most prevalent special forms involved exponential terms such
as e c w , but other cases arise. We note which results have been proved earlier and
which have not. The results are summarized in a 4 = 4 array which shows the
effects of each Žtheory, special condition. combination.
Sections 4 through 7 cover the proofs of new results. Section 4 focuses on rank
dependent utility, Sections 5 and 6 consider the one-switch rule for skew-symmetric
bilinear utility and weighted linear utility, respectively, and Section 7 deals with a
contextual uncertainty condition.
We have mentioned the Bernoullian tradition of intensive utility to clarify what
our study is not about as well as what it is about. To explain this further, we first
recall the expected utility model of both interpretations.
Throughout the paper, a risky prospect is defined as a random function w
˜ with
.
Ž
.
finite support SŽ w
on
levels
of
wealth.
When
w
has
probability
p
w
for
w
g
SŽ w
˜
˜
˜ .,
we have ÝSŽ w˜ . pŽ w . s 1 and refer to p as the lottery associated with w.
Under
the
˜
obvious convention that pŽ w . s 0 for w f SŽ w
˜ ., each lottery is a simple probability distribution on all wealth levels. With respect to a utility function u on levels of
wealth, the expected utility of w
˜ with associated lottery p is defined by
Eu Ž w
˜. s
Ý
p Ž w . uŽ w . .
Ž 1.
wgS Ž w̃ .
This form is used by Bernoulli and by von Neumann and Morgenstern, and in both
cases risky prospect w
˜ is regarded at least as good as w⬘
˜ if and only if EuŽ w
˜. G
EuŽ w⬘
˜ .. However, the two approaches differ radically in their interpretation and
assessment of u.
As indicated earlier, Bernoulli’s u is intended to reflect a person’s intensities of
preference for levels of wealth. It is riskless and logically precedes, or is at least
independent of, probability considerations. On the other hand, von Neumann and
Morgenstern’s u reflects attitudes toward risk and is inextricably tied to preference
UTILITY FUNCTIONS FOR WEALTH
7
and indifference judgments between lotteries. The difference between the two
interpretations impacts our study in at least three ways.
First, the special forms for u and related functions specified later should not be
viewed as intrinsic, intensive measures of utility. They are merely consequences of
Ž1. in the expected utility case, or other representational models in generalized
cases, and of our special conditions.
Second, the logarithmic form of u favored by Bernoulli, which can be written as
u Ž w . s log Ž 1 q cw .
for w G 0
Ž 2.
for some positive constant c, will not be found among our special forms. In
particular, our special conditions do not permit this form even though it would be
allowed by other conditions such as a relative-risk-constant condition in Harvey
Ž1990, p. 1485.. Bernoulli argued for Ž2. with an early expression of the law of
diminishing marginal utility which says that the increment of utility for the next bit
of wealth ought to be inversely proportional to the amount of wealth prior to the
incremental increase. Support for Ž2. as an intensive measure was obtained by
Allais in 1952 from questionnaire responses that are reported in Allais Ž1979..
Although Savage was definitely not a Bernoullian in the sense used here, he says in
Savage Ž1954, p. 94. that ‘‘To this day, no other function has been suggested as a
better prototype for Everyman’s utility function.’’ Perhaps Savage would have
reconsidered had he known about results in Bell Ž1988, 1995a, 1995b.. The reason
is that, within the von Neumann-Morgenstern interpretation that Savage favored,
u Ž w . s aw y beyc w
Ž 3.
for constants a G 0 and b, c ) 0, is the only increasing Ž u⬘ ) 0. and risk averse
Ž u⬙ - 0. form that satisfies the one-switch and contextual uncertainty conditions in
the context of Ž1..
Our third point concerns the treatment of special conditions in Ž1. and its
generalizations. In our approach, we begin with Ž1. or a generalization such as rank
dependent utility or weighted linear utility on the basis of preference axioms which
correspond to the theory under consideration. We then ask how the special
conditions constrain the functions involved in the representation. Although there
are also generalizations of Ž1. in the Bernoullian approach, as described for
example in Allais Ž1953, 1979. and Hagen Ž1972, 1979., our approach is inadmissible there because u exists independently of risk or special conditions on preferences between risky prospects. It follows that special conditions in Bernoullian
theory can only be accounted for by modifying the expectation rule of Ž1. for
combining probabilities and utilities, not by modifying u. This is a perfectly
legitimate route of inquiry, but it is not pursued here apart from alternatives to Ž1.
that generalize von Neumann-Morgenstern expected utility prior to imposition of
our special conditions.
8
BELL AND FISHBURN
1. Utility theories and special conditions
The numerical utility representations of our study apply to a strict preference
relation % on the set Wq of all risky prospects w,
˜ ˜x, ˜y, . . . whose supports are
subsets of the set ⺢q of nonnegative real numbers. We could also allow negative
wealth with some fixed lower bound since this would not entail any substantial
changes under the translation that treats the lower bound in the way we treat zero
wealth in what follows. The indifference relation ; and preference-or-indifference
relation ' induced by % are defined for ˜
x, ˜
y g Wq by
˜x ; ˜y if neither ˜x % ˜y nor ˜y % ˜x,
˜x ' ˜y if ˜x % ˜y or ˜x ; ˜y.
We denote the set of simple probability distributions on ⺢q by Pq and let w
˜lp
mean that p g Pq is the lottery associated with w
˜ g Wq. Preference on Pq is
defined in the obvious way from preference on Wq by
p % q if ˜
x %˜
y when ˜
x l p and
˜y l q.
The degenerate risky prospect that assigns probability 1 to wealth w is denoted
simply by w, so the assumption that more wealth is preferred to less can be
expressed as
x%y
when x ) y G 0.
Ž 4.
We assume Ž4. throughout.
Our presumption that wealth is nonnegati¨ e has a practical appeal, but it is also
important in our derivations and a few results. For example, one special form for u
that arises later is the quadratic form in which
u Ž w . s aw 2 q bw for w G 0,
with a G 0, b G 0 and a q b ) 0 to satisfy Ž4. for constants a and b. If wealth
were unbounded below, with x % y m x ) y, then the quadratic case would reduce
to the linear case of uŽ w . s w because any nonzero a would violate monotonicity.
On the other hand, factors such as borrowing and catastrophic events can render
net wealth negative. To account for this, it is natural to replace ⺢q, Wq, and Pq in
the preceding definitions by ⺢, the set W of all risky prospects whose supports are
subsets of ⺢, and the set P of simple probability distributions on ⺢, respectively.
Most of the results in the next section for % on Wq hold also for the
unconstrained case of % on W, and when they do not we will say so. But proofs
will be given only for % on Wq, which tends to be the more delicate case because
of the w G 0 constraint.
9
UTILITY FUNCTIONS FOR WEALTH
Expectations
Our four utility representations for % on Wq are identified by EU Žexpected
utility., RDU Žrank dependent utility., WLU Žweighted linear utility., and SSBU
Žskew-symmetric bilinear utility.. Each is based on a notion of expected value,
which is Ž1. for EU and WLU. For WLU, we work with a ratio Eu1Ž w
˜ .rEu2 Ž w
˜.
with u 2 ) 0. This reduces to the EU representation when the weighting function
u 2 is constant.
For RDU, we define ⌸ as the set of increasing functions from w0, 1x onto w0, 1x.
Members of ⌸ are viewed as transformations of objective probabilities of lotteries
into psychologically equivalent subjective probabilities. Each ␲ g ⌸ is continuous
and increasing with ␲ Ž0. s 0 and ␲ Ž1. s 1. When SŽ w
˜ . has n members ordered as
w 1 - w 2 - ⭈⭈⭈ - wn , and w
˜ l p, the rank dependent expectation of u with respect
to ␲ g ⌸ is defined by
n
E␲ u Ž w
˜. s
Ý u Ž wk .
ks1
½
k
␲
Ý p Ž wj .
ky1
y␲
js1
Ý p Ž wj .
js1
5
.
Ž 5.
This reduces to EuŽ w
˜ . when ␲ is the identity transform with ␲ Ž ␭. s ␭.
For SSBU, let ⌽ denote the set of skew-symmetric functions ␾ from ⺢q= ⺢q
into ⺢ for which ␾ Ž x, y . ) 0 when x ) y G 0. The latter constraint corresponds to
Ž4. since the SSBU representation has ␾ Ž x, y . ) 0 m x % y. Skew-symmetry for
␾ g ⌽ means that
␾ Ž x, y . q ␾ Ž y, x . s 0 for all x, y g ⺢q.
Ž 6.
In particular, ␾ Ž w, w . s 0. The bilinear expectation of ␾ g ⌽ for the ordered pair
Ž˜
x, ˜
y . of risky prospects ˜
x and ˜
y with ˜
x l p and ˜
y l q is defined by
E␾ Ž ˜
x, ˜
y. s
Ý
Ý
p Ž x . q Ž y . ␾ Ž x, y . .
Ž 7.
xgS Ž ˜
x . ygS Ž ˜
y.
This reduces to the EU form when ␾ Ž x, y . has the decomposition ␾ Ž x, y . s uŽ x .
y uŽ y ., for then E␾ Ž ˜
x, ˜
y . s EuŽ ˜
x . y EuŽ ˜
y .. More generally, given the representation ˜
x %˜
y m E␾ Ž ˜
x, ˜
y . ) 0, it reduces to the WLU form when ␾ Ž x, y . can be
written as u1Ž x . u 2 Ž y . y u1Ž y . u 2 Ž x . with u 2 ) 0, for then E␾ Ž ˜
x, ˜
y . ) 0 is equivalent to Eu1Ž ˜
x .rEu2 Ž ˜
x . ) Eu1Ž ˜
y .rEu 2 Ž ˜
y ..
Four representations
We now specify our four numerical representations for % on Wq along with
admissible transformations for their functions, which are used later to obtain the
simplest forms for those functions under the special conditions described shortly.
10
BELL AND FISHBURN
Although there is no need here to go into the details of axioms which correspond
to the representations, a few words about axioms may provide perspective.
Representations like those that follow are usually axiomatized in terms of the
behavior of % on a convex set P* of lotteries or probability distributions on a set
of outcomes, which in our particular setting are wealth levels. Convexity means
that ␭ p q Ž1 y ␭. q is in P* whenever p, q g P* and 0 F ␭ F 1, where the convex
combination assigns probability ␭ pŽ x . q Ž1 y ␭. q Ž x . to outcome x.
Three axioms are typically used for EU: see, for example, Herstein and Milnor
Ž1953., Jensen Ž1967. and Fishburn Ž1970, 1988.. They are an ordering axiom, an
independence condition, and an Archimedean axiom to ensure real-valued utilities.
The independence condition, which in one version says that
p % q « ␭ p q Ž 1 y ␭. r % ␭ q q Ž 1 y ␭. r
Ž 8.
whenever p, q, r g P* and 0 - ␭ - 1, is central to the derivation of representation
by expected utilities.
Common violations of Ž8. described in Allais Ž1953, 1979., MacCrimmon and
Larsson Ž1979., and Kahneman and Tversky Ž1979., among others, motivated the
other representations considered here, each of which weakens Ž8. to accommodate
some of the observed violations. Axioms for RDU are discussed in Quiggin Ž1982,
1993., Quiggin and Wakker Ž1994., and Segal Ž1989, 1993.; axioms for WLU appear
in Chew and MacCrimmon Ž1979., Chew Ž1982., Fishburn Ž1983, 1988., and
Nakamura Ž1984.; SSBU is axiomatized in Fishburn Ž1982, 1988.. The last of these
also weakens the ordering axiom to allow preference cycles such as p % q % r % p.
However, if transitivity of ; on P* is added to the SSBU axioms, then SSBU
reduces to WLU.
Our four representations are: for all ˜
x, ˜
y g Wq,
EU:
˜x % ˜y m Eu Ž ˜x . ) Eu Ž ˜y .
RDU:
˜x % ˜y m E␲ u Ž ˜x . ) E␲ u Ž ˜y .
WLU:
˜x % ˜y m Eu1 Ž ˜x . rEu2 Ž ˜x . ) Eu1Ž ˜y . rEu2 Ž ˜y .
SSBU:
˜x % ˜y m E␾ Ž ˜x, ˜y . ) 0,
where u, u1 and u 2 map ⺢q into ⺢, u is strictly increasing, u 2 ) 0, u1Ž x .ru 2 Ž x . )
u1Ž y .ru 2 Ž y . when x ) y, ␲ g ⌸ and ␾ g ⌽. The theorems in the next section
invoke additional smoothness conditions for u, u1 , u 2 and ␾ because they are
either implied by our special conditions or facilitate derivation of the special forms,
for example by the use of differential equations. The following uniqueness properties, or admissible transformations, do not presume the smoothness conditions.
For EU, u is unique up to a positi¨ e affine transformation: ¨ satisfies the
representation in place of u if and only if there are constants a ) 0 and b such
11
UTILITY FUNCTIONS FOR WEALTH
that
¨ Ž w . s au Ž w . q b
for all w G 0.
Ž 9.
If ¨ does not satisfy Ž9. with a ) 0, then E¨ Ž ˜
x . F E¨ Ž ˜
y . for some ˜
x and ˜
y for
which the given % has ˜
x %˜
y.
For RDU, ␲ is unique and u is unique up to a positive affine transformation.
For WLU, u1 and u 2 ) 0 are unique up to the transformation pair
¨ 1 Ž w . s au1 Ž w . q bu 2 Ž w .
for all w G 0
¨ 2 Ž w . s cu1 Ž w . s du 2 Ž w .
for all w G 0
Ž 10 .
with ad ) bc and cu1Ž w . q du 2 Ž w . ) 0 for all w G 0: see Fishburn Ž1988, p. 132..
Under smoothness assumptions Žcontinuity suffices . we can have ¨ 1 ) 0 as well as
¨ 2 ) 0 for our w G 0 context by taking a s 1, b suitably large, c s 0 and d s 1.
For SSBU, ␾ is unique up to a positi¨ e multiplicati¨ e transformation: ␺ satisfies
the representation in place of ␾ if and only if there is an a ) 0 such that
␺ Ž x, y . s a␾ Ž x, y .
for all x, y g ⺢q.
Ž 11 .
Every other transformation in ⌽ has E␺ Ž ˜
x, ˜
y . F 0 for some ˜
x and ˜
y for which
x˜ % ˜
y.
Four special conditions
A few new definitions are involved in our special conditions. When w
˜ g Wq,
q
q Ž
x g ⺢ and w
˜ ; x, x is the certainty equi¨ alent of w.
˜ For x, y g ⺢ , x, 1r2, y . is
the e¨ en-chance prospect that assigns probability 12 to each of x and y when x / y,
and assigns probability 1 to x when x s y.
The next definition is stated in the unrestricted W mode. Given ˜
x j g W, ␣ j g ⺢,
and ˜
x j l pj for j s 1, . . . , m, the weighted con¨ olution Ý ␣ j ˜
x j is defined as the risky
prospect associated with p g P given by
pŽ y . s
Ý
Ž x1, . . . , x n.
½
m
p1 Ž x 1 . p 2 Ž x 2 . ⭈⭈⭈ pm Ž x m . :
Ý ␣j xj s y
js1
5
,
Ž 12 .
for all y g ⺢. Each ˜
x j in this definition, including identical ˜
x j ’s with different
indices, is treated as an independent random function for the probability calculation. Some examples follow.
12
BELL AND FISHBURN
Suppose ˜
x has probability
2˜
x has pr.
1
3
1
3
for 2 and pr.
for x s 1 and probability
2
3
2
3
for x s 3. Then
for 6;
˜x q ˜x has pr. 91 for 2, pr. 94 for 4 and pr. 94 for 6.
Note in particular that ␣ ˜
x q ␤˜
x is not generally equal to Ž ␣ q ␤ . ˜
x. In addition, if
y˜ has probability 34 for y s 2 and probability 14 for y s 4, then
˜x q ˜y has pr.
3
12
for 3, pr.
˜x q ˜y y 3 has pr.
3
12
7
12
for 5 and pr.
for 0, pr.
7
12
2
12
for 7;
for 2 and pr.
2
12
for 4.
In terms of Ž12., the latter case has Ž ˜
x1, ␣ 1 . s Ž ˜
x, 1., Ž ˜
x2 , ␣2 . s Ž ˜
y, 1. and Ž ˜
x3, ␣3.
s Ž3, y1..
We also let ˜
xq for ˜
x g W denote the set of all w g ⺢ for which ˜
x q w g Wq.
Equivalently,
˜xqs w g ⺢: min x q w : x g S Ž ˜x . 4 G 0 4 .
Then ˜
xql ˜
yq is the set of w g ⺢ for which both ˜
x q w and ˜
y q w are in Wq.
Our first special condition, C1, is a limited version of risk neutrality which says
that some risky prospects with identical mean wealths are indifferent.
C1. For all x, y, z g ⺢q, Ž x, 12 , y q z . ; Ž x q y, 12 , z ..
An even simpler condition, Ž x q y .r2 ; Ž x, 12 , y ., is equivalent to C1 when ; is
transitive, but not otherwise ŽFishburn, 1998.. C1 is very strong and seems plausible
only when x and z are approximately equal or y is comparatively large. For EU
and RDU, it implies uŽ x . s ax q b, a ) 0, or uŽ x . s x under affine rescaling. Its
effects for WLU and SSBU are less restrictive. For example, under SSBU, it says
that ␾ Ž x, y . depends only on the difference x y y, and it is consistent with the
existence of preference cycles.
Our next two special conditions focus on how preference between w q ˜
x and
w q˜
y in Wq changes as w varies with ˜
x and ˜
y fixed. The more restrictive of the
two, C2, is tantamount to Pfanzagl’s consistency axiom ŽPfanzagl, 1959. when all
risky prospects have certainty equivalents. Pfanzagl’s axiom says that if x is the
certainty equivalent of w
˜ g Wq, then x q y is the certainty equivalent of w
˜qy
q
for y g w
˜ . It is also similar to conditions used in Rothblum Ž1975. and Farquhar
and Nakamura Ž1987.. We refer to C2 as the zero-switch condition ŽBell, 1988..
C2. For all ˜
x, ˜
y g W, if w q ˜
x ; w q˜
y for some w g ˜
xql ˜
yq, then w q ˜
x;w
q
q
q˜
y for all w g ˜
x l˜
y .
This also is fairly strong. For example, it implies uŽ x . s x or uŽ x . s ae a x , a / 0,
under EU or RDU. Its companion, Bell’s one-switch condition, says that preference
between w q ˜
x and w q ˜
y can reverse as w increases, but if this happens then it
UTILITY FUNCTIONS FOR WEALTH
13
happens only once with no further changes beyond the point of reversal. A closely
related condition that has a similar effect under EU is discussed in Farquhar and
Nakamura Ž1987, 1988..
C3. For all ˜
x, ˜
y g W, if w 1 q ˜
x % w1 q ˜
y and w 2 q ˜
y ' w2 q ˜
x for some w 1 , w 2
g˜
xql ˜
yq, then there is a unique w* g ˜
xql ˜
yq such that either
Ži.
Žii.
w q˜
x % w q˜
y for all w ) w*; w q ˜
y %w q˜
x for all w - w* in ˜
xql ˜
yq, or
q
w q˜
y %w q˜
x for all w ) w*; w q ˜
x % w q˜
y for all w - w* in ˜
x l˜
yq.
Let ˜
x s $3000 and ˜
y s Ž$0, 12 , $8000., and suppose that a person of modest
wealth w 0 has w 0 q ˜
x % w0 q ˜
y and w 0 q $5,000,000 q ˜
y % w 0 q $5,000,000 q ˜
x.
Then C3 asserts that, as t increases, preference for w 0 q t q ˜
x over w 0 q t q ˜
y
switches to the opposite preference and stays the same thereafter. Apart from the
issue of determining the change point precisely, we regard C3 as very appealing. It
is the least restrictive of our four special conditions and allows the most variety for
the functions in our four representations.
Even more liberal conditions than C3 would allow two or more reversals, as
when there are w 1 - w 2 - w 3 with
w1 q ˜
x % w1 q ˜
y
w2 q ˜
y % w2 q ˜
x
w3 q ˜
x % w3 q ˜
y.
We do not consider two-switch and higher-switch conditions further, but note that
the two-switch case is related to the doubly-inflected utility function in Friedman
and Savage Ž1948., and that examples of double switches arise later in our proofs
for C3. In addition, Bell Ž1988. demonstrates that the logarithmic function Ž2.
violates C3; indeed it permits any finite number of switches Žhis Proposition 9..
Our final special condition, C4, is based on a version of what Bell Ž1995b. refers
to as a contextual uncertainty condition ŽCUC. in the EU context. Its thrust is that
if you can resolve one of two independent subprospects or ‘‘contextual uncertainties’’ that are scale multiples of each other prior to making a decision, then you
would just as soon resolve the subprospect with the larger spread. Suppose for
example that the subprospects are ˜
z s Ž$0, 12 , $100. and 100 ˜
z s Ž$0, 12 , $10000..
Then, stated in the negative, CUC says that if resolution of 100 ˜
z would not affect
your choice between main prospects that include 100 ˜
zq˜
z, resolution of ˜
z rather
than 100 ˜
z also would not affect your choice. The version of CUC that we adopt
here is:
C4. For all k ) 1 and all ˜
x q kz˜ q ˜
z, ˜
y q kz˜ q ˜
z g Wq, if there is no a g SŽ ˜
z.
such that
˜x q ka q ˜z % ˜y q ka q ˜z,
14
BELL AND FISHBURN
then there is no a g SŽ ˜
z . for which
˜x q kz˜ q a % ˜y q kz˜ q a.
It is easy to see that C4 and CUC are equivalent for EU when CUC is defined by
the requirement that if ˜
x '˜
y when neither contextual uncertainty is resolved, and
if ˜
x '˜
y for every possible resolution of the larger contextual uncertainty, then
˜x ' ˜y for every possible resolution of the smaller contextual uncertainty. Then, by
definition, CUC implies C4. Moreover, if ˜
x '˜
y for all resolutions of the larger
contextual uncertainty, then it follows by taking expectations that ˜
x '˜
y if neither
contextual uncertainty is resolved. Hence C4 implies CUC.
The same argument shows that C4 and CUC are equivalent for SSBU because
its expectation mode is similar to that of EU. We shall see, however, that C4 and
CUC are not equivalent for RDU and WLU.
It was proved in Bell Ž1995b. that for EU, CUC is equivalent to the function V,
defined by
V Ž w . s Eu Ž w q ˜
x . y Eu Ž w q ˜
y. ,
being strictly monotonic whenever ˜
x and ˜
y switch in w. An equivalent result holds
for SSBU, namely that C4 holds if and only if V Ž w . s E␾ Ž w q ˜
x, w q ˜
y . is strictly
monotone in w whenever ˜
x and ˜
y switch. For RDU the implications are more
stark. The only cases for RDU that satisfy C4 require ␲ Ž p . s p, which reduces
RDU to EU. In other words, rank-dependent utility is identical to expected utility
when C4 is imposed. Finally, for WLU we show that C4 entails C3, and that two of
the four functions that satisfy C3 also satisfy C4.
2. Theorems
We organize our results into four theorems, one for each basic utility representation. Each theorem notes the effects of C1 through C4 in terms of the functional
form or forms allowed by the special conditions. Constants are denoted by a, b, c,
d and f, and restrictions on their values that are implied by Ž4. are noted in
brackets, e.g. w a / 0x and w a G 0, b G 0, a q b ) 0x. We say that a function is
smooth if it has derivatives of all orders. For convenience, continuity or smoothness
assumptions are stated at the beginning of a theorem and apply to each special
condition, but in many cases such properties are implied by Ž4. and the special
condition. For example, in Theorems 1 and 2, C1 or C2 and Ž4. imply continuity for
u, and C3 or C4 and Ž4. imply that uŽ w . is continuous for all w ) 0. For Theorems
3 and 4, the smoothness hypothesis is used in our proofs with C3 and C4, but
continuity suffices for the proofs with C1 and C2 as noted in Fishburn Ž1998..
The functional forms stated in each theorem have important implications for risk
attitudes and risky choice behavior in many applications, but we do not discuss
UTILITY FUNCTIONS FOR WEALTH
15
these here. They are described for the EU representation in some detail in Bell
Ž1988, 1995a, 1995b., and to a lesser extent for WLU and SSBU in Fishburn Ž1988,
Chapter 6. and other references given there. See also Quiggin Ž1993. for further
analyses of RDU.
The functional forms in the theorems are unique up to the admissible transformations noted in Ž9. ᎐ Ž11..
Theorem 1. Suppose % on Wq satisfies EU and u is continuous. Then:
C1 « uŽ w . s w;
C2 « Ži. uŽ w . s w, or
Žii. uŽ w . s ae b w w ab ) 0x;
C3 « Ži. uŽ w . s aw 2 q bw w a G 0, b G 0, a q b ) 0x, or
Žii. uŽ w . s ae b w q ce d w w ab ) 0, ab q cd G 0, d F b if cd - 0,
bŽ a q c . ) 0 if b s d x, or
Žiii. uŽ w . s aw q be c w w a q bc G 0, a ) 0 if bc s 0, b G 0 if c ) 0x, or
Živ. uŽ w . s Ž aw q b . e c w w ac G 0, a q bc G 0, a q bc ) 0 if ac s 0x;
C4 « C3Ži., or C3Žii. with bd F 0, or C3Žiii..
Proofs for C1 and C2 are included in Bell Ž1995a. and Fishburn Ž1998., C3 is
proved in Bell Ž1988, 1995a., and C4 in Bell Ž1995b.. Precisely the same forms hold
for the unrestricted case of % on W, except that a must then be set at 0 in the
quadratic form of C3Ži.. Generalizations of C3Žii. and C3Žiii. are characterized in
Nakamura Ž1996., and Harvey Ž1981, 1990. axiomatizes families of special u
functions that are closely related to those in Theorem 1.
It is easily seen for % on Wq that Ž4. and C3 Žone-switch. or C4 imply that u is
continuous on Ž0, ⬁.. If continuity at w s 0 were not presumed, two more EU
forms that are discontinuous at the origin arise for C3 as follows:
Žv.
Žvi.
uŽ0. s 0, uŽ w . s a q w for w ) 0 w a ) 0x,
uŽ0. s 0, uŽ w . s 1 y a q ae b w for w ) 0 w ab ) 0x.
The parts of these for w ) 0 are similar to C2Ži. and C2Žii., respectively, so Žv. and
Žvi. satisfy the zero-switch condition on the strictly positive domain.
Theorem 2. Suppose % on Wq satisfies RDU and u is continuous. Then C1 through
C4 ha¨ e precisely the same implications for u as in Theorem 1. In addition, C1 implies
␲ Ž 12 . s 12 , C4 requires ␲ Ž p . s p for all p g w0, 1x, but C2 and C3 place no
restrictions on ␲ g ⌸.
The initial axiomatization for RDU in Quiggin Ž1982. presumed ␲ Ž 12 . s 12 , but
this was removed in subsequent axiomatizations. Wu and Gonzalez Ž1996. and
Prelec Ž1998. provide nice discussions about shapes for ␲ .
Our proof of Theorem 2 is presented in the next section.
16
BELL AND FISHBURN
Theorem 3. Suppose % on Wq satisfies WLU and u1 and u 2 are smooth. Then:
C1 « Ži.
Žii.
C2 « Ži.
Žii.
C3 « Ži.
Žii.
u1Ž w . s w, u 2 Ž w . s 1, or
u1Ž w . s ae aw , u 2 Ž w . s eya w w a / 0x;
u1Ž w . s we aw , u 2 Ž w . s e aw , or
u1Ž w . s ae b w , u 2 Ž w . s e c w w aŽ b y c . ) 0x;
u1Ž w . s Ž aw 2 q bw . e c w , u 2 Ž w . s e c w w a G 0, b G 0, a q b ) 0x, or
u1Ž w . s ae b w q ce d w , u 2 Ž w . s e f w w aŽ b y f . ) 0, aŽ b y f . q cŽ d y f .
G 0, d F b if cŽ d y f . - 0x, or
Žiii. u1Ž w . s awe b w q ce d w , u 2 Ž w . s e b w w a2 q c 2 ) 0, a q cŽ d y b . G 0,
a ) 0 if b s d, c ) 0 if d ) b x, or
Živ. u1Ž w . s Ž aw q b . e c w , u 2 Ž w . s e d w w a q bŽ c y d . G 0, aŽ c y d . G 0,
a q bŽ c y d . ) 0 if aŽ c y d . s 0x;
Ž
.
C4 « C3 i with ac s 0, C3Žii. with bd F 0, C3Žiii. with bd F 0, or C3Živ. with
c s d.
Parts C1 and C2 of Theorem 3 are proved in Fishburn Ž1998., but because those
proofs leave u 2 implicit we sketch their Ž u1 , u 2 . proofs here in Section 6. Part C3 is
also proved in Section 6, and part C4 is proved in Section 7. When ⺢q is replaced
by ⺢, we need to take a s 0 in C3Ži., but no changes are required for C3Žii. ᎐ Živ.. It
is worth noting that in applying a special form of the theorem, expectations are
taken separately for u1 and u 2 . For example, in C2Ži., u1Ž x .ru 2 Ž x . s x, but
Eu1Ž ˜
x .rEu2 Ž ˜
x . is not generally equal to Ex˜ unless ˜
x is degenerate or a s 0, in
which case C2Ži. reduces to C1Ži..
For our final theorem, which involves functions of ⌬ s x y y, we define h:
⺢ ª ⺢ as
e¨ en if h Ž ⌬ . s h Ž y⌬ .
odd if h Ž ⌬ . s yh Ž y⌬ .
for all ⌬ g ⺢, and
for all ⌬ g ⺢.
In Theorem 4, ␣ and ␤ denote e¨ en functions, and g denotes an odd function. These
functions are unrestricted except for the presumption of smoothness and the
requirement of monotonicity in Ž4.. We emphasize that expressions like ␣ Ž x y y .
and g Ž x y y . denote the value of the function ␣ or g at argument ⌬ s x y y.
Theorem 4. Suppose % on Wq satisfies SSBU and ␾ is smooth. Then:
C1 « ␾ Ž x, y . s g Ž x y y . w g Ž ⌬ . ) 0 if ⌬ ) 0x;
C2 « Ži. ␾ Ž x, y . s g Ž x y y . w g Ž ⌬ . ) 0 if ⌬ ) 0x, or
Žii. ␾ Ž x, y . s ␣ Ž x y y .Ž e a x y e a y . w a ␣ Ž ⌬ . ) 0 if ⌬ / 0x;
C3 « Ži. ␾ Ž x, y . s ␣ Ž x y y .Ž x 2 y y 2 . q ␤ Ž x y y .Ž x y y . w ␣ Ž ⌬ . G 0 and ⌬ ␣ Ž ⌬ .
q ␤ Ž ⌬ . ) 0 for ⌬ ) 0x, or
Žii. ␾ Ž x, y . s ␣ Ž x y y .Ž e a x y e a y . q ␤ Ž x y y .Ž e b x y e b y . w ␣ Ž ⌬ .Ž e a⌬ y
1. e a y q ␤ Ž ⌬ .Ž e b⌬ y 1. e b y ) 0 for all ⌬ ) 0, y G 0x, or
17
UTILITY FUNCTIONS FOR WEALTH
␾ Ž x, y . s ␣ Ž x y y .Ž x y y . q ␤ Ž x y y .Ž e a x y e a y . w ⌬ ␣ Ž ⌬ . q ␤ Ž ⌬ .Ž e a⌬
y 1. ) 0 for ⌬ ) 0; ␣ Ž ⌬ . G 0 if a - 0, ␤ Ž ⌬ . G 0 if a ) 0 for ⌬ ) 0x,
or
Živ. ␾ Ž x, y . s ␣ Ž x y y .Ž xe a x y ye a y . q ␤ Ž x y y .Ž e a x y e a y . w a ␣ Ž ⌬ . G 0
and ⌬ ␣ Ž ⌬ . e a⌬ q ␤ Ž ⌬ .Ž e a⌬ y 1. ) 0 for ⌬ ) 0x;
Ž
.
C4 « C3 i , or C3Žii. with ab - 0, or C3Žiii..
Žiii.
Parts C1 and C2 of Theorem 4 are proved in Fishburn Ž1998., part C3 is proved
in Section 5, and part C4 is proved in Section 7. None of the special forms of the
theorem precludes preference cycles. Examples for cycles under C1 are given in
Fishburn Ž1984; 1988, pp. 74᎐75..
Table 1 provides a quick reference to the results of Theorems 1 through 4. It
omits the restrictions needed to satisfy Ž4., but includes the additional constraints
imposed by C4.
3. RDU proofs
We assume RDU with u continuous and increasing in w and ␲ g ⌸. Let ␭ s ␲ Ž 12 ..
C1 proof. It is easily seen that C1 implies Ž x q y .r2 ; Ž x, 12 , y . for x, y G 0, so
x-y«u
ž
xqy
2
/
s ␭u Ž x . q Ž 1 y ␭. u Ž y . .
Hence
u Ž x . s ␭u Ž x y d . q Ž 1 y ␭. u Ž x q d .
for all x G d G 0.
We have ␭w uŽ x . y uŽ x y d .x s Ž1 y ␭.w uŽ x q d . y uŽ x .x for x G d G 0. Suppose
␭ / 12 . Let ␶ s ␭rŽ1 y ␭., so ␶ / 1. Then for every x G 0 and n G 1,
ny1
u Ž x q 1. y u Ž x . s
u Ž x q Ž i q 1 . rn . y u Ž x q irn .
Ý
is0
␭
ny1
s
Ý
is0
s
ž
ž
i
1y␭
␶ny1
␶y1
/
/
u Ž x q 1rn . y u Ž x .
u Ž x q 1rn . y u Ž x . .
Then
u Ž x q 1rn . y u Ž x .
1rn
s
u Ž x q 1. y u Ž x . n Ž ␶ y 1.
␶ny1
.
18
BELL AND FISHBURN
19
UTILITY FUNCTIONS FOR WEALTH
Since uŽ x q 1. y uŽ x . ) 0, it follows that if ␶ ) 1 then the right derivative of u at
x equals 0, and if ␶ - 1 then the right derivative of u at x equals ⬁. Since this is
impossible for every x G 0, we conclude that ␶ s 1, or ␭ s 12 . Hence uŽ x . y uŽ x y
d . s uŽ x q d . y uŽ x . for all x G d G 0, so u⬘Ž w . s a ) 0 for all w G 0. This
implies that uŽ w . s aw q b, a ) 0, or uŽ w . s w with affine rescaling. Moreover,
␲ g ⌸ is unrestricted except for ␲ Ž 12 . s 12 since C1 holds for any such ␲ .
C2 and C3 sufficiency. It is easily checked that the EU forms for C2 and C3 also
satisfy C2 and C3 for RDU without restrictions on ␲ g ⌸. We illustrate this for
the quadratic form of C3Ži.: see also Bell Ž1988, p. 1418.. Suppose uŽ w . s aw 2 q bw
for w G 0. Then for w q ˜
x, x q ˜
y g Wq,
E␲ u Ž w q ˜
x . s aw 2 q 2 awE␲ ˜
x q aE␲ ˜
x 2 q bw q bE␲ ˜
x
E␲ u Ž w q ˜
y . s aw 2 q 2 awE␲ ˜
y q aE␲ ˜
y 2 q bw q bE␲ ˜
y.
Therefore
E␲ u Ž w q ˜
x . y E␲ u Ž w q ˜
y . s w2 a E␲ ˜
x y E␲ ˜
y q K,
where K does not involve w. We vary w with ˜
x and ˜
y fixed. If E␲ ˜
x s E␲ ˜
y, there
are no switches; otherwise E␲ uŽ w q ˜
x . s E␲ uŽ w q ˜
y . for at most one w, so the
quadratic form is one-switch ŽC3. for RDU.
The restrictions of Ž4. on the utility functions for these cases are also easily
verified. We illustrate this for C3Žii., where
u⬘ Ž w . s abe b w q cde d w .
u⬘Ž0. G 0 requires ab q cd G 0, and uŽ w . ) 0 for w ) 0 requires at least either
ab ) 0 or cd ) 0. Assume ab ) 0 without loss of generality. If cd - 0, then
abrŽycd . ) e Ž dyb.w for all w ) 0, and this requires d F b. If b s d then u⬘Ž w . s
Ž a q c . be b w , and u⬘Ž w . ) 0 requires bŽ a q c . ) 0.
Deri¨ ation for C2. Assume C2. Let ␭ satisfy ␲ Ž ␭. s 12 , and let Ž x, ␭, y . denote
the risky prospect with probability ␭ for x and probability 1 y ␭ for y, x / y. Then
Ž x, ␭, y . with 0 F x - y has a unique certainty equivalent cŽ x, y . strictly between x
and y:
c Ž x, y . ; Ž x, ␭ , y .
and
c Ž x, y . s uy1
ž
uŽ x . q uŽ y .
2
/
.
By the zero-switch condition C2, for every w with min w q x, w q y4 G 0 we have
cŽ w q x, w q y . ; Ž w q x, ␭, w q y . and
20
BELL AND FISHBURN
c Ž w q x, w q y . s w q c Ž x, y . s uy1
ž
uŽ w q x . q uŽ w q y .
2
/
.
Therefore
uy1
ž
uŽ w q x . q uŽ w q y .
2
/
s w q uy1
ž
uŽ x . q uŽ y .
2
/
.
It then follows from Aczel
´ Ž1966, p. 153. that u has one of the two forms for C2 in
Theorem 1.
Deri¨ ation for C3. Assume C3. The proof of Result 2 in Bell Ž1995a. for C3
under EU also implies that C3Ži. ᎐ Živ. of Theorem 1 are the only u solutions for C3
under RDU, given Ž4.. Note in particular that p and q in Bell’s proof can be
replaced by ␲ values under our continuity assumption for ␲ , so the EU proof
holds also for RDU.
Deri¨ ation for C4. The comprehensive impact of C4 on RDU stems from the
re-ranking of outcomes that occur for the particular contextual uncertainties
involved.
We will demonstrate a class of counterexamples to C4 in which u is linear. Each
specific counterexample thus applies to all utility functions that are differentiable
in a closed interval. Since all utility functions that satisfy C4 for EU are differentiable everywhere, and since all other functions violate C4 for RDU even when ␲
is the identity function, we are done.
Let ˜
x s Ž x, p, 0. and ˜
y s Ž y, q, 0.. The contextual uncertainties for our coun1
Ž
.
terexamples are 2, 2 , y2 and Ž1, 12 , y1.. We use specific numbers for clarity only.
Without loss of generality, we take 2 - x - 4, 2 - y - 4, p - 1r2 and q - 1r2.
Suppose Ž2, 12 , y2. has been resolved with payoff outcome 2. The evaluation of
˜x, combined with Ž1, 21 , y1., proceeds as follows:
payoffs, highest to lowest:
3 q x, 1 q x, 3, 1
probabilities, respectively:
pr2, pr2, Ž1 y p .r2, Ž1 y p .r2
probability weights, respectively:
1 y ␲ Ž1 y pr2., ␲ Ž1 y pr2. y ␲ Ž1 y p ., ␲ Ž1 y p . y ␲ Ž 12 y pr2., ␲ Ž 12 y
pr2..
The RDU utility is therefore
Ž 3 q x . 1 y ␲ Ž 1 y pr2. q Ž 1 q x . ␲ Ž 1 y pr2. y ␲ Ž 1 y p .
q3 ␲ Ž 1 y p . y ␲
ž
1
2
y pr2
/
q␲
ž
1
2
y pr2
/
s x 1 y ␲ Ž1 y p. q 3
y 2 ␲ Ž 1 y pr2. y ␲ Ž 1 y p . y ␲
ž
1
2
y pr2
/
.
21
UTILITY FUNCTIONS FOR WEALTH
Thus ˜
x %˜
y if and only if
x 1 y ␲ Ž1 y p. y y 1 y ␲ Ž1 y q.
) 2 ␲ Ž 1 y pr2. y ␲ Ž 1 y qr2. y ␲ Ž 1 y p . q ␲ Ž 1 y q .
q␲
ž
1
2
y pr2 y ␲
/ ž
1
2
y qr2
/
.
The evaluation of ˜
x and ˜
y when Ž2, 12 , y2. is resolved at y2 proceeds similarly
and leads to exactly the same inequality because u is linear.
Consider now the resolution of Ž1, 12 , y1.. When this smaller contextual uncertainty is resolved at 1, the evaluation of ˜
x goes as follows:
payoffs, highest to lowest:
3 q x, 3, x y 1, y1
probabilities:
pr2, 12 y pr2, pr2, 12 y pr2
probability weights:
1 y ␲ Ž1 y pr2., ␲ Ž1 y pr2. y ␲ Ž 12 ., ␲ Ž 12 . y ␲ Ž 12 y pr2., ␲ Ž 12 y pr2..
In this case, we find that ˜
x %˜
y if and only if
x 1 y ␲ Ž 1 y pr2. q ␲
1
1
y␲
y pr2
ž / ž
/
.
ž / ž
2
) y 1 y ␲ Ž 1 y qr2 q ␲
2
1
2
y␲
1
2
y qr2
/
.
Again, because u is linear, resolution of Ž1, 12 , y1. at y1 leads to exactly the same
inequality for ˜
x %˜
y.
There are many ways to construct a counterexample to C4 from the preceding
inequalities for ˜
x %˜
y. For instance, assume that ␲ Ž p . s p except locally near p
s 12 , where ␲ Ž 12 . s 12 q ⑀ . Then the preceding two inequalities are
xp ) yq
and
xp q Ž x y y . ⑀ ) yq.
We need only choose x, p, y, q and ⑀ so that
xp ) yq ) xp q Ž x y y . ⑀
22
BELL AND FISHBURN
to contradict C4. A specific example is Ž x, p, y, q, ⑀ . s Ž3, 1r3, 7r2, 1rŽ7r2 q
␦ ., 3 ␦rŽ7r2 q ␦ .. with ␦ small and positive.
4. One-switch for SSBU
Assume that % on Wq satisfies SSBU with ␾ smooth, and let
␾Ž w q˜
x, w q ˜
y . s E␾ Ž w q ˜
x, w q ˜
y.
as defined in Ž7.. We first prove that the forms for ␾ in C3Ži. ᎐ Živ. of Theorem 4
satisfy C3.
For fixed ˜
x and ˜
y, we obtain
␾Ž w q˜
x, w q ˜
y . s A q Bw for C3 Ž i .
s Ae aw q Be b w
s A q Be aw
for C3 Ž ii .
for C3 Ž iii .
s w A q Bw x e aw
for C3 Ž iv . ,
where A and B are expectations with respect to ˜
x and ˜
y that do not involve w. To
verify C3 Žone-switch., it suffices to note that ␾ Ž w q ˜
x, w q ˜
y . s 0 for no w, or
exactly one w, or for all w. This is obvious for C3Ži., C3Žiii., and C3Živ., where
e aw ) 0 for all w. Suppose C3Žii. obtains. If AB G 0, then Ae aw q Be b w vanishes
for no w or for all w. If AB - 0, equality to 0 requires e Ž bya.w s yArB, which is
true for no w or all w if a s b, and for no w or exactly one w if a / b.
We show next that Ž4., i.e., ␾ Ž x, y . ) 0 when x ) y G 0, corresponds to the
restrictions in brackets for the four ␾ functions in C3 of Theorem 4. Let
⌬ s x y y ) 0. For C3Ži.,
2
␾ Ž y q ⌬, y. s ␣ Ž ⌬. Ž y q ⌬. y y2 q ␤ Ž ⌬. ⌬
s ⌬ ⌬ ␣ Ž ⌬ . q 2 y␣ Ž ⌬ . q ␤ Ž ⌬ . 4 .
This is positive for all y G 0 and ⌬ ) 0 only if ␣ Ž ⌬ . G 0, for if ␣ Ž ⌬ . - 0 then
large y will make the entire expression negative. Given ␣ Ž ⌬ . G 0, at y s 0 we
require ⌬ ␣ Ž ⌬ . q ␤ Ž ⌬ . ) 0. Conversely, if ␣ Ž ⌬ . G 0 and ⌬ ␣ Ž ⌬ . q ␤ Ž ⌬ . ) 0 for
all ⌬ ) 0, it is easily seen that ␾ Ž y q ⌬, y . ) 0 for all y G 0 and all ⌬ ) 0. For
C3Žii.,
␾ Ž y q ⌬ , y . s e a y ␣ Ž ⌬ . Ž e a⌬ y 1 . q e b y ␤ Ž ⌬ . Ž e b⌬ y 1 . ,
23
UTILITY FUNCTIONS FOR WEALTH
so this must be positive for all y G 0 and ⌬ ) 0. For C3Žiii.,
␾ Ž y q ⌬ , y . s e a y ␣ Ž ⌬ . Ž e a⌬ y 1 . q e b y ␤ Ž ⌬ . Ž e b⌬ y 1 . ,
so this must be positive for all y G 0 and ⌬ ) 0. For C3Žiii.,
␾ Ž y q ⌬ , y . s ⌬ ␣ Ž ⌬ . q e a y ␤ Ž ⌬ . Ž e a⌬ y 1 . .
At y s 0, this requires ⌬ ␣ Ž ⌬ . q ␤ Ž ⌬ .Ž e a⌬ y 1. ) 0 for all ⌬ ) 0. If a - 0 then
the second term goes to 0 as y ª ⬁, so we need ␣ Ž ⌬ . G 0 in this case. We have
already required ␣ Ž ⌬ . ) 0 when a s 0. If a ) 0, we also need ␤ Ž ⌬ . G 0. For
C3Živ.,
␾ Ž y q ⌬ , y . s e a y ␣ Ž ⌬ . Ž y q ⌬ . e a⌬ y y q ␤ Ž ⌬ . w e a⌬ y 1 x 4 .
For y s 0, we require ⌬ ␣ Ž ⌬ . e a⌬ q ␤ Ž ⌬ .Ž e a⌬ y 1. ) 0 for all ⌬ ) 0. The other
part within braces is y␣ Ž ⌬ .Ž e a⌬ y 1., so we need ␣ Ž ⌬ .Ž e a⌬ y 1. G 0 for all ⌬ ) 0.
This is true if a s 0, or if ␣ Ž ⌬ . G 0 if a ) 0, or if ␣ Ž ⌬ . F 0 if a - 0, so it holds if
and only if a ␣ Ž ⌬ . G 0 for all ⌬ ) 0.
We now assume C3 and derive the forms for C3 in Theorem 4. Let ␾ 1Ž i., 2Ž j.Ž w, w .
or ␾ 1 ⭈ ⭈ ⭈ 12 ⭈ ⭈ ⭈ 2 w i 1⬘s, j 2⬘s x denote the derivative of ␾ Ž w q x, w q y . i times with
respect to x and j times with respect to y, evaluated at x s y s 0. Then a Taylor
series expansion for a function of two variables ŽTaylor, 1955, p. 228. yields
␾ Ž w q x, w q y . s ␾ Ž w, w . q
⬁
Ý
ns1
n
1
n!
Ý
is0
n ny i i
x
y ␾ 1Ž nyi., 2Ž i. Ž w, w . .
i
ž /
Skew-symmetry gives ␾ Ž w, w . s 0 and ␾ Ž w q x, w q y . q ␾ Ž w q y, w q x . s 0.
The preceding equation then yields
0 s Ž x q y . Ž ␾1 q ␾2 . q
q
q
q
1
6
24
2
2
Ž x 2 q y 2 . Ž ␾ 11 q ␾ 22 . q 2 xy␾ 12
Ž x 3 q y 3 . Ž ␾ 111 q ␾ 222 . q
1
1
1
1
2
xy Ž x q y . Ž ␾ 112 q ␾ 122 .
Ž x 4 q y 4 . Ž ␾ 1111 q ␾ 2222 . q
x 2 y 2␾ 1122 q ⭈⭈⭈ .
1
6
xy Ž x 2 q y 2 . Ž ␾ 1112 q ␾ 1222 .
24
BELL AND FISHBURN
We divide this by sums of powers of x and y and let x and y go to zero to
conclude that
␾1 q ␾2 s 0
Ž set x s y, divide by x, let x ª 0 .
␾ 11 q ␾ 22 s 0
Ž set
y s 0, divide by x 2 , let x ª 0 .
␾ 12 s 0
Ž set
x s y, divide by x 2 , let x ª 0 .
␾ 111 q ␾ 222 s 0
␾ 112 q ␾ 122 s 0
and so forth. It follows that
␾ Ž w q x, w q y .
ny1
s
⬁
ns1
s
2
1
Ý
n!
n i i ny2 i
x y Žx
y y ny2 i . ␾ 1Ž nyi., 2Ž i. Ž w, w .
i
ž /
Ý
is0
⬁
⬁
1
ns2 iq1
n!
Ý Ž xy . i Ý
is0
n
i
ž /Ž
x ny 2 i y y ny2 i . ␾ 1Ž nyi., 2Ž i. Ž w, w . .
Ž 13 .
Equivalently,
␾ Ž w q x, w q y . s Ž x y y . ␾ 1 q
q
1
2
1
2!
Ž x 2 y y 2 . ␾ 11 q
1
3!
Ž x 3 y y 3 . ␾ 111
Ž x 2 y y xy 2 . ␾ 112 q ⭈⭈⭈ ,
so, for lotteries ˜
x and ˜
y, bilinear expansion gives
␾Ž w q˜
x, w q ˜
y.
s
⬁
⬁
Ý Ý
is0 ns2 iq1
1
n!
n
i
ž /
s Ex˜ y Ey˜ ␾ 1 q
q
1
6
1
2
Ex˜ny i Ey˜i y Ey˜nyi Ex˜i ␾ 1Ž nyi., 2Ž i. Ž w, w .
Ex˜2 y Ey˜2 ␾ 11
Ex˜3 y Ey˜3 ␾ 111 q
1
2
Ex˜2 Ey˜ y Ey˜2 Ex˜ ␾ 112 q ⭈⭈⭈ .
Ž 14 .
25
UTILITY FUNCTIONS FOR WEALTH
If any ␾ term here after ␾ 1 and ␾ 11 is not linearly dependent on ␾ 1 and ␾ 11 as a
function of w, then for suitably small Žnear 0. outcomes it is is possible to construct
˜x and ˜y to contradict C3 by a two-switch example. Thus, for n G 3 and 0 F i
F Ž n y 1 . r2 , there are constants c1Ž nyi., 2Ž i. and d1Ž nyi., 2Ž i. such that
␾ 1Ž nyi., 2Ž i. Ž w, w . s c1Ž nyi., 2Ž i. ␾ 1 Ž w, w . q d1Ž nyi., 2Ž i. ␾ 11 Ž w, w .
for all w, i.e., for all w G 0. Let
a Ž w . s ␾ 1 Ž w, w . ,
b Ž w . s ␾ 11 Ž w, w . .
Then, by Ž13.,
␾ Ž w q x, w q y .
s Ž x y y . aŽ w . q
1
2
Ž x 2 y y2 . bŽ w . q
⬁
Ý
Ž xy .
i
⬁
Ý
is0
ns2 iq1
Žexclude ns1, 2 for is0 .
1
n!
n
i
ž /
= Ž x ny 2 i y y ny2 i c1Ž nyi., 2Ž i. a Ž w . q d1Ž nyi., 2Ž i. b Ž w .
½
s xyyq
q
⬁
Ý Ž xy . i
ai Ž x . y ai Ž y .
is0
½
1
⬁
Ž x 2 y y 2 . q Ý Ž xy . i
2
5
aŽ w .
bi Ž x . y bi Ž y .
is0
5
bŽ w .
s A Ž x, y . a Ž w . q B Ž x, y . b Ž w . ,
Ž 15 .
where the a i , bi and A, B are defined in context, with A and B skew-symmetric.
When x and y are suitably small with x ) y, the leading terms x y y in A and
1
2.
Ž 2
in B dominate to give AŽ x, y . ) 0 and B Ž x, y . ) 0. For lotteries ˜
x and
2 x yy
˜y we have
␾Ž w q˜
x, w q ˜
y . s E AŽ ˜
x, ˜
y . aŽ w . q E B Ž ˜
x, ˜
y . bŽ w . ,
and it follows from Ž14. that there is a positive-radius disk with center Ž0, 0. such
that every point in the disk has an ˜
x, ˜
y at which that point equals
Ž Ew AŽ ˜
x, ˜
y .x, Ew B Ž ˜
x, ˜
y .x..
We assume without loss of generality that neither aŽ w . nor bŽ w . in Ž15. is
uniformly zero. For example, if ␾ 11 ' 0, then x ) y implies AŽ x, y . aŽ w . ) 0 for
all w; hence a has constant sign and ␾ is zero-switch.
To determine forms for aŽ w . and bŽ w ., let u s ␾ Ž w q x, w q y ., let u1 Ž u 2 .
denote the first derivative of u with respect to x Ž y . ᎏwith similar notation for
26
BELL AND FISHBURN
higher-order partial derivatives, and series-expand ␾ Ž w q x q z1 , w q y q z 2 . to
obtain
␾ Ž w q x q z1 , w q y q z 2 .
s u q z 1 u1 q z 2 u 2
q
q
1
2
Ž z12 u11 q 2 z1 z2 u12 q z 22 u 22 .
1
3!
Ž z13 u111 q 3 z12 z 2 u112 q 3 z1 z 22 u122 q z 23 u 222 . q ⭈⭈⭈ .
Ž 16 .
Now let ˜
z1 and ˜
z 2 be lotteries with the same first two moments:
m1 s Ez˜1 s Ez˜2
m 2 s Ez˜12 s Ez˜22 .
Bilinear expansion and Ž16. yield
␾Žw q x q ˜
z1 , w q y q ˜
z2 .
s u q m1 Ž u1 q u 2 . q
q m12 u12 q
1
3!
1
2
m 2 Ž u11 q u 22 .
Ez˜13 u111 q 3m1 m 2 Ž u112 q u122 . q Ez˜23 u 222 q ⭈⭈⭈ .
Consider x ) y with x y y small. If u11 q u 22 and u12 are not both linearly
dependent on u1 q u 2 and u, then it will be possible to construct ˜
z1 and ˜
z 2 with
outcomes near zero which satisfy the noted moment equalities and contradict C3.
It follows for the given x and y that there are constants ␣ and ␤ such that
u11 q 2 u12 q u 22 s ␣ Ž u1 q u 2 . q ␤ u
for all w. In terms of the total differentials ␾ ⬘ and ␾ ⬙ of ␾ Ž w q x, w q y . with
respect to w, this is precisely
␾ ⬙ Ž w q x, w q y . s ␣␾ ⬘ Ž w q x, w q y . q ␤␾ Ž w q x, w q y . .
Ž 17 .
For convenience let f Ž w . s ␾ Ž w q x, w q y ., and write the preceding equation as
the linear differential equation
f ⬙ Ž w . y ␣ f ⬘Ž w . y ␤ f Ž w . s 0
27
UTILITY FUNCTIONS FOR WEALTH
with auxiliary equation r 2 y ␣ r y ␤ s 0. Let r 1 and r 2 be the roots of the
auxiliary equation. Then ŽPhillips, 1951. we have
f Ž w . s c1 e r 1 w q c 2 e r 2 w
if r 1 / r 2
f Ž w . s Ž c1 q c 2 w . e r w
if r 1 s r 2 s r ,
where c1 and c 2 are constants. Actually, because ␣ and ␤ in Ž17. might depend on
x y y, r 1 and r 2 can be written as continuous functions of x y y. Similarly, c1 and
c 2 can depend on Ž x, y ., so we write them as functions of Ž x, y .. The preceding f
solutions for variable x and y with x y y small then take the forms
␾ Ž w q x, w q y . s c1 Ž x, y . e w r 1Ž xyy . q c 2 Ž x, y . e w r 2 Ž xyy .
and
␾ Ž w q x, w q y . s c1 Ž x, y . q wc 2 Ž x, y . e w r Ž xyy . .
We compare these to Ž15., i.e., to
␾ Ž w q x, w q y . s A Ž x, y . a Ž w . q B Ž x, y . b Ž w . .
Taking account of the comment after Ž15. for a positive-radius disk, it follows that
r 1Ž x y y . and r 2 Ž x y y ., or r Ž x y y ., can be presumed Žor must be, when the
coefficients are nonzero. to be constant for small x and y with < x y y < small and
nonzero. Hence, for such x and y, either
␾ Ž w q x, w q y . s c1 Ž x, y . e r 1w q c 2 Ž x, y . e r 2 w
s A Ž x, y . a Ž w . q B Ž x, y . b Ž w .
for all w,
Ž 18 .
for all w,
Ž 19 .
or
␾ Ž w q x, w q y . s c1 Ž x, y . q wc 2 Ž x, y . e r w
s A Ž x, y . a Ž w . q B Ž x, y . b Ž w .
where c1 and c 2 , like A and B, are easily seen to be skew-symmetric.
Suppose Ž18. applies. We can then suppose that
a Ž w . s ␣ 1 e r 1w q ␣ 2 e r 2 w q ␣ 3 U Ž w . ,
␣ 3 / 0,
b Ž w . s ␤ 1 e r 1w q ␤ 2 e r 2 w q ␤ 3V Ž w . ,
␤ 3 / 0,
where U and V have no additive term like e r i w and each is either uniformly zero or
linear independent of e r 1w and e r 2 w. When these are substituted into the AB
28
BELL AND FISHBURN
version of Ž18. and compared to its c1 c 2 version, it follows from the remarks
following Ž15. that U s V ' 0. Then
␾ Ž w q x, w q y .
s ␣ 1 A Ž x, y . q ␤ 1 B Ž x, y . e r 1w q ␣ 2 A Ž x, y . q ␤ 2 B Ž x, y . e r 2 w
for all x, y and w. Let C Ž x, y . s ␣ 1 AŽ x, y . q ␤ 1 B Ž x, y . and DŽ x, y . s ␣ 2 AŽ x, y .
q ␤ 2 B Ž x, y .. Then
␾ Ž w q x, w q y . s C Ž x, y . e r 1w q D Ž x, y . e r 2 w ,
Ž 20 .
where C and D are skew-symmetric because A and B are skew-symmetric.
Suppose Ž19. applies. We can then presume that
a Ž w . s ␣ 1 e r w q ␣ 2 we r w q ␣ 3 U Ž w . ,
␣ 3 / 0,
b Ž w . s ␤ 1 e r w q ␤ 2 we r w q ␤ 3V Ž w . ,
␤ 3 / 0,
with stipulations on U and V similar to those above. We again get U s V ' 0, so
␾ Ž w q x, w q y . s ␣ 1 A Ž x, y . q ␤ 1 B Ž x, y . e r w
q ␣ 2 A Ž x, y . q ␤ 2 B Ž x, y . we r w
s C Ž x, y . e r w q D Ž x, y . we r w .
Ž 21 .
We now proceed from Ž20. and Ž21. to our final forms for ␾ Ž x, y .. There are
three cases for Ž20.. One of these, which is the case of complex conjugate roots, is
inapplicable because it cannot satisfy the basic monotonicity property of x ) y « x
% y. The other two are the real Ž r 1 , r 2 . cases, one for r 1 s 0 and the other for
r 1 r 2 / 0:
I.
II.
␾ Ž w q x, w q y . s C Ž x, y . q D Ž x, y . e r 2 w
r2 / 0
␾ Ž w q x, w q y . s C Ž x, y . e r 1w q D Ž x, y . e r 2 w
r 1 r 2 / 0.
We also consider the two cases for Ž21., for r s 0 and r / 0:
III.
␾ Ž w q x, w q y . s C Ž x, y . q wD Ž x, y .
IV.
␾ Ž w q x, w q y . s C Ž x, y . e r w q D Ž x, y . we r w
r / 0.
29
UTILITY FUNCTIONS FOR WEALTH
Each case has the form C Ž x, y . cŽ w . q DŽ x, y . dŽ w .:
cŽ w.
ebw
b/0
e aw
ebw
ab / 0
1
w
I. 1
II.
III.
dŽ w .
IV. e b w
we b w
b / 0.
Given ␾ Ž w q x, w q y . s cŽ w .C Ž x, y . q dŽ w . DŽ x, y ., we replace w, x and y by
w q Ž x q y .r2, Ž x y y .r2 and Ž y y x .r2, respectively, to obtain
␾ Ž w q x, w q y . s c w q
ž
xqy
2
s x y y. q d w q
/Ž
ž
xqy
2
t x y y. ,
/Ž
Ž 22 .
where sŽ x y y . s C ŽŽ x y y .r2, Ž y y x .r2., t Ž x y y . s DŽŽ x y y .r2, Ž y y x .r2.,
and s and t are odd because C and D are skew-symmetric.
I. With cŽ w . s 1 and dŽ w . s e b w , b / 0, Ž22. gives
␾ Ž w q x, w q y . s s Ž x y y . q e b w e bŽ xqy .r2 t Ž x y y .
s sŽ x y y . q ebw
ž
ebx y eb y
e bŽ xyy .r2 y e bŽ yyx .r2
/
tŽ x y y.
s s Ž x y y . q w e bŽ wqx . y e bŽ wqy . x t 1 Ž x y y . ,
where t 1 is even. We have assumed here that x / y, and for x s y can let t 1Ž0. be
any finite value, e.g. t 1Ž0. s 0. It follows that
␾ Ž x, y . s s Ž x y y . q Ž e b x y e b y . t 1 Ž x y y . ,
with s odd and t 1 even. This matches the form of C3Žiii. in Theorem 4.
II. With cŽ w . s e aw and dŽ w . s e b w , ab / 0, a calculation like that for t in
case I in Ž22. gives
␾ Ž x, y . s Ž e a x y e a y . s1 Ž x y y . q Ž e b x y e b y . t 1 Ž x y y . ,
with s1 and t 1 even. This is C3Žii. in Theorem 4.
30
BELL AND FISHBURN
III. With cŽ w . s 1 and dŽ w . s w, Ž22. gives Žfor x / y .
␾ Ž w q x, w q y .
s sŽ x y y . q w q
ž
xqy
t x y y.
/Ž
2
s Ž w q x . y Ž w q y . s1 Ž x y y .
2
q Ž w q x. y Ž w q y.
2
xqy
tŽ x y y.
2
Ž2 w Ž x y y . q x 2 y y 2 .
ž
wq
/
2
s Ž w q x . y Ž x q y . s1 Ž x y y . q Ž w q x . y Ž w q y .
tŽ x y y.
2
2
2Ž x y y .
2
s Ž w q x . y Ž w q y . s1 Ž x y y . q Ž w q x . y Ž w q y . t 1 Ž x y y .
where s1 and t 1 are even. Therefore
␾ Ž x, y . s Ž x y y . s1 Ž x y y . q Ž x 2 y y 2 . t 1 Ž x y y . ,
which is C3Ži. in Theorem 4.
IV. With cŽ w . s e b w and dŽ w . s we b w , b / 0, Ž22. gives
␾ Ž w q x, w q y .
s e b w e bŽ xqy .r2 s Ž x y y . q w q
ž
s e b w e bŽ xqy .r2
ž
xqy
2
tŽ x y y.
e
bŽ xyy .r2
y e bŽ yyx .r2
/½
/
e b w e bŽ xqy .r2 t Ž x y y .
w Ž e bŽ xyy .r2 y e bŽ yyx .r2 .
q
q
xyy
ye
¡t Ž x y y . ž
~
¢
e
b w bŽ xqy .r2
e
2
/Ž
bŽ xyy .r2
ž
ž
xqy
2
xyy
2
/Ž
/
e bŽ xyy .r2 y e bŽ yyx .r2 .
e bŽ xyy .r2 q e bŽ yyx .r2
e bŽ xyy .r2 q e bŽ yyx .r2 .
y e bŽ yyx .r2
s t 1 Ž x y y . e b w w Ž e b x y e b y . q xe b x y ye b y 4
q e b w e bŽ xqy .r2 s1 Ž x y y . w s1 odd, t 1 even x
5
¦¥
§
y sŽ x y y .
UTILITY FUNCTIONS FOR WEALTH
31
s w Ž e bŽ wqx . y e bŽ wqy. . q xe bŽ wqx . y ye bŽ wqy . t 1 Ž x y y .
q w e bŽ wqx . y e bŽ wqy . x s2 Ž x y y . w s2 even x
s Ž w q x . e bŽ wqx . y Ž w q y . e bŽ wqy. t 1 Ž x y y .
q w e bŽ wqx . y e bŽ wqy . x s2 Ž x y y . .
Therefore
␾ Ž x, y . s Ž xe b x y ye b y . t 1 Ž x y y . q Ž e b x y e b y . s2 Ž x y y . ,
which matches C3Živ. in Theorem 4.
5. Derivations for WLU
Assume that % on Wq satisfies WLU with u1 and u 2 smooth. We focus here on
C1᎐C3 in Theorem 3, beginning with outlines of the derivations of the forms for
C1 and C2. We omit the straightforward proofs that the forms given in Theorem 3
for C1, C2, and C3 imply these conditions. Verification of this for C3 is essentially
the same as for C3 in the second paragraph of preceding section because ␾ Ž x, y .
defined by
␾ Ž x, y . s u1 Ž x . u 2 Ž y . y u1 Ž y . u 2 Ž x .
Ž 23 .
gives a case of SSBU. We also omit verification of the conditions in brackets that
ensure Ž4..
C1 proof. Given Ž23., Fishburn Ž1998. proves that C1 implies for all x G y G 0
that, up to a positive multiplicative transformation, either
Ž i . ␾ Ž x, y . s ␾ Ž x y y, 0 . s x y y
or
Ž ii . ␾ Ž x, y . s ␾ Ž x y y, 0 . s aw e aŽ xyy . y eyaŽ xyy . x , a / 0.
Alternative Ži. gives ␾ Ž ˜
x, ˜
y . s Ex˜ y Ey,
˜ and it suffices to take u1Ž w . s w and
u 2 Ž w . s 1 for all w G 0. This gives C1Ži. in Theorem 3.
When Žii. holds, we have
␾Ž ˜
x, ˜
y . s aEe a x̃ Eeya ỹ y aEe a ỹ Eeya x̃ ,
so for ˜
x %˜
y m u1Ž ˜
x . u2Ž ˜
y . ) u1Ž ˜
y . u2Ž ˜
x . it suffices to take u1Ž w . s ae aw and u 2 Ž w .
ya w
se
, a / 0, which matches C1Žii. in Theorem 3.
32
BELL AND FISHBURN
C2 proof. The two WLU forms implied by C2 in Fishburn Ž1998. can be
written, for all x G y G 0, as follows:
Ž i . ␾ Ž x, y . s Ž x y y . ␤ y␤ Ž xyy .r2 s Ž x y y . e aŽ xqy .
Ž ii . ␾ Ž x, y . s u Ž x y y . ␤ y s aw e dŽ xyy . y eydŽ xyy . x ␤ y␤ Ž xyy .r2
s aw e b xqc y y e b yqc x x .
Skew-symmetry shows that both forms hold for x - y as well as x G y. For Ži.,
␾Ž ˜
x, ˜
y . s EŽ ˜
xe a x̃ . Ee a ỹ y E Ž ˜
ye a ỹ . Ee a x̃ ,
so for ␾ Ž ˜
x, ˜
y . s u1Ž ˜
x . u2Ž ˜
y . y u1Ž ˜
y . u2Ž ˜
x . it suffices to take u1Ž w . s we aw and
aw
u 2 Ž w . s e , which are the forms for C2Ž i . in Theorem 3.
For Žii.,
␾Ž ˜
x, ˜
y . s aEe b x̃ Ee c ỹ y aEe b ỹ Ee c x̃ ,
so we can take u1Ž w . s ae b w and u 2 Ž w . s e c w , as in C2Žii. of Theorem 3.
Deri¨ ation for C3. We now assume C3 and derive the forms for C3 shown in
Theorem 3. As remarked after Ž10., we assume without loss of generality that
u1 ) 0 and u 2 ) 0. We shall also let
X
X
¨ Ž w . s u1 Ž w . u 2 Ž w . y u1 Ž w . u 2 Ž w .
for all w G 0.
Because x ) y « u1Ž x .ru 2 Ž x . ) u1Ž y .ru 2 Ž y ., u1Ž w .ru 2 Ž w . increases in w, and
therefore
¨ Ž x.
u2 Ž w .
2
d Ž u1 Ž w . ru 2 Ž w . .
s
dw
) 0,
so ¨ Ž w . ) 0 for all w. We note also that
Y
Y
¨ ⬘ Ž w . s u1 Ž w . u 2 Ž w . y u1 Ž w . u 2 Ž w . ,
Z
Z
Y
X
X
Y
¨ ⬙ Ž w . s u1 Ž w . u 2 Ž w . y u1 Ž w . u 2 Ž w . q u1 Ž w . u 2 Ž w . y u1 Ž w . u 2 Ž w . .
Several steps in the ensuing proof are similar to steps in the SSBU proof for C3 in
the preceding section.
33
UTILITY FUNCTIONS FOR WEALTH
We begin our derivation for C3 with a Taylor-series expansion of u i Ž w q x .
around w:
ui Ž w q x . s ui Ž w . q
⬁
1
Ý
ns1
n!
x n uŽi n. Ž w . ,
Ž n.
Ž n. Ž .
Ž .
where uŽi n. Ž w . s d n u i Ž w q x .rdx n < xs0 . With uŽ0.
w , we
i s u i w and u i s u i
have
u1 Ž w q x . u 2 Ž w q y . s
⬁
⬁
Ý Ý u1Ž j. uŽ2k . x j y kr Ž j!k! . .
js0 ks0
Therefore
u1 Ž w q x . u 2 Ž w q y . y u1 Ž w q y . u 2 Ž w q x .
s
Ý Ž x k y j y x j y k . Ž u1Ž k . uŽ2j. y u1Ž j. uŽ2k . . Ž j!k! .
0Fj-k
s Ž x y y . Ž uX1 u 2 y u1 uX2 . q
q
q
q
1
6
1
2
1
6
1
2
Ž x 2 y y 2 . Ž uY1 u 2 y u1 uY2 .
Ž x 3 y y 3 . Ž uZ1 u 2 y u1 uZ2 . q ⭈⭈⭈
Ž x 2 y y xy 2 . Ž uY1 uX2 y uX1 uY2 .
Ž x 3 y y xy 3 . Ž uZ1 uX2 y uX1 uZ2 . q ⭈⭈⭈ .
Let u1Ž ˜
x . s Eu1Ž ˜
x . and u 2 Ž ˜
x . s Eu 2 Ž ˜
x .. Taking expectations for ˜
x and ˜
y, we have
u1 Ž w q ˜
x . u2 Ž w q ˜
y . y u1 Ž w q ˜
y . u2 Ž w q ˜
x.
s Ex˜ y Ey˜ Ž uX1 u 2 y u1 uX2 . q
q
q
1
6
1
2
1
2
Ex˜2 y Ey˜2 Ž uY1 u 2 y u1 uY2 .
Ex˜3 y Ey˜3 Ž uZ1 u 2 y u1 uZ2 . q ⭈⭈⭈
Ex˜2 Ey˜ y ExEy
˜ ˜2 Ž uY1 uX2 y uX1 uY2 . q ⭈⭈⭈ .
34
BELL AND FISHBURN
C3 implies that every parenthetical u term beyond ¨ Ž w . s uX1 u 2 y u1 uX2 and
Y
Y
¨ ⬘Ž w . s u1 u 2 y u1 u 2 must be linearly dependent on those two to avoid a two-switch
situation. One of the consequences of this is ¨ ⬙ Ž w . s a¨ ⬘Ž w . q b¨ Ž w . for real a
and b because
Z
Z
Y X
X
Y
¨ ⬙ s Ž u1 u 2 y u1 u 2 . q Ž u1 u 2 y u1 u 2 .
s Ž a1¨ ⬘ q b1¨ . q Ž a2¨ ⬘ q b 2¨ .
s Ž a1 q a2 . ¨ ⬘ q Ž b1 q b 2 . ¨ .
The solutions of the linear differential equation ¨ ⬙ y a¨ ⬘ y b¨ s 0 for real roots
of the auxiliary equation r 2 y ar y b s 0 are
¨ Ž w . s ae b w q ce d w ,
b/d
Ž 24 .
¨ Ž w . s Ž a q bw . e c w .
Ž 25 .
We do not consider the case of complex conjugate roots because it does not lead to
viable monotonic one-switch functions u1 and u 2 . We return to Ž24. and Ž25. after
we develop general solution forms for Ž24. and Ž25. based on the linear dependence implied by C3.
By linear dependence, there are real ␭1 and ␭2 such that
uZ1 u 2 y u1 uZ2 s ␭1 Ž uY1 u 2 y u1 uY2 . q ␭2 Ž uX1 u 2 y u1 uX2 . .
Therefore, with u1 ) 0 and u 2 ) 0 as noted earlier, we have
uZ1 y ␭1 uY1 y ␭2 uX1
u1
s
uZ2 y ␭1 uY2 y ␭2 uX2
u2
s ␪ Ž w. ,
where ␪ is defined by the common ratio. Thus
uZi s ␭1 uYi q ␭2 uXi q ␪ u i
for i s 1, 2.
Ž 26 .
We will prove that ␪ Ž w . is a constant, independent of w, so the preceding
equations for i s 1, 2 become order-3 linear differential equations that yield
general solution forms for u1 and u 2 subject to Ž24. and Ž25..
When the preceding expression is differentiated with respect to w, we obtain
Z
Y
X
uXXXX
i s ␭1 u i q ␭ 2 u i q ␪ u i q ␪ ⬘u i
for i s 1, 2.
By linear dependence, there are real ␮ 1 and ␮ 2 such that
XXXX
Y
Y
X
X
uXXXX
1 u 2 y u1 u 2 s ␮ 1 Ž u1 u 2 y u1 u 2 . q ␮ 2 Ž u1 u 2 y u1 u 2 . .
UTILITY FUNCTIONS FOR WEALTH
35
Substitutions in this from the preceding two expressions give
Ž ␭12 q ␭2 y ␮1 . ¨ ⬘ q Ž ␪ q ␭1 ␭2 y ␮ 2 . ¨ s 0,
so that there are constants a1 and b1 for which
␪ Ž w . s a1 q b1
ž
¨ ⬘Ž w .
¨ Ž w.
/
.
X
X XXXX
Linear dependence also gives ␲ 1 and ␲ 2 such that uXXXX
1 u 2 y u1 u 2 s ␲ 1¨ ⬘ q ␲ 2 ¨ ,
and by substitutions we obtain
␭1 Ž ␣ 1¨ ⬘ q ␤ 1¨ . q ␭2 Ž ␣ 2¨ ⬘ q ␤ 2¨ . y ␪ ⬘¨ s ␲ 1¨ ⬘ q ␲ 2¨ ,
so that there are constants a2 and b 2 such that
␪ ⬘ Ž w . s a2 q b 2
ž
¨ ⬘Ž w .
¨ Ž w.
/
.
We therefore have the following equation pair for ␪ Ž w . and ␪ ⬘Ž w .:
␪ s a1 q b1 Ž ¨ ⬘r¨ .
␪ ⬘ s a2 q b 2 Ž ¨ ⬘r¨ . .
One solution to this pair is ␪ s a1 with b1 s a2 s b 2 s 0, in which case ␪ is
constant and there are no implications for ¨ ⬘r¨ beyond what we already know
from Ž24. and Ž25.. There are also solutions when b1 / 0. Given b1 / 0, it follows
without difficulty from Ž24. and Ž25. that the only solution to the Ž ␪ , ␪ ⬘. pair is
␪ s constant with ¨ ⬘r¨ also constant, in which case ¨ s ae b w . Because this ¨ is a
specialization of Ž24. and Ž25. Žand in fact characterizes the zero-switch situation
under C2., we proceed with the more general solution that leaves Ž24. and Ž25.
intact.
Equation Ž26. with constant ␪ yields the following forms for u1 and u 2 when the
auxiliary equation r 3 y ␭1 r 2 y ␭ 2 r y ␪ s 0 has three real roots:
I. All roots identical:
u1 Ž w . s Ž aw 2 q bw q c . e d w
u2 Ž w . s Ž ␣ w 2 q ␤ w q ␥ . e d w
¨ Ž w. s
Ž a ␤ y ␣ b . w 2 q 2 Ž a␥ y ␣ c . w q Ž b␥ y ␤ c . e 2 d w ;
36
BELL AND FISHBURN
II. Two roots identical:
u1 Ž w . s Ž aw q b . e c w q de f w ,
c/f
u2 Ž w . s Ž ␣ w q ␤ . e c w q ␦ e f w
¨ Ž w . s Ž a ␤ y ␣ b . e 2 c w q e Ž cqf .w
= Ž c y f . Ž a␦ y ␣ d . w
q Ž c y f . Ž b␦ y ␤ d . q Ž a ␦ y ␣ d . ;
III.
All three roots different:
u1 Ž w . s ae r w q be s w q ce t w ,
r/s/t/r
u2 Ž w . s ␣ e r w q ␤ e s w q ␥ e t w
¨ Ž w . s e Ž rqs.w Ž a ␤ y ␣ b . Ž r y s .
q e Ž rqt .w Ž a␥ y ␣ c . Ž r y t .
q e Ž sqt .w Ž b␥ y ␤ c . Ž s y t . .
We do not consider the auxiliary solution with complex conjugate roots because it
does not yield admissible forms for u1 and u 2 .
The following restrictions are required for ¨ in I, II and III so that it satisfies the
necessary Ž24. or Ž25.:
I.
II.
III.
a␤ y ␣ b s 0
a ␤ y ␣ b s 0, or a ␦ y ␣ d s 0
a ␤ y ␣ b s 0, or a␥ y ␣ c s 0, or b␥ y ␤ c s 0.
We complete the proof of the C3 forms in Theorem 3 by showing how u1 and u 2
of I᎐III reduce to C3Ži. ᎐ Živ. under the preceding restrictions.
I. Given u1 and u 2 for I and a ␤ s ␣ b, we have
u1 Ž ˜
x . u2 Ž ˜
y . y u1 Ž ˜
y . u2 Ž ˜
x.
s EŽ ˜
x 2 e d x̃ . Ee d ỹ w a␥ y ␣ c x
y EŽ ˜
y 2 c d ỹ . Ee d x̃ w a␥ y ␣ c x
q EŽ ˜
xe d x̃ . Ee d ỹ w b␥ y ␤ c x
y EŽ ˜
ye d ỹ . Ee d x̃ w b␥ y ␤ c x
s EŽ ˜
x 2 e d ˜x . Ž a␥ y ␣ c . q E Ž ˜
xe d ˜x . Ž b␥ y ␤ c . Ee d ˜y
y EŽ ˜
y 2 e d ˜y . Ž a␥ y ␣ c . q E Ž ˜
ye d ˜y . Ž b␥ y ␤ c . Ee d ˜x .
37
UTILITY FUNCTIONS FOR WEALTH
With ␣ 1 s a␥ y ␣ c and ␣ 2 s b␥ y ␤ c, precisely the same conclusion obtains
when we redefine u1 and u 2 by
u1 Ž w . s Ž ␣ 1 w 2 q ␣ 2 w . e d w
u2 Ž w . s e d w ,
which give the forms in C3Ži. of Theorem 3.
II. Given u1 and u 2 for II, we have
u1 Ž ˜
x . u2 Ž ˜
y . y u1 Ž ˜
y . u2 Ž ˜
x . s Ž a␤ y ␣ b . EŽ ˜
xe c ˜x . Ee c ˜y y E Ž ˜
ye c ˜y . Ee c ˜x
q Ž a␦ y ␣ d . E Ž ˜
xe c ˜x . Ee f ˜y y E Ž ˜
ye e ˜y . Ee f ˜x
q Ž b␦ y ␤ d . Ee c ˜x Ee f ˜y y Ee c ˜y Ee f ˜x .
when a ␤ s ␣ b, this can be written as
c1 E Ž ˜
xe c ˜x . q c 2 Ee c ˜x Ee f ˜y y c1 E Ž ˜
ye c ˜y . q c 2 Ee c ˜y Ee f ˜x ,
which is the form for u1Ž ˜
x . u2Ž ˜
y . y u1Ž ˜
y . u2Ž ˜
x . obtained when we redefine u1 and
u 2 by
u1 Ž w . s Ž c 1 w q c 2 . e c w
u2 Ž w . s e f w ,
as in C3Živ. of Theorem 3.
When a ␦ s ␣ d, the initial expression can be written as
c0 E Ž ˜
xe c ˜x . y c 2 Ee f ˜x Ee c ˜y y c 0 E Ž ˜
ye c ˜y . y c 2 Ee f ˜y Ee c ˜x ,
which is the form for u1Ž ˜
x . u2Ž ˜
y . y u1Ž ˜
y . u2Ž ˜
x . obtained when we redefine u1 and
u 2 by
u1 Ž w . s c 0 we c w y c 2 e f w
u2 Ž w . s e c w ,
as in C3Žiii. of Theorem 3.
III. Given u1 and u 2 for III, it suffices from the symmetry of terms to suppose
that a ␤ s ␣ b. Then
u1 Ž ˜
x . u2 Ž ˜
y . y u1 Ž ˜
y . u2 Ž ˜
x . s Ž ␤ 1 Ee r x̃ q ␤ 2 Ee s x̃ . Ee t ỹ
y Ž ␤ 1 Ee r ỹ q ␤ 2 Ee s ỹ . Ee t x̃ ,
38
BELL AND FISHBURN
which is the same as the form obtained when we redefine u1 and u 2 by
u1 Ž w . s ␤ 1 e r w q ␤ 2 e s w
u2 Ž w . s e t w
as in C3Žii. of Theorem 3.
6. Contextual uncertainty under WLU and SSBU
We conclude our proofs with results for WLU and SSBU that involve C4. We begin
under the conditions of Theorem 3.
Proof for WLU that C4 « C3. Let
V Ž w . s Eu1 Ž w q ˜
x . Eu 2 Ž w q ˜
y . y Eu1 Ž w q ˜
y . Eu 2 Ž w q ˜
x. .
If w is itself uncertain, say w
˜ s Ž w1 , p, w 2 ., then
VŽw
˜ . s pV Ž w1 . q Ž 1 y p . V Ž w 2 . q p Ž 1 y p .
=
Ž Eu1 Ž w1 q ˜x . y Eu1 Ž w 2 q ˜x . .Ž Eu2 Ž w 2 q ˜y . y
Eu 2 Ž w 1 q ˜
y. .
q Ž Eu1 Ž w 1 q ˜
y . y Eu1 Ž w 2 q ˜
y . .Ž Eu 2 Ž w 1 q ˜
x . y Eu 2 Ž w 2 q ˜
x. . .
So long as V Ž w 2 . / 0, V Ž w
˜ ., Ž1 y p .V Ž w 2 . and pV Ž w1 . q Ž1 y p .V Ž w 2 . will have
the same sign for sufficiently small p. This fact allows us to adapt the argument
used to show that C4 « C3 for EU in Bell Ž1995b, p. 1149.. That proof made use
of the fact that, for EU at least, V Ž w
˜ . s pV Ž w1 . q Ž1 y p .V Ž w 2 ., but in fact the
proof only requires that V Ž w
˜ . and pV Ž w1 . q Ž1 y p .V Ž w 2 . have the same sign for
small p. Hence Bell’s argument also applies to WLU. Thus C4 « C3.
Proof of Theorem 3 for C4. Recall that four WLU families satisfy C3. The
following table sets up an analogy between these families and closely related EU
families.
WLU Family
EU Family
Ži. u1 s ae b w q ce d w , u 2 s e f w
u s ae b w q ce d w
Žii. u1 s awe b w q ce d w , u 2 s e b w
u s ae b w q ce d w
Žiii. u1 s Ž aw q b . e c w , u 2 s e d w
u s Ž aw q b . e c w
Živ.
u1 s Ž aw 2 q bw . e c w , u 2 s e c w
u s aw 2 q bw.
UTILITY FUNCTIONS FOR WEALTH
39
To begin, consider a WLU function with a general u1 but specific u 2 , namely
u 2 s e c w . To satisfy C4, we must show that the relative preference between
xq˜
z q kz˜ q ˜
x and w q ˜
z q kz˜ q ˜
y cannot depend on the particular resolution of
˜z if it does not depend on the particular resolution of kz.
˜ The difference in their
expected utilities is
Eu1 Ž w q ˜
z q kz˜ q ˜
x . e c w Ee c ỹ Ee c z̃ Ee ck z̃
y Eu1 Ž w q ˜
z q kz˜ q ˜
y . e c w Ee c x̃ Ee c z̃ Ee ck z̃ ,
which has the same sign as
Eu1 Ž w q ˜
z q kz˜ q ˜
x . Ee c ỹ y Eu1 Ž w q ˜
z q kz˜ q ˜
y . Ee c x̃ .
Ž 27 .
Since all the C3 families have u 2 exponential, it will suffice to think about the
properties of Ž27. when ˜
z or kz˜ is resolved.
Case (i). When u1 s ae b w q ce d w , equation Ž27. becomes
ae b w Ee b z̃ Ee b k z̃ Ee b x̃ Ee f ỹ q ce d w Ee d z̃ Ee d k z̃ Ee d x̃ Ee f ỹ
y ae b w Ee b z̃ Ee b k z̃ Ee b ỹ Ee f x̃ y ce d w Ee d z̃ Ee d k z̃ Ee d ỹ Ee f x̃ .
This may be written as
ae b w Ee b z̃ Ee b k z̃A q ce d w Ee d z̃ Ee d k z̃ B,
Ž 28 .
where A and B are functions of ˜
x and ˜
y but not of ˜
z or kz.
˜ Note what happens
when we consider C4 for the EU family ae b w q ce d w . The difference in expected
utility between w q ˜
z q kz˜ q ˜
x and w q ˜
z q kz˜ q ˜
y is
ae b w Ee b z̃ Ee b k z̃ C q ce d w Ee d z̃ Ee d k z̃ D,
Ž 29 .
where C and D are functions of ˜
x and ˜
y but not of ˜
z or kz.
˜ Although A / C and
B / D, it is the case that if for some ˜
z and kz˜ there is a counterexample to C4 for
some choice of ˜
x and ˜
y in Ž28., then there will also exist a counterexample for the
same ˜
z and kz˜ in Ž29., albeit with a different ˜
x and ˜
y, but chosen so that
AD s BC.
Now ae b w q ce d w satisfies C4 for EU if and only if bd F 0. Hence u1 s ae b w q
dw
ce , u 2 s e f w satisfies C4 for WLU if and only if bd F 0.
Case (ii). When u1 s awe b w q ce d w and u 2 s e b w , expression Ž27. is
E aŽ w q ˜
z q kz˜ q ˜
x . e b w e b ˜z e b k ˜z e b ˜x Ee b ˜y q ce d w Ee d ˜z Ee d k ˜z Ee d ˜x Ee b ˜y
y E aŽ w q ˜
z q kz˜ q ˜
y . e b w e b ˜z e b k ˜z e b ˜y Ee b ˜x
y ce d w Ee d z̃ Ee d k z̃ Ee d ỹ Ee b x̃ ,
40
BELL AND FISHBURN
which may be written as
ae b w Ee b z̃ Ee b k z̃A q ce d w Ee d z̃ Ee d k z̃ B.
Ž 30 .
As before, A and B depend on ˜
x and ˜
y but not on ˜
z or kz.
˜ Expression Ž30. is
analogous to Ž29., which was derived from the EU family ae b w q ce d w . Thus
u1 s awe b w q ce d w , u 2 s e b w satisfies C4 if and only if bd F 0.
Case (iii). When u1 s Ž aw q b . e c w and u 2 s e d w , Ž27. is
aŽ w q ˜
z q kz˜ q ˜
x . q b e c w e c ˜z e ck ˜z e c ˜x Ee d ˜y
E
yE
aŽ w q ˜
z q kz˜ q ˜
y . q b e c w e c ˜z e ck ˜z e c ˜y Ee d ˜x ,
or
aŽ w q ˜
z q kz˜. q b e c w e c ˜z e ck ˜z A q ae c w Ee c ˜z Ee ck ˜z B,
E
Ž 31 .
where A and B depend on ˜
x and ˜
y but not on ˜
z or kz.
˜ In fact, A s Ee c x̃ Ee d ỹ y
c ỹ
d x̃
c x̃ .
d ỹ
c ỹ . Ž d x̃ .
Ž
Ž
Ee Ee and B s E ˜
xe Ee y E ˜
ye E e .
If c s d then Ž31. has constant sign for all resolutions of ˜
z or kz,
˜ and so this
case satisfies C4. It is, however, a special case of case Žii.. The EU family
Ž aw q b . e c w leads to an expression identical to Ž31. except that A is replaced by
C s Ee c x̃ y Ee c ỹ and B by D s EŽ ˜
xe c x̃ . y EŽ ˜
ye c ỹ .. By analogy with this case, no
cases with c / d satisfy C4.
Case (i¨ ). When u1 s Ž aw 2 q bw . e c w and u 2 s e c w , expression Ž27. is
2
E aŽ w q ˜
z q kz˜ q ˜
x . q bŽ w q ˜
z q kz˜ q ˜
x . e c w e c ˜z e ck ˜z e c ˜x Ee c ˜y
ž
/
2
y E aŽ w q ˜
z q kz˜ q ˜
y . q bŽ w q ˜
z q kz˜ q ˜
y . e c w e c ˜z e ck ˜z e c ˜y Ee c ˜x ,
ž
/
which may be written Žafter dividing by e c w . as
Ž 2 aw q b . Ee c z̃ Ee ck z̃ E Ž Ž ˜x y ˜y . e c x̃ e c ỹ . q 2 aE Ž Ž ˜z q kz˜. e c z̃ e ck z̃ .
= EŽ Ž ˜
x y˜
y . e c x̃ e c ỹ .
qaEe c z̃ Ee ck z̃ E Ž Ž ˜
x2 y ˜
y 2 . e c x̃ e c ỹ .
s Ee c z̃ Ee ck z̃ Ž Ž 2 aw q b . A q aB .
q2 aE Ž Ž ˜
z q kz˜. e c ˜z e ck z . A,
Ž 32 .
where A s EŽŽ ˜
x y˜
y . e c x̃ e c ỹ . and B s EŽŽ ˜
x2 y ˜
y 2 . e c x̃ e c ỹ .. Comparing this to aw 2
q bw in EU, we find that
VŽw q˜
xq˜
z q kz˜. y V Ž w q ˜
yq˜
z q kz˜.
s Ž 2 aw q b . C q aD q 2 a Ž k q 1 . E Ž ˜
z . C,
Ž 33 .
41
UTILITY FUNCTIONS FOR WEALTH
where C s EŽ ˜
x y˜
y . and D s EŽ ˜
x2 y ˜
y 2 .. Though the expressions are close, they
are no longer identical but for A, B, C, D. While Ž33. is, in effect, independent of z̃
and kz˜ Žthere is no loss in assuming EŽ ˜
z . s 0. and so all quadratics satisfy C4 in
Ž
.
EU, we can show that 32 fails C4 for all c Žexcept c s 0, or a s 0.. To see this,
rewrite Ž32. as
Ž 2 aw q b . A q aB q 2 aAZ,
Ž 34 .
where
Zs
E Ž˜
z q kz˜. e c ˜z e ck ˜z
Ee c ˜z Ee ck ˜z
.
If a s 0, then Ž34. is independent of the ˜
z’s and trivially satisfies C4. If a / 0,
then the value of the terms Ž2 aw q b . A q aB may be selected at will by appropriate choice of w Žwhen A / 0.. There is no loss in assuming aA ) 0 Ž ˜
x and ˜
y may
be reversed at will.. In order then to ensure that Ž34. is compatible with C4, it must
be that the extremes of Z occur when kz˜ is resolved Žas kz min and kz max . rather
than when ˜
z is resolved Žas z min or z max .. We require then, that
Ž Eze
˜ c z̃ . e ck z mi n q Ž Ee c z̃ . kz min e ck z min
Ee c ˜z e ck z mi n
-
E Ž kze
˜ ck z̃ . e c z min q z min e c z min Ž Ee ck z̃ .
or
EŽ ˜
ze c z̃ .
Ee c z̃
q kz min -
kE Ž ˜
ze ck z̃ .
q z min .
Ee ck z̃
Similarly, we require
EŽ ˜
ze c z̃ .
Ee c z̃
q kz max )
k Ž Eze
˜ ck z̃ .
q z max .
Ee ck z̃
Substitute ˜
z s Ž1, 12 , y1. to get the requirements
e c y eyc
e c q eyc
yk-
k Ž e ck y eyc k .
Ž e ck q eyck .
y1
and
e c y eyc
e c q eyc
qk)k
ž
e ck y eyc k
e ck q eyck
/
q 1,
e c z min Ee ck z̃
42
BELL AND FISHBURN
so that
1yk-
k Ž e ck y eyc k .
Ž e ck q eyck .
y
Ž e c y eyc .
- k y 1.
Ž e c q eyc .
If c ) 0, then as k gets large the right-hand inequality is violated. If c - 0, the
left-hand inequality is violated. Thus u1 s Ž aw 2 q bw . e c w and u 2 s e c w satisfies
WLU only in trivial cases that are already covered.
This completes the proof of Theorem 3. We conclude with the completion of
Theorem 4.
C4 for SSBU. Assume the hypotheses of Theorem 4 along with C3. The
argument in Bell Ž1995b. for EU carries over precisely for SSBU, and so we omit
the details here. In outline, we first show that if C3 is false then so is C4. One can
also show that if V Ž w . s E␾ Ž w q ˜
x, w q ˜
y . is zero for exactly one value of w, but
is not strictly monotonic Žor equal to zero everywhere., then there exists a violation
of C4. Thus C4 for SSBU is equivalent to the requirement that V Ž w . is strictly
monotonic Žor constant. whenever it has at least one zero. ŽThere are no restrictions on V if it never changes sign..
It remains to establish which SSBU families have monotonic V functions. The
forms for C3 in the theorem imply
Ž i . V Ž w . s a1 q b1w
Ž ii . V Ž w . s a2 e aw q b 2 e b w
Ž iii . V Ž w . s a3 q b 3 e aw
Ž iv. V Ž w . s Ž a4 w q b4 . e aw
where the a j and bj do not depend on w. Cases Ži. and Žiii. satisfy C4 without
further restriction.
For Žii. we have a2 s Ew ␣ Ž ˜
x y˜
y .Ž e a x̃ y e a ỹ .x and b 2 s Ew ␤ Ž ˜
x y˜
y .Ž e b x̃ y e b ỹ .x.
If ␣ ' 0 or ␤ ' 0 or ab s 0, Žii. reduces to Žiii.. Otherwise, Žii. satisfies C4 if and
only if ab - 0.
For Živ. we have a4 s Ew ␣ Ž ˜
x y˜
y .Ž e a x̃ y e a ỹ .x and b4 s Ew ␣ Ž ˜
x y˜
y .Ž ˜
xe a x̃ y
a
x̃
a
ỹ
a ỹ .x
˜ye q Ew ␤ Ž ˜x y ˜y .Ž e y e .x. If either ␣ ' 0 or a s 0, Živ. reduces to Žiii..
Otherwise, it does not satisfy C4.
This completes the proof of Theorem 4.
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