One-Switch Utility Functions and a Measure of Risk

INFORMS is collaborating with JSTOR to digitize, preserve and extend access to Management Science.

http://www.jstor.org

One-Switch Utility Functions and a Measure of Risk Author(s): David E. Bell Source: Management Science, Vol. 34, No. 12 (Dec., 1988), pp. 1416-1424Published by: INFORMSStable URL: http://www.jstor.org/stable/2632032Accessed: 18-06-2015 14:53 UTC

REFERENCESLinked references are available on JSTOR for this article:

http://www.jstor.org/stable/2632032?seq=1&cid=pdf-reference#references_tab_contents

You may need to log in to JSTOR to access the linked references.

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

This content downloaded from 143.106.81.78 on Thu, 18 Jun 2015 14:53:51 UTCAll use subject to JSTOR Terms and Conditions

MANAGEMENT SCIENCE Vol. 34, No. 12, December 1988

Printed in U.S.A.

ONE-SWITCH UTILITY FUNCTIONS AND A MEASURE OF RISK*

DAVID E. BELL Harvard Business School, Soldiers Field, Boston, Massachusetts 02163

Consider the relative attractiveness to a decision maker of two financial gambles as the wealth of that individual varies. It may seem reasonable that either one alternative should be preferred for all wealth levels or that there exists a unique critical wealth level at which the decision maker switches from preferring one alternative to the other. Decreasing risk aversion is not sufficient for this property to hold: we identify the small class of utility functions for which it does. We show how the property leads naturally to a measure of risk.

The results of this paper apply equally well to discounting functions for cash flows: one-switch discount functions permit at most one change in preference between cash flows as all payoffs are deferred in time. (DECISION ANALYSIS; UTILITY THEORY; RISK, DISCOUNTING)

1. Introduction

You are making choices among gambles and wish to abide by the axioms of expected utility. In addition, you are risk averse and, indeed, decreasingly risk averse. For two particular gambles A and B you prefer A to B. If you had more money, say $20,000 more, you feel you would then prefer B to A. Could it be that with even more money, say another $20,000, you might switch back to preferring A to B? I suspect that you think this shouldn't happen: that you would confine your choice of utility function to one that does not allow such switching back and forth, that you think all well behaved decreasingly risk-averse utility functions probably qualify.

DEFINITION 1. A decision maker obeys the one-switch rule if, for every pair of alternatives whose ranking is not independent of wealth level, there exists a wealth level above which one alternative is preferred, below which the other is preferred. A one-switch utility function permits an expected utility maximizer to satisfy the one-switch rule.

In general, an n-switch utility function permits at most n switches of preference between a pair of alternatives. For expositional convenience an (n - 1)-switch automatically qualifies as an n-switch. The class of zero-switch utility functions is well known.

PROPOSITION 1. Linear and exponential functions are the only zero-switch utility functions.

A linear utility function is unsatisfactory if a decision maker is risk averse, preferring less uncertainty in a gamble for any given expected value. An exponential utility function is unsatisfactory if a decision maker is decreasingly risk averse since, due to its zero- switch property, no amount of wealth can make the risk in a once rejected gamble acceptable.

One purpose of this article is to propose the one-switch rule as a reasonable property for utility functions to satisfy. Though many people find the rule intuitively appealing, the alternatives shown in Figure 1 may help to explain how a violation might occur.

If you are poor, you may wish to avoid alternative B since it has the worst payoff. On the other hand, if you are rich you should prefer A since it offers a higher expected value,

* Accepted by Robert L. Winkler; received August 28, 1987. This paper has been with the author 2! months for 1 revision.

1416 0025-1909/88/3412/1416$01.25

Copyright C) 1988, The Institute of Management Sciences


ONE-SWITCH UTILITY FUNCTIONS 1417

FIGURE 1. Two Alternatives That May Switch Twice.

Alternative A Alternative B

$10 $3

2 .9

$1 $-2

$2.80 as opposed to $2.50 for B. At moderate levels of wealth, however, you may prefer B on the grounds of its lower variability (variance of $2.25 compared to $12.96 for A).

Is such behavior consistent with expected utility? The logarithmic utility function is risk averse, decreasingly risk averse and generally well behaved, but switches from A to B and back to A as wealth increases:

Wealth Level Choice Informal Criterion u(w) = log w

Poor A Max Min 2 < w < 3.4 Moderate B Variance 3.5 < w < 7 Rich A Expected Value 7 < w

How could our intuition lead us so far astray? I believe that our informal thinking about the one-switch rule is as follows. If B becomes preferred to A as wealth increases, it must be that B is riskier than A. Being decreasingly risk averse means that as wealth rises, risk diminishes in importance so that further increases in wealth can only reinforce our choice of B. The mistake occurs with the assumption that one alternative can unilaterally be declared more risky than another, even for one individual. At least that's where our intuition and expected utility part company.

In the next section of the paper we identify the class of one-switch utility functions. In ?3 and ?4 we pursue two applications of the one-switch rule. The first is based on the close association between the one-switch rule and our notion of riskiness. For as a corollary to my explanation of how the intuition of single switching comes about, we know that only people with one-switch utility functions can unambiguously rank order gambles by the property of riskiness.'

The second application is to the domain of discounted cash flows.2 The analogous rule is that as two cashflows are deferred in time, preference between them should switch at most once. ?5 identifies the class of n-switch utility functions.

Finally ?6 contains a proof that no assumptions of continuity and differentiability are needed to prove our main result. That is, there are no "trick solutions" to the one- switch rule.

2. One-Switch Utility Functions

The following proposition identifies the class of one-switch utility functions. The class is identical to that previously discovered by Farquhar and Nakamura (1987) as a result of their condition called augmented constant exchange risk. As with the one-switch rule

'The one-switch rule developed during discussions with Vijay Krishna after he questioned when such an ordering was possible.

2 Drazen Prelec and Michael Rothkopf pointed out this connection to me.


1418 DAVID E. BELL

their condition is a property holding for all wealth levels, but otherwise the two conditions are not obviously equivalent. In Farquhar and Nakamura ( 1988 ) they provide constructive assessment procedures for this class. Their representation theorem assumes unlimited differentiability of the utility function and we will too, at least in this section. ?6 provides a proof that does not use either continuity or differentiability.

PROPOSITION 2. A utility function satisfies the one-switch rule if and only if it belongs to one of thefollowingfamilies:

(i) the quadratics, u(w) = aw2 + bw + c; (ii) the sumex functions, u(w) = aebw + cedw, (iii) linear plus exponential, u(w) = aw + becw, (iv) linear times exponential, u(w) = (aw + b)ecw. The first and fourth of these have the disadvantage, for practical applications, of being

increasingly risk averse for all choices of parameters a, b and c. The sumex function is discussed by Schlaifer (197 1) as being convenient for assessment purposes and shown to be increasing, risk averse and decreasingly risk averse so long as all its coefficients are negative. The third class of one-switch utility functions, u( w) = aw + becw is a limiting case of a sumex function in which one of the exponentials has converged to a linear function.

PROPOSITION 3. If a decision maker (a) prefers more money to less, (b) wishes to obey the axioms of expected utility, (c) is risk averse at all wealth levels, (d) is decreasingly risk averse at all wealth levels, (e) wishes to obey the one-switch rule, (f) will approach risk neutrality for small gambles when extremely rich,

then the only feasible utility function is u( w) = w - be-cw for some positive parameters b and c.

PROOF OF PROPOSITION 2. We begin by demonstrating that the named functions are one-switch functions.

Quadratics: If u(w) = aw2 + bw + c and x is a gamble, then Eu(w + x) = aw2 + 2aw, + blu + bw + c + au2 + aA2 where ,u = E(x) and a2 = E(x - ,)2. For two gambles xl and x2 we have E(u(w +.k1) - u(w + x2)) = 2aw(til - A2) + b(zl - A2) + a( 2 - 2) + a(2 - 2_ ). So long as A1 * A2, preferences between these gambles will switch exactly once. If ti = ,U2 they will not switch at all.

Sumex: If u(w) = aebw + cedw let a = E(ebxl - ebi2) and f3 = E(edxl - edi2) then E(u(w +.k1) - u(w + x2)) = agaebw + fcedw. Thus preferences for xl and x2 switch whenever e (b-d)W =-c/aa. This equation has exactly one solution if b # d.

Linear plus exponential: If u(w) = aw + becw, let a = E(x1 - x2) and: = E(edx - ed2) and proceed as before.

Linear times exponential: If u(w) = (aw + b)ecw let a = E(1ecx' - g2ecx2) and: - E(ecx' -ec2). Then E(u(w +.k1) - u(w + X2)) = f(aw + b)ecw + aaecw. This equals zero only when w = -(oaa + bo)/fa.

Now to show that these are the only one-switch functions. Let x and y be any distinct gambles with suitably small spread. We may write:

c0

Eu(w + x) - Eu(w + E) = z mkUk(W) k=l

where mk = (E(gk) - E(yk))/k! and uk is the kth derivative of u. If u is to be one- switch then the system of equations



00

I mkuk(wi) = ai, i = 1, 2, 3, k= 1

cannot hold for w1 < W2 < W3 if ala2 < 0 and a2a3 < 0. But if we can identify some set of W1 < W2 < W3 such that the 3 X 3 matrix (Uk(Wi)) (k = 1, 2, 3) has a nonzero determinant, then there will exist for any choice of a's a solution in terms of the m's which will satisfy the equations. (The relevant gambles x and y are deduced from the equations E(gk) - E( 5k) = Mkk! where Mk = 0 for k > 3.) Thus a necessary condition for a one-switch function to satisfy is that the first 3 derivatives be collinear i.e. for some coefficients b1, b2 we have U3(W) = blul(w) + b2u2(w). This differential equation has the solutions stated in the proposition (see for example Piaggio 1965, p. 25). In general, the solutions could involve imaginary coefficients. These can easily be dismissed in the context of utility functions except in the case of sumex functions. For if u(w) = ae(c+id)w + be(c-id)w where i = V-T then if a = b, u becomes real and equal to 2aecw cos dw. This function, however, is an infinite switch utility function since it alter- nates sign indefinitely.

PROOF OF PROPOSITION 3. Pratt (1964) has shown that a necessary and sufficient condition for a doubly differentiable utility function to be risk averse is that r(w) =

-u"(w)/u'(w) is positive. A necessary and sufficient condition for decreasing risk aversion is that r be monotonically decreasing. The quadratic u( w) = aw2 + bw + c has u'= 2aw + b, u" = 2a and r' = 4a2/(2aw + b)2 which shows that r is always increasing. Similarly for u(w) = (aw + b)ecw we have u' = (a + bc + acw)ec', u" = (2ac + bc2 + ac2w)ecw, r = -c - ac/(a + bc + acw), r' = a2c2/(a + bc + acw)2 which is always positive. The sumex functions u(w) = aebw + ce dw are increasing, risk averse and decreasingly risk averse so long as each of a, b, c, and d is negative but r(oo) = min(-b, -d) which is not zero, as required by condition (f) of this proposition. This is a somewhat artificial condition since a utility function is usually of interest only over some bounded interval since infinite sums of money do not occur in practice.

For u(w) = aw + becw we have u' = a + bcecw, u" + bc2ecw so r(w) = -bc2/(ae-cw + bc) and r'(w) = -abc3e-cw/(ae-cw + bc)2. For r' < 0 we require abc > 0. For u' > 0 we require a > 0 and bc > 0. For r > 0 we require b < 0. Hence the conditions a > 0, b < 0, c < 0 are necessary and sufficient for aw + becw to satisfy the conditions of the proposition.

3. Measuring Risk

Many people wish to regard decisions under uncertainty as problems of trading off risk against return. In this section we establish the conditions under which such a view is compatible with expected utility and derive the relevant measure of risk.

The one-switch rule (Definition 1) does not incorporate a rather basic element of the intuition that supports it. In the opening paragraph of this paper you were asked to imagine that an increase in wealth of $20,000 caused you to switch from choosing A to choosing B. A common interpretation of this switch is that B must have been riskier than A. Hence we could strengthen the one-switch rule by requiring, not only that only one switch may occur, but also that it be in favor of the more risky alternative. While intuitively satisfying, this strengthening is rather vague without a definition of risk. How- ever if we assume that our intuition presupposes some system by which risk is traded off against return, then we may finesse this difficulty by recognizing that if the switch is in favor of the alternative with the higher risk it must also be in favor of the alternative with the higher return. Though a definition of risk may not be standard, a definition of return most certainly is, namely the expected value E(x).


1420 DAVID E. BELL

DEFINITION 2. A decision maker is risk consistent if, whenever a preference reversal occurs due to an increase in wealth, the newly favored alternative has a higher expected value.

PROPOSITION 4. Risk consistency implies the one-switch rule.

PROOF. To switch from x to y implies E(y) > E(x). Hence a switch back is not risk consistent.

PROPOSITION 5. An expected utility maximizer who is risk consistent and risk averse is also decreasingly risk averse (or constantly risk averse).

PROOF. If c(w) is the certainty equivalent of a gamble x at wealth level w, then by risk aversion c(w) < E(xf). By risk consistency, c(w) must be increasing (or constant) in w.

PROPOSITION 6. If a decision maker (i) prefers more money to less, (ii) wishes to obey the axioms of expected utility, (iii) is risk averse at all wealth levels, (iv) is risk consistent,

then the only feasible utility function is u( w) = aw - be-cw where a 2 0 and b and c are positive.

PROOF. From the proof of Proposition 3 we know that the only one-switch utility functions that are increasing, risk averse and decreasingly risk averse are

(a) aw - be-cw for a, b, c positive and (b) -ae-bw - ce-dw for a, b, c, d positive. To see that (a) satisfies risk consistency note that for large w, gambles are ranked by

expected value, hence any switches that occur must be in the direction of the one with the higher expected value. To see that (b) does not, it suffices to note that two exponentials will not agree on the ranking of all gambles with a given mean. However there are counterexamples even for gambles with different means. For example, using the alternatives in Figure 1, alternative A is ranked lower than alternative B only when the coef- ficient of risk aversion is between 0.067 and 0.529.

Proposition 6 holds even when the assumptions are required to hold only on some open interval of wealth levels (because the range of gambles is unrestricted), so that the implications cannot be avoided simply by restricting the utility function's domain to some small range.

To interpret Proposition 6, let us suppose that a decision maker wishes to order all gambles x by their risk (or riskiness), R(x) and then to make decisions about relative preference between gambles based solely on their levels of risk, R (x), and return, E(x). What the proposition tells us is that such a scheme is compatible with expected utility (and behavioral assumptions such as risk aversion) if and only if u( w) = w - be-cw (b, c > 0) or the special case u(w) = -e-cw (c > 0).

We achieved this result without relying on any definition of risk, other than that implicit in the concept of risk aversion, indeed without any presumption (except in the motivation behind Definition 2) that a risk ordering existed at all.

But as a result of Proposition 6 we may infer a measure of risk for any decision maker satisfying its assumptions. We have

Eu(w +xk) = w + E(x) - be-cwE(ecx).

A rather appealing definition of risk would be R(x) = E(ecx) since this leads to a deCiSion criterion Of



E(x-,) - k(w)R(x) (1) where k( w) is a tradeoff constant declining in w. With this definition of risk, the zero- switch utility functions have a simple interpretation. The linear u(w) = w yields Eu(w + .) = w + E(x), a case of maximizing return, while the exponential u(w) = -e-'w yields Eu(w + x) = -e -cwE(e`cx) a case of minimizing risk.

However it does not seem sensible to regard a person with constant risk aversion as being interested only in minimizing risk. For this person will take a risk if the returns are adequate.

I think it is better to regard risk and return as "orthogonal" concepts, that is the riskiness of a gamble should be independent of its expected value.

ASSUMPTION. (Independence of risk and return). Two alternatives whose distribution of payoffs differs only by a translation of mean, have the same risk.

PROPOSITION 7. The only definition of risk which is compatible with an increasing, risk averse utility function, with risk consistency and which is independent of return, is R(x5) = E(e-c(x-E(x))) or a monotonic transformation thereof.

PROOF. We may write Eu(w + x) = E(w + x - bec(v+x)) as w + E(xZ) - be-e`E(x) X Ee-c(x-E(x)). It is clear that R(xk) = Eec(xE(x)) is independent of E(x.) and w. By assumption the expected utility for a given w depends solely on risk and return; since E(x) is the return, R(xk) must be the risk.

Note that the exponential case has now a more appealing interpretation. Eu( w + x) may be written as-e-cwecE(x)R(x). Taking logarithms we see it is equivalent to evaluating gambles with the formula E() - c-' log R(x) which of course does not depend on w. The corresponding formula for u( w) = w - be-cw is

E(x)- k(w + E(xZ))R(x&) (2) which makes some sense if one thinks of w + E(x) as the expected wealth upon taking the gamble. The aesthetic value of criterion (1) must be balanced against the independence of risk and return which led to (2).

Because both definitions of risk depend on the individual through the parameter c, it is not possible to talk about the risk of a gamble. This is unfortunate for those who would wish to separate the measurement of risk from an individual's aversion to it. In retrospect however it does seem sensible to permit disagreement on the relative importance of the tails of a distribution: people with high c's place more emphasis on the possibility of bad outcomes. For a given c, however, the parameter b does control the degree of aversion to risk.

4. Discounting Cash Flows Let x be a cashflow which pays off an amount $xi at time ti, i = 1, . , k. Analogous

axioms to those leading to expected utility for gambles give us discounted cashflows: there exists a discount function d(t) such that cashflow x is preferred to cashflow y if and only if

k k

2: xid(ti) > yid(ti). i=l I=l

Since the scale of d is arbitrary, we may take d(O) = 1. Receiving money at t = oo is worthless so d(oo) = 0. It is also safe to assume that people are impatient, preferring income sooner rather than later, so d(t) is decreasing in t.

One way to think about properties of the discount function is to consider how behavior changes if all cashflows are delayed by a fixed interval of time, say T. As viewed from time T the discount function may be regarded as d( T + t)/d( T) .


1422 DAVID E. BELL

We will say that a decision maker is increasingly, decreasingly, or constantly impatient according as d( T + t)/d( T) is less than, greater than, or equal to d(t) for all T and t. (The relationship will not be in the same direction for all T and t in general.)

Koopmans (1960) showed that if a fixed delay never causes preference to change, then the decision maker must be constantly impatient. The relation d( T + t) = d(T)d(t) leads, of course, to the conclusion that d( t) = e-Ct for some c. In our terminology Koop- mans' proposal was for a no-switch rule for cashflows. (The linear case is invalid because d becomes negative.)

It is well known that if money is lent and borrowed at the same rate, then a rational decision maker should use exponential discounting, but in the absence of such efficient money markets, the no-switch rule may not be so attractive. It seems perfectly plausible that someone would choose to receive $1,000 today over $2,000 in one year's time, but prefer $2,000 in 6 years' time over $1,000 in 5 years' time.

PROPOSITION 8. A discountfunction d(t) satisfies the one-switch rule (and d(0) = 1, d(oo) = 0 and d(t) decreasing) but is not exponential if and only if d(t) = ae-bt + (1 - a)e-(b+c)t where a, b, c are positive and a < 1 + b/c. It is decreasingly impatient if a < 1 and increasingly impatient if 1 < a < 1 + b/c.

PROOF. We need only check the four functional forms of Proposition 2. The quadratic can never satisfy d(oo) = 0 and d(0) = 1. The linear plus exponential cannot both be decreasing and always positive. This applies to the linear times exponential also.

For the sumex function we may as well write d(t) = ae-bt + ce-dt with b, d positive so that d(oo) = 0. Now d(O) = 1 means a + c = 1. Without loss we may assume d > b. Since d'(t) = -(bae-b1 + cde-dl) we must have ba + cd > 0. Also, for no positive t must bae-bt + cde-dt = 0 or e(d-b)t = -cd/ba. This will be so only if-cd/ba < 1. If a > 0 this is ba + cd > 0 as before. If a < 0 this is ba + cd < 0. Thus we require a > 0 andba+(l -a)d>Oora d( T)d(t): ae-bTe-bt + (1 -a)e-(b+c)Te-(b+c)t

> (ae-bT + (1 - a)e-(b+c)T)(ae-bt + (1 - a)e-(b+c)t) if a(l - a)[e-bT- e-(b+c)TI[e-bt -e-(b+c)t] > 0.

Thus the sumex is decreasingly impatient if a(l - a) is positive, and increasingly impatient if it is negative. O

It is not implausible for multiple switching to occur. If I need money urgently today and again when my children go to college, one can easily imagine (again, in the absence of borrowing/lending) that I would prefer

($1000 at t = 0) over ($2000 at t = 1), ($2000 at t = 6) over ($1000 at t = 5), ($1000 at t = 10) over ($2000 at t = 11), ($2000 at t = 16) over ($1000 at t = 15),

if my children graduate from college in year 10. This is modest justification for the next section of the paper, which identifies the class

of n-switch utility functions.

5. Multiple Switching

The following result for n-switch utility functions is proved in identical fashion to that of Proposition 2. Unlike Proposition 2, I do not have a proof that eliminates the need for an assumption of continuity, but since the one-switch functions are continuous and differentiable, it seems reasonable to conjecture that this is not an issue.



PROPOSITION 9. A utility function that is infinitely differentiable is an n-switch utility function if and only if it may be written as fo(w) + L1t-= fi(w)e'iw where fi(w) is a polynomial of order ni such that z k=o ni < n + 1 - k. (If this last inequality is an equality then u is not an (n - 1)-switch utility function .)

Among the class of n-switch utility functions are (i) (n + 1 )th order polynomials, and (ii) a sum of (n + 1) exponentials.

The only one-switch utility functions that were decreasingly risk averse were sums of two exponentials (counting a linear function as an exponential for this purpose). It is also the case that for two-switch utility functions the only decreasingly risk averse functions are sums of exponentials. However this is not true for all n. For example, the four-switch function u(w) = -el/2w _ (w2 + 4)e-w - e-3/2"' is increasing, risk averse and decreasingly risk averse everywhere.

6. Removing Trick Solutions We have noted that our results are not altered if attention is confined to a subinterval

of the real line. Another possible way around the restrictiveness posed by the one-switch rule would be if it were possible to construct, say, piecewise linear functions, that obeyed the rule. In this section we establish that no function, no matter how perversely con- structed, can satisfy the one-switch rule, except as allowed by Proposition 2.

LEMMA 1. To satisfy the one-switch rule, a utility function must be continuous.

PROOF. It suffices to consider the case u(w) = 0 for w < w*, u(w) = 1 for w ? w*. Any pair of gambles whose cumulative probability functions cross at least twice is a counterexample.

LEMMA 2. The certainty equivalent of any gamble is monotonic in w for any one- switch utility function.

PROOF. If, for three wealth levels w1 < W2 < W3, the certainty equivalent c(w) of a gamble x satisfies, say, c(w1) < c(w2) > c(W3) then for any value c* such that c(w2) > c* > max(c(wi), C(W3)) the sure payoff c* is preferred to x at w1, less preferred at w2 and preferred once again at W3, a contradiction.

LEMMA 3. If u is one-switch, and X1 I-X2 at two wealth levels wo and w1, then xl x2 for all w.

PROOF. Clearly x I x2 throughout the interval (wo, w1), for if xg1 > x2 at some w2 E (wo, w1) then a slight improvement in x2 would produce a double switch. But suppose there were some value w2 > w1 such that x1 > 2 say. In that case since u is not zero- switch there must exist a pair of gambles -1 and -2 and a value w* such that -2 and at w2 we have, for small enough E, z1 > z2. At w1, of course, z

z2. Now make z2 slightly more attractive to achieve a double switch.

The proof will proceed by demonstrating that for some pair of wealth levels w, and w2, and for any increment x less than some positive upper bound, there exist probabilities p and q (depending on x) such that the two gambles

a p chance at 2x and a (1 - p) chance at 0 and a q chance at 3x and a (1 -q) chance at x

are indifferent at both w, and w2 and hence, by Lemma 3, at all w. If we let u, denote u(wo + nx) for some base value wo then we have

pUn+2 + (1 - p)un = qun+3 + (1 q)u+

for all n. We now rely on a well known result (e.g., Gray 1967, p. 126).


1424 DAVID E. BELL

LEMMA 4. If {u u} is a sequence such that for all n Un+3 = au,+2 + bu,+1 + (1 - a - b)u, for some constants a and b then if zI, z2, z3 are the roots of the equation z3 = az2 + bz + (1 - a - b) where z3 may be assumed to be equal to one, then un is given by

(i) b1n2 + b2n + b3 if ZI = z2 = 1, (ii) b1zn + b2Zn + b3 if zl # z2 and neither equals 1, (iii) b1z n + b2n + b3 if zI # z2 = 1, (iv) (bln + b2)zl + b3 if z1 = z2 = 1

for various constants b1, b2 and b3. As x goes to zero, these formulas become the familiar functions of Proposition 2. We

will choose wo and x so that the range (wo, wo + 4x) lies entirely within a strictly monotonic range of u. We may as well assume u is increasing in this range. Let w1 = wo, w2 = wo + x.

We must show that if p and q are chosen so that pu2 + (1 - p)uo = qu3 + (1-q)u1 andpu3 + (1 -p)u1 = qu4 + (1 - q)u2 then O uO, U2 < pIU4 + (1 - pi)ul) can be reversed without affecting the conclusion. The argument for q is similar.

References FARQUHAR, P. H. AND Y. NAKAMURA, "Constant Exchange Risk Properties," Oper. Res., 35 (1987), 206-

214. AND , "Utility Assessment Procedures for Polynomial-Exponential Functions," Naval Res. Logist.

Quart., (1988) (to appear). GRAY, J. R., Probability, Oliver & Boyd, Edinburgh, 1967. KOOPMANS, T. C., "Stationary Ordinal Utility and Impatience," Econometrica, 28 (1960), 297-309. PIAGGIO, H. T. H., Differential Equations, G. Bell and Sons, London, 1965. PRATT, J. W., "Risk Aversion in the Small and in the Large," Econometrica, 32 (1964), 122-136. SCHLAIFER, R. O., Computer Programs for Elementary Decision Analysis. Division of Research, Harvard Business

School, Boston, 1971.


Article Contentsp. 1416p. 1417p. 1418p. 1419p. 1420p. 1421p. 1422p. 1423p. 1424

Issue Table of ContentsManagement Science, Vol. 34, No. 12, Dec., 1988Volume InformationFront MatterInducing Rules for Expert System Development: An Example Using Default and Bankruptcy Data [pp. 1403 - 1415]One-Switch Utility Functions and a Measure of Risk [pp. 1416 - 1424]An Object-Oriented World-View for Intelligent, Discrete, Next-Event Simulation [pp. 1425 - 1440]Approximating Expected Warranty Costs [pp. 1441 - 1449]Dynamically Updating Relevance Judgements in Probabilistic Information Systems via Users' Feedback [pp. 1450 - 1459]Selection and Design of Heuristic Procedures for Solving Roll Trim Problems [pp. 1460 - 1471]Computing Block-Angular Karmarkar Projections with Applications to Stochastic Programming [pp. 1472 - 1479]The Single Machine Problem with a Quadratic Cost Function of Completion Times [pp. 1480 - 1488]Towards a Heuristic Theory of Problem Structuring [pp. 1489 - 1506]NotesOn Piecewise Reference Technologies [pp. 1507 - 1511]Erratum: "The Theory of Ratio Scale Estimation: Saaty's Analytic Hierarchy Process" [p. 1511]

Back Matter [pp. 1512 - 1513]

Documents

One-Switch Utility Functions and a Measure of Risk