Structuring public utility functions

Volume 3, Number 1

OPERATIONS RESEARCH LETTERS

April 1984

STRUCII.IRING PUBLIC UTILITY FUNCTIONS

Ralph L. KEENEY Systems Science Department, University of Southern California, Los Angeles, CA 90089, USA

Received December 1983 Revised February 1984 Public yon Neumann-Morgenstem utifity functions are constructed using assumptions about individual and subgroup sovereigntyand equality of individuals. These public utility functions require knowledgeof one value parameter about equity and the average of individual utility functions. The latter can be obtained by sampling. utility theory*decision analysis*public decisions*multiattribute utifity

Many important decision problems have consequences that significantly affect the public. Decisions on national policy concerning the environment, energy, immigration, foreign affairs, the death penalty and taxes are examples. When legislators, the executive branch, .the legal system, or regulators fece such problems, presumably one of their aims is to make a decision that will have desirable--perhaps even the best--consequences on the public. A key issue is, of course, what is desirable to the public. This is an issue of value, and it is obvious that different individuals may have very different values regarding the consequences of a decision. For most decision problems, the 'public servants" facing a decision try to account for the public values in an informal manner. They receive letters from constituents, talk to citizens, review newspapers, and perhaps even have opinion poll information. This information about values is useful, but less precise than need be for two basic reasons. First, it is often expressed in a manner that confounds values for consequences with perceptions of alternatives available, judgments about the likelihoods of possible consequences, and implications of other decision problems. Second, the values most often expressed are from the vocal, well-financed, articulate members of society who have an understanding of the political system. For some decision problems, it may be worthwhile to obtain public values that are both more focused on

the potential consequences of the problem being considered and more representative of the pubfic. The methodology and procedures of utility theory offer one approach to carefully quantify an individual's values for consequences (see Keeney and Raiffa [5]). This note suggests and motivates a procedure for integrating such individually assessed values into a representation of pubfic preferences. Because of the inherent uncertainties in most important decision problems, we want the public values to be expressed in terms of a v o n Neumann-Morgenstern utility function which is sirnply referred to as a utility function in this paper. Then the expected public utility can be used as a guide for evaluating alternatives (see yon Neumann and Morgenstern, 1947). The basic idea for structuring a public utility function would involve (i) sampling members of the public to obtain individual von Neumann-Morgenstern utility functions for public consequences, and (ii) combining them into an expression of public value. Section 1 provides the theory necessary to combine such individual utility functions and Section 2 comments on potential use of the results.

I. Structuring a puldk utility f u n c ~ n To define our problem, let us suppose that a decision is to be made that wi.ll affect a large

0167-6377/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North.Holland)

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OPERATIONS RESEARCH LETTERS

number of individuals (perhaps all the citizens of the United States). We label these individuals I~, i= 1. . . . . N where N can be in the millions. Let u~ be the utility function of individual I~ over consequences described by the set of attributes X which may be single- or multidimensional. With x representing a consequence, the utility u~(x) is the utility of x to individual I~. We are interested in a public utility function u e over the consequences x. To obtain u e, we will structure a utility function over the set of attributes U~, i = l . . . . . N, which are measured by the respective u~. Specifically, up(x) =f[ul(x ). . . . . uN(x)] where f is some function. Two basic assumptions are necessary. Individual and subgroup sovereignty. If the u i are constant for all but a subgroup of individuals, the evaluation of alternatives should only depend on the preferences of the individuals in this subgroup. Equality of individuals. The importance assigned to each individual's range of preferences is the same. The sovereignty assumption is technically equivalent to stating that for a utility function over attributes measured by individual preferences, any subset of these attributes is utility independent of the other attributes. To be precise, if J is the subgroup of individuals whose ui are not fixed and .I is the subgroup of other individuals, the sovereignty assumption is

ue(ul ..... uN) = gj(ui) + f j ( u y ) ~ ( u j )

for all J,

(1) where u.z and uj represent the sets of u~ in .7 and J respectively, fj is positive, and fj = u~ when J consists only of individual I~. Using the sovereignty assumption and results in Keeney and Raiffa [5], we know the public utility function must either be the additive form N

u,~(x) = E k,u,(x)

(2)

i=l

tO N

1 + k = 1-I(1 + kk,).

(4)

i=1 Distinctions between the additive and muitiplicative form are discussed in Section 2. To scale (2) and (3), let us define consequences x ? and x ° such that all x are less desirable than x? and more desirable than x,0 for individual 1~. Then set

u,(x:)=1, u,(~°)=o,

i = i .....

N,

(5)

so the scale of up ranges from up--1, when all u~ = 1, to up ffi 0, when all u~ = 0. Of course, because different individuals may order their preferences for the x differently, there may be no actual x leading to either Up = 1 or up--0. This would only occur when there is a desirable consequence x* and an undesirable consequence x ° such that all possible x are less preferred than x* and preferred to x ° for each individual. It is important to recognize that the interpersonal comparison of preferences is done by the scaling in (5). We recognize the historical controversy on this topic (see Arrow [1], Harsanyi [4], and Luce and Raiffa [7]), but also recognize that such interpersonal comparison is necessarily made either implicitly or explicitly for the public problews of concern in this paper. By exploiting the flexibility afforded in choosing scales, the analyst can likely render the 'equality of individuals' assumption as appropriate. On the other hand, this opportunity for manipulation requires careful scrutiny in any application of the theory. Finally, as a practical assumption, we assume that the u~ are each one of a finite set of M utility functions urn, m ffi 1 . . . . . M, and that the proportion of the public having function u,, is Pro" Then of course, EPm = 1 and the number of individuals with u,. is Npm. Results. It is easy to show that when up is additive,

or the multiplicative form

l + kue(x )= ~ [l + kk,u,(x)],

April 1984

uAx) = ~(x) - l
where k~ = 1/N in (2) because of the equality assumption and k~ is a positive constant (not equal to 1/N) in (3) with k being the nonzero solution

(6)

where M

r,(x) ffi E pmum(x)

(7)

mml

is the average of the individual utility functions. However, an analogous result when us is multiplicative is not so transparent. Specifically, an

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OPERATIONS RESEARCH LETrERS

excellent representation of up in this case is

+ k)°"'-l]

Using (13) in (12), we find

(s)

1 + kl,~v_.~limup(x)]=,

M

I~.! eknl'-"-~x~

where ~(x) is again defined by (7). ffi et~r.M-~p,.J..~x))

Proof of (8). The proof of (8), which uses ideas similar to a result of Meyer [8] concerning preferences over time, requires that we show [¢ + k)U,x~_ I]. u~(x) ffi~1 ix1 Define D=ffi~.N.tk~where D ¢ 1 or

(9)

lim

N-"* ao

the additive utility function (2) would hold. Then, because of the equality assumption, k~ ffi D/N. Substituting this into (3) yields

kD l + kup(x)= ~l [l +----~-u,(x)],

= e `Du~x).

(14)

Evaluating (14) when fi --- 1, and hence up ffi 1, we find 1 + k = e *°, SO

D = In(l + k ) / k .

(15)

Substituting (15) into (14) yields

, + q Jim .p(x))ffi

(lO)

(l+k) "x' (16)

which proves our result.

or

I + k..(x) =

1 + -~-u~tx)]

(11)

when like u, are collected. Taking the limit of (11) as N increases, lim (1 +

N--. oo

:

kup(x)) M kD

i n [, ÷

N'-*~,~ ~ m - I L

oq

)

which implies 1 + k( lim

/V--~ oO

up(x)l

) since first the limit of a sum is the sum of a limit and the limit of a product is the product of a fimit (Taylor [10]), and then the limit of a function of a sequence is the function of the limit of the sequence if the function is continuous at the limit of the sequence (Salas and Hille [9]). Also from calculus, it is known that lim 1 +

ffi e'.

(13)

2. Interpretation and imtential use d the results The main results are (6) and (8) which provide a reasonable manner to obtain a utility function for the public given the basic assumptions of 'individual and subgroup sovereignty' and 'equality of individuals'. These public utility functions are special cases of either the additive utility function (2) or the multipficative utility function (3). From Keeney and Raiffa [5], we know that the additive utifity function is actually a limiting case of the multiplicative utility function when k in (3) approaches zero (which occurs when ,Y.k~approaches one). In the additive case, there are no interaction terms in the utility function to account for concerns like equity. The k in (3) and (8) does allow one to account for equity--a desire to evenly satisfy all--when k is positive, or its opposite--a desire to at least please some--when k is negative. The important point about results (6) and (8) is that the pubfic utifity function can be expressed with two pieces of information, the average t~tility function (7) of the individuals making up the public and the parameter k concerning equity. A reasonable representation of the former can be obtained by sampling the public to determine a number of individual utility functions. As suggested in Baecher [2], the information from such sampling can be combined with any other information on public preferences, such as market data,

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OPERATIONSRESEARCHLETTERS

for evaluating alternatives. The parameter k can be determined based on one value judgment expressed by a choice between two options with different equity implications. In any use of the results here, key questions would involve whose utility functions to assess and how to assess them. The answers depend on the specific problems, but to illustrate one case, consider the situation in Tacoma, Washington where a smelter provides employment for many but emits toxic arsenic as one of its pollutants. The Environmental Protection Agency estimates that one statistical life is lost per year due to the pollution, but that closing the facility would result in the direct loss of 570 facility jobs (see Los Angeles Times [6]). EPA proposes that public values be used to evaluate the alternatives of closing the facility or not. Even in this 'local' problem, many citizens have a legitimate voice. It is clear that all these citizens cannot directly interact in the decision process in a manner that would allow a well thought out decision. Also, the average citizen would probably have little to contribute to the technical aspects (i.e. dose-response effects) of the problem. Hence, citizen input should focus on values for the possible impacts and sampling of some sort is necessary. One might randomly sample citizens and assess utility functions over the impacts and an appropriate level for parameter k. If even one hundred citizens' values were carefully assessed, the insights from analysis of the options using (6) or (8) could prove useful. Another assessment strategy might first identify the stakeholders in the problem. Two obvious stakeholders might be smelter workers and citizens living in the vicinity of the smelter. Stakeholder utility functions might be constructed with sampling using ideas in this paper. Analysis of options would then use each of. the separate stakeholder utility functions or combine them in some manner into a public utility function (see, for example, Gardiner and Edwards [3] and Keeney and Raiffa [5]). For many decision problems involving the public, public input about values for the consequences of action could be useful. In obtaining public values, there will always be a tradeoff between the number of individuals involved and the quality of information from any particular involvement. Util-

April 1984

ity functions offer an operational procedure to describe carefully, and relatively completely, the values of an individual. The results in this note prc,vide a reasonable procedure to integrate those individual values into a public utility function. Especially for problems where simple expressions of value (i.e. do you like alternative A) simply do not provide useful information because they avoid the complex value issues in the problem, the results here could be helpful.

Acknowledgement The work described here was supported by contract N00014-81-C-C0536 with the Office of Naval Research. The author thanks David Bell of Harvard, Gary Smith of Woodward-Clyde Consultants, and Detlof yon Winterfeldt of the University of Southern California for helpful discussions on this paper.

References [ll K.J. Arrow, Social Choice and Individual Values (Wiley, New York, 1951, 2nd ed., 1963). [2] G.B. Baecher, "Sampling for group utility", Report RM75-39, International Institute for Applied Systems Analysis, Laxenburg,Austria, 1975. [3] P.C. Gardiner and W. Edwards, "Public values: Multi-attribute utility measurement for social decision making", in: M.F. Kaplan and S. Swartz,eds., Human Judgment and Decision Processes (AcademicPress, New York, 1975). 14] J.C. Harsanyi, "Cardinal welfare, individualistic ethics and interpersonal comparisonsof utility", Journal of Political Economy 63, 309-321 (1955). [5] R.L. Keeneyand H. Raiffa, Decisions with Multiple Objectives (Wiley, New York, 1976). [6] Los AngelesTimes, "What cost a life? EPA asks Tacoma", August 13,1983. [7] R.D. Luce and H. Raiffa, Games and Decisions (Wiley, New York, 1957). [8] R.F. Meyer, "On the relationship among the utility of assets, the utility of consumption and investment strategy in an uncertain, but time invariant world", in OR 69: Proceedings of the Fifth International Conference on Operational Research, J. Lawrence, ed., Tavistock Publications,

London, 1970. 19] S.L. Salas and E. Hille, Calculus: One and Several Variables (Wiley, New York, 1978). [10] A. Taylor, Calculus (Prentice-Hall, EnglewoodCliffs, NJ, 1959).