Objective interpersonal comparisons of utility

Objective interpersonal comparisons of utilityAuthor(s): Kevin RobertsSource: Social Choice and Welfare, Vol. 14, No. 1 (1997), pp. 79-96Published by: SpringerStable URL: http://www.jstor.org/stable/41106196 .

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Soc Choice Welfare (1997) 14: 79-96 ~ - I

Social Choice © Springer-Verlag 1997

Objective interpersonal comparisons of utility Kevin Roberts

London School of Economics, London, United Kingdom

Received: 16 February 1995/Accepted: 13 October 1995

Abstract. This paper examines the problem of distilling conflicting interpersonal comparisons into a single set of interpersonal comparisons. The mapping that achieves this has a richer co-domain than all social choice problems (and a richer domain than most social choice problems). The set of mappings satisfying a mild set of restrictions is very small. If interpersonal comparisons embody ratio-scale comparability then the mapping is Cobb-Douglas in form; if interpersonal comparisons embody no more than level and difference comparability then the mapping must be dictatorial and it is impossible to combine different interpersonal comparisons.

1. Introduction

Much has been written abut the existence or nonexistence of interpersonal comparisons of utility. From a pragmatic viewpoint, it is clear that interpersonal comparisons exist, in the sense that everybody makes such comparisons, and they form part of everyday discourse (Little 1957). To deny existence in the usual sense of this word, one would need to define interpersonal comparisons in a way that ruled as inadmissible these comparisons made by us all (Hausman 1994).

In fact, the standard existence debate is not about whether the interpersonal comparisons exist but the fact that there may be a multiplicity of such comparisons. For Robbins (1935), the problem is that there may be many different views or subjective comparisons but there is no "scientific" way of choosing between them. Thus, the fundamental question is how to proceed when there are many different views and, in particular, how can a set of objective interpersonal comparisons be found in such a situation? This is the subject of the present paper.

The author is grateful to seminar participants, particularly those at the 3rd Osnabrück Seminar on Individual Decisions and Social Choice, June 1995, and also to a referee for their useful comments and suggestions.



80 K. Roberts

For Robbins, the problem was one of starting with a number of subjective comparisons and then choosing one of them as the objective comparison. This paper considers the distillation of many subjective comparisons into a single objective comparison but does not impose the restriction that the distillation process must be one of choice. The notion of objective comparisons should not be misconstrued: it is not necessary to assume that objective comparisons have an independent existence which can be glimpsed only by looking at individuals' opinions; for the present purpose, it is sufficient to view them as an aggregation or distillation of individual opinions given by subjective comparisons. However, if an independent existence is assumed then this will influence the way in which individual opinions are aggregated. For instance, Harsanyi (1955) has proposed the idea that opinions differ only because of differences in experience and that underlying different opinions is a common basis; objective interpersonal comparisons may be interpreted as this common basis. If the mechanism by which opinions evolve from the common basis is known then it is natural to look at the inversion of this mechanism to move back from opinions to the basis.

In abstract terms, we are interested in mappings from different sets of (subjective) interpersonal comparisons to a single set of (objective) comparisons. The problem has a family resemblance to the Arrow (1963) problem of social choice where mappings from different orderings to a single ordering are investigated. Like Arrow, we adopt an axiomatic approach which investigates the class of mappings that satisfy a number of "reasonable" restrictions; the principal difference is that a set of interpersonal comparisons will be much richer than a single ordering so that both the domain and the co-domain of the mappings under investigation will be richer than in the Arrow problem.

The present problem also differs from the social choice analyses of the last two decades that have examined the problem of social choice, in the sense of determining a ranking of social states, given information based upon interpersonal comparisons. The original Arrow problem may be viewed as the problem of social choice when intrapersonal comparisons of utility across social states can be made but interpersonal comparisons are ruled out. This problem can be opened up to investigate what is possible when interpersonal comparisons can be used. The mappings to be investigated, which may be thought of as social welfare functions, take a single set of interpersonal comparisons to a single ordering. This is the problem faced by an individual making ethical judgements using his own subjective interpersonal comparisons or that faced by a society determining best policy using objective interpersonal comparisons. The extent to which the problem differs from that studied by Arrow is governed by the richness of interpersonally comparable information that is available and, following the construction of the appropriate framework by Sen (1970, 1974), a number of Arrow-like classification results for different information structures have been obtained by Hammond (1976), d'Aspremont and Gevers (1977), Deschamps and Gevers (1977), Maskin (1978), Gevers (1979) and Roberts (1980a, b) amongst others.

The principal difference compared to the literature just mentioned is that the mappings under investigation take several different sets of interpersonally comparable information, instead of one set, to a set of interpersonally comparable information, instead of to a single ordering. The problem is close to that studied in Roberts (1980a, 1994) where the aggregation of a number of different sets of interpersonal comparisons into a single social ordering is examined; in Roberts (1994) this is termed "double-aggregation" because there is both an aggregation of different sets of interpersonal comparisons and there is an aggregation of different



Objective interpersonal comparisons of utility 81

individuals' well-being, as captured by interpersonal comparisons, into an ordering capturing societal well-being. One particular example of double-aggregation oc- curs when different sets of interpersonal comparisons are first mapped to a single (objective) set of interpersonal comparisons and then this single set of comparisons is used in a social welfare function to create a social ranking. The first step of this exercise is the subject of the present paper, the second step is the subject of a literature referred to above. In the double-aggregation problem one is able to utilize the requirement that the co-domain of the mapping must be a single ordering and this dictates the method of analysis. The examination of the problem of creating objective comparisons is side-stepped in this analysis and the nature of objective interpersonal comparisons is not fully revealed.

The general nature of the present problem is close to that encountered in Measurement Theory (see Roberts (1979) for an introduction) and applied princi- pally in the field of psychology. In that literature, it is common to examine problems involving the distillation of opinions into a single opinion or view. However, the standard approach is to impose stronger general and mathematical restrictions on the "distillation" process than those one would meet in social choice problems; for instance, an opinion is conventionally taken to be a single judgement and the function that brings about the "distillation" is assumed to be continuous with pre-specified arguments. In this paper, as in social choice theory, the existence and structure of such a function must be derived from general restrictions on the distillation process. This also explains why it is not possible to appeal to standard representation theorems (Aczel 1966).

In the next section, the framework and axioms are laid down. Section 3 looks at general properties of mappings from different sets of interpersonal comparisons to a single set of comparisons and Sect. 4 examines how the set of possible mappings is influenced by the information structure underlying interpersonal comparisons. The principal results of the paper are contained in this section. It is shown that if interpersonal comparisons embody very rich information as captured by ratio-scale comparability - they must embody information that says, for example, that one individual has a utility which is some multiple of the utility of another individual - then there is a small class of mappings that can take different sets of interpersonal comparisons into a single set. However, with a weaker informational structure, and this includes so-called full comparability (Sen 1970) where both levels and differences in utility are interpersonally comparable, then there is no way different sets of interpersonal comparisons may be combined and the only possible mappings are dictatorial. In this case, one individual's subjective interpersonal comparisons must be utilized as "objective" interpersonal comparisons.

The results of Sect. 4 are based upon the assumption that the informational basis underlying subjective interpersonal comparisons is inherited by the objective set of comparisons formed from these subjective comparisons. In principle, one may wish to permit the informational content of objective comparisons to differ from that of the subjective comparisons; given the difficulty of forming objective comparisons alluded to above, it is most natural to demand a weaker informational content for the set of objective comparisons. This issue is examined in Sect. 5. If objective comparisons embody a weaker information structure then it is shown that this is equivalent to demanding an identical structure to the subjective comparisons and then discarding information. If, on the other hand, the objective comparisons must embody a richer structure then there is no way that objective comparisons may be determined using subjective comparisons.



82 K. Roberts

The axiomatic results of Sects. 3-5 make use of an assumption, an unrestricted domain condition, ensuring that the domain of the mapping is large with many different interpersonal comparisons possible and different individuals' subjective interpersonal comparisons not necessarily related closely. But if subjective comparisons have a common basis, as in Harsanyi's (1955) approach, then this could act like a restricted domain and the common basis may be retrievable from the subjective comparisons. The difficulty with the Harsanyi argument (see Kaneko 1984) is that objective comparisons are retrievable in an obvious way only when it is assumed that over an appropriate set of choices, which includes the choice of being one individual rather than another, subjective comparisons are identical. Section 6 explores a model where subjective comparisons differ because individuals make "errors of judgement" in comparing people very different from themselves. But because the reason for different comparisons is specified, the route to retrieve objective comparison is also specified. Although different subjective comparisons can arise it is shown that the domain restriction of these comparisons is very strong - almost all sets of subjective comparisons will violate the restriction. This suggests that models which explain differences in subjective comparisons, and through this explanation allow objective comparisons to be retrieved, must impose a strong restricted domain on subjective comparisons with the implication that models will be rejected when faced with examples of subjective opinions. Concluding remarks to the paper are contained in Sect. 7.

2. The framework

The principal problem to be addressed is the aggregation or distillation of the "subjective" interpersonal comparisons of individuals in society into a single set of "objective" interpersonal comparisons. The interpersonal comparisons will relate to different individuals in different social states. Let X be the set of social states (X = (•••, w,x,y,z, ...)) and assume that the cardinality of X is at least three. N = { 1, . . . , i, j, k, . . . , n} is the set of individuals in society and n is assumed finite. Individuals have two roles: as the providers of subjective interpersonal comparisons - their opinions - and as that group about which comparisons are made. Let U be the set of real-valued functions defined on X x N. Elements of U may be viewed as sets of interpersonal comparisons: Uj(x,i)y where UjeU, isy's opinion of the utility of i in state x (it embodies j 's subjective interpersonal comparisons). Define an opinion aggregator (OA) to be a mapping v from a set D ç U- the domain of the aggregator - to I/, i.e. v: D -> U.

The function v maps n different interpersonal comparisons into a single comparison. The problem to be analyzed is trivial if D is the set where all opinions are identical. To be useful, D should be large but it may not be necessary to aggregate all opinions. In particular, it may be reasonable to restrict attention to the aggregation of nonpaternalistic opinions - if it is f s opinion that he is better off in state x than in state y then; 's opinion will be nonpaternalistic if he makes the same comparison, i.e. Uj(x9 i) > Uj{y, i). This leads to

Nonpaternalistic unrestricted domain (NPU).

D = {(m, ...,«„): u,e U VieN & VxjeX, ViJeN: ti,(x, i) > ut(y9 i)

=>Uj(x,i)Ûj(y,i)}.




If this condition is considered to be too strong a domain restriction then, given that the class of possible aggregators under NPU can be determined, one can check which members continue to work in a larger domain. Corresponding to nonpaternalism of opinions, the nonpaternalism of the aggregator is perhaps more compelling:

Nonpaternalistic aggregator (NPA). For all u = (ui, ..., un) e D, ufa, i) ̂ u¡(y, î) => vu(x, i) ̂ vu(y9 i) where v is the O A derived from u.

The next condition is an independence of irrelevant alternatives condition: to determine objective comparisons involving a pair of social states, it is not necessary to know the subjective comparisons involving different states. This may be stated as

Independence (I). For any u, u'ei), if for some x, y,eX and for all i, je N:

ui(x,i) = ui(xJl Ui(y,i) = u'i(yJl

then vu(x, k) = vM<(x, k) and vu(y, k) = vM'(y» k) for all keN. This independence condition allows the aggregated comparisons over a pair of

social states to depend upon which pair of states is being examined. A stronger and easier to use condition is:

Neutrality (N) For any u, u'e£>, if for some w, x, y, zeX and for all iJeN:

Ui(xJ) = u'i(wj), Ui(yJ) = ufa]),

then vu(x, k) = vu.(w, k) and vu(y, k) = vu>(z, k) for all ksN. In social choice problems, nonneutral features of social choice rules get diluted

by welfarist conditions like the Pareto condition. In the present framework, a similar role is played by the NPA condition.

Lemma 1. Assume that v satisfies NPU, NPA and I. Then it satisfies N.

Proof. Consider u,ur sD such that for some w, x, y, zeX and for all i,jeN:

Ui(xJ) = u'¡(wj), Ui(yJ) = u'i(zj). Take the case where w, x, y, z are distinct - if not, standard social choice

techniques (d'Aspremont and Gevers 1977) can be applied to cope with particular cases. Consider u" such that for all UjeN:

u"(wj) = u'i(wj) = Ui(x,j) = uï(xj' u>/(yj) = ui(yj) = u!i(zj) = uf/(zj). If u and u' satisfy NPU then u" is in the domain of v. By I, vM»(w,;) = vM(w,;)

and vM-(z,;) = vtt>(z,;); also vu>,(xj) = vM(x,;) and vu..(yJ) = Vu(yJ)- By NPA, V(w,7) = vM"(x,;) and vu..(zj) = vu»(yj). Combining, gives vM(x, k) = vM,(w, k) and vu(y, k) = vtt-(z, k) for all k. □

Given the conditions that have been imposed, the easiest method of aggregation would be to construct objective comparisons from utilities which are the average of utilities underlying the subjective comparisons. But this fails to take account of informational content of comparisons as captured by invariance



84 K. Roberts

conditions which make different sets of utilities (elements of U) equivalent. The most demanding invariance condition would rule out interpersonal comparisons in the sense that information involving the same intrapersonal comparisons would be treated as identical. This is the informational structure of the original Arrow problem. In the present context, the invariance condition would require that if u,u'eUn contained the same intrapersonal information then vM and vu< should contain the same intrapersonal information. Built into this is the notion that objective comparisons should not be expected to inherit a different information structure compared to the information from which they derive.

If interpersonal comparisons are ruled out then each individual's opinion will be an ordering of utility for each individual. The condition NPU will then require that aggregation is necessary only when everybody's ordering of fs utility is the same as i's ordering: all individuals will then have the same opinion. The condition NPA will then ensure that this common opinion is what is chosen by the opinion aggregator.

So much for cases where opinions embody little information. The class of opinion aggregators will be made larger by admitting richer informational structures. A very rich information structure comes from

Ratio-scale comparability (R-SC). If u, w'eDaresuch that u^xj) = seu¡(x,7')for all x, ij and for some (su...9 sn)y s¡ > 0 for all i, then there exists an s > 0 such that vu(x,yj = svu.(xj) for all x,j.

With R-SC, the doubling of utilities in one individual's opinion can do no more than cause a scaling of the utilities underlying the objective comparison. Thus R-SC permits the aggregator to be sensitive to an opinion of the form that ¿'s utility in state x is double that of fc's utility in state y, i.e. such an opinion is meaningful information which may be utilized by the aggregator. Under R-SC, a standard interpretation would be that the lowest possible utility is zero and, for simplicity, we will assume utilities must be non-negative. Characterization results with negative utilities can be obtained directly from the results to be presented and a general result can then be obtained by patching together results applicable to different domains.

A weaker informational condition permits the utilization of information which compares the level of utility across individuals and the utility difference between social states across invidividuals but, for instance, does not make use of a concept like zero utility which is inherent in R-SC. This condition is

Full comparability (FC). If u,u'eD are such that ufaj) = sf + s¡u¡(x, ;') for all x, i, j and for some (s?,S2, ...,s°, 5i,s'2,...,siX s' '> 0 for all i, then there exist s°,s's' > 0, such that vu(xj) = s° + s'vu>(xj) for all xj.

If FC is satisfied then R-SC is implied directly. However, some aggregators that satisfy R-SC may fail to satisfy FC.

3. Opinion aggregators: general structure

In this section, the general structure of opinion aggregators is examined. In particular, the implications of imposing NPU, NPA and I will be investigated. The next section will examine the implications of also imposing invariance transforms.

It will be convenient to concentrate on a two-person society though the arguments that will be presented extend directly to the general case. The advantage of the two-person case is that opinions can be depicted easily. An




opinion configuration is shown in Configuration 1. Here I and II denote the opinions of the two persons.

Configuration 1

I II Aggregate

x: Wl(x,l) u, (x, 2) u2(x,l) u2(jc,2) vw(x, 1) vu(x92) Y- «i(y, 1) «i(*2) u2(*l) u2(y,2) vM(y,l) v„(*2)

We will consider a number of different configurations which permit the form of the function v to be determined. Configuration 2 gives an example of possible opinions.

Configuration 2

I II Aggregate

x 0 0 0 0 y' y" y 0 0 0 0 y' y" z «! 0 ßl 0 yAoLußi) y"

Taking the pair {x, y}, N implies that the aggregate opinion must give the same utility to an individual in each state. This is defined to be / and /'. Looking over {y,z}9 NPA shows that individual 2's utility should be the same in both states according to the aggregate opinion. Over {y, z}} cl' and ß' are subject to variation and their value can influence individual Ts utility in state z: this is written as

Next consider Configuration 3.

Configuration 3

I II Aggregate

x: 0 a2 0 ß2 y' y2(^ ßi) y: 0 0 0 0 y' y" z: a, 0 ft 0 yAOLußi) y"

The aggregate opinion over {y, z} is given by Configuration 2 and the application of condition I. The aggregate opinion over {x, y) is given by identical arguments which allowed the aggregate opinion to be pinned down over { y, z) in Configuration 2. Looking now at Configuration 4:

Configuration 4

I II Aggregate

*: 0 a2 0 ß2 y' yi(*i,ßi) y- «i ä2 ßi ßi yi(ai,0i) T2(a2,02) z: at 0 ßx 0 7i(a„0i) y"



86 K. Roberts

where the aggregate opinion over{x, z] is given by Configuration 3 and I and then over the triple {x, yy z) by applying NPA. This, in turn, allows the aggregate opinion over Configuration 5 to be determined - over {y, z) using Configuration 4 and I. A similar procedure to that used to obtain Configuration 5 can be used to obtain Configuration 5' (where the roles of states y and z are reversed).

Configuration 5

I II Aggregate

x: 0 0 0 0 y y" y- «i «2 ßi ßi yiiûßi) yi(*2,ßi) z: «, 0 & 0 7i(«i,j?i) f

Configuration 5'

I II Aggregate

x: 0 0 0 0 / y" * «Ì «i ß' ß'i yi(«i,Ä) 72(«i,Ä) y: «i 0^0 yi(QL'u ß') f

Finally, given Configuration 5 over {x, y} and Configuration 5' over {x, z}, application of/ gives Configuration 6. Given Lemma 1, the ranking over {y, z) in Configuration 6 is determined solely by the utilities underlying opinions in states y and z, rather than other utilities or the particular state y and z under considera- tion. This gives rise to our general characterization result.

Configuration 6

I II Aggregate

x: 0 0 0 0 y' y" y- «i «2 ßi ßi yAaußi) y i (*i*ßi) *'• «Í «i ß' ß'2 yi(«i,Ä) yiMJ'i)

Theorem 1. If the opinion aggregator v satisfies NPU, NPA and I then there exist n real-valued functions, y u ... ,y, ji'.R" -+R, such that the utilities underlying aggregate opinions are given by vu(x, i) = y¡{Ui(x9i)9 ...,m„(x, i)) for all u, x, i.

The argument presented above did not depend upon n = 2 and the theorem is presented for the general case. The value of the result is that it tells us that aggregate opinion utilities are formed in a simple way from the "subjective" utilities of individuals - most importantly, the only information relevant to determine the "objective*' utility of an individual in some state is the identity of the individual and the "subjective" utilities assigned to that individual/state pair in everyone's opinions.

4. Opinion aggregators: invariance transformations

Given the assumptions of Theorem 1, the y functions can take on any form compatible with NPA. This will require each y¡ to be increasing with an increase in




all arguments. To determine further restrictions on these functions it is necessary to examine the consequences of applying invariance transforms.

4.1 Ratio-scale comparability

We start with ratio-scale comparability (R-SC) which imposes minimal restrictions on the opinion aggregator. Consider Configurations 7 and 8 where Theorem 1 has been applied to obtain "objective" utilities. We again concentrate for explanatory purposes on the n = 2 case.

Configuration 7

I II Aggregate

x: 1 0 1 0 yi(l,l) /' y: <x 0 ß 0 yi(*,ß) y"

Configuration 8

I II Aggregate

*: * 0 n 0 y ¿X.H) y" y: ¿a 0 fiß 0 Jifanfi) Y'

Assume that Configuration 8 is obtained from Configuration 7 by scaling of the utilities underlying Ts and 2's opinions by X and & respectively. Applying C-RS, the net effect upon aggregated opinions should be that they are altered only by an invariance transformation - a common scaling v to underlying utilities. We thus have

(i)7l(A,/¿) = vyi(l,l),

(ii) y1(A,^) = vy1(a,A

(iii) y" = v/'.

If A, fi > 1 then NPA implies that v > 1 which is incompatible with (ii) if /' ^ 0. Thus y" = 0. Eliminating v from (i) and (ii) gives:

yi(A,l)y1(Aa,^) = y1(a,j5)y1(A,^ (t)

To see the implication of this more clearly, define Fi(A9 B) = logyê"*, e3) and let A = loga, B = log/?, A = log A and M = log// (recall that utilities are assumed nonnegative). Then

A(o, o) + rx{A + a,b + m) = r,(A, B) + rM, M) and, if

F(A,B) = r1(x,B)~r1(o,o),

F(A + A,B + M) = F(A, B) + F (A, M).



88 K. Roberts

If

B = A = 0 then

F(A,M) = F(A,0) + F(0,M), so F is additively separable. To get further it is necessary to invoke NPA which ensures that F is weakly increasing in its arguments. Let us concentrate on F (A, 0) and define this to btf{A). Then f(A) is weakly increasing and

f(A + A)=f{A)+f{A). Thus

/(0) = 0 and f(A)^0ifA^0. Let J(x) and J(x) be the smallest (respectively largest) integer at least as great

(respectively no greater than) x. Straightforwardly,

HnA)f(^j >f{A) > IJM)f(^' and

„/(i) -/<!>, where n is integer valued. As nA + 1 ̂ T(nA) and UnA) ̂ nX - 1, we have

so, letting n -► oo, we have

/M) = Af('' which means that/is linear; in consequence, F is linear, Tt is an affine transformation and y i is Cobb-Douglas in form

Fl=-alA-'-blB + Cl or yi(a,j?) = c^ß01 (where Ci = logci). A similar exercise can be carried through to obtain "objective" utilities for the other individual; aggregated opinions are then determined as in Configurations 9 and 10.

Configuration 9

I II Aggregate

x 1 1 1 1 c, c2 y «i «2 íi 02 e!*?/*?' c2a&fà

Configuration 10

I II Aggregate

x: X k 1 1 c,/la' c2Aa'




Now assume that Configuration 10 is formed by an invariance transformation involving a common scaling X to 1's "subjective" utilities. Then by R-SC, the utilities underlying aggregate opinions must differ in the two configurations only by a common scaling factor. From Configuration 10, XOi = Xa2 for all strictly positive X which gives aY = a2. A similar exercise applies to show that bi = b2 and a straightforward amendment of the configurations presented gives restrictions on the y functions in the n-person situation. We have:

Theorem 2. If the opinion aggregator v satisfies NPU, NPA, I and R-SC then there exist numbers cu c2, . . . , cn, al9 a2i..., an, c¡ ̂ 0, a, ̂ 0 such that

VxeX,ieN: vM(x, i) = c, f[ (ufa iff*.

Given the weakness of the conditions imposed upon v, Theorem 2 is a strong result with the functional form of the aggregator being Cobb-Douglas. If NP-U is relaxed, to allow the aggregator to apply when ¿'s opinion of ;'s intrapersonal ordering fails to coincide withy's opinion, then to satisfy NP-A, it will be necessary for a} ̂ 0 but ax = 0. For this to be true for all ¿,y, one requires a¡ = 0 for all i; this implies that v is independent of individual's opinions. This supports the view that if individual opinions fail to be nonpaternalistic then it is overdemanding to require nonpaternalism of the aggregator.

A surprising feature of Theorem 2 relates to another aspect of nonpaternalism. In the spirit of nonpaternalism, it may be thought desirable to give extra weight in the aggregator to somebody's opinions about their own utility. The theorem shows that this is impossible - if ¿'s opinions are given more weight than j 's, a, > a,, then this holds with respect to the aggregate opinion concerning everybody's utility. This suggests that the imposition of an anonymity condition will further severely limit the class of possible aggregators. Consider the following condition:

Anonymity (A). Let a : N -► N be a permutation function. If u.u'eD are such that

Vi,;eN, Vx: u'(xj) = uff(l)(x, <r(j)),

then

V;gN, Vx: vu,(x,j) = vu(x, a(j)).

Notice that this condition does not say that if everyone's opinion about i and j are swapped then the aggregate opinion about i and; are swapped. For this to be the case, A requires that individuals i and; also swap their opinions: A permits ¿'s opinion about ¿'s utility to be given preferential weight in the aggregator. Condition A can be applied to the aggregator defined in Theorem 2 by letting ¿ and j swap positions. The condition implies that

clMl(x, îfuj(x9 if' = c,w,(x, if 'ufa ifi

and as this must hold for all ux(x, ¿), ufa ¿), we have c¡ i = Cj (letting u,(x, ¿) = ufaj) = 1) and a¡ = a, (letting u,(x, ¿) = 1, ufa i) # 1). Thus almost all members of the class of aggregators in Theorem 2 possess nonanonymous aspects and we have:



90 K. Roberts

Theorem 3. // the opinion aggregator v satisfies NPU, NPA, I, R-SC and A then:

VxeX, ieN: vtt(x, z) « c( fj (u,(x, i))j, where c and a are nonnegative constants.

In this result, the constant c plays no role as utilities underlying interpersonal comparisons are unique only up to a ratio-scale invariance transformation. If it was required that identical opinions should be reflected in the aggregate opinion then the constant a would be l/n and the aggregator function would be uniquely determined: objective interpersonal comparisons would have to be based upon the geometric mean of utilities underlying subjective interpersonal comparisons.

4.2 Full comparability

Though ratio-scale comparability has turned out to admit a small class of aggregator functions, it is generally viewed as too stringent a condition in terms of the richness of interpersonal comparisons that it is assumed possible to make. We now investigate full comparability where comparisons of levels and differences in utility are possible, but, for instance, it is not possible to say that one individual's utility is some multiple of the utility of someone else.

As ratio-scale invariance transforms are invariance transforms under full comparability, FC implies R-SC so that the class of possible aggregator functions under FC is a subset of those under R-SC. The theorems of the last subsection may be applied to partially restrict the utilities underlying aggregate opinions. Assuming NPU, NPA and I, Theorem 2 can be applied to Configurations 11 and 12 where a, /? ̂ 1. However, Configuration 12 can be obtained from Configuration 11 by invariance transforms under FC. In particular, individual l's opinions are transformed by the addition of unity and scaling by a factor (a - 1). Individual 2's opinions are transformed by the addition of unity and scaling by a factor iß - 1). Thus the utilities underlying the aggregate opinion also must be transformed by at most a common addition to each utility and by a common scaling. Comparing the two configurations, the state x utilities tell us that the common addition must be Ci and it must be c2' Thus, under FC, cx = c2 and this extends directly to the

Configuration 11

I II Aggregate

x: 0 0 0 0 0 0 y: 1 0 1 0 d 0 z: a 0 ß 0 C|flta«/?fl* 0

Configuration 12

I II Aggregate

x: 1 11 1 cj c2 y: a 1 ß 1 c^'ß** 0 z: a(a-l)+l 1 ß(ß - 1) + 1 1 Ci(a(a - 1) + lTl(ßiß ~ 1) + W* c2




n individual case. Looking at the state y utilities, the scaling factor must be (a*1/?02 - 1) but applying this to the state z utilities gives the restriction:

1 + 0^(0«»/^ - 1) = (a(a - 1) + irO?O? - 1) + If2.

Let a = 2 and /? -> 1. Then this gives

1 + 2fl«(2fll-l) = 3fl«,

which has two solutions: ax = 0 or ax = 1. Similarly, if ß = 2 and a -+ 1, we obtain a2 = 0 or a2 = 1. Finally, letting a = ß = 2 gives

1 _l_ 2(ai + a2)(2ifl» + flJ - l) = 3(fl* + fla)

which has the solutions ay + a2 = 0 or ax + a2 = 1. Putting these three sets of possibilities together gives three possible solutions:

(1) ax = a2 = 0 or

(2) a! = 1, a2 = 0 i?r

(3) fll = 09 a2 » 1.

This argument extends to the n individual case by considering the n situations where a utility of 2 is assigned by one individual and close to unity is assigned by all others and the situation where a utility of 2 is assigned by all individuals. Corres- ponding to Theorem 2, we have:

Theorem 4. If the opinion aggregator v satisfies NPU, NPA, I and FC then there exists an individual deN such that (for some constant c ̂ 0):

V xeX, isN: vM(x, i) = cud(x, i).

In this result, if c = 0 we have the degenerate solution where objective interpersonal comparisons declare universal indifference; otherwise, a scaling factor of c is a invariance transform under FC so that Theorem 4 tells us that objective interpersonal comparisons cannot combine different subjective comparisons and one individual's opinions must be dictatorial in determining objective interpersonal comparisons. This dictatorship result is reminiscent of Arrow's (1963) famous impossibility theorem dealing with the aggregation of orderings and the two results are not entirely independent. What is surprising is that the dictatorship result of the Arrow theorem is often blamed on the informational parsimony which is enforced by his framework and assumptions. Here, we are considering the problem of aggregation where the informational content of the entities being aggregated is very much richer but still one of the entities is dictatorial. The sequence of results that has led to Theorem 4 provide an explanation for this strong conclusion. Theorem 1 shows that the "objective" utility of individual i can be constructed by combining different subjective utilities but all these utilities must relate to opinions concerning fs utility. This is not unappealing in itself but an implication is that interpersonal comparisons underlying opinions, which relate an individual's opinion concerning utilities across individuals, have little power. In particular, because subjective utilities across opinions are not themselves inter personally comparable, there is little opportunity to trade-off one individual's opinion about i's utility with another individual's opinion. The reason that some trade-off is possible under



92 K. Roberts

ratio-scale comparability is that the ratio of two subjective utilities in an individual's opinion is a pure number, unaltered by invariance transforms, which can be compared across different opinions. But with full comparability, though levels and differences are comparable within an opinion, no information beyond a basic ranking is comparable across opinions - the connection with the Arrow theorem is clear.

The foregoing discussion suggests the existence of a "metatheorem" concerned with interpersonally comparability. Different opinions could be comparable with each other if they emanated from the mind of a single observer. The entity v¡(x,j) could be this observer's view of how somebody with the life experiences of individual i would judge being individual j in state x. However, if one attempts to combine the views of different observers then noncomparability enters the picture. Of course, there could be several stages of introspection - a person could form a view about how observer k would view how individual i would judge) in state x and so on - but noncomparability would again be a problem when one wished to combine information emanating from different minds. In this light, the Arrow impossibility theorem is perhaps the simplest version of this metatheorem.

The above discussion also throws light on the main result in Roberts (1994). In that paper extensive social welfare functions were examined, the domain being a number of different sets of interpersonal comparisons, as in this paper, but the co-domain was taken to be orderings of the state space. One type of extensive social welfare function could take the form of first mapping, through an opinion aggregator, to a single set of interpersonal comparisons, and then using this information in a social choice rule mapping interpersonally comparable utility information into orderings. Theorem 4 shows that, taking this route, the extensive social welfare function must involve dictatorial opinions in the construction when the invariance transform is one of full comparability. Whatever route is taken, this was shown to be the case in Roberts (1994). That paper imposed conditions on the extensive social welfare functions which related to the particular co-domain and the conditions giving rise to theorem 4 are weaker in the sense that they embody less power when the co-domain is richer. Furthermore, they do not embody a strong mono- tonicity (Pareto) condition which is not unreasonable when social orderings are being examined. The nature of the problem investigated in that paper meant that the proof threw little on the dictatorial opinion result and this is now clarified. Indeed, Theorem 2 suggests that it may be possible to characterize extensive social welfare functions for the invariance transform underlying ratio-scale comparability.

5. Mixed invariance transforms

Thus far, we have investigated invariance transforms where it has been assumed that the informational content of objective comparisons has the same degree of richness as the subjective comparisons from which they are created. Given the restrictive class of aggregation possibilities as demonstrated in the last section, one way of making fewer demands on the aggregator is to require objective comparisons to embody less information as determined by the underlying informational basis. In this section we look at the possibilities where the basis underlying objective comparisons may be both weaker and richer than the basis underlying subjective comparisons. For concreteness, we concentrate on the possibilities under ratio-scale and full comparability.




Corresponding to R-SC and FC we have:

Ratio-scale to full comparability (R-S/FC). If u, u'eD are such that Ui(*J) = s¡u'i(xj) for all x, i, ; and for some (su ... ,sn), sf > 0 for all i, then there exist s°, 5', s' > 0, such that vu(xj) = s° + s'vfoj) for all x, ;.

Fw// io ratio-scale comparabiity (F/R-SC). If w, u'ei) are such that Wf(^j) = s? + sjiiííx,;) for all x, ¿,; and for some (5?,sS:... ,s?,5i,5'2, ... ,siX 5i > ° for all i, then there exists an 5 > 0 such that vM(x,;) = svtt<(x,;) for all xj.

It is useful to note that R-SC implies R-S/FC and F/R-SC implies FC. Condi- tion F/R-SC is perverse in the sense that the input into creating objective comparisons embodies jess information than the created comparisons. As F/R-SC implies FC, Theorem 4 can be applied: there exists a constant c and an individual d such that vM(x, i) = cud(x, i). However, let m,(x, j) = s? + s'-u'^x, j) for all x, i, j where s'i > 0. Then vM(x, i) = cud(xy i) = cs° + cs'dud(x, i) = csd + 5ÍvM(x, i) which violates F/R-SC if sd is chosen nonzero and c is nonzero: the constant c must therefore be zero and we have: Theorem 5. // the opinion aggregator v satisfies NPU, NPA, I and F/R-SC then vu(x, 0 = Ofor all x, i (as vu satisfies R-SC it can in fact be set equal to any constant).

The aggregated opinion is independent of individuals' opinions and universal indifference is the only way that nonpaternalism in a weak form can be retained. Notice that whatever the relevant informational basis, if objective comparisons should equate with subjective comparisons when all opinions coincide then it is impossible for objective comparisons to have a richer informational basis.

The more interesting case is where information richness is given up in the aggregation process. As R-SC implies R-S/FC, the opinion aggregators given in Theorem 2 may be used under R-S/FC. To examine what else is possible, it is necessary to amend the proof of Theorem 2. Returning to that proof, Configura- tion 8 is obtained from Configuration 7 by a scaling of opinions and the application of R-S/FC gives, instead of (i)-(iii) of Section 4.1.

(i') y1(l,//) = HvTi(U), (ii') yi(Aa,^) = ^ + vy!(a,)?), (iii')y" = £ + v/',

where g is some constant. Combining these equations gives ¿1(l,l)¿1(Aa,//i?) = ¿1(a,j?)¿1(A,//),

where ox - yl - /'. Thus the restriction satisfied by yx under R-SC (see (f) Section 4.1) is satified by ¿, under R-S/FC: òx has a Cobb-Douglas form and following through the rest of steps of Theorem 2 we obtain: Theorem 6. // the opinion aggregator v satisfies NPU, NPA, I and R-S/FC then either there exist numbers aua2,.--9 an, cu c2, . . . , cn, ax ̂ 0, c¡ ̂ 0, and a d such that

VxeX, i e N: vtt(x, i) = d + c, ¿ (v,-(x, i))a<

or there exist numbers du d2,...,dn such that

VxeX,ieN: vu(x, i) = d¡.



94 K. Roberts

The characterization of this theorem comes from considering Configurations 9 and 10 and applying R-S/FC. The second possibility is the case where objective comparisons are independent of subjective comparisons. The first possibility, when compared to Theorem 2, tells us that by relaxing ratio-scale comparability of objective comparisons to full comparability, the possibilities are as under the requirement of ratio-scale comparability with some information then discarded. In essence, the implication of demanding less information content to objective comparisons is that there is a discarding of information - there is no gain in terms of the size of the class of feasible opinion aggregators.

6. Restricted domains

Given that both full and ratio-scale comparability make large demands on the interpersonal comparisons that are meaningful and can be made, the results of the last two sections suggest that it may be impossible to combine different subjective judgements. As in the conventional social choice literature, the problem is less acute if different subjective comparisons are not independent in which case the domain of the opinion aggregator can be restricted. We have from the outset ruled out subjective comparisons that embody paternalism and this domain restriction has not prevented negative results. Here, we look briefly at stronger domain restrictions.

Different subjective comparisons will be related if one can impose structure on how such comparisons are formed. An appealing structure is one where subjective comparisons imperfectly reflect some "objective" comparisons which have an independent existence. Any model explaining this which gives rise to limited domain conditions also suggests the form of the appropriate opinion aggregator; namely, one that gives rise to comparisons as close as possible to the original "objective" comparisons.

The most well known example of a model of the above type is Harsanyi's (1955) where subjective comparisons differ because individuals' personal experiences differ. Harsanyi suggested that if these different personal experiences were themselves part of individual choice then an underlying set of objective comparisons could be determined. However, as Kaneko (1984) and others have shown, this is possible only if subjective comparisons are, in essence, identical. This is a very strong domain restriction which makes the problem uninteresting. It is interesting to see whether there are appealing models where the domain is reasonably unrestricted but where knowledge of the model allows one to reconstruct "objective" comparisons.

We will consider a minor amendment of the Harsanyi structure: subjective comparisons differ because individuals make "mistakes" in making comparisons involving individuals very different from themselves. However, comparisons of individuals with similar characteristics to oneself reflect the "objective" comparisons and these can be used in the reconstruction process. The question to be examined is whether reconstruction is possible with a weak domain restriction on subjective comparisons.

We will look at only one social state. Individuals are indexed by a characteristic keK. We wish to develop the idea of individuals being close to each other so K is taken to be a metric space and, for concreteness, assume that K = [0, l]w, a subset of Euclidean m-space. Assume that individual k assigns utility u(fc, fc*) to individual k* and this utility underlies fc's subjective interpersonal comparisons. It is assumed




that individuals with similar characteristics have similar opinions and their opinions reflect "objective" comparisons about themselves. Thus, assume that u(fc, fc*) is twice-differentiable. Individual fc's opinions about the marginal utility with respect to the characteristic at k is given by

u2(Kk) and let us assume that this corresponds with an "objective" marginal utility. Thus, if w(fc) is "objective" utility then:

wk = u2{k,k) (*)

Objective utility can be recovered from subjective utility by integrating (*). If m = 1 then fc is a scalar and this integration is straightforward. However, if m > 1 then one needs to check the appropriate integrability condition - if w is a well- defined twice-differentiable function then wkk must be symmetric. Thus, if w exists when defined by (*), u21 + w22 must be a symmetric mxm matrix. As u22 is symmetric, the integrability conditions boils down to symmetry of the mxm matrix u21 which is a strong condition to impose on the subjective utility functions. At least in this example, we can draw the conclusion that even though different subjective comparisons are possible, sufficient specificity to permit recoverability of the "objective" comparisons imposes very strong restrictions on the domain of subjective comparisons. Robustness of this conclusion is an interesting topic for further research.

7. Concluding remarks

Given the negativism of the Arrow impossibility theorem, the desire to make use of interpersonal comparisons in social choice is clear. However, the interpersonal comparisons that are available are the opinions of individuals in society. Indi- viduals can combine their own opinions and their ethical beliefs to determine what is best, but to guide policy it would seem desirable not to rely upon the opinions of one individual and, instead, to combine opinions to create what may be mislead- ingly called objective comparisons. This paper has constructed a framework to look at this aggregation problem and has characterized the form of reasonable opinion aggregators. Concentration on this problem has permitted an understand- ing of results that have been obtained dealing with the use of many different opinions to determine a social ordering.

It has been shown that under mild assumptions, objective comparisons between some individual i in state x and another individual; in state y must be based solely on subjective comparisons of this pair. Because opinions themselves are not comparable across the individuals making them, the class of possible aggregation mechanisms is severely restricted. With very rich information underlying comparisons (ratio-scale comparability) it is possible to combine opinions but the opinion aggregator must be Cobb-Douglas in form. Under the still rich information structure of full comparability, the only opinion aggregator is dictatorial in the sense that objective comparisons must coincide with one individual's opinions. For deciding what is best for society, this dictatorship result may be more disturbing than the original Arrow result.

Finally, we note that the problem of combining different sets of interpersonal comparisons has close affinities to other problems, notably the problem of



96 K. Roberts

aggregating subjective probability judgements. To reinterpret our problem, u¡(xj) is to be viewed as f s judgement of the probability of state x conditional on event £, (in our analysis, j was drawn from the set of individuals but this is easily gener- alized). Conditions NPU and NPA ensure that individuals have the same ordering of the likelihood of states and conditions I and R-SC impose the restriction that the relative subjective probabilities between two states is determined by the relative probabilities between these states. Theorem 2 then shows that objective probabilities are a Cobb-Douglas function of subjective probabilities (unlike in our analysis, probabilities need to be scaled to sum to unity). This is in essence the characterization (Proposition 2.4) in Barrett and Pattanaik (1987) and the references cited therein where there is an unrestricted domain and only single events are examined: our results show that the characterization extends to a model of multiple events where there is agreement about the ordering of the likelihood of events.

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1237-1241 d'Aspremont C, Gevers L (1977) Equity and informational basis of collective choice.

Rev Econ Studies 44: 199-209 Deschamps R, Gevers L (1978) Leximin and utilitarian rules: a joint characterisation. J Econ

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Objective interpersonal comparisons of utility

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