Directory UMM :Data Elmu:jurnal:M:Mathematical Social Sciences:Vol37.Issue3.May1999:

Mathematical Social Sciences 37 (1999) 211–233

Collective judgement: combining individual value judgements
a,
´
*, Gyula Maksa a , Robert J. Mokken b
¨
Akos
Munnich
a

Institute of Mathematics and Informatics, Kossuth Lajos University, Egyetem ter 1, 4032 Debrecen,
Hungary
b
Department of Statistics and Methodology, PSCW-University of Amsterdam, O.Z. Achterburgwal 237,
1012 DL Amsterdam, The Netherlands
Received 7 August 1997; received in revised form 7 June 1998; accepted 8 July 1998

Abstract
This paper addresses a fundamental problem of collective decision making: how to derive a
collective value judgement from the individual value judgements of the members of a committee.

Three structural conditions will be introduced, which correspond to certain consistency requirements for the collective judgement. It will be shown that the formula for the collective value
judgement, based on these consistency conditions, is a quasilinear mean of the individual
judgements, and moreover, that the generating function of the corresponding quasilinear mean is
independent of the number of people in the committee. Some uniqueness properties are considered
and, finally, it is shown that the quasilinear mean is suitable as a social choice function satisfying
six Arrowian conditions.  1999 Elsevier Science B.V. All rights reserved.
Keywords: Collective judgement; Combining individual

1. Introduction
The theory of collective (or social) choice belongs to several disciplines. It concerns
the possibility of making a choice or a judgement that is in some way based on the
subjective evaluations or preferences of the individuals involved. Its aim is to establish
specific satisfactory combinations of such individual views, in order to produce a definite
social evaluation or choice. As such it is a crucial aspect of economics (welfare
economics, planning theory and public economics), and it also relates closely to political
science, public oriented social choice theory and the theory of decision procedures. It
*Corresponding author. Tel.: 136 52 316666; fax: 136 52 431216.
´ Munnich)
¨
E-mail address: munnich@tigris.klte.hu (A.

0165-4896 / 99 / $ – see front matter  1999 Elsevier Science B.V. All rights reserved.
PII: S0165-4896( 98 )00030-4

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et al. / Mathematical Social Sciences 37 (1999) 211 – 233
A´ . Munnich

also has important philosophical aspects, related to ethics and especially to the theory of
justice.
Decision-making can be defined as making choices between alternatives after an
evaluation of their effectiveness for achieving the decision-maker’s objectives. For the
purpose of studying collective choice-making, a committee is defined as a set of
individuals who share the common duty to prepare a final collective choice or decision.
Each member of the committee has the capability of making that choice alone, but each
of them in doing so is committed to joint decision-making according to the objectives of
the committee. For example, consider the following elementary collective decision
problem. A committee has to decide who is the best among a group of candidates for a
given job or, in an other situation, which scientific project should be granted from a

number of proposals. Each of the candidates or the project proposals are evaluated
beforehand by the members of the committee and then, in a following step, from the
individual evaluations an ‘‘overall’’ evaluation or judgement is synthesised by the
committee.
Sometimes, collective choice depends not merely on individual preferences, but also
on their intensities of preference, hence cardinal preference functions for individuals
may be considered (e.g. the most commonly used and easily appreciated measure of
benefits and costs in modern society is monetary value). In using individual preferences
as input for collective evaluation and choice we are faced with the problems of
measurability of (cardinal) preferences, interpersonal comparability of individual preferences, and the form of a function which specifies a social preference relation given
individual preferences and the comparability assumptions. Arrow (1951) using individual group members’ preference orderings, stated in a formal way a set of seemingly
reasonable constraints for social choice such as ‘‘unrestricted domain’’, ‘‘weak Pareto
principle’’, ‘‘independence of irrelevant alternatives’’ and ‘‘nondictatorship’’, and proved
that these are inconsistent. (Chilchilinsky, 1980, 1982) has shown that preference
orderings can not be transformed into social orderings by a continuous aggregation rule
satisfying ‘‘anonymity’’ and ‘‘respect unanimity’’. There exist also possibility and
impossibility results for cardinal preferences (utilities). Keeney (1976) established a
cardinal utility variant of Arrow’s conditions and gave a necessary and sufficient
condition for consistency. Montero (1987) proved Arrowian possibility theorems on the
basis of fuzzy opinion relations and fuzzy rationality. Skala (1978); Ovchinnikov (1991)

showed that Arrow’s conditions are consistent if the society uses Lukaszewicz logic for
modelling its preference. For cardinal preferences (preferences represented by numerical
functions which are invariant under positive linear transformations), Kalai and Schmei´
dler (1977); Hylland (1980) gave cardinal impossibility theorems. Harsanyi
(1955,
1986), utilitarian approach, in contrast to the Arrowian approach, provides the weighted
sum of individual utilities for the social welfare function. For a standard reference of
social choice theory, see Sen (1986). There are probabilistic versions of collective
choice, e.g. (Blokland-Vogelesang, 1990; Fishburn, 1975; Fishburn and Gehrlein, 1977;
¨
Intrilligator, 1973; Munnich
et al., submitted; Pattanaik and Peleg, 1986; Williamson and
Sargent, 1967), among others. The social choice theory in a conflicting situation and the
fairness of social choice rules are discussed by (Rawls, 1971; Sen, 1982, 1986) and
Bezembinder (1989), among others.

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213

Generally speaking, voting is used for combining individual judgements, and so it is a
mechanism of social evaluation and choice. The best known procedure is the majority
voting rule (see, e.g., Black, 1958; May, 1952; Sen, 1970), but there are two other
approaches which can take into consideration a sort of ‘‘intensity’’ of individual
preferences. One is the approval voting (Brams and Fishburn, 1978, 1983), the other is
the SPAN (successive proportionate additive numeration) method of weighted voting
(MacKinnon, 1966; Willis et al., 1969). A systematic analysis of specific conditions of
social choice functions is studied by (Richelson, 1975, 1978). For a good summary of
popular aggregation methods (see Ferrel, 1985; Sen, 1986), among others. We should
also refer to the vast literature dealing with group decision processes concerning voting
bodies and qualitative processes for group consensus making (Allison, 1971; Janis, 1989;
Mokken and Stokman, 1985; Mokken and De Swaan, 1980; Poole and Roth, 1989).
There are methods for aiding qualitative consensus making (see, e.g., Dalkey, 1975;
Delbeq et al., 1975), there are also studies describing real-life processes (see, e.g.,
French, 1986; Gallhofer et al., 1994; Maoz, 1990; Tversky and Kahneman, 1981), but
these kind of approaches are beyond this paper’s scope.
This paper addresses a well-known and fundamental problem of collective decision
making: how to derive a satisfactory and unique collective value judgement from the

individual value judgements of the members of the collective. The situation which will
be considered in this paper is as follows: let the committee consist of (finite) n members
producing a set of n quantifiable judgements x 1 , . . . , x n concerning a choice or decision
making task. The question then is how these judgements can be synthesised into a single
overall quantitative judgement Bn (x 1 , . . . , x n ). Three formal structural conditions will be
introduced, which correspond to certain fairness and consistency requirements for an
adequate collective evaluation of the candidates, both from the point of view of the
objects or candidates being rated and the collectivity the committee is supposed to
represent. It will be shown that the collective judgement of the candidates then proves to
be a quasilinear mean of the individual judgements and that, moreover, the generating
function of that quasilinear mean is independent of the number of the people in the
committee.

2. Conditions for a fair and consistent procedure
Any social evaluation rule (as, in this paper, given by the function B) should be based
on some requirements (which in our case are given by some conditions concerning
properties of B). These are designated to determine some minimal conditions (or
axioms) for a fair and consistent decision rule, according to which individual judgements
should be combined. These conditions should be applicable to any fixed number of
judges, but it is not supposed a priori that the combination rule is ‘‘independent’’ of the

number of the judgements.
In what follows, it will be assumed that the individual value judgements x 1 , . . . , x n
are elements of a real interval (finite or infinite) I, and that the overall judgement Bn
(depending on the group-size n of the committee) is a function of the individual
judgements with its range in I. By definition, B1 (x) 5 x. Three consistency conditions

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214

will be specified for combining individual value judgements. These requirements will
specify the nature of the aggregation procedure only for some special situations: the first
condition will be the reflexivity property of n-variable functions, the second one refers to
a certain type of recursivity, and the third one is a generalisation of bisymmetry to n
variables.
Condition of reflexivity (it is also known as idempotency): This condition refers to the
case when all individual judgements are the same, requiring the overall judgement then
to be equal to that common value

Bn (x, . . . , x) 5 x.

(1)

Reflexivity embodies the idea that the collective evaluation rule should reproduce every
unanimous outcome, which is why it is also known as ‘‘respect unanimity’’ (e.g.
Chilchilinsky, 1980, 1982). The reflexivity condition, although rather plausible, has
important consequences, such as ruling out associative solutions for Bn such as
summation.
Condition of subgroup consistency: The value Bn (x 1 , . . . , x n ) denotes the overall
evaluation of the committee when each of the committee members’ evaluation is taken
into consideration simultaneously. But sometimes the committee is partitioned into two
or more smaller groups (e.g., one consisting of experts of the government parties and the
other of those of the opposition). These subgroups make their own evaluations separately
and prior to the final decision, which then is to be based on the evaluations produced by
each of the groups. So in order to get the final overall judgement (that is the judgement
involving each member of all the subgroups together in the judgmental process), the
judgements of the subgroups are further evaluated by a groupwise collective evaluation
process. Formally, the subgroup consistency condition assumes that the groupwise
collective evaluation can also be performed by means of an intermediary step involving

g subgroups, with sizes k 1 , . . . , kg , . . . , k g , respectively, such that 1#kg ,n; g : 1,2, . . . ,
g and og 51 g kg 5n. Each subgroup evaluates candidates independently according to Bkg .
In the next step the overall evaluation is then performed by an unknown function
Dk 1 ,k 2 , . . . ,k g ;n (which refers to the groupwise collective evaluation and may depend on
the sizes of the total and subgroups), based on the subgroups’ evaluations. The condition
of subgroup consistency then requires that the final, collective objective evaluation of
the candidate should not depend on whether the committee is groupwise divided into
subgroups. That is, if n (.2 integer) fixed, then for any g subgroups (2#g#n21) of
size kg (n#kg ,n); og 51 g kg 5n; g, kg are positive integers, and let indices sg be defined
as:

O k ; g 5 1,2, . . . , g; s 5 0, s 5 O k 5 n;
g

sg 5

n 51

g

n

0

n

g

n 51

there is a function Dk 1 ,k 2 , . . . ,k g ;n :I g →I, such that
Bn (x 1 , . . . , x n ) 5 Dk 1 , . . . k g ;n (Bk 1 , . . . , Bkg , . . . , Bk g )

(2)

where Bkg 5 Bkg (x 11sg 21 , . . . , x sg ); g 5 1, . . . , g. Condition Eq. (2) formalises the two

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215

step groupwise evaluation procedures, as well as its specific requirements. Because the
D functions are arbitrary at this stage, additional assumptions about Bn will be necessary
to determine and solve the system.
Similar assumptions have been employed in other publications, such as ‘‘replicative
invariance’’ by (Kolmogorov, 1930; Nagumo, 1930); and ‘‘strong decomposability’’ by
Marichal et al., in press.
Condition of bisymmetry (cross-sectional consistency) : The condition of subgroup
consistency is a kind of consistency for partitioning the whole group into ‘‘parallel’’
possible non-equal-sized subgroups, while the next condition concerns consistency of
rearranging a partition into ‘‘cross-sections’’. The condition of bisymmetry concerns the
case where the committee is subdivided into possibly more than two equal-sized
subgroups. Let us suppose (similarly to the condition of subgroup consistency) that
collective evaluation can be performed in subgroups, the same procedure being used for
the collective evaluation per subgroup as for the final aggregation over subgroups, in
order to reach the final collective decision. Let us more specifically assume the
committee to be partitioned in a fixed number of equally sized subgroups. This implies
that we can represent the value judgements of the members of the committee in terms of
an n3m matrix form as follows:
x 11
x 21
:
:
x m1

x 12
x 22

....
....

....
....

x m2

....

....

x 1n
x 2n
:
:
x mn

This matrix representation suggests either a row wise partitioning into m

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¨
et al. / Mathematical Social Sciences 37 (1999) 211 – 233
A´ . Munnich

Fig. 1. Condition of bisymmetry.

The generalised concept of n-variable bisymmetry was also discussed by the authors
¨
in another context (Munnich
et al., submitted). Bisymmetric operations have been
´ (1946), (1948), (1966); Coombs et al. (1981); Falmagne (1985); Fodor
studied by Aczel
and Marichal (1997); Krantz et al. (1971); Luce et al. (1990); Marichal and Mathonet,
submitted; Pfanzagl (1968) among others. The problem of finding the general solution of
´ 1966, 1989; Krantz et al., 1971;
Eq. (4) for two variables has been studied by (Aczel,
Pfanzagl, 1968), among others. In economics, in the context of aggregation, a more
general form of Eq. (3) has been discussed, e.g., by (Green, 1964; Gorman, 1968;
Pokroff, 1978), among others, but a different theoretical orientation made them assume
other conditions of solvability than here, excluding even the simple arithmetic mean
from the possible joint solutions of Eq. (1), Eq. (2) or Eq. (4).

3. Combining individual value judgements: solutions
In this section, the system of functional Eqs. (1), (2), (4) is solved, and we discuss
also the consequences of various additional possible assumptions concerning ‘‘symmetries’’ of Bn and
Dk 1 ,k 2 , . . . ,k g ;n
Theorem 1. (The main theorem). Let I be a real interval, Bn :I n →I (n52, 3, . . . ) are
functions which are continuous and monotone strictly increasing in each of their
arguments; and Dk 1 ,k 2 , . . . ,k g ;n :I g →I (n52,3, . . . ; 2#g#n21; 1#kg ,n are integers;
og 51 g kg 5n), are functions. Then for any fixed n ($2) integer, Eqs. (1), (2), (4) hold

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217

simultaneously if and only if there is a real valued, continuous and strictly monotone
(n)
n
increasing function w :I →R, and constants (a (n)
1 , . . . , a n ) []0,1[ , (n52,3, . . . ) such
n
(n)
that o i 51 a i 51 (n52,3, . . . ), and

SO

D

n

Bn (x 1 , . . . , x n ) 5 w 21

a i(n) w (x i ) ; x i [ I, i 5 1, . . . , n, n 5 2,3, . . . ,

(5)

S O v w(z )D z , . . . ,z [ I, n 5 2,3, . . .

(6)

i51
g

Dk 1 ,k 2 , . . . ,k g ;n (z 1 , . . . ,z g ) 5 w 21

g

g

1

g

g 51

and so

S O v w(B
g

Dk 1 ,k 2 , . . . ,k g ;n (Bk 1 , . . . , Bkg , . . . , Bk g ) 5 w 21

g

g 51

kg

D

) ,

where
sg

vg 5

O

i511sg 21

(n)
i

a ; Bkg 5 Bkg (x 11sg 21 , . . . , x sg ) 5 w

21

S

sg

O

i 511sg 21

D

g]
a [k
w (xi )
i

(7)

and the relationships between the coefficients of the and Bkg , and Bn respectively, are as
follows:
[kg ]

ai

a (n)
i
5 ]; g 5 1, . . . , g; n 5 2,3, . . . .
vg

Proof: The proof of the ‘‘if’’ part of the theorem is almost immediate, as the reader can
verify by the substitution of the vg and Bkg and a i[kg ] in Eq. (6). The proof of the ‘‘only
if’’ part is given in Appendix A, and is based on induction on n (for g52), followed by
induction on g, for any n.
The right member in Eq. (5) is known as a quasilinear mean and the function w is
called its generating function. Theorem 1 therefore establishes that the general solution
for Bn is the quasilinear mean of the individual judgements and that its generating
function w is independent of the size of the committee. The coefficients a (n)
can be
i
regarded as weights of the individual judges, when these are assumed to be labelled
accordingly, the weights reflecting their varying importance or contributions in the
evaluation process.
´ 1966; Pfanzagl, 1968), among
Quasilinear means were studied extensively by (Aczel,
others. Examples of quasilinear means are the weighted arithmetic mean, the harmonic
mean, the root-mean power-mean, and the exponential mean, among others. The practice
of taking the average or weighted average of judgements as an overall judgement, is
widely used, e.g., in the expected utility theory Neumann and Morgenstern (1944); the
´
cardinal welfare function of Harsanyi
(1955); the subjective expected utility theory (see
Savage, 1954); averaging vote probabilities (see Maas et al., 1990); and averaging test
scores for getting an overall evaluation of students. Also, in statistical inference, in many
respects the simple arithmetic (or geometric) mean is a ‘‘best’’ estimator. Some of these

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A´ . Munnich

218

theories are based on specific axioms (or conditions) regarding the specific objectives of
the theories in question, and some use averages as a plausible method, but frequently
with no rigid foundation in the form of a set of axiomatic premises.
Expression Eq. (6) of Dk 1 ,k 2 , . . . ,k g ;n shows how to combine evaluations of the
subgroups. The evaluation of the first subgroup is given in the first argument of
Dk 1 ,k 2 , . . . ,k g ;n and based on the first k 1 individual’s judgement, and the evaluation of the
second subgroup is given in the second argument of Dk 1 ,k 2 , . . . ,k g ;n and based on
judgements of the second group, and so on. Expressions in Eq. (7) determine a
correspondence between the coefficients of the judges in the whole group and those of
the subgroups. If for example, n55, k 1 52 and k 2 53, then it is easy to verify that
Bn (x 1 ,x 2 ,x 3 ,x 4 x 5 ) 5 w

21

(5 )

(5)

(5)

(5)

(a 1 w (x 1 ) 1 a 2 w (x 2 ) 1 a 3 w (x 3 ) 1 a 4 w (x 4 )

1 a (5)
5 w (x 5 )),
D2,3;5 (x, y) 5 w

21

(5)

(5)

(5)

(5)

(5 )

(a 1 w (x) 1 a 2 w (x) 1 a 3 w (x) 1 a 4 w (x) 1 a 5 w (x))

where
[2]
x 5 B2 (x 1 , x 2 ) 5 w 21 (a [2]
1 w (x 1 ) 1 a 2 w (x 2 ))

y 5 B3 (x 3 , x 4 , x 5 ) 5 w

21

[3]

[3]

[3]

(a 3 w (x 3 ) 1 a 4 w (x 3 ) 1 a 5 w (x 5 ))

and
a (15 )
a (25 )
[2]
[2]
]]]
a 1 5 ]]]
,
a
5
2
(5)
( 5)
(5)
( 5) ,
a1 1 a2
a1 1 a2
a

[3]
3

a (35 )
a 4(5 )
a (55 )
[3]
[3]
5 ]]]]]
, a 4 5 ]]]]]
, a 5 5 ]]]]]
a 3( 5 ) 1 a 4( 5 ) 1 a 5( 5 )
a 3( 5 ) 1 a 4( 5 ) 1 a (55 )
a 3( 5 ) 1 a 4( 5 ) 1 a 5( 5 )

The judges’ evaluations are the input variables of Bn , and as such Bn may depend on the
ordering of the judges, when these are correspondingly labelled or indexed. The order of
the judges and consequently the corresponding weights attached to the arguments of Bn
(see the solution given by Eq. (5)), may then reflect a kind of importance or power of the
judges. For instance, in a committee of share-holders of a firm, the power of the
members of the committee is represented by the percentages of their own shares.
Nevertheless, in many other practical situations (e.g., when a university board has to
evaluate different scientific programs) it is assumed ‘‘a priori’’ that the judges have equal
weight, so that any ordering of the judges should lead to the same conclusion, and only
the judges’ opinions (and not their particular position or status) are taken into
consideration. The same problem can arise with combining the evaluations of the
subgroups by the functions Dk 1 ,k 2 , . . . ,k g ;n (see Eq. (6)), that is the subgroups can have ‘‘a
priori’’ different (that is Dk 1 ,k 2 , . . . ,k g ;n is not necessarily ‘‘symmetric’’) or equal weights
(that is Dk 1 ,k 2 , . . . ,k g ;n is ‘‘symmetric’’) through the groupwise collective evaluation. For
example, regard a parliamentary committee consisting of experts of the government and

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A´ . Munnich

219

the opposition parties. In this case, one of the subgroups consist of experts of the
government parties, and the other groups consist of experts of the oppositions. A natural
weighting (or order) of the subgroups, given proportional representation, corresponds to
the percentages of votes the parties received in the last election. But in some particular
cases the judges may decide to equate the power of the subgroups a priori, which
happened in 1996 in Hungary, when the experts of the parties tried to prepare a new
constitutional law, and in the preparatory process both expert groups of the government
and the opposition parties were allotted the same number of votes.
The next two corollaries are concerned with these additional assumptions, that is
special symmetric cases of Theorem 1 where the combination rule Bn and Dk 1 ,k 2 , . . . ,k g ;n
are permutation invariant. The results in both cases boil down to the obvious equal
weights solution.
Corollary 1. If in Theorem 1, for a fixed n the combination rule Bn is permutation
invariant, that is, for any permutation ( r1 , . . . , r2 ) of (1,2, . . . , n)
Bn (x 1 , . . . , x n ) 5 Bn (xr 1 , . . . , xrn ),

(8)

(n)
(n)
then a (n)
.
1 5a 2 5 . . . 5a n

Proof: See Appendix A.
Corollary 2. If in Theorem 1, for a fixed n and k 1 , . . . , k g , Dk 1 ,k 2 , . . . ,k g ;n is permutation
invariant, that is, for any permutation ( r1 , . . . , r2 ) of (1,2, . . . , n),
Dk 1 ,k 2 , . . . ,k g ;n (x 1 , . . . , x n ) 5 Dk 1 ,k 2 , . . . ,k g ;n (xr 1 , . . . , xrn )

(9)

g
then v1 5 v2 5 . . . 5 vg , where vg 5 o si 511s
a i(n)
g 21

Proof: See Appendix A.
In Theorem 1, we searched for solutions of Eqs. (1), (2), (4), which are applicable for
any sizes of the committees, and it was shown that the quasilinear mean Eq. (5) is that
solution. However, Theorem 1 does not answer the question of the solution of Eq. (3)
for fixed integer n, m$2. At the moment we can not provide the general solution of Eq.
(3), but in the next proposition it is shown that quasilinear means are solutions of the
general Eq. (3).
Proposition. If Bn :I n →I and Bm :I m →I (n, m$2 fixed integers) are quasilinear means
with continuous and strictly monotone generating function w, and weights a i , i51, . . . ,
n; b j , j51, . . . , m respectively, then for all x ij [I ( i51, . . . , n; j51, . . . , m)
Bm (Bn (x 11 , . . . , x 1n ), . . . , Bn (x m1 , . . . , x mn ))
5 Bn (Bm (x 11 , . . . , x m1 ), . . . , Bn (x 1n , . . . , x mn ))
Proof: See Appendix A.

(10)

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et al. / Mathematical Social Sciences 37 (1999) 211 – 233
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220

Both in Eqs. (3) and (4), it is supposed that the row and column wise evaluations can
be expressed by the same family of functions Bn (n51,2,3, . . . ), and in Theorem 1 it is
proved that applying Eq. (4), Bn is the quasilinear mean and its generating function is
the same for each n. In the Proposition we proved that the solutions of Eq. (4), that is the
quasilinear means Bn (n51,2,3, . . . ) with equal generating functions, are also solutions
of Eq. (3).
In Theorem 2 a type of uniqueness property of quasilinear means is proved. We
consider there the case where in Eq. (3), and hence in Eq. (4) also, for fixed n and m, the
row and column wise evaluations (see Fig. 1 and Eq. (13) below) are expressed by
quasilinear means A (substituting Bn ) and B (substituting Bm ), respectively, such that
their generating functions are different. We establish then that, even if A and B are two
possible different quasilinear means satisfying Eq. (13) below, their generating functions
must be equal, and only their weights can be different,.
Theorem 2. Let I ,R be an open interval, w, c :I →R be continuous and strictly
monotone increasing functions, n,m[N, n$ 52, m$2, ak , bj []0,1[, k51, . . . , n,
j51, . . . , m such that o kn51 ak 5o jm51 bj 51. Define the functions A and B on I n and I m ,
respectively by

SO

D

(11)

S O b c ( y )D

(12)

n

A(x 1 , . . . , x n ) 5 w 21

ak w (xk )

k51

and
m

B( y 1 , . . . ,y m ) 5 c 21

j

j

j 51

Suppose that
B(A(x 11 , . . . , x 1n ), . . . , A(x m1 , . . . , x mn )) 5 A(B(x 11 , . . . , x m1 ), . . . , B(x 1n , . . . , x mn ))
(13)
holds for all x kj [I, k51, . . . , n, j51, . . . , m. Then

SOb w( y )D y , . . . ,y
m

B( y 1 , . . . ,y m ) 5 w 21

j

j

1

m

[I

(14)

j 51

Proof: See Appendix A.

4. The quasilinear mean as an Arrowian social choice function
Until now, we focused on the group evaluation of a single object and showed that the
quasilinear mean satisfies the consistency conditions Eqs. (1), (2), (4). We shall discuss
now how can we apply this result to the situation of social choice: the committee has to
choose from a set of alternatives based on the individual judgements of the committee’s
members. According to (Arrow, 1951, pp. 23), a social welfare or choice function is ‘‘a
process or rule which, for each set of individual orderings for alternative social states
(one ordering for each individual), states a corresponding social ordering of alternative

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221

social states’’. He postulated conditions for a socially acceptable social choice function
and showed that they are inconsistent. His conditions can be summarised in terms of
collective or group choice rule as follows (see e.g., French, 1986):
• ‘‘non-triviality’’: There are at least two members of the group, and three alternatives.
• ‘‘ordering’’: It is supposed that each individual holds a preference (that is a weak
order: transitive and comparable) concerning the alternatives.
• ‘‘universal domain’’: The collective choice function should be defined for any finite
set of orderings.
• ‘‘independence of irrelevant alternatives’’: The collective ‘‘preference’’ of i to j
should not depend on any other alternatives.
• ‘‘Pareto principle’’: If i is ‘‘preferred’’ to j for everybody, then the collective will
‘‘prefer’’ i to j also.
• ‘‘No dictatorship’’: There should not be a person such that whenever he ‘‘prefers’’ i
to j, i is collectively ‘‘preferred’’ to j.
Arrow’s approach is based on individual orderings of the alternatives and in order to be
consistent with his approach, we introduce for each member of the committee an
ordering on the alternatives based on their individual judgements. Our approach
´
presupposes, similar to that of utilitarians (see Harsanyi,
1955, 1986), that individual
judgements are cardinally measurable (hence implies infinite number of possible
preferenses) and admit a meaningful interpersonal comparison. Utilitarians mostly use
the arithmetic mean as a social utility function, while we extend it to the quasilinear
means of the subjective evaluations. Let C (with elements 1,2, . . . , n) be a subset of
subjects (that is the committee), A be a set of alternatives, x si [I be the subjective
evaluation of alternative i by subject s, B the quasilinear mean of the subjective
evaluations (that is the solution of Eqs. (1), (2), (4)) of the committee’s members, and
the relations R(s), R*(C) on A defined as follows:
(i, j) [ R(s) iff x si $ x sj ;
(i, j) [ R*(C) iff B(x 1i , . . . , x ni ) $ B(x 1j , . . . , x nj ).
If in a committee there is a person with extremely large ‘‘power’’, then of course that
single dominant person can be a dictator very easily, so the ‘‘no dictatorship’’ condition
cannot hold together with the other conditions. By not weighting the individual
orderings, Arrow assumes (implicitly) that all of the individuals have equal ‘‘power’’ in
the decision process. In our approach we assume a somewhat weaker condition, which
nevertheless implies non-dictatorship. This condition, to be called that of countervailing
pairs, allows attachment of different ‘‘powers’’ to different individuals (expressed by
their weights in Eq. (5)), but it requires the presence of ‘‘counterparts’’, in the sense that
for each subject s there is an other subject s 1 , whose ‘‘power’’ is equal to that of s. For
that purpose we introduce the concept of equally weighted pairs, that is, let s, s 1 [C,
then the two-element subgroup hs, s 1 j of C is called a pair of equally weighted subjects
(that is s and s 1 are counterparts of each other), if, regarding the two subjects as a
two-element collective, the corresponding weights in Eq. (5) are equal. Again,

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et al. / Mathematical Social Sciences 37 (1999) 211 – 233
A´ . Munnich

intuitively, if in a committee there is a person with the highest power but without
‘‘counterpart’’ (as it happens frequently in strictly hierarchical organizations), then this
person can be a dictator easily by beating each other persons step-by-step in pairwise
battles.
But, even if any person has a ‘‘counterpart’’ regarding ‘‘power’’, one of them (or each
of the highest, but equally ‘‘powered’’ persons) can be a dictator when there is not a
person, who has equal ‘‘power’’, but ‘‘reverse’’ preferences. The consequence of the
‘‘universal domain’’ condition is that for any preference ordering of subject s there is an
other subject whose preference ordering is the ‘‘reverse’’ of the ordering of s. Again, our
approach is less restrictive, as we will assume that if subject s prefers alternative i to j,
that is x si $x sj , then there is another subject s9 such that x s 9 j $x si $x sj .x s 9i (the set hs,
s9j is called countervailing pair). The meaning of this assumptions is that s9 prefers j to i
more intense than s prefers i to j. The strict inequality is necessary to exclude ‘‘group of
dictators’’, because equality let exist more than one ‘‘parallel’’ dictator, but without loss
of generality, it can be at either end of the inequality chain. The requirement of
countervailing pairs does not imply restrictions on the finite set of committee members,
which can be verified easily for 2 and 3 persons and 3 alternatives, and as a consequence
is applicable to any number of committee members (by simply partitioning a large group
to 2 and / or 3 elements subgroups).
In the next theorem, we show that the solution of Eqs. (1), (2), (4), that is the
quasilinear mean, is suitable for a kind of Arrowian social choice function.
Theorem 3. For every collective C, s[C,
1.
2.
3.
4.

R*(C) is defined for any number of subject and alternatives; ‘‘non triviality’’
R*(C) is a weak ordering on A; ‘‘ordering’’
R*(C) is defined whenever R(s) is defined for all s[C; ‘‘universal domain’’
R*(C) satisfies the ‘‘independence of irrelevant alternatives’’ condition, that is, (i,
j)[R*(C) does not depend on any k[ A, k±i, j;
5. R*(C) satisfies the ‘‘Pareto principle’’, that is, if (i, j)[R(s) for all s[C, then (i,
j)[R*(C);
6. (countervailing pairs) If for every s[C, there is s 1 [C, i, j [ A, such that hs, s 1 j is a
pair of equally weighted subjects, and x s 1 j $ x si $ x sj . x s 1 i , then the ‘‘no dictatorship’’ is fulfilled. (Note that under condition 6, s can not be a dictator: it will be
shown that if for every s[C there is s 1 [C, i, j [ A, such that (i, j) [ R(s), then it
implies (i, j)[
⁄ R*(hs, s 1 j), hence the ‘‘no dictatorship’’ condition is fulfilled.)
Proof. See Appendix A.

5. Conclusions
According to the related topics discussed in this paper, some conclusions can be
mentioned. It might be useful for understanding the process of social choice when a
prior evaluative stage, where the judges individually evaluate the candidates, is

¨
et al. / Mathematical Social Sciences 37 (1999) 211 – 233
A´ . Munnich

223

distinguished from the actual choice-making stage, where the committee has to make the
final single selection (probably according to some predetermined rules). The conditions
of consistency give the opportunity for a different approach of the classical Arrowian
problem of the existence of an axiomatised social welfare function. The conditions of
reflexivity and subgroup consistency can be interpreted as formalisations of criteria of
fairness and justice. The bisymmetry condition seems to be a technical extension of the
subgroup condition, but it is accepted and used in our daily life (e.g., when we check the
total of crosstabulated data) and it is well known in many psychological phenomena
(e.g., bisection, axiomatized by bisymmetry, is a standard psychophysical measurement
procedure). The general solution of our system Eqs. (1), (2), (4), which is the quasilinear
mean, is a direct generalisation of some utility based aggregation formulas, e.g. the
linear or the weighted linear models. Further restrictions on the special form of the
generating function w of an appropriate quasilinear mean may be imposed on the basis
of substantive reasons such as some specific psychological consideration of perception
or cognition which may govern the evaluation of the alternatives.

Appendix A
Lemma. Let J be a real interval, and let the function H: J 2 →R be real valued,
continuous and strictly monotone increasing in each of its arguments. Let H(x, x)5x if
x[ J. Let the positive integers n and real numbers a1 , . . . , an be such that 2#n,
(a1 , . . . , an )[]0,1[ n , o i 51 n ai 51. Let us suppose that for each x 1 , . . . , x n , y 1 , . . . ,
yn [ J

O a HSx , O a y D 5O a HSO a x , y D
n

n

i

i51

i

n

j

j

n

j

j 51

j 51

i i

i

(15)

i 51

2

Then there is ( l, m )[]0,1[ , such that l 1 m 51, such that l 1 m 51, and for each x,
y[ J
H(x, y) 5 lx 1 m y.

(16)

Proof: Let x, u, y, v[ J and x 1 5 . . . 5x n 21 5x, x n 5u, y 1 5 . . . 5y n 21 5y, y n 5v and
12 an 5 a. Then from Eq. (15) it follows that

a H(x, a y 1 (1 2 a )v) 1 (1 2 a )H(u, a y 1 (1 2 a )v) 5 a H(a x 1 (1 2 a )u, y)
1 (1 2 a )H(a x 1 (1 2 a )u, v).

(17)

´ 1966, Theorem 2, pp.
From Eq. (17), for fixed y 5 v [ J applying a result of (Aczel,
67.), it follows that there are real valued continuous functions c, d:J →R such that for
each x, y[ J, H(x, y)5c( y)x1d( y). Resubstituting this form of H into Eq. (17) and
substituting u with x, after some algebraic manipulations (factorizing both sides of Eq.
(17) according to x), we get that for each y, v [ J

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et al. / Mathematical Social Sciences 37 (1999) 211 – 233
A´ . Munnich

c(a y 1 (1 2 a )v) 5 a c( y) 1 (1 2 a )c(v) and d(a y 1 (1 2 a )v)
5 a d( y) 1 (1 2 a )d(v).
´ 1966, Theorem 2, pp. 67.) it follows that there are l, l9, m, m 9[R
Again from (Aczel,
such that for each x, y[ J, H(x, y)5 l9yx1 lx1 m y1 m 9. Since H is strictly monotone
increasing in each of its arguments, and H(x, x)5x if x[ J, it follows that l95 m 950,
l .0, m .0 and l 1 m 51, hence Eq. (16) holds and the proof is complete.
Proof of Theorem 1. Suppose that Eqs. (1), (2) and (4) hold. The proof will be based
on double induction on n and g. Assume first that g52 (two subgroups with members k n
and n2k n , respectively), and denote Dk n ,n 2k n ;n by Dk n , for simplicity. It then will be
´ 1966, Sect.
shown that for each fixed n52,3, . . . the function is bisymmetric (Aczel,
6.4). From Eqs. (2) and (1) it follows that
1

k n k n 11

n

˘ . . . , x˘ , y˘ , . . . , y)
˘ x, y [ I
Dk n (x, y) 5 Bn (x,

(18)

Let now x, y, u, v[I, then applying Eqs. (18) and (4) and again Eq. (18) we get
Dk n (Dk n (x, y), Dk n (u, v))
1

kn

k n 11

n

˘ y), . . . , Dk (˘ x, y), Dk (u,v),
˘
˘ v))
5 Bn (Dk n(x,
. . . , Dk n(u,
n
n
5 Bn (Bn (x, . . . , x, y, . . . , y), . . . , Bn (x, . . . , x, y, . . . , y),
3 Bn (u, . . . , u, v, . . . , v), . . . , Bn (u, . . . , u, v, . . . , v))
5 Bn (Bn (x, . . . , x, u, . . . , u), . . . , Bn (x, . . . , x, u, . . . , u),
3 Bn ( y, . . . , y, v, . . . , v), . . . , Bn ( y, . . . , y, v, . . . , v))
5 Bn (Dk n (x,u), . . . , Dk n (x,u), Dk n ( y,v), . . . , Dk n ( y,v)) 5 Dk n (Dk n (x,u), Dk n ( y,v)),
that is Dk n is bisymmetric. From Eq. (18) it is obvious that Dk n is continuous and strictly
´ 1966,
monotone increasing in both of its arguments, and (x, x)5x(x[I). From (Aczel,
Sect. 6.4) it follows that there is a continuous and strictly monotone increasing function,
and constants such that and
Dk n (x, y) 5 f k21
(a k n fk n (x) 1 b k n fk n ( y)) x, y [ I.
n

(19)

´ 1966, Sect. 6.4), hence there are a
Because B2 satisfies all the conditions of (Aczel,
continuous and strictly monotone increasing function w :I →R and constants (a (12 ) ,
)
a 2( 2 ) )[]0,1[ 2 , such that a (12 ) 1a (2
and
2
B2 (x 1 , x 2 ) 5 w 21 (a 1( 2 ) w (x 1 ) 1 a 2( 2 ) w (x 2 )) x 1 , x [ I.

(20)

¨
et al. / Mathematical Social Sciences 37 (1999) 211 – 233
A´ . Munnich

225

In order to prove that for each n52, 3, . . . there are constants such that and also Eq. (5)
is true, the method of induction on n will be used. It is already proved for n52.
Now suppose that n.2 and Eqs. (5)–(7) hold for each 2#j ,n. Then there are
constants
kn

(a

(n)
1

,...,a

(n)
kn

kn

) [ ]0,1[ ,

O a 5 1 and ( b
, O b 51
(n)
i

i 51

(n)
k n 11

, . . . , b n(n) )

n

[ ]0,1[

n 2k n

(n)
i

i 5k n 11

such that for each x 1 , . . . , x k n , x k n 11 , . . . , x n )[I n ,

SO
SO

D

kn

Bk n (x 1 , . . . , x k n ) 5 w 21

a i(n) w (x i )

i 51

and Bn 2k n (x k n 1 1, . . . , x n )

n

5 w 21

b i(n) w (xi )

i 5k n 11

D

(21)

Let J5 w (I) (which is an interval), h n 5fk n + w 21 and for each x, y[ J Hn (x, y)5
h n 21 (a k n h n (x)5b k n h n ( y)). Then h n is strictly monotone increasing and continuous, and
Hn is continuous, bisymmetric, strictly monotone increasing in each of its arguments and
Hn (x, x)5x (x[I). Let x 1 , . . . , x n [ J. Then from Eqs. (2), (21) and (19) it follows that

S SO D S O DD
S SO D
S O DD
S SO D S O DD
SO O D
kn

w + Bn (w

21

(x 1 ), . . . , w

21

(x n )) 5 w + Dk n w

n

21

a j(n) xj , w 21

j 51

b j(n) x j

j 5k n 11

kn

5 w + f 21
a k n fk n + w 21
kn

a (n)
j xj

j 51

n

1 b k n fk n w 21

b j(n) xj

j 5k n 11
kn

5h

21
n

ak n hn

j 51

kn

5 Hn

n

a j(n) x j 1 bk n h n

b j(n) xj

j 5k n 11

n

a j(n) xj ,

j 51

b j(n) xj ,

j 5k n 11

and so, for each x 1 , . . . , x n [ J

SO
kn

w + Bn (w

21

(x 1 ), . . . ,w

21

(x n )) 5 Hn

j 51

O .b
n

a j(n) xj ,

j 5k n 11

(n)
j
j

D

x .

(22)

From Eqs. (22) and (4) we can establish an equation for Hn . Let x ij [ J, i, j51, 2, . . . , n
and change x ij with w 21 (x ij ) in Eq. (4). Then from Eq. (22) and after some computations
we get:

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et al. / Mathematical Social Sciences 37 (1999) 211 – 233
A´ . Munnich

226

S O SO
O
SO
S O SO
O
SO
kn

kn

(n)
p

p 51

O b x D,
a x , O b x DD
a x , O b x D,
a x , O b x DD.
n

a s(n) x ps ,

a Hn

Hn

s 51

kn

n

(n)
p

n

(n)
s
ps

b Hn

p5k n 11

(n)
s
ps

s 5k n 11

s 51

kn

(n)
s
ps

s 5k n 11

kn

a p(n) Hn

5 Hn

p 51

n

(n)
s
sp

(n)
s
sp

s 51

kn

n

bp (n)Hn

p 5k n 11

s 5k n 11

n

(n)
s
sp

s 51

(n)
s
sp

s 5k n 11

(23)
kn

Let now x 1 , . . . , x k n , y 1 , . . . , y 1 , . . . , y k n , x[ J, y 5 o aj (n) y j and
j 51

x ps 5 x p , if p, s 5 1, 2, . . . ,k n
that is in a matrix form:
x ps 5 x sp 5 x, if p, s 5 k n 1 1, . . . , n
x ps 5 y s , if s 5 1, . . . , k n , p 5 k n 1 1, . . . , n
x ps 5 y, if p 5 1, . . . , k n , s 5 k n 1 1, . . . , n,
that is in
x1
.
.
.
xk n
y1
.
.
.
y1

a matrix form:
. . . x1 y .
. . . .
. .
.
.
. .
. . . xk n y .
. . . yk n x .
. . . .
. .
.
.
. .
. . . yk n x .

. . y
. . .
.
.
. . y
. . x
. . .
.
.
. . x

Then from Eq. (23) it follows that:

SO a

D

kn

Hn

(n)
p

p 51

SO a
kn

Hn (x p , y), Hn ( y, x) 5 Hn

p 51

SO a
kn

(n)
p

Hn

(n)
s

(x s , y p ), Hn ( y, x)

j 51

DD

.

Because Hn is strictly monotone increasing in its first argument, it then follows that:
kn

Oa

p 51

(n)
p

S

Hn x p ,

kn

Oa

j 51

(n)
j

D

kn

yj 5

Oa

j 51

SO
kn

(n)
j

Hn

s 51

D

a s(n) x s , y j .

Similarly, let now x k n 11 , . . . , x n , y k n 11 , . . . , y n , x [ J, y 5 o sn5k n 11 b s(n) y s and
x ps 5 x p , if p, s 5 k n 1 1, . . . , n

(24)

¨
et al. / Mathematical Social Sciences 37 (1999) 211 – 233
A´ . Munnich

227

that is in a matrix form:
x ps 5 x sp 5 x, if p, s 5 1, . . . , k n
x ps 5 y s , if p 5 1, . . . , k n , s 5 k n 1 1, . . . , n
x ps 5 y, if p 5 k n 1 1, . . . , n, s 5 1, . . . , k n ,
that is in a
x .
. .
.
.
x .
y .
. .
.
.
y .

matrix form:
. . x y k n 11
. . .
.
.
.
.
.
. . x y k n 11
. . y x k n 11
. . .
.
.
.
.
.
. . y
xn

. . .
. . .

yn
.
.
.
yn

. . .
. . . x k n 11
. . .
.
.
.
. . .
xn

Then from Eq. (23) it follows that:

S O b H ( y, x )D
5 H SH (x, y) O b H S y , O
n

(n)
p

Hn Hn (x, y),

n

p

p 5k n 11
n

n

n

(n)
p

n

n

b s(n) x s )

p

p5k n 11

s 5k n 11

DD

Because Hn is strictly monotone increasing in its first argument, it then follows that:

O
n

p 5k n 11

SO

D

n

b (n)
p Hn

O

s 5k n 11

S

n

b s(n) ys ,xp 5

O
n

b p(n) Hn y p ,

p 5k n 11

s 5k n 11

D

b s(n) x s .

(25)

Since n5k n 1(n2k n ).2, it is clear that k n $2, or n2k n $2. In the first case Eq. (24), in
the second case Eq. (25) can be applied in the Lemma, and accordingly we get that there
are constants ( ln , mn )[]0,1[ 2 , ln 1 mn 51, such that for each x, y[ J, Hn (x, y)5 ln x1
mn y. Hence, from Eq. (22) it follows that for each x 1 , . . . , x n [I

SO a w(x ), O b w(x )D
SO l a w(x ), O m b w(x )D.
kn

Bn (x 1 , . . . , x n ) 5 w 21 + Hn

n

(n)
i

i 51

n

n

i 51

i

i 5k n 11

kn

5 w 21

(n)
i

i

(n)
i

i

n

(n)
i

i

i 5k n 11

Then with the coefficients a i (n) 5 ll ai (n) if i51, 2, . . . , k n and a i (n) 5 mn ai (n) if i5k n 1
1, . . . , n we get Eq. (5). By now, Eq. (6) follows from Eqs. (18), (5), (7) follows from
Eqs. (2), (5), (6). This establishes the theorem for g52 and for n52, 3, . . . .
Now assume, for given n, g subgroups of size k1 , . . . , kg , . . . , k g ; og 51 g 5n, and
suppose the theorem to be true for g21. Combine, without loss of generality, the first

¨
et al. / Mathematical Social Sciences 37 (1999) 211 – 233
A´ . Munnich

228

two subgroups into one subgroup of size k c 5k1 1k2 , so that n is subdivided into g21
subgroups. By assumption the theorem is true for g21. Hence Eqs. (1), (2), (4) imply
that there are a real valued, continuous and strictly monotone increasing function
w :I →R, and constants
(n)
n
(a (n)
1 , . . . , a n ) [ ]0,1[ , (n 5 2,3, . . . )

such that

Oa
n

(n)
i

5 1 (n 5 2,3, . . . ),

i51

and

SO

D

n

Bn (x 1 , . . . ,x n ) 5 w 21

a i(n) w (x i ) ; x i [ I, i 5 1, . . . , n, n 5 2,3, . . . ;

i 51

as well as

S

O v w(B
g

Dk c ,k 3 , . . . ,k g ;n (Bk c , Bk 3 , . . . ,Bk g ) 5 w 21 vc w (Bk c ) 1

g

g 53

kg

D

) ,

(26)

where
k1

vc 5

Oa

k2

(n)
i

1

i 51

O

a i(n) ;

i 511k 1

SO a
k1

Bk c 5 Bk c (x 1 , x 2 , . . . , x k 1 , x 11k 1 , . . . , x k 2 ) 5 w 21

k2

(k c )
i

i 51

w (xi ) 1

O

i 511k 1

D

a i(k c ) w (x i )

and where the other vg , are defined as before. Moreover from the induction hypothesis
c)
we also have: a (k
5 [(a (n)
i
i ) / vc ]. Appropriate substitution in Bk c gives

SO ]av w(x ) 1 O
5 O a w (x ) 1 O a w (x ).

vc w (Bk c ) 5 vc ww 21

k1

(n)
i

i 51

c

k2

i

k1

D

a (n)
i
]
w (x i )
i 511k 1 vc

k2

(n)
i

(n)
i

i

i 51

i

i 511k 1

This we can write as

SO
k1

v1 ww 21

D

a i(n)
]
w (xi ) 1 v2 ww 21
i51 v1

so that we have:

vc w (Bk c ) 5 v1 w (Bk 1 ) 1 v2 w (Bk 2 )

SO
k2

D

a i(n)
]
w (xi ) ,
i 511k 1 v2

¨
et al. / Mathematical Social Sciences 37 (1999) 211 – 233
A´ . Munnich

229

. Substitution in Eq. (26) leads to the expression:

S O v w(B
g

Dk 1 ,k 2 , . . . ,k g ;n (Bk 1 , Bk 2 , . . . , Bk g ) 5 w 21

g

kg

g 51

D

) ,

which with Eqs. (1) and (2), completes the proof for general g, (2 #g#n21), and any
n52, 3, . . .
Proof of Corollary 1. It will be shown that for any integers i, j from 1, 2, . . . , n, it
follows that a i (n) 5a j (n) . Let, for simplicity, i51 and j52, and x1 ±x2 . Then from Eq.
(8) it follows that

S

(n)
i

D

Oa

(n)
i

S

w (x i ) 5 w 21 a 1(n) w (x 2 ) 1 a 2(n) w (x 1 )

i 53

D

n

1

Oa
n

w 21 a 1(n) w (x 1 ) 1 a 2(n) w (x 2 ) 1
w (x i ) ,

i 53

hence
(n)
(a (n)
1 2 a 2 )(w (x 1 ) 2 w (x 2 )) 5 0,
(n)
which implies that a (n)
and the proof is complete.
1 5a 2

Proof of Corollary 2. The proof is similar to the proof of Corollary 1.
Proof of Proposition. We have to prove that

SO

D SO

m

w

21

n

b j w (Bn (x j 1 , . . . , x jn )) 5 w

j 51

21

a i w (Bm (x 1i , . . . , x mi ))

i 51

Eq. (27) holds iff

O b wSw SO a w(x )DD 5O a wSw SO b w(x )DD iff
m

n

n

m

21

21

j

i

ji

i

i 51

j51

i

j 51

i51

O b SO a w(x )D 5O a SO b w(x )Diff
m

n

j

n

i

ji

i

i 51

j51

m

i 51

j

ji

j 51

O O b a w(x ) 5O O a b w(x ) and
m

n

n

j i

m

ji

j51 i51

the proof is complete.

i j

i51 j51

ji

ji

D

(27)

¨
et al. / Mathematical Social Sciences 37 (1999) 211 – 233
A´ . Munnich

230

Proof of Theorem 2. Let u, v, x, y[I, a 5 a1 , b 5 b1 and apply Eq. (13) for the m by n
matrix:
u
x
.
.
x

v
y
.
.
y

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

v
y
.
.
y

Then, taking into consideration Eqs. (11) and (12), we have

c 21 ( bc + w 21 (aw (u) 1 (1 2 a )w (n )) 1 (1 2 b )c + w 21 (aw (x) 1 (1 2 a )w ( y)) 5
w 21 (aw + c 21 ( bc (u) 1 (1 2 b )c (x)) 1 (1 2 a )w + c 21 ( bc (n ) 1 (1 2 b )c ( y)).

(28)

21

Let c (I)5 J. Then J is an open interval again. Let p,q,r,s[ J and u5 c ( p), n 5
c 21 (q), r5 c 21 (x), s5 c 21 ( y) in Eq. (28). Then, introducing the notation w + c 21 5h,
Eq. (28) goes over into
h( b h 21 (a h( p) 1 (1 2 a )h(q)) 1 (1 2 b )h 21 (a h(r) 1 (1 2 a )h(s)) 5 a h( b p 1 (1 2

b )r) 1 (1 2 a )h( b q 1 (1 2 b )s).

(29)

Obviously, h:J →R is strictly monotone increasing, therefore the function H is well2
defined on J 3 J by
H( p, q) 5 h 21 (a h( p) 1 (1 2 a )h(q))

(30)

and so Eq. (29) can be written in the form

b H( p, q) 1 (1 2 b )H(r, s) 5 H( b p 1 (1 2 b )r, b q 1 (1 2 b )s) ( p, q, r, s [ J).
¨
Similarly to the proof of the Lemma (the proof is given in Munnich
et al., submitted),
this implies that H( p, q)5 l p1(12 l)q for some l []0,1[and for all p, q[ J,
consequently, by Eq. (30) h( l p1(12 l) q5 a h( p)1(12 a ) h(q) for all p, q[ J. It
´ 1966, Theorem 2, pp. 67.) that l 5 a and there exist real numbers
follows from (Aczel,
a, b such that a.0 and h( p)5ap1b for all p[ J.
According to the definition of h, this implies that
1
c (u) 5 ] (w (u) 2 b) for all u [ I and
a

(31)

c 21 ( p) 5 w 21 (ap 1 b) for all p [ J.

(32)

Finally, Eq. (14) follows from Eqs. (12), (31), (32).
Proof of Theorem 3. (i) – (iv) Trivial.
(v) If i, j [R(s), s[C, then x si $x sj , s[C, and because B is strictly monotone in each
of its arguments, (i, j)[R*(C).

¨
et al. / Mathematical Social Sciences 37 (1999) 211 – 233
A´ . Munnich

231

(vi) Let s, s 1 [C and i, j [ A, such that hs, s 1 j is a countervailing pair, hence
satisfying (without loss of generality) x s 1 j $x si $x sj . x s 1 i . Then i, j [R(s), and x si 2
x sj ,x s 1 j 2 x s 1 i . Because the generating function w of B is strictly monotone increasing
w (x si )2 w (x sj ), w (x s 1j ) 2 w (x s 1i ) and s and s 1 are equally weighted subjects, it follows
that B(x si , x s 1i ),B(x sj , x s 1j ), hence (i, j)[
⁄ R*(hs, s 1 j) and therefore s cannot be a
dictator.

References
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´ J., 1966. Lectures on Functional Equations and Their applications. Academic Press, New York.
Aczel,
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Aczel,
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