Collective judgement: combining individual value judgements
a , a b
´ ¨ *
Akos Munnich , Gyula Maksa , Robert J. Mokken
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 require-ments 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. ´ ¨
E-mail address: [email protected] (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 prefer-ences, and the form of a function which specifies a social preference relation given individual preferences and the comparability assumptions. Arrow (1951) using in-dividual 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;
¨
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 , . . . , x concerning a choice or decision1 n
making task. The question then is how these judgements can be synthesised into a single overall quantitative judgement B (x , . . . , x ). Three formal structural conditions will ben 1 n
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 , . . . , x1 n
are elements of a real interval (finite or infinite) I, and that the overall judgement Bn
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
B (x, . . . , x)n 5x. (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 B (x , . . . , x ) denotes the overalln 1 n
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 , . . . , k , . . . , k , respectively, such that 11 g g #kg,n;g: 1,2, . . . ,
g
g andog 51 kg5n. Each subgroup evaluates candidates independently according to B .kg
In the next step the overall evaluation is then performed by an unknown function
Dk ,k , . . . ,k ;n1 2 g (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
g
size k (ng #kg,n);og 51 kg5n; g, k are positive integers, and let indices s be definedg g
as:
g g
sg5
O
k ;n g 51,2, . . . , g; s050, sg5O
kn5n;n 51 n 51
g
there is a function Dk ,k , . . . ,k ;n1 2 g :I →I, such that
step groupwise evaluation procedures, as well as its specific requirements. Because the
D functions are arbitrary at this stage, additional assumptions about B will be necessaryn
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:
x11 x12 .... .... x1n
x21 x22 .... .... x2n
: :
: :
xm1 xm 2 .... .... xmn
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 ´
studied by Aczel (1946), (1948), (1966); Coombs et al. (1981); Falmagne (1985); Fodor 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
´
Eq. (4) for two variables has been studied by (Aczel, 1966, 1989; Krantz et al., 1971; 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 ‘‘symmet-ries’’ of B andn
Dk ,k , . . . ,k ;n1 2 g
n
Theorem 1. (The main theorem). Let I be a real interval, B :In →I (n52, 3, . . . ) are
functions which are continuous and monotone strictly increasing in each of their
g
arguments; and Dk ,k , . . . ,k ;n1 2 g :I →I (n52,3, . . . ; 2#g#n21; 1#kg,n are integers;
g
simultaneously if and only if there is a real valued, continuous and strictly monotone
(n) (n) n
increasing functionw:I→R, and constants (a1 , . . . , an ) []0,1[ , (n52,3, . . . ) such
n (n)
that oi51 ai 51 (n52,3, . . . ), and
n
21 (n)
B (x , . . . , x )n 1 n 5w
S
O
ai w(x ) ; xiD
i[I, i51, . . . , n, n52,3, . . . , (5)i51 g
21
Dk ,k , . . . ,k ;n1 2 g (z , . . . ,z )1 g 5w
S
O
v wg (z )gD
z , . . . ,z1 g[I, n52,3, . . . (6)g 51
and so
g
21
Dk ,k , . . . ,k ;n1 2 g (B , . . . , B , . . . , B )k1 kg kg 5w
S
O
v wg (B ) ,kgD
g 51
where
sg sg
(n) 21 [k ]g
v 5g
O
ai ; Bkg5B (xkg 11sg 21, . . . , x )sg 5wS
O
ai w(x )iD
(7)i511sg 21 i511sg 21
and the relationships between the coefficients of the and B , and B respectivelykg n , are as
follows:
(n)
ai [k ]g ]
ai 5 v ; g 51, . . . , g; n52,3, . . . .
g
Proof: The proof of the ‘‘if’’ part of the theorem is almost immediate, as the reader can
[k ]g
verify by the substitution of thevg and Bkgand ai 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 B is the quasilinear mean of the individual judgements and that its generatingn
(n)
function w is independent of the size of the committee. The coefficients ai can be 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.
´
Quasilinear means were studied extensively by (Aczel, 1966; Pfanzagl, 1968), among 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
´
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 ,k , . . . ,k ;n1 2 g shows how to combine evaluations of the subgroups. The evaluation of the first subgroup is given in the first argument of
Dk ,k , . . . ,k ;n1 2 g and based on the first k individual’s judgement, and the evaluation of the1
second subgroup is given in the second argument of Dk ,k , . . . ,k ;n1 2 g 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, k152 and k253, then it is easy to verify that
21 ( 5 ) ( 5 ) ( 5 ) ( 5 )
B (x ,x ,x ,x x )n 1 2 3 4 5 5w (a1 w(x )1 1a2 w(x )2 1a3 w(x )3 1a4 w(x )4 ( 5 )
1a5 w(x )),5
21 ( 5 ) ( 5 ) ( 5 ) ( 5 ) ( 5 )
D2,3;5(x, y)5w (a1 w(x)1a2 w(x)1a3 w(x)1a4 w(x)1a5 w(x)) where
21 [2] [2]
x5B (x , x )2 1 2 5w (a1 w(x )1 1a2 w(x ))2
21 [3] [3] [3]
y5B (x , x , x )3 3 4 5 5w (a3 w(x )3 1a4 w(x )3 1a5 w(x ))5
and
( 5 ) ( 5 )
a1 a2
[2] [2]
]]] ]]]
a1 5 ( 5 ) ( 5 ), a2 5 ( 5 ) ( 5 ),
a1 1a2 a1 1a2
( 5 ) ( 5 ) ( 5 )
a3 a4 a5
[3] [3] [3]
]]]]] ]]]]] ]]]]]
a3 5 ( 5 ) ( 5 ) ( 5 ), a4 5 ( 5 ) ( 5 ) ( 5 ), a5 5 ( 5 ) ( 5 ) ( 5 )
a3 1a4 1a5 a3 1a4 1a5 a3 1a4 1a5
The judges’ evaluations are the input variables of B , and as such B may depend on then n
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
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 B and Dn k ,k , . . . ,k ;n1 2 g
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 B is permutationn
invariant, that is, for any permutation (r1, . . . , r2) of (1,2, . . . , n)
B (x , . . . , x )n 1 n 5B (x , . . . , x ),n r1 rn (8)
(n) (n) (n)
then a1 5a2 5. . .5an .
Proof: See Appendix A.
Corollary 2. If in Theorem1, for a fixed n and k , . . . , k , D1 g k ,k , . . . ,k ;n1 2 g is permutation invariant, that is, for any permutation (r1, . . . , r2) of (1,2, . . . , n),
Dk ,k , . . . ,k ;n1 2 g (x , . . . , x )1 n 5Dk ,k , . . . ,k ;n1 2 g (x , . . . , x )r1 rn (9)
sg (n)
then v 5v 51 2 . . .5vg, where v 5g oi511sg 21ai
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).
n m
Proposition. If B :In →I and B :Im →I(n, m$2 fixed integers) are quasilinear means
with continuous and strictly monotone generating functionw, and weights a , ii 51, . . . ,
n; b , jj 51, . . . , m respectively, then for all xij[I (i51, . . . , n; j51, . . . , m)
B (B (x , . . . , x ), . . . , B (xm n 11 1n n m1, . . . , xmn))
5B (B (x , . . . , xn m 11 m1), . . . , B (x , . . . , xn 1n mn)) (10)
Both in Eqs. (3) and (4), it is supposed that the row and column wise evaluations can be expressed by the same family of functions B (nn 51,2,3, . . . ), and in Theorem 1 it is proved that applying Eq. (4), B is the quasilinear mean and its generating function isn
the same for each n. In the Proposition we proved that the solutions of Eq. (4), that is the quasilinear means B (nn 51,2,3, . . . ) with equal generating functions, are also solutions of Eq. (3).
In Theorem 2 a type of uniqueness property of quasilinear means is proved. We consider there the case where in Eq. (3), and hence in Eq. (4) also, for fixed n and m, the row and column wise evaluations (see Fig. 1 and Eq. (13) below) are expressed by quasilinear means A (substituting B ) and B (substituting B ), respectively, such thatn m
their generating functions are different. We establish then that, even if A and B are two possible different quasilinear means satisfying Eq. (13) below, their generating functions must be equal, and only their weights can be different,.
Theorem 2. Let I,R be an open interval, w, c:I→R be continuous and strictly
monotone increasing functions, n,m[N, n$ 52, m$2, ak, bj[]0,1[, k51, . . . , n,
n m n m
j51, . . . , m such thatok51 a 5k oj51 b 5j 1. Define the functions A and B on I and I ,
respectively by
n
21
A(x , . . . , x )1 n 5w
S
O
a wk (x )kD
(11)k51
and
m
21
B( y , . . . , y )1 m 5c
S
O
b cj ( y )jD
(12)j51
Suppose that
B(A(x , . . . , x ), . . . , A(x11 1n m1, . . . , xmn))5A(B(x , . . . , x11 m1), . . . , B(x , . . . , x1n mn)) (13)
holds for all xkj[I, k51, . . . , n, j51, . . . , m. Then
m
21
B( y , . . . , y )1 m 5w
S
O
b wj ( y )jD
y , . . . , y1 m[I (14)j51
Proof: See Appendix A.
4. The quasilinear mean as an Arrowian social choice function
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, xsi[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 xsi$x ;sj
(i, j )[R*(C ) iff B(x , . . . , x )1i ni $B(x , . . . , x ).1j nj
If in a committee there is a person with extremely large ‘‘power’’, then of course that single dominant person can be a dictator very easily, so the ‘‘no dictatorship’’ condition cannot hold together with the other conditions. By not weighting the individual orderings, Arrow assumes (implicitly) that all of the individuals have equal ‘‘power’’ in the decision process. In our approach we assume a somewhat weaker condition, which nevertheless implies non-dictatorship. This condition, to be called that of countervailing
pairs, allows attachment of different ‘‘powers’’ to different individuals (expressed by
their weights in Eq. (5)), but it requires the presence of ‘‘counterparts’’, in the sense that
1
for each subject s there is an other subject s , whose ‘‘power’’ is equal to that of s. For
1
that purpose we introduce the concept of equally weighted pairs, that is, let s, s [C,
1
then the two-element subgrouphs, s jof C is called a pair of equally weighted subjects
1
intuitively, if in a committee there is a person with the highest power but without ‘‘counterpart’’ (as it happens frequently in strictly hierarchical organizations), then this person can be a dictator easily by beating each other persons step-by-step in pairwise battles.
But, even if any person has a ‘‘counterpart’’ regarding ‘‘power’’, one of them (or each of the highest, but equally ‘‘powered’’ persons) can be a dictator when there is not a person, who has equal ‘‘power’’, but ‘‘reverse’’ preferences. The consequence of the ‘‘universal domain’’ condition is that for any preference ordering of subject s there is an other subject whose preference ordering is the ‘‘reverse’’ of the ordering of s. Again, our approach is less restrictive, as we will assume that if subject s prefers alternative i to j, that is xsi$x , then there is another subject ssj 9 such that xs9j$xsi$xsj.xs9i (the seths, s9jis called countervailing pair). The meaning of this assumptions is that s9prefers 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. R*(C ) is defined for any number of subject and alternatives; ‘‘non triviality’’ 2. R*(C ) is a weak ordering on A; ‘‘ordering’’
3. R*(C ) is defined whenever R(s) is defined for all s[C; ‘‘universal domain’’
4. 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 );
1 1
6. (countervailing pairs) If for every s[C, there is s [C, i, j[A, such thaths, s jis a pair of equally weighted subjects, and xs1j$xsi$xsj.xs1i, then the ‘‘no
dictator-ship’’ is fulfilled. (Note that under condition 6, s can not be a dictator: it will be
1
shown that if for every s[C there is s [C, i, j[A, such that (i, j )[R(s), then it
1
implies (i, j )[⁄ R*(hs, s j), hence the ‘‘no dictatorship’’ condition is fulfilled.)
Proof. See Appendix A.
5. Conclusions
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 functionw 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
2
Lemma. Let J be a real interval, and let the function H:J →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,
n n
(a1, . . . , an)[]0,1[ , oi51 a 5i 1. Let us suppose that for each x , . . . , x , y , . . . ,1 n 1
yn[J
n n n n
O
aiH x ,S
iO
ajyjD
5O
ajHS
O
ai ix , yiD
(15)i51 j51 j51 i51 2
Then there is (l, m)[]0,1[ , such that l1m 51, such thatl1m 51, and for each x,
y[J
H(x, y)5lx1my. (16)
Proof: Let x, u, y, v[J and x15. . .5xn215x, xn5u, y15. . .5yn215y, yn5v and
12a 5an . Then from Eq. (15) it follows that
aH(x,ay1(12a)v)1(12a)H(u,ay1(12a)v)5aH(ax1(12a)u, y)
1(12a)H(ax1(12a)u,v). (17)
´
From Eq. (17), for fixed y5v[J applying a result of (Aczel, 1966, Theorem 2, pp.
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
c(ay1(12a)v)5ac( y)1(12a)c(v) and d(ay1(12a)v)
5ad( y)1(12a)d(v).
´
Again from (Aczel, 1966, Theorem 2, pp. 67.) it follows that there arel,l9,m, m9[R
such that for each x, y[J, H(x, y)5l9yx1lx1my1m9. Since H is strictly monotone increasing in each of its arguments, and H(x, x)5x if x[J, it follows that l95m950,
l.0, m .0 and l1m 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 kn
and n2k , respectively), and denote Dn k ,nn 2k ;nn by D , for simplicity. It then will bekn
´
shown that for each fixed n52,3, . . . the function is bisymmetric (Aczel, 1966, Sect. 6.4). From Eqs. (2) and (1) it follows that
k k 11
1 n n n
˘ ˘ ˘ ˘
D (x, y)kn 5B (x, . . . , x , y , . . . , y)n x, y[I (18) Let now x, y, u, v[I, then applying Eqs. (18) and (4) and again Eq. (18) we get
D (D (x, y), D (u,v)) kn kn kn
k k 11
1 n n n
˘ ˘ ˘ ˘
5B (D (x, y), . . . , D ( x, y), D (u,v), . . . , D (u,v))
n kn kn kn kn
5B (B (x, . . . , x, y, . . . , y), . . . , B (x, . . . , x, y, . . . , y),n n n
3B (u, . . . , u,v, . . . ,v), . . . , B (u, . . . , u,v, . . . ,v))
n n
5B (B (x, . . . , x, u, . . . , u), . . . , B (x, . . . , x, u, . . . , u),n n n
3B ( y, . . . , y,v, . . . ,v), . . . , B ( y, . . . , y,v, . . . ,v))
n n
5B (D (x,u), . . . , D (x,u), D ( y,v), . . . , D ( y,v))5D (D (x,u), D ( y,v)),
n kn kn kn kn kn kn kn
that is Dkn is bisymmetric. From Eq. (18) it is obvious that Dkn is continuous and strictly ´ monotone increasing in both of its arguments, and (x, x)5x(x[I ). From (Aczel, 1966,
Sect. 6.4) it follows that there is a continuous and strictly monotone increasing function, and constants such that and
21
D (x, y)kn 5fkn(a f (x)kn kn 1b f ( y)) x, ykn kn [I. (19) ´
Because B satisfies all the conditions of (Aczel, 1966, Sect. 6.4), hence there are a2 ( 2 )
continuous and strictly monotone increasing function w:I→R and constants (a1 ,
( 2 ) 2 ( 2 ) ( 2 )
a2 )[]0,1[ , such that a1 1a2 and
21 ( 2 ) ( 2 )
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
(n) (n) kn (n) (n) (n)
(a1 , . . . ,akn)[]0,1[ ,
O
ai 51 and (bkn11, . . . ,bn )i51 n n2kn (n)
[]0,1[ ,
O
bi 51i5kn11
n
such that for each x , . . . , x , x1 kn kn11, . . . , x )n [I ,
kn
21 (n)
B (x , . . . , x )kn 1 kn 5w
S
O
ai w(x )iD
and Bn2kn(xkn11, . . . , x )n i51n
21 (n)
5w
S
O
bi w(x )iD
(21)i5kn11
21
Let J5w(I ) (which is an interval), hn5fk +w and for each x, y[J H (x, y)n 5
n
21
hn (a h (x)kn n 5b h ( y)). Then h is strictly monotone increasing and continuous, andkn n n
H is continuous, bisymmetric, strictly monotone increasing in each of its arguments andn
H (x, x)n 5x (x[I ). Let x , . . . , x1 n[J. Then from Eqs. (2), (21) and (19) it follows that
kn n
21 21 21 (n) 21 (n)
w+B (nw (x ), . . . ,1 w (x ))n 5w+Dkn
S
wS
O
aj xjD
,wS
O
bj xjD
D
j51 j5kn11kn
21 21 (n)
5w+fkn
S
a fkn kn+wS
O
aj xjD
j51 n21 (n)
1b fkn knw
S
O
bj xjD
D
j5kn11kn n
21 (n) (n)
5hn
S
a hkn nS
O
aj xjD
1b hkn nS
O
bj xjD
D
j51 j5kn11kn n
(n) (n)
5Hn
S
O
aj x ,jO
bj xjD
,j51 j5kn11
and so, for each x , . . . , x1 n[J
kn n
21 21 (n) (n)
w+B (nw (x ), . . . ,1 w (x ))n 5Hn
S
O
aj x ,jO
.bj xjD
. (22) j51 j5kn11From Eqs. (22) and (4) we can establish an equation for H . Let xn ij[J, i, j51, 2, . . . , n
21
and change x withij w (x ) in Eq. (4). Then from Eq. (22) and after some computationsij
kn kn n
(n) (n) (n)
Hn
S
O
ap HnS
O
as x ,psO
bs xpsD
,p51 s51 s5kn11 k
n n n
(n) (n) (n)
O
bp HnS
O
as x ,psO
bs xpsDD
p5kn11 s51 s5kn11kn kn n
(n) (n) (n)
5Hn
S
O
ap HnS
O
as x ,spO
bs xspD
,p51 s51 s5kn11 k
n n n
(n) (n)
O
bp(n)HnS
O
as x ,spO
bs xspDD
. (23)p5kn11 s51 s5kn11
kn (n)
Let now x , . . . , x , y , . . . , y , . . . , y , x1 kn 1 1 kn [J, y5o aj y andj j51
xps5x , if p, sp 51, 2, . . . ,kn
that is in a matrix form:
xps5xsp5x, if p, s5kn11, . . . , n
xps5y , if ss 51, . . . , k , pn 5kn11, . . . , n
xps5y, if p51, . . . , k , sn 5kn11, . . . , n, that is in a matrix form:
x1 . . . x1 y . . . y
. . . .
. . . .
. . . .
xkn . . . xkn y . . . y y1 . . . ykn x . . . x
. . . .
. . . .
. . . .
y1 . . . ykn x . . . x
Then from Eq. (23) it follows that:
kn kn kn
(n) (n) (n)
Hn
S
O
ap H (x , y), H ( y, x)n p nD S
5HnO
ap HnS
O
as (x , y ), H ( y, x)s p nDD
.p51 p51 j51
Because H is strictly monotone increasing in its first argument, it then follows that:n
kn kn kn kn
(n) (n) (n) (n)
O
ap HnS
x ,pO
aj yjD
5O
aj HnS
O
as x , ys jD
. (24)p51 j51 j51 s51
n (n)
Similarly, let now xkn11, . . . , x , yn kn11, . . . , y , xn [J, y5os5kn11bs y ands
that is in a matrix form:
xps5xsp5x, if p, s51, . . . , kn
xps5y , if ps 51, . . . , k , sn 5kn11, . . . , n
xps5y, if p5kn11, . . . , n, s51, . . . , k ,n
that is in a matrix form:
x . . . x ykn11 . . . yn
. . . .
. . . .
. . . .
x . . . x ykn11 . . . yn
y . . . y xkn11 . . . xkn11
. . . .
. . . .
. . . .
y . . . y xn . . . xn
Then from Eq. (23) it follows that:
n (n)
Hn
S
H (x, y),nO
bp H ( y, x )n pD
p5kn11n n
(n) (n)
5Hn
S
H (x, y)nO
bp HnS
y ,pO
bs x )sDD
p5kn11 s5kn11Because H is strictly monotone increasing in its first argument, it then follows that:n
n n n n
(n) (n) (n) (n)
O
bp HnS
O
bs y ,xs pD
5O
bp HnS
y ,pO
bs xsD
. (25)p5kn11 s5kn11 p5kn11 s5kn11
Since n5kn1(n2k )n .2, it is clear that kn$2, or n2kn$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
2
are constants (ln, mn)[]0,1[ , l 1m 5n n 1, such that for each x, y[J, H (x, y)n 5lnx1 mny. Hence, from Eq. (22) it follows that for each x , . . . , x1 n[I
kn n
21 (n) (n)
B (x , . . . , x )n 1 n 5w +Hn
S
O
ai w(x ),iO
bi w(x )iD
i51 i5kn11kn n
21 (n) (n)
5w
S
O
l an i w(x ),iO
m bn i w(x ) .iD
i51 i5kn11(n) (n) (n) (n)
Then with the coefficients ai 5l al i if i51, 2, . . . , k and an i 5m an i if i5kn1
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, . . . .
g
Now assume, for given n, g subgroups of size k , . . . , k , . . . , k ;1 g g og 51 5n, and
two subgroups into one subgroup of size kc5k11k , so that n is subdivided into g2 21 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) n
(a1 , . . . , an )[]0,1[ , (n52,3, . . . ) such that
n (n)
O
ai 51 (n52,3, . . . ),i51
and
n
21 (n)
B (x , . . . ,x )n 1 n 5w
S
O
ai w(x ) ; xiD
i[I, i51, . . . , n, n52,3, . . . ;i51
as well as
g
21
Dk ,k , . . . ,k ;nc 3 g (B , B , . . . ,B )kc k3 kg 5w
S
v wc (B )kc 1O
v wg (B ) ,kgD
(26)g 53
where
k1 k2
(n) (n)
v 5c
O
ai 1O
ai ;i51 i511k1
k1 k2
21 (k )c (k )c
Bkc5B (x , x , . . . , x , xkc 1 2 k1 11k1, . . . , x )k2 5w
S
O
ai w(x )i 1O
ai w(x )iD
i51 i511k1and where the other vg, are defined as before. Moreover from the induction hypothesis
(k )c (n)
we also have: ai 5[(ai ) /vc]. Appropriate substitution in Bkc gives
k1 (n) k2 (n)
ai ai
21
] ]
v wc (B )kc 5v wwc
S
O
v w(x )i 1O
v w(x )iD
c c
i51 i511k1
k1 k2
(n) (n)
5
O
ai w(x )i 1O
ai w(x ).i i51 i511k1This we can write as
k1 (n) k2 (n)
ai ai
21 21
] ]
v ww1
S
O
v w(x )iD
1v ww2S
O
v w(x ) ,iD
1 2
i51 i511k1
so that we have:
. Substitution in Eq. (26) leads to the expression:
g
21
Dk ,k , . . . ,k ;n1 2 g (B , B , . . . , B )k1 k2 kg 5w
S
O
v wg (B ) ,kgD
g 51
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
(n) (n)
follows that a 5a . Let, for simplicity, i51 and j52, and x ±x . Then from Eq.
i j 1 2
(8) it follows that
n
21 (n) (n) (n) 21 (n) (n)
w
S
a1 w(x )1 1a2 w(x )2 1O
ai w(x )iD
5wS
a1 w(x )2 1a2 w(x )1 i53n (n)
1
O
ai w(x ) ,iD
i53hence
(n) (n)
(a1 2a2 )(w(x )1 2w(x ))2 50,
(n) (n)
which implies that a1 5a2 and the proof is complete.
Proof of Corollary 2. The proof is similar to the proof of Corollary 1.
Proof of Proposition. We have to prove that
m n
21 21
w
S
O
bjw(B (x , . . . , x ))n j 1 jnD
5wS
O
aiw(B (x , . . . , x ))m 1i miD
(27)j51 i51
Eq. (27) holds iff
m n n m
21 21
O
bjw wS S
O
aiw(x )jiDD
5O
aiw wS S
O
biw(x )jiDD
iffj51 i51 i51 j51
m n n m
O
bjS
O
aiw(x )jiD
5O
aiS
O
bjw(x ) iffjiD
j51 i51 i51 j51m n n m
O O
b aj iw(x )ji 5O O
a bi jw(x ) andji j51 i51 i51 j51Proof of Theorem 2. Let u, v, x, y[I,a 5a1,b 5b1 and apply Eq. (13) for the m by n matrix:
u v . . . v
x y . . . y
. . . . . . . .
x y . . . y
Then, taking into consideration Eqs. (11) and (12), we have
21 21 21
c (bc+w (aw(u)1(12a)w(n))1(12b)c+w (aw(x)1(12a)w( y))5
21 21 21
w (aw+c (bc(u)1(12b)c(x))1(12a)w+c (bc(n)1(12b)c( y)). (28)
21
Let c(I )5J. Then J is an open interval again. Let p,q,r,s[J and u5c ( p), n 5
21 21 21 21
c ( q), r5c (x), s5c ( y) in Eq. (28). Then, introducing the notation w+c 5h,
Eq. (28) goes over into
21 21
h(bh (ah( p)1(12a)h( q))1(12b)h (ah(r)1(12a)h(s))5ah(bp1(12
b)r)1(12a)h(bq1(12b)s). (29)
Obviously, h:J→R is strictly monotone increasing, therefore the function H is well2
defined on J3J by
21
H( p, q)5h (ah( p)1(12a)h( q)) (30) and so Eq. (29) can be written in the form
bH( p, q)1(12b)H(r, s)5H(bp1(12b)r,bq1(12b)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)5lp1(12l)q for some l[]0,1[and for all p, q[J,
consequently, by Eq. (30) h(lp1(12l) q5ah( p)1(12a) h( q) for all p, q[J. It
´
follows from (Aczel, 1966, Theorem 2, pp. 67.) thatl5a and there exist real numbers
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)2b) for all u[I and (31)
a
21 21
c ( p)5w (ap1b) 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 xsi$xsj, s[C, and because B is strictly monotone in each
1 1
(vi ) Let s, s [C and i, j[A, such that hs, s j is a countervailing pair, hence satisfying (without loss of generality) xs1j$xsi$xsj.xs1i. Then i, j[R(s), and xsi2
xsj,xs1j2xs1i. Because the generating functionw of B is strictly monotone increasing
1
w(x )si 2w(x )sj ,w(xs1j)2w(xs1i) and s and s are equally weighted subjects, it follows
1
that B(x , xsi s1i),B(x , xsj s1j), hence (i, j )[⁄ R*(hs, s j) and therefore s cannot be a dictator.
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