Mathematical Social Sciences 39 2000 81–107 www.elsevier.nl locate econbase
Explicit form of neutral social decision rules for basic rationality conditions
Lev A. Sholomov
Institute of Systems Analysis , 9 Prospect 60-Letiya Octyabrya, Moscow 117312, Russia
Received 1 October 1995; received in revised form 1 October 1997; accepted 1 July 1998
Abstract
n
The paper deals with aggregation operators F : 5 →
5 which satisfy the classical require-
1 2
ments of binariness independence, neutrality to alternatives, non-imposition, and which transform any n-tuple of individual relations of the given class 5 into a collective relation of the
1
given class 5 . We consider as 5 and 5 the classes: + of linear orders, 0 of weak orders, 6
2 1
2
of semiorders, of interval orders, 3 of partial orders, 7 of transitive relations, of acyclic relations. For all 27 possible pairs 5 ,5 [
h+,0,6,, 3,7, j such that 5 5 , we bring the
1 2
1 2
n
explicit form of operators 5 →
5 . The results are obtained on the basis of the following
1 2
approach. Using a logical form of operators, we associate to each F a so-called ordinal binary
n n
relation
r on R for any x,y [ R one is uniquely determined by signs of coordinate differences
F n
x 2 y , 1 i n. We prove that if 5 satisfies some mild conditions then F maps 0 into 5 if
i i
2 2
n
and only if r [ 5 . So the description of the operators 0
→ 5 amounts to the description of
F 2
2
the ordinal relations of 5 . The approach can be adapted to some classes 5 ± 0.
2000
2 1
Elsevier Science B.V. All rights reserved.
Keywords : Social choice; Social decision rules; Normative constraints; Rationality conditions; Aggregation
operator; Arrow theorem; Ordinal relation
1. Introduction
The theory of social choice deals with different ways of aggregating individual opinions into collective decisions. In widespread models, each individual i, 1 i n, is
associated with a binary relation r describing his preferences between alternatives. A
i
social decision rule SDR determines a way to aggregate the collection of individual
Tel.: 1095-3725831; fax: 1095-9382209. E-mail address
: sholomovcs.isa.ac.ru L.A. Sholomov 0165-4896 00 – see front matter
2000 Elsevier Science B.V. All rights reserved.
P I I : S 0 1 6 5 - 4 8 9 6 9 9 0 0 0 0 2 - 5
82 L
.A. Sholomov Mathematical Social Sciences 39 2000 81 –107
relations profile r , . . . ,r into a collective relation r which gives a basis for the
1 n
resulting choice. Systematic study of SDR’s was initiated by Arrow 1950, 1963 who formulated a
number of reasonable requirements on SDR’s and proved their inconsistency this fact is referred to as Arrow’s impossibility theorem. Usually, formal requirements on SDR are
subdivided into two classes: normative constraints and rationality conditions. The former are related to properties of SDR while the latter deal with domains and ranges of SDR.
The main normative constraint is binariness independence of irrelevant alternatives Arrow, 1950; Arrrow, 1963, the quasilocality condition Aleskerov and Vladimirov,
1986. Another example of an important normative constraint is neutrality to alter- natives. Rationality conditions are described by means of classes 5 and 5 of binary
1 2
relations. If the profile r , . . . ,r consists of relations of the class 5 then the social
1 n
1
relation r has to belong to 5 . Usually, the condition 5 5 is supposed to be valid.
2 1
2
The modern study of problems concerning the Arrow’s impossibility theorem is made in a more constructive form. Different consistent collections of both normative
constraints and rationality conditions are considered and explicit descriptions of suitable SDR’s Aleskerov and Vladimirov, 1986; Brown, 1975; Danilov, 1983; Fishburn, 1975;
Levchenkov, 1990; Morkjalunas, 1985; Vladimirov, 1987 or their complete characteri- zations Aizerman and Aleskerov, 1983; Blau and Brown, 1989; Ferejohn and Fishburn,
1979 are found. The present paper is a survey of results obtained by the author in the field of explicit descriptions of SDR’s. It is written on the basis of Sholmov 1990a,
1990b, 1994, 1996, 1998a, 1998b. In most cases, proofs are omitted, but sometimes, if it is possible and useful, proofs or their sketches are given.
To explain our approach and results, it is more convenient to use the term ‘aggregation
operator’ instead
of ‘SDR’. We
consider aggregation
operators
n
F : 5 →
5 which satisfy the classical normative constraints of binariness, neutrality
1 2
to alternatives, non-imposition see, for instance, Aleskerov and Vladimirov, 1986, and which transform any profile r , . . . ,r of relations of the class 5 to a collective
1 n
1
relation of 5 . Any operator F, provided mentioned normative constraints, can be
2 n
n
associated with a function in P , the set of all maps
h 2 1,0, 1 1j →
h0,1j. In turn, the
3,2 n
set P corresponds bijectively with a set of binary relations defined on the n-
3,2 n
n
dimensional real space R , the set of the so-called ordinal relations a relation r on R is
n
ordinal if for all x,y,z,v in R the conditions x p y
⇔ z p v , p [
h . , , , 5 j, 1 i n,
i i
i i
i i
i
imply x ry
⇔
z rv. So with every operator F we can associate an ordinal relation r . The
F
central result motivating our approach is the following: if 5 is the set of weak orders
1 n
and if 5 satisfies some mild conditions, F maps 5 into 5 if and only if r [ 5 .
2 1
2 F
2 n
So in this case, the description of operators 5 →
5 amounts to the description of all
1 2
ordinal relations belonging to 5 . Some particular case of it has been used by Fishburn
2
1975. The approach is also applicable in a modified form to classes 5 differing from the
1
set of weak orders. We consider as 5 and 5 the following classes: + of linear orders,
1 2
0 of weak orders, 6 of semiorders, of interval orders, 3 of partial orders, 7 of transitive relations, of acyclic relations. For all 27 possible rationality conditions
5 ,5 , 5 ,5 [ h+,0,6,,3,7, j such that 5 5 , we found the explicit form
1 2
1 2
1 2
n
of operators 5 →
5 . It has provided differing proofs for a number of known results
1 2
L .A. Sholomov Mathematical Social Sciences 39 2000 81 –107
83
detailed references will be given and has brought some new outcomes. Especially, we
n n
mention the cases of mapping 6 →
and 7 →
7 which led to some kinds of operators not encountered in the scientific literature before. This approach also helps to
construct the SDR for a number of specific choice problems for example, to take into account a priority of individuals; see Section 13.
The revealed correspondence between aggregation operators and ordinal relations allows us to find connections between results on aggregation and known facts of the
theory of relations and to explain the observed analogies. The proposed approach simplifies the study of SDR by reduction of fairly complicated objects operators on a set
of relations to the simpler objects ordinal relations. This allows us to use the geometric intuition in SDR search.
By the way, we worked out effective tools for operating with ordinal relations. The means are of great interest for the theory of multicriteria choice which widely uses
ordinal relations Berezovsky et al., 1989; Makarov et al., 1987. All proofs are based on the special developed techniques for two-valued functions in
three-valued variables. Other logic methods were used in Aleskerov and Vladimirov, 1986; Levchenkov, 1990; Murakami, 1968; Vladimirov, 1987 for aggregation problems,
in Berezovsky et al., 1989; Makarov et al., 1987 for ordinal relations analysis, and in Sholomov, 1989 for different problems relating to discrete models of choice.
2. Classes of relations
