Directory UMM :Data Elmu:jurnal:M:Mathematical Social Sciences:Vol39.Issue1.Jan2000:

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 :(51) →52 which satisfy the classical require-ments of binariness (independence), neutrality to alternatives, non-imposition, and which transform any n-tuple of individual relations of the given class51into a collective relation of the given class5₂. We consider as5₁and5₂the classes:+of linear orders, 0of weak orders,6 of semiorders, (of interval orders, 3 of partial orders, 7 of transitive relations,! of acyclic relations. For all 27 possible pairs5₁,5₂[h+,0,6,(,3,7,!jsuch that5₁#5₂, we bring the

n

explicit form of operators (51) →52. The results are obtained on the basis of the following approach. Using a logical form of operators, we associate to each F a so-called ordinal binary

n n

relationrF onR (for any x,y[R one is uniquely determined by signs of coordinate differences n

x_i2y , 1_i #i#n). We prove that if5₂satisfies some mild conditions then F maps0 into5₂if n

and only ifrF[52. So the description of the operators0 →52amounts to the description of the ordinal relations of 5₂. The approach can be adapted to some classes 5₁±0.  2000 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. Ai

social decision rule (SDR) determines a way to aggregate the collection of individual

*Tel.: 1095-3725831; fax:1095-9382209.

E-mail address: sholomov@cs.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

relations (profile) (r , . . . ,r ) into a collective relation r which gives a basis for the1 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 51 and52 of binary relations. If the profile (r , . . . ,r ) consists of relations of the class1 n 51 then the social relation r has to belong to52. Usually, the condition51#52is supposed to be valid. 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 :(51) →52 which satisfy the classical normative constraints of binariness, neutrality to alternatives, non-imposition (see, for instance, Aleskerov and Vladimirov, 1986), and which transform any profile (r , . . . ,r ) of relations of the class1 n 51 to a collective relation of 52. Any operator F, provided mentioned normative constraints, can be

(n) n

associated with a function in P3,2, the set of all mapsh21,0,11j →h0,1j. In turn, the

(n)

set P3,2 corresponds bijectively with a set of binary relations defined on the

n-n n

dimensional real spaceR , the set of the so-called ordinal relations (a relationronR is

n

ordinal if for all x,y,z,v inR the conditions x p y⇔z pv, p [h.,,,5j, 1#i#n,

i i i i i i i

imply xry⇔zrv). So with every operator F we can associate an ordinal relationrF. The central result motivating our approach is the following: if 51 is the set of weak orders

n

and if52 satisfies some mild conditions, F maps (51) into 52 if and only ifrF[52.

n

So in this case, the description of operators (51) →52amounts to the description of all ordinal relations belonging to52. Some particular case of it has been used by Fishburn (1975).

The approach is also applicable (in a modified form) to classes51differing from the set of weak orders. We consider as51and52the following classes:+ of linear orders,

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 (51,52), 51,52[h+,0,6,(,3,7,!jsuch that 51#52, we found the explicit form

n

(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

2

Let r be a binary relation on a given set A (r#A ). For x, y[A, the notations xry ¯

and xry mean (x, y)[r and (x, y)[⁄ r, respectively. The relation r is referred to as

(19) reflexive if xrx holds,

¯

(29) irreflexive if xrx holds,

¯

(39) asymmetric if xry⇒yrx,

¯

(49) antisymmetric if x±_y∧_xry⇒_yrx,

(59) complete if xry∨yrx holds,

(69) connected if x±_y⇒_xry∨_yrx,

(79) transitive if xry∧yrz⇒xrz,

¯ ¯ ¯

(89) negative transitive if xry∧yrz⇒xrz.

]_{21 2 21 21}

*

The relation r 5r 5A \r is called dual to r (r is the relation inverse to r:

21

(79)–(89) are pairwise dual i.e., for example, if r satisfies the condition of asymmetry

*

then r is a complete relation and vice versa.

A transitive, irreflexive relation is called a (strict) partial order (PO); by a linear

order (LO) we mean a connected PO; a negative transitive, asymmetric relation is called

a weak order (WO). A PO r provided that

xry∧zrv⇒xrv∨zry (1)

is referred to as an interval order (IO); an IO r restricted by the condition

xry∧yrz⇒xrv∨vrz (2)

is called a semiorder (SO). A relation r which has no cycles x rx1 2∧x rx2 3∧? ? ?∧

xs21rxs∧x rx , ss 1 $1, is said to be acyclic.

We denote by3,+,0,(,6,!and7the sets of all PO’s, LO’s, WO’s, IO’s, SO’s, acyclic relations (AR’s) and transitive relations (TR’s), respectively, on arbitrary sets A. In the case of relations on the given set A we will use the notations 3(A),+(A), . . . . The following strict inclusions

+,0,6,(,3,7! (3)

are valid (the classes 7 and ! are incomparable and both of them contain 3). All principal types of relations used in the choice theory are present in (3).

Usually, classes of relations are described by means of axioms. One can consider any binary relation as the binary predicate r(x, y)5xry on A. In the majority of the cases,

axioms which describe meaningful properties of relations can be written in the form

;x . . .1 ;x P(x , . . . , x ),s 1 s (4)

for some integer s. Here the formula P includes, besides logic operations, the single predicate symbol r. According to terminology of Malcev (1973), we will call a class of relations universally axiomatizable if it can be described by means of axioms (4). A set of axioms is not supposed to be finite. For example, the class! of acyclic relations is defined by the following countable set of axioms

]

;x . . .1 ;x (x rxy 1 2nx rx2 3n. . .nxy21rxy⇒x rx ),y 1 y51, 2, . . .

It is essential that P does not contain equality signs, i.e. (4) is a formula of the Pure predicate calculus (Malcev, 1973). Let us indicate a property of universally axiomatiz-able classes which will be needed later on. It will clarify partly why equalities are inadmissible.

2

We say that a relation r9 is formed by a fission of the given relation r#A at the

2

element a[A if r9#(A9) where A9 5A<ha9j for some a9[⁄ A, and

r9 5r<h(a9, y)uaryj<h(z,a9)uzraj<ra9 (5)

¯

where ra95h(a9,a9)j if ara, and ra955 if ara. The fission operation is inverse to a identification: the relation r can be formed from r9by identifying a and a9. If a relation is obtained from r by finite number of successive fissions (at some elements), we will

say that it is formed by fission of r (any WO on a finite set can be formed by fission of some LO).

A class relations5 will be called closed with respect to fission if for each r[5 it contains all relations formed by fission of r. Any universally axiomatizable class 5 is closed with respect to fission. To make sure of that, consider a formula (4), used in the definition of5. For any given x , . . . , x1 s[A9, examine the value of P on the relation r9

(5), formed by fission of r[5. According to (5), replacements in P of all r9and a9by r and a do not change its value. The new value is true as it relates to r[5. Hence, the original value of P on r9is also true. It gives r9[5.

All properties of relations introduced above except for (49) and (69) which use inequalities (negations of equalities) can be described in terms of axioms (4), and all classes in (3) except for+are universally axiomatizable. Note, the class+is nonclosed with respect to fission.

3. Aggregation operators

One of the principal goals of the theory of social choice is the study of operators

r5F(r , . . . , r ) which aggregate profiles (r , . . . , r ) of individual relations into the1 n 1 n

corresponding collective relations r. Relations r , . . . , r , r are supposed to be defined1 n

on the same finite set A elements of which are called alternatives.

Let us formulate some normative requirements on operators F(r , . . . , r ) (for1 n

example, see Aleskerov and Vladimirov, 1986). In the formulation below we suppose

9 9

that x, y, x9, y9[A are arbitrary alternatives, (r , . . . , r ) and (r , . . . , r ) are arbitrary1 n 1 n

9 9

profiles, r5F(r , . . . , r ), r1 n 9 5F(r , . . . , r ).1 n

(B) The condition of binariness

9 9

(;i, 1#i#n)(xr yi ⇔xr yi ∧yr xi ⇔yr x)i ⇒(xry⇔xr9y).

(nI) The condition of non-imposition

¯

9 9

;x;y'(r , . . . , r )1 n '(r , . . . , r )(xry1 n ∧xr9y).

(N) The condition of neutrality to alternatives

(;i, 1#i#n)(xr yi ⇔x9r yi 9∧yr xi ⇔y9r xi 9)⇒(xry⇔x9ry9). Sometimes we will assume that the following requirement holds.

(M) The monotonicity condition

9 9

(;i, 1#i#n)(xr yi ⇒xr yi ∧yr xi ⇒yr x)i ⇒(xry⇒xr9y).

The binariness condition isolates such procedures for which the preference of an alternative x over y in the social relation depends only on how these alternatives are

related in individual relations and does not depend on some other alternatives. The condition of non-imposition guarantees for any pair (x, y) of alternatives the possibility to be or not to be included in the social relation. The neutrality condition means equal rights for all alternatives, the monotonicity condition signifies for any alternative that an improvement of its role in individual relations leads to the same effect in the social relation.

2

As every relation r is a subset of A , it is possible to consider set-theoretical (s.-t.)

operations over relations. Let us recall some notions concerning this topic. By a s.-t.

2 B k B

operation on a set B (in our case, B5A ) we mean a mappingF: (2 ) →2 (for some

k) which satisfy the condition

(x[Xi⇔y[Y , 1i #i#k)⇒(x[F(X , . . . , X )1 k ⇔y[F(Y , . . . , Y ))1 k (6) for any x, y[B, X , Yi i#B, 1#i#k. We say that the operationF is trivial ifF;5or

F;B, and F is monotone if

(Xi$Y , 1i #i#k)⇒(F(X , . . . , X )1 k $F(Y , . . . , Y )).1 k

With k-ary s.-t. operation F, it can be associated one-to-one a Boolean function

k

w(x1, . . . , xk) (i.e a mapw:h0,1j →h0,1j). To find the value ofw on k-tuple (x1, . . . ,

xk), consider any x[B, Xi#B, 1#i#k, such that x[X is true or false depending oni

x 5i 1 or x 5i 0, and put

1 if x[F(X , . . . , X ),1 k

w(x₁, . . . ,x_k)5

H

0 if x[⁄ F(X , . . . , X ).1 k

The definition is correct by virtue of (6). The correlation between F and w can be written as

w(x[X , . . . , x1 [X )k 5(x[F(X , . . . , X ))1 k

where the values 1 and 0 are used instead of TRUE and FALSE.

It is known that any s.-t. operation F can be presented as a superposition of the ¯

operations >, < and (complement). The corresponding function w can be formed by means of the replacement inF of the operations by the relevant Boolean operations of

¯ _¯

conjunction, disjunction and negation (for example, ifF 5X₁>(X₂<X ) then₃ w 5 x₁∧

(x2∨x3)). If F is monotone then w is monotone also, i.e. w satisfied the condition (x $hi i, 1#i#k)⇒w(x1, . . . ,xk)$w(h1, . . . ,hk). Such a function w, if it nontrivial (i.e. w;⁄ 0, w;⁄ 1), can be presented as a superposition of the operations ∧ and ∨ (without negations), and the relevant s.-t. operationsF can be presented via > and <. Consider the social relation r for given r , . . . , r . The binariness condition allows us1 n

to write

21 21

xry5wxy(xr y, . . . , xr y, xr1 n 1 y, . . . , xrn y) (7) wherewxy is a Boolean function. The neutrality means thatw 5 wxy is independent on x,

y, the non-imposition implies nontriviality ofw. Henceforth, it will be more convenient

]

21 21 *

to use not ri , but (ri )5r . After some modification ofi w, we can write (7) in following manner:

* *

xF(r , . . . , r )y1 n 5w(xr y, . . . , xr y, xr y, . . . , xr y).1 n 1 n (8)

The equality can be rewritten as

* *

F(r , . . . , r )1 n 5w(r , . . . , r , r , . . . , r ).1 n 1 n (9)

*

Here the symbol r (and r ) is understood as a Boolean variable the value of which on ai i *

pair (x, y) is 1 and 0 depending on truth of xr y (of xr y). The equivalent writing of (9)i i

in terms of s.-t. operations is

* *

F(r , . . . , r )1 n 5F(r , . . . , r , r , . . . , r )1 n 1 n (10)

2 *

whereF corresponds tow. Here r , ri i are treated as subsets of A . Note thatF in (10) andw in (8), (9) are nontrivial. Depending on what is more convenient, we will consider

2 *

r and ri i as subsets of A (and use (10)) or as Boolean variables (and use (9)) or as two-place predicates (and use (8)).

According to what has been said, operators satisfying the conditions B, nI and N have representations in the form (10) (also in forms (8), (9)). It is not difficult to see that the inverse also holds. The set of all operators representable in the form (10) will be denoted by F. We will work only with operators of F, i.e. with operators which satisfy the requirements of binariness, neutrality to alternatives and non-imposition.

The additional condition M is equivalent to the monotonicity of bothF in (10) andw

1

in (9). The set of such monotone operators will be denoted by F .

The requirements B, nI, N, M are normative constraints on operators. Along with them, the so-called rationality conditions are imposed. For this aim, the classes51and

52 (51#52) are introduced and requirements are put to transform any profiles (r , . . . ,r ) of individual relations of1 n 51 to collective relations r of 52. Operators, satisfying the constraints B, nI, N and the rationality conditions (51(A), 52(A)) for any

n

finite A, will be called operators of the type (51) →52 (monotone operators of the

n

type (51) →52 if the condition M is satisfied, in addition). The sets of all operators

n

(all monotone operators) of the type (51) →52 for all n will be denoted byF(51,52)

1

(by F (51,52)). Note, if the class 51 consists of irreflexive only (or reflexive only) relations then operators ofF(51,52) generate irreflexive or reflexive relations in52. It is consequence of the conditions B and N.

The problem of describing aggregation operators in an explicit form provided that the given requirements are valid is referred to as the problem of operators (of SDR’s)

synthesis. The setting of the operators synthesis problem requires some more precise definitions. We say that operators F (r , . . . , r ) and F (r , . . . , r ) are equivalent on the1 1 n 2 1 n

set5if r , . . . , r1 n[5 implies F (r , . . . , r )1 1 n 5F (r , . . . , r ). For example, operators2 1 n

21

r1>r2 and r1>(r2<r1 ) are equivalent on the set A of acyclic relations but are nonequivalent on the set of arbitrary relations. The synthesis problem for operators of the classF(51,52) can be formulated in the following way. It is necessary to find (and to describe explicitly) a subset ^#F(51, 52) which, for any operator F9[F(51,

52), contains an operator F[^, equivalent to F9on51. Note, if different sets^ and

^9 with ^#^9 give a solution of the same synthesis problem then it is natural to prefer ^.

The synthesis problem for operators turns out to parallel to the problem of description of so-called ordinal relations. These relations are considered in Section 4.

4. Ordinal relations

n

LetR be the n-dimensional real space. To every pair of points x5(x , . . . , x ) and1 n n

y5( y , . . . , y ) in1 n R we associate the n-tuple

n

D(x, y)5(sgn(x12y ), . . . , sgn(x1 n2x ))n [h21, 0,11j ,

n

where sgn(z) is 11, 21, 0 if z.0, z,0, z50, respectively. A relation r on R is called ordinal if for all x, y, z, v

D(x, y)5D(z, v)⇒(xry⇔zrv).

An ordinal relation r is referred to as regular if xry∧z$x⇒zry where z$x⇔z1$

x1∧? ? ?∧zn$x .n

Ordinal relations are often used in multicriteria choice models in which alternatives are described by tuples x5(x , . . . , x ) of estimates according to n criteria. The1 n

following examples of regular relations are well known: the lexicography: xly⇔'i(xi.y , xi j5y for jj ,i ),

the Pareto relation: xpy⇔x$y∧x±_y.

˜

Let us denote by P3,2 the set of all two-valued functions g(u , . . . , u )1 n 5g(u ) in

n

three-valued variables, g:h21,0,11j →h0,1j. Any ordinal relationr can be uniquely

˜

presented by its describing function g (u )r [P3,2: g (rD(x, y))51⇔xry. And conversely,

any function g[P3,2 describes the single ordinal relation r for which gr5g. It is easy ˜ ˜

to see that r is a regular relation iff the function g is monotone (i.e. u$v implies r

˜ ˜

g (u )$g (v)). We denote by M the set of all monotone functions in P .

r r 3,2 3,2

We will use some special presentations of functions g[P3,2. Let us introduce functions p(u), p9(u)[P3,2 in a single (three-valued) variable by

1 if u51, 1 if u50 or u51,

p(u)5

H

p9(u)5

H

0 if u51 or u50, 0 if u5 21.

¯ ¯

It is obvious that p(u)∨p9(u)5p9(u), p(u)∧p9(u)5p(u), p(2u)5p9(u), p9(2u)5p(u),

¯

where ∨, ∧ and are Boolean operations.

Any function g[P3,2 can be presented in the form

g(u , . . . ,u )1 n 5w( p(u ), . . . , p(u ), p1 n 9(u ), . . . , p1 9(u )),n (11)

˜

where w is a Boolean function. To this end, associate to each n-tuple s 5(s1, . . . ,

n

˜

sn)[h21, 0, 11j the conjunction K (u )s˜ 5q (u )s1 1 ∧? ? ?∧q (u ) where q (u) issn n s

¯ ¯

p(u), p(u)∧p9(u), p9(u), if s 51, s 50, s 5 21, respectively. It is easy to see that

˜ ˜ ˜

˜ ˜ ˜

as the disjunction of K (u ) for alls˜ ssuch that g(s)51 (if g;0 putw;0). It is easy to prove for monotone g that w can be taken to be monotone. Denoting p(u )i 5p ,i

˜ ˜

9 9 9

p9(u )i 5p , pi 5( p , . . . , p ), p1 n 9 5( p , . . . , p ), we will write (11) as1 n

˜ ˜ ˜

g(u )5w(p, p9). (12)

As examples, let us present the describing functions of both the lexicography and the Pareto relation:

9 9 9 9

gl5p1∨p p1 2∨? ? ?∨p p1 2? ? ?pn21p ,n (13)

9 9 9

gp5p p1 2? ? ?p ( pn 1∨p2∨? ? ?∨p )n

(the symbols ∧ of conjunction are omitted).

¯ ¯ ¯

9 9 9

A conjunction K5q . . . q is called elementary if q1 n i[hp , p , p , p , p p , 1i i i i i i j, 1#i#n ( qi51 indicates the absence of ith factor). A disjunction of elementary conjunctions is called a disjunctive normal form (DNF). The formula (13) is an example of DNF. Any function g[P3,2can be presented by some DNF (for instance, as described above).

5. Techniques for study of ordinal relations

It is possible to use, instead of operations with ordinal relations, operations with their describing functions (Sholomov, 1990a; Sholomov, 1994).

5.1. Set-theoretical operations on ordinal relations

Ifr 5 F(r1, . . . , rk), r, r1, . . . , rk are ordinal relations,Fis a s.-t. operation, then

˜ ˜ ˜

g (u )r 5w( g (u ), . . . , g (u )) where the Boolean functionr1 rk w corresponds to F. This fact is evident.

¯

As an example, let us consider the relation l\p 5 l>p (the complement of the

2

Pareto relation with respect to the lexicography) onR . We have

]]]] ] ]

¯ 9 9 9 9 9 9 ¯ ¯ ¯9

g_l\p5g gl p5( p1∨p p )(p p ( p1 2 1 2 1∨p ))2 5( p1∨p p )(p1 2 1∨p2∨p p )1 2 5p p .1 2

Thus, x(l\p)y⇔x1.y1∧x2,y .2

5.2. Inversion of ordinal relation

21

˜ The inverse r of an ordinal relation r is described by the function g_r21(u )5

˜ ˜

g (_r 2u ), 2u5(2u , . . . ,1 2u ). It arises fromn

21

g_r21(D(x,y))51⇔xr y⇔yrx⇔g (_rD(y,x))51⇔g (_r2D(x,y))51.

ˆ

The relation r, formed from the ordinal relationr by the inversion of directions for

n 21

¯ ¯

of p(2u)5p9(u) and p9(2u)5p(u), a presentation of grˆ can be obtained from the

¯9 ¯ 9

presentation (12) of g by substitutions of p_r i and p instead of p and p , respectively,i i i

for variables corresponding to inverted axes.

2

For example, describing function of the Pareto relation inverse on R is g_p215

ˆ ¯ ¯ ¯9 ¯9

p p (p1 2 1∨p ), while the relation2 l formed from the lexicography lby inversion of the

¯

9 9

second axis is described by the function g_lˆ5p1∨p p .1 2

5.3. Dual ordinal relation

¯ ¯ ¯

*

By analogy to the notion of dual Boolean functionw (x1, . . . , xn)5w(x1, . . . ,xn),

˜ ˜ ˜ ¯ ˜

* *

we define the dual function g (u ) for g(u )_] [P3,2 setting g (u )5g(2u ). From 21

*

Sections 5.1, 5.2 and r 5r it follows

21

˜ ¯ ˜ ¯ ˜ ˜

* *

g (u )r 5gr (u )5g (r2u )5g (u ).r

˜

*

So the describing function of a dual relationr is dual to g . If g(u ) is presented in ther

˜ ˜ ˜ ˜ ˜

* *

form (12) then g (u )5w (p9, p ) (p and p9change places). It arises from

˜ ¯ ˜ ¯

* 9 9

g (u )5g(2u )5w( p (12u), . . . , p (n2u), p (12u), . . . , p (n2u)) ˜ ˜

¯ ¯9 ¯9 ¯ ¯ *

5w(p (u), . . . , p (u), p (u), . . . , p (u))1 n 1 n 5w (p9, p ).

Recall, if the functionw is given by a formula in the basish∨, ∧, ∨¯jthen a formula for

*

w can be formed by replacements of all ∨ and ∧ by ∧ and ∨, respectively.

2

For example, the describing functions of relations onR dual to p and l are

* 9 9 * 9 9

g_p5( p p ( p1 2 1∨p ))2 5p1∨p2∨p p ,1 2

* 9 * 9 9 9 9

gl 5( p1∨p p )1 2 5p ( p1 1∨p )2 5p1∨p p .1 2

5.4. Product of ordinal relations

A relationris the product of relationsr1 andr2(r 5 r r1 2) if xry⇔'z(xr1z∧zr2y)

holds for all x, y. The product operation on ordinal relations can be described in terms of

ˆ ˆ

a binary operation of composition g+g on functions g, g[P3,2. We will describe the

ˆ 9 ¯ ¯9 9¯

composition operation sequentially: for factors q , qi i[hp , p , p , p , p p , 1i i i i i i j at first,

ˆ _ˆ

for conjunctions K, K after that, and for functions g, g in the form of DNF at last. For

ˆ

factors, the operation qi+q is defined by Table 1, rows and columns of which arei

Table 1

¯ ¯ ¯

p p9 p p9 p9p 1

p p p 1 1 p 1

p9 p p9 1 1 p9 1

¯ ¯ ¯ ¯

p 1 1 p p9 p 1

¯ ¯ ¯ ¯

p9 1 1 p9 p9 p9 1

¯ ¯ ¯ ¯

p9p p p9 p p9 p9p 1

ˆ

associated with factors q and q , respectively (subscripts i in the table are omitted). Fori i

ˆ _{ˆ ˆ}

elementary conjunctions K5q , . . . , q and K1 n 5q , . . . , q we put1 n

ˆ _{ˆ ˆ}

K+K5( q1+q )1 ∧? ? ?∧( qn+q )n (14)

ˆ ˆ

¯ 9¯ 9 9 ¯ ¯ 9 9¯

(for example, if K5p p p p , K1 2 2 3 5p p p p , then K1 2 3 4 +K5(p1+p )1 ∧( p p2 2+p )2 ∧

¯ ˆ

9

( p₃+p )₃ ∧(1+p )₄ 51∧p₂∧p₃∧15p p ). For functions g_{2 3} 5K₁∨? ? ?∨K and g_u 5

ˆ ˆ

K1∨ ? ? ?∨K , presented in the form of DNF, we define the operation asv

ˆ ˆ

g+g5 k Ks+K .t (15)

1#s#u, 1#t#v

ˆ

Formally, the result of the operation g+g depends on the used DNF’s. But it can be ˆ

shown that different DNF of functions g and g lead to different DNF of the same

ˆ

function g+g, i.e. the dependence is fictitious. The following assertion explains the role of the operation. The describing function gr r_{1 2} of the product r r1 2 can be found as gr₁+g .r₂

2 _ˆ

For example, let us find the product of the Pareto relation on R and the relation l

9 9

from 5.2. Taking into account that DNF of g is p pp 1 2∨p p , we have1 2

¯ ¯

9 9 9 9 9 9 9 9

g_{p l}ˆ5g_p+g_lˆ5( p p1 2∨p p )1 2 +( p1∨p p )1 2 5( p p )1 2 +p1∨( p p )1 2 +( p p )1 2

¯

9 9 9 9 9 9

∨( p p )1 2 +p1∨( p p )1 2 +( p p )1 2 5p1∨p1∨p1∨p15p .1

Thus, xg y_{p l}ˆ ⇔x1$y .1

¯ ¯ ¯

Now, let us give a sketch of the proof. Associate with the factors p, p9, p, p9, p9p, 1

the symbols ., $, #, ,, 5, [, respectively. Create Table 2 by replacement in Table 1 of each q[hp, p9, . . . , 1jby the correspondingn[h., $, . . . ,[j. It easy to verify directly the following fact. If for some quantities x, y, z we take in Table 2 some

ˆ ˆ ˆ ˚

rown and columnnsuch that xnz and zny, then the cell isolated byn andncontainsn ˚

such that xny, where[ signifies that correlation between quantities can be arbitrary. In addition, any correlation between x and y, presented in the cell, is attainable for suitably chosen z.

Let r1 and r2 be ordinal relations, x and y be such that gr r1 2(D(x,y))51 (i.e.

x(r r1 2)y). Then xr1z and zr2y hold for some z, from where g (r1D(x,z))51 and

ˆ _{ˆ ˆ}

g (r2D(z,y))51. There are conjunctions K5q , . . . , q and K1 n 5q , . . . , q of DNF’s g1 n r1

ˆ

and g_r such that K(D(x,z))51 and K(D(z,y))51. It is not difficult to understand that

2

ˆ ˆ ˆ

xi i inz and zi iny take place withi niandnicorresponded to q and q , 1i i #i#n. Hence, the ˆ

˚ ˚

correlations xi iny , 1i #i#n, hold where ni is corresponded to qi+q . It leads toi

Table 2

. $ # , 5 [

. . . [ [ . [

$ . $ [ [ $ [

# [ [ # , # [

, [ [ , , , [

5 . $ # , 5 [

ˆ ˜

(K+K )(D(x,y))51 and gives ( g_r +g )(_r D(x,y))51. As a result, we obtain g_{r r} (u )5

1 2 1 2

˜ ˜ ˜

1⇒( gr₁+g )(u )r₂ 51. The inverse proposition ( gr₁+g )(u )r₂ 51⇒gr r_{1 2}(u )51 can be derived in a somewhat similar way.

The composition operation has a number of good properties. It is symmetric, associative, monotone ( g9 $g⇒g9+g0 $g+g0).

5.5. Transitive closure of ordinal relations

Because of associativity of the composition operation, the many-placed operation

s

g1+? ? ?+g can be introduced. Let g be a s-tuple composition gs +g+? ? ?+g. It is not

2 s

˜

difficult to prove that g#g # ? ? ? #g # . . . (the notation g9 #g0 means g9(u )#

n s

˜ ˜

g0(u ) for all u[h21,0,11j ). Since the sequence hg j is increasing and bounded

s11 s s9

above, it will be g 5g beginning with some s9. Denote by [ g] the result g of the stabilization. The following fact is fairly evident. If [r] is the transitive closure of an

3

ordinal relationrthen g[r]5[ g ]. As an example, consider onr R the relation r given

2 3

9 9

by the function gr5p p1 2∨p p . One can verify directly that g1 3 r5p1∨p p1 35g .r

9

Hence, g[r]5p1∨p p i.e. [1 3 r] is the lexicography on 1st and 3rd components.

˜ _{¯9 ¯9 ˜ ˜} ˜

Put 05(0, . . . , 0) and D 5p ∨p ∨? ? ?∨p ∨p (note, D (u )51⇔u±_{0 ). The}

0 1 1 n n 0

next assertion characterizes properties of ordinal relations r in terms of functions g ._r

An ordinal relationr is (a) reflexive, (b) irreflexive, (c) asymmetric, (d)

antisymmet-˜ ric, (e) complete, ( f ) connected, ( g) transitive, (h) negative transitive iff (a) g (0 )_r 51,

˜ _{* * * *}

(b) g (0 )_r 50, (c) g_r#g , (d ) D g_r 0 r#D g , (e) g0 r r$g , ( f ) D gr 0 r$D g , ( g)0 r

* * *

g_r+g_r#g , (h) g_{r r} +g_r #g ._r

˜

Item (a) follows from xrx⇔g (_rD(x,x))51⇔g (0 )_r 51; (b) is similar to (a). Note, (a) and (b) imply each ordinal relation to be reflexive or irreflexive. Let us prove (c). Ifris asymmetric then

¯ *

g (rD(x,y))51⇒g (rD(y,x))50⇒g (r2D(x,y))51⇒g (r D(x,y))51.

* *

As x, y are arbitrary, gr $g holds. Conversely, gr r $g impliesr

¯

*

g (rD(x,y))51⇒g (r D(x,y))51⇒g (r2D(x,y))51⇒g (rD(y,x))50.

It signifies that r is asymmetric. Item (e) is dual to (c). As a distinction between

˜ ˜

asymmetry and antisymmetry appears at u50 only, the statements (c) and (e) imply (d)

2

and (f). Taking into account the equivalence of transitivity ofrtor #r, we obtain (g). Item (h) is dual to (g).

The described techniques allow us to solve a number of principal problems in the design and the analysis of multicriteria choice models (Sholomov, 1990a; Sholomov, 1990b; Sholomov, 1994; Sholomov, 1996). Earlier, similar problems were considered in Berezovsky et al. (1989) and Makarov et al. (1987) where some qualitative results were obtained but no efficient methods were worked out. Also, tools of this section make it possible to find the explicit description of ordinal relations for all classes in (3).

6. Characterization of ordinal relations

The ordinal relation l9, described by the function

9 9 9

gl95pi₁∨p pi₁ i₂∨? ? ?∨pi₁? ? ?pi_k21p ,i_k (16)

0#k#n, i ±_{? ? ?}±_{i , will be called a partial lexicography (PL ). In the case k50,}

1 k

this relation is empty, by definition. The lexicography l, also called the total

lexicography (TL ), corresponds to the case k5n. It is possible to check directly that the

*

describing function of the dual relation (l9) can be formed by the replacement of the

9 9 9

last monomial in (16) by pi₁? ? ?pik21p (the case of kik 5n52 see in Section 5.3). A relation similar to the lexicographyl(to the PLl9) is called a generalized lexicography

ˆ ˆ

– GL (a generalized PL – GPL) and is denoted byl (byl 9).

Now, we present the explicit description of ordinal relations for all classes (3) (see Sholomov, 1990b, 1994). Recall, we use the abbreviation LO, WO, PO, IO, SO, TR and AR for linear, weak, partial, interval orders, semiorders, transitive and acyclic relations, respectively.

ˆ

a) An ordinal relationr is a linear order iffrcoincides with a GLl. Any non-strict

ˆ ˆ _*

(reflexive) LO r is dual to a GL l, i.e. r 5(l) .

ˆ

b) An ordinal relationris a weak order iffris a GPLl 9. For regular relations, this observation is made in Fishburn (1975).

c) In the case of ordinal relations, the classes0, 6 and( are identical. Note that

this is false for non-ordinal relations (Fishburn, 1970).

d ) An ordinal relation r is a partial order iff r is an intersection of GL’s, i.e.

ˆ

r 5>1#i#kli( gr5∧1#i#k glˆiin terms of describing functions). Therefore, one has an analog of Dushnik–Miller theorem (Ore, 1962) about presentability of any PO as an intersection of LO’s.

J

Let J5hj , . . . , j1 kj be a subset of the set h1, . . . , nj, R be the corresponding

n J

subspace of R . An ordinal relation r is said to be a generalized lexicography on R

J n

(GL onR ) ifr is a GPL on R and the set of all indices of essential variables for gr

coincides with J (we say that a variable u is unessential for the function g if anyi

variation of u , at fixed values of other variables, does not change value of g). We will_Ji J

ˆ

denote by l a GL on R .

e) A reflexive ordinal relationris transitive iffr can be presented as an intersection of relations which are dual to some GL’s on a common set J, i.e. r 5>

J

ˆ _{* *}

(l ) ( g 5 ∧ g )ˆJ in terms of describing functions). Note that any

1#i#k i r 1#i#n li

irreflexive TR is a PO, and this case is treated in (d ).

J9 J0

Relationsr9onR andr0onR will be called equivalent if g_{r 9}5g_{r 0}up to unessential variables. Along with strict PO’s, nonstrict (reflexive) ones will be considered. Taking into account (d ), (e) and (a), we obtain that any ordinal TR is equivalent to some PO (strict or nonstrict). Recall, any ordinal relation is reflexive or irreflexive.

f ) An ordinal relation r is acyclic iff r can be completed to a GL. In terms of ˆ

describing functions the fact is formulated as g_r5( g )_lˆ ∧g, where g[P3,2is arbitrary,l

analog of the statement, equivalent to Szpilrajn’s lemma (Ore, 1962; Szpirlajn, 1930), about the extentendability of any AR to an LO.

In addition, in Sholomov (1990b) a general form of describing functions for t-acyclic (having no cycles of the length s, s#t) ordinal relations was obtained. In particular case t52, it gives that the relation r is asymmetric iff the function g is expressible in the_r

*

form g_r5g∧g , g[P3,2. By duality of completeness and asymmetry, the relationr is

*

complete iff g can be present as g_{r r}5g∨g , g[P3,2.

Remark. To adapt these assertions to the case of regular relations r, one needs to replace

• – all generalized lexicographies (total, partial, on J ) by suitable lexicographies (total, partial, on J )

• – arbitrary functions g[P3,2 by monotone ones.

This works for all assertions except for (d). In (d) PL’s are to be used instead of GL’s. It is possible to limit oneself to total lexicographies iff r satisfies the inclusion r$p

wherep is the Pareto relation. To clarify the fact, notice that any TL contains the Pareto relation.

ˆ

Let us prove the statement (f), for instance. One can verify directly that a GL l

ˆ

satisfies the items (b), (g) of the assertion in Section 5. Hence,l is a PO. It is acyclic, and any smaller relation r is also acyclic. Conversely, let r be an ordinal AR. Its

˜

transitive closure [r] is irreflexive and, according to (b) of Section 5, [ g ] (0 )_r 50 holds. Let g_r5K1∨ . . .∨K be a DNF, Ks 5q . . . q1 n be the composition K1+? ? ?+K .s

s _˜

Because of [ g ]_r $g , K_r #[ g ] holds, and we have K(0 )_r 50. It means that there exists

¯9 ¯9

some index i such that qi5p or qi i5p . Suppose that ii 51 and q15p (the case of p1 1

is reduced to it by inverting the axis). It follows from Table 1 that the first factor of each

¯

9 9

K , 1j #j#s, is p or p or p p . By grouping and factoring out, we obtain1 1 1 1

˜ ˜ 9 ˜ 9¯ ˜

g (u )r 5p g (u1 1 9)∨p g (u1 2 9)∨p p g (u1 1 3 9) (17)

˜

for some g , g , g , where u1 2 3 9 5(u , . . . , u ). The substitution u2 n 150 in (17) gives g (0,r

n21

˜ ˜ ˜ ˜

u9)5g (u2 9)∨g (u3 9). Introducing the ordinal relation r9 on R by g (ur 9 9)5g (0,r ˜u9), we deduce from (17)

˜ 9 ˜ ˜ 9 ˜ 9 ˜

g (u )r #p1∨p ( g (u1 2 9)∨g (u3 9))5p1∨p g (0,u1 r 9)5p1∨p g (u1 r 9 9). (18)

9 9 9

The relation r9 is acyclic since any cycle x1r9. . .r9xmr9x1 in r9 gives the cycle

9

x1r. . .rxmrx1 in r, where xi5(0, x ), 1i #i#m. Having applied to r9 the same

˜ ˜

arguments, we get a similar inequality g (ur 9 9)#p2∨p9g2r 0(u0), which combined with (18) leads to

˜ 9 9 ˜ 9 9 9 ˜

g (u )r #p1∨p ( p1 2∨p g (u2 r 0 0))5p1∨p p1 2∨p p g (u1 2 r 0 0),

9 9 9

side can be supplemented to TL. Since the consideration was led up to reindexing and inverting of axes, we have g_r#g , in the general case._lˆ

The assertion (a) is a simple corollary of (f). According to (f), a LOrcan be extended

ˆ

to some GLl. As a connected relationrcannot be increased without loss of asymmetry,

ˆ

r coincides withl. Conversely, any GL meets the conditions (b), (c), (f) of Section 5, and therefore arbitrary GL is a LO.

n

The results of this section show that the structure of ordinal relations overR is fairly simple. Namely, all LO’s are reduced to lexicographies up to directions of coordinate

n

axes; WO’s are reduced to lexicographies on subspaces of R ; proper SO’s and proper IO’s are missing; each PO is an intersection of lexicographies; TR’s are equivalent to

n

PO’s strict or not on subspaces of R ; each AR is a subset of some lexicography.

7. Connection between aggregation operators and ordinal relations

All classes of relations in (3) except for 7 consist of asymmetric relations only. We

will suppose (up to Section 11 devoted to7) individual relations to be asymmetric. The

21

*

condition of asymmetry ri>ri 55 _{is equivalent to the inclusion r}i#r , i.e. to thei *

inequality xr yi $ xr y.i

* *

Let F(r , . . . , r )1 n 5F(r , . . . , r , r , . . . , r ) be an operator of1 n 1 n F. Associate with F

ˆ ˆ

the function gF[P3,2 as below. To find the value g for some (u , . . . , u ), considerF 1 n

any x, y[A and any profile of asymmetric relations (r , . . . , r ) such that1 n

ˆ

yr xi if ui5 21,

¯ ¯ ˆ

xr yi ∧yr xi if ui50,

6

, (19)

ˆ xr yi if ui51, and put

1 if (x, y)[F(r , . . . , r ),1 n

ˆ ˆ

g (u , . . . , u )F 1 n 5

H

_{0 if (x, y)[⁄ F(r , . . . , r ).} (20)

1 n

It is easy to see that the definition is correct i.e. independent of chosen x, y and (r , . . . ,1

r ) provided (19). The function g was used in Aleskerov and Vladimirov (1986).n F

It is not difficult to understand that the mapF→P is injective (F±_F9⇒_g ±_{g ).}

3,2 F F9

Let us prove that it is also surjective (and hence bijective). Consider any g[P3,2,

9 9

g(u , . . . , u )1 n 5w( p , . . . , p , p , . . . , p ).1 n 1 n (21) Let F be the operator (8) with the functionw from (21). Let us verify that gF5g. The

ˆ ˆ *

equalities (20) and (21), with regard to the correlations p(u )i 5xr y and pi 9(u )i 5xr y,i

equivalent to (19), give

ˆ ˆ * *

g (u , . . . , u )F 1 n 51⇔(x, y)[F(r , . . . , r )1 n ⇔w(xr y, . . . , xr y, xr y, . . . , xr y)1 n 1 n 51

ˆ ˆ ˆ ˆ ˆ ˆ

⇔w( p(u ), . . . , p(u ), p1 n 9(u ), . . . , p1 9(u ))n 51⇔g(u , . . . , u )1 n 51. The correspondence between F and g in the explicit form isF

* * 9 9

F5F(r , . . . , r , r , . . . , r )1 n 1 n ⇔gF5w( p , . . . , p , p , . . . , p ).1 n 1 n (22) Different presentations (21) of the same function g led to different ones of the sameF

operator F. Therefore, equivalent transformations in the classF are similar to them in

P3,2.

Associate with each operator F the ordinal relation rF by gr_F5gF (the corre-spondence is one-to-one). The following statement (Sholomov, 1990b; Sholomov, 1994)

n

helps to find the explicit form of operators0 →5 for different 5. A particular case

n

of the approach (for operators 0 →0) was used in Fishburn (1975) (and in Morkjalunas, 1985).

Theorem. If 5 is an universally axiomatizable class then an operator F[F has the

n

type0 →5 iffrF[5.

A proof of the fact uses the following lemma where by A we mean a finite set.

Lemma. If F is an operator in the form (8) then

n

(a) (;x . . .1 ;xs[_R )('A)('r . . .1 'rn[0(A))('x . . .1 'xs[A) x rxi j⇔xirFx , 1j #i, j#s, r5F(r , . . . , r ),1 n

n

(b) (;A)(;r . . .1 ;rn[0(A))(;x . . .1 ;xs[A)('x . . .1 'xs[_R )

x rxi j⇔xirFx , 1j #i, j#s, r5F(r , . . . ,r ).1 n

(i ) (i )

Proof of Lemma. (a) Let xi5(x , . . . , x ), 11 n #i#s. Associate x with an element xi i

(e.g. an integer) and put A5hx , . . . , x1 sj. Introduce the relations r (1t #t#n) on A by

(i ) ( j )

ˆ

setting x r xi t j⇔xt .xt , 1#i, j#s. Consider any i, j, and denoteD(x , x ) by (u , . . . ,i j 1

ˆ ˆ ˆ ˆ9 ˆ

u ). Introducing pn t5p(u ) and pt t5p9(u ), we havet (i ) ( j )

ˆ x r xi t j⇔xt .xt ⇔pt51,

(i ) ( j )

¯ ˆ

* 9

x r xi t j⇔x r xj t i⇔xt $xt ⇔pt 51. With regard to (8) and (22), this gives

ˆ ˆ ˆ ˆ

* * 9 9

x rxi j⇔w(x r x , . . . , x r x , x r x , . . . , x r x )i 1 j i n j i 1 j i n j 51⇔w(p , . . . , p , p , . . . , p )1 n 1 n 51

ˆ ˆ

⇔g (u , . . . , u )F 1 n 51⇔g (FD(x , x ))i j 51⇔xirFx .j

(b) Since rt[0(A) (1#t#n), the finite set A can be subdivided into levels such that

(i )

xr y holds iff x’s level is higher than y’s level (Fishburn, 1970). Denote by xt t the

(i ) (i )

9

number of the x s level in r , and put xi t i5(x , . . . , x ), 11 n #i#s. We have

(i ) ( j )

x r xi t j⇔xt .xt , i.e. x and x connect with x and x by the same way as in (a). Iti j i j

makes it possible to finish the proof similarly to (a), and to obtain x rxi j⇔xirFx , 1j #i, j#s.

n

Proof of Theorem. Let F be an operator 0 →5 for an universally axiomatizable class5. Consider some axiom (4) used in definition of5. Let x , . . . , x be any points1 s

n

inR . Let us take both relations r , . . . , r1 n[0 and elements x ,.., x1 s[A guaranteed

by (a) of the Lemma. The relation r5F(r , . . . , r ) of1 n 5 satisfies the formula P(x , . . . ,1

x ) in (4). According to (a), xs irFxj⇔x rx and, therefore, P(x , . . . , x ) is true fori j 1 s rF. Since points x , . . . , x are arbitrary,1 s rF meets the axiom (4). As this reasoning can be applied to any axiom used in the definition of5, it gives rF[5.

Conversely, let us show thatrF[5 implies r5F(r , . . . , r )1 n [5 for any r , . . . ,1 n

rn[0. Consider an axiom (4) and any x ,.., x1 s[A. Let x , . . . , x be points of1 s R

guaranteed by (b) of the Lemma. Because rF meets the formula P(x , . . . , x ), the1 s

relation r satisfied P(x , . . . , x ). To complete the proof, it is sufficient to remark about1 s

arbitrariness of both considered axiom and elements x ,.., x .1 s

n

As the result, the synthesis problem for operators 0 →5 is reduced to the description problem for ordinal relations of5 studied in Section 6.

n 8. Explicit form of operators 0 →5

The aggregation operator, associated with a lexicography l, is called a linear

hierarchy (LH) and denoted byL. By (13), its representation (9) is

* * * *

L 5r1∨r r1 2∨? ? ?∨r r1 2? ? ?rn21 nr . (23)

ˆ

Analogously, a partial LH (PLH) L9, a generalized LH (GLH) L, and a generalized

ˆ ˆ ˆ

PLH (GPLH) L 9 are defined using suitable lexicographies l9,l and l 9. The GLH is

21

formed from LH by replacement of some relations r by their inverses ri i . The same is

ˆ 9 9_¯ ˆ

true for GPLH’s. For example, if GPLl 9is given by g_{] l 9}ˆ 5p1∨p p then GPLH1 3 L 9_]is

21 _ˆ 21 21 _˜ 21 _˜ 21

˜ ˜

¯ ¯

*

r1∨r r1 3 . It can be transformed toL 9 5r1∨r r1 1 r3 5r1∨r r1 3 , where r15r r1 1

ˆ

is the incomparability relation for r . The social relation r1 5L 9(r , r , r ) is described1 2 3

˜˜

by xry⇔xr y1 ∨(xr y1 ∧yr x).3

For operator F(r , . . . , r ) we will denote by J the set of all indices j of its essential1 n F

variables r (a variable r is unessential if any variation of r preserves the value of thei i i J

operator). A PLH L9 with J_L95J is referred to as LH on J and is denoted by L. A

J

ˆ 9 9

GLHL on J is defined analogously. If gF5w( p , . . . , p , p , . . . , p ) is the function1 n 1 n

* *

corresponding to F then the dual operator F can be introduced by gF_*5g .F

Applying the theorem from Section 7 to the results of Section 6, we can obtain the

n

explicit form of operators0 →5 for all classes5 $0 from (3) (Sholomov, 1990b; Sholomov, 1994). Statements (b), (c) of Section 6 lead to the proposition

1 ˆ

a) F[F(0, 5),5[h0, 6,(j, iff F is equivalent to a GPLHL 9. If F[F (0,

5) then F is equivalent to some PLH L9.

For the class 550 and monotone operators, the result was published by Fishburn (1975). The article (Wilson, 1972) allows us to predict the fact for the case of 550

and of general type operators (strict proofs for the case are in Aleskerov and Vladimirov (1986), Levchenkov (1990) and Morkjalunas (1985). For 5 [h6, (j and monotone

operators the result could be predicted on the basis of Blair and Pollak (1979) and Blau (1979), the explicit formulation for556 and monotone operators is given by Danilov (1983).

Complete description of all mappings into the class of partial orders results from (d) of Section 5.

ˆ

b) F[F(0,3) iff F is equivalent to an intersection of GLH’s >1#i#kLi(of PLH’s

1

9

>1#i#kLi if F[F (0, 3)).

This result for monotone operators is obtained by Danilov (1983). In nonmonotone case the synthesis problem was solved by Vladimirov (1987) and Levchenkov(1990), but our solution gives a more restricted set containing for each operator of the type

n

0 →3 an equivalent operator. Note that it is impossible to strengthen this result for monotone operators by using LH’s instead of PLH’s. That can be done iff F satisfies the Pareto condition (Danilov, 1983); also see Remark in Section 6). Some analogy between

n

the form of operators0 →3 and Dushnik–Miller theorem (see Ore, 1962) was noted in Danilov (1983) (for the monotone case).

A general form of aggregation operator for the class of transitive relations follows from (d) and (e) of Section 6.

ˆ

c) F[F(0,7) iff F is equivalent to an operator in either of two forms >₁#i#kLi

J J 1 J

ˆ _{* 9 *} ˆ

or >1#i#k(Li) (of >1#i#kLi or >1#i#k(Li) if F[F (0,7)) where allLi are GLH’s on the same J.

The synthesis problem for F(0, 7) was solved by Vladimirov (1987) and Levchenkov (1990), but we indicate a smaller set containing for each F[F(0, 7) an equivalent operator.

The statement (g) of Section 6 makes it possible to indicate all operators which map collections of weak orders to acyclic relations.

ˆ

d) F[F(0,!) iff F is equivalent to an operator expressible in the formL>G (in

1 1 1 1

the form L>G if F[F (0, !)) where G[F (G [F ) is arbitrary.

In an equivalent form, this result is in Vladimirov (1987). For monotone operators, the result is contained in Blau and Deb (1977). In the case of nonmonotone operators, some weaker assertion is obtained in Kelsey (1984).

Denote by!

‡

and#, respectively, the classes of asymmetric and complete relations. From results of Section 6 it follows that operators of F(0, !

‡

) and F(0, #) are

* *

equivalent to operators, expressible in the forms F5G>G and F5G<G ,

respectively, where G is any operator (any monotone operator, in the monotone case). In

(t )

addition, in Sholomov (1990b) a general presentation for operators of F(0, ! ) is is

(t )

found, where ! is the class of t-acyclic relations.

The approach, based on the theorem from Section 6, allows us to simplify proofs of

n

some known results concerning neutral operators 0 →5. As an illustration, let us

n

derive the characterization of operators0 →! which was obtained by Ferejohn and Fishburn (1979) and somewhat simplified in Aleskerov and Vladimirov (1986). In our terms, the result has the following formulation: F(r , . . . , r )1 n [⁄ F(0,!) iff there exist

n

˜ ˜

k ands1, . . . , sk[h21, 0, 11j for which

˜ ˜

g (F s1)5 ? ? ? 5g (F sk)51, (24)

1 2

˜ ˜

1 2

˜ ˜

where v (s)5hsus 5 1s 1j, v (s)5hsus 5 2s 1j. According to the theorem of Section 6, it is sufficient to prove that (24)–(25) are necessary and sufficient conditions

˜

of a cycle in rF. Let x1rF. . .rFxkrFx1 be a cycle in rF. Putting s 5 Di (x , xi i11),

2

˜ ˜

1#i#k, xk115x , we derive g (1 F si)5g (r_FD(x , xi i11))51. If s[v (si) holds for some s and i then the point x_i11 exceeds x in the component s. Since (x_{i 1}2x )₂ 1 ? ? ? 1(xk212x )k 1(xk2x )1 50, there exists j for which x exceeds xj j11in the component

1 1 2

˜ ˜ ˜

s and, consequently, s[v (sj). Analogously, s[v (si) implies s[v (sj) for some

j. Thus, (24)–(25) obtains. The inverse statement is simple as well.

n

All assertions of Aizerman and Aleskerov (1983) relating to operators0 →5 for different5 also can be derived on the basis of our approach. But it should be mentioned that in Aizerman and Aleskerov (1983) and Ferejohn and Fishburn (1979) results were obtained under weaker assumptions (without the neutrality condition).

n 9. Explicit form of operators + →5

Using equivalent transformations, it is possible to convert any operator (10) to the form

] ]

21 21 21 21

¯ ¯

F 9(r , . . . , r , r1 n 1 , . . . , rn , r , . . . , r , r1 n 1 , . . . , rn )

21 21

¯ ¯ * *

5F 9(r , . . . , r , r1 n 1 , . . . , rn , r , . . . , r , r , . . . , r )1 n 1 n (26)

( 2 )

ˆ

whereF 9is a nontrivial, monotone s.-t. operation. For example, the GLHL obtained

( 2 )

*

from the LH L 5r1<r1 >r by inverting of r can be presented as2 1

( 2 ) 21 21 21 21

ˆ _{* ¯ ¯}

L 5r1 <(r1 ) >r25r1 <r1>r25F 9(r1 , r , r ),1 2

where the operation F 9(X, Y, Z )5X<Y>Z is monotone and nontrivial. Applied to

relations of+, the operator (26) generates only irreflexive or only reflexive relations. Let

21

us consider at first the case of irreflexive relations. Substituting ri and r instead of alli

¯ *

r and ri i in (26), we get an operator

21 21

F(r , . . . , r )1 n 5C(r , . . . , r , r1 n 1 , . . . , rn ) (27) where the s.-t. operationC is nontrivial and monotone. The operator F is equivalent on

21

¯

+ to (26) as, first, F gives irreflexive relations and, second, xr yi 5xri y and

*

xr yi 5xr y hold for xi ±_{y, r}i[+. In case (26) generates reflexive relations, we, by

21

¯

*

means of substitutions ri and r instead of all r and ri i i in (26), come to the operator

¯ ¯

* *

F(r , . . . , r )1 n 5C(r , . . . , r , r , . . . , r )1 n 1 n (28) which is also equivalent on+ to (26) (C is nontrivial and monotone). Operators (27) and (28) are called uniform. The above procedure for obtaining them is referred to as a

reduction.

( 2 ) 21

ˆ _¯

For example, consider the foregoing operator L 5r1 <r1>r . The result of its2 ( 2 )

21 21 21 _ˆ 21

* *

reduction is r₁ <r₁ >r₂5r₁ . Applied to the dual operator (L ) 5(r₁ ) >

21

¯ * * ¯ * ¯ ¯ * ¯

t5(w11 ? ? ? 1w ) / 2. After the normalization of (w , . . . , w , t), the realization isn 1 n

reduced to (w , . . . , w , 1 / 2) where w1 n 11 . . . 1wn51. To each monotone Boolean function

f5 k x . . . x ,i₁ i_s

hi , . . . ,i1 sj

put in correspondence the function

9

u 5f k n pi n pj

hi , . . . , i1 sji[hi , . . . , i1 sj j[⁄hi , . . . , i1 sj

of P3,2, and denote by Qf the aggregation operator associated withuf. Also denote by

o

7'1 / 2the set of operatorsQffor all f[Th1 / 2, and by7'1 / 2the set of operators similar

o

to operators of 7'1 / 2 (a similar operator comes out the initial one by replacements of

21 o

some r by ri i ). The sense of operatorsQf[7'1 / 2 can be explained in the following manner. Each individual i, 1#i#n, has w votes, and the alternative x is preferred overi y in the collective decision iff none of voters prefers y and at least one half of votes

o

support x. According to that, operators of the class 7'1 / 2 of 7'1 / 2 will be called (generalized) weighted majority operators with right of veto.

The following statement is valid.

(d) F[F(6,!) iff F is equivalent to an operatorQ>G whereQ[7'_{1 / 2}, G[F

o 1 1

(Q[7'1 / 2, G[F , if F[F (6, !)).

n

11. Explicit form of operators 7 →7

Our description of the classF(7, 7) (Sholomov, 1998b) uses the following special operators which have not appeared in the scientific literature before:

( 0 )

*

Tn (r , . . . , r )1 n 5

<

S

ri >

>

rj

D

, n$1 (29)

1#i#n 1#j#i

( 1 )

*

Tn (r , . . . , r )1 n 5

<

S

ri >

>

rj

D

<

>

r ,j n$1. (30)

1#i#n21 1#j#i 1#j#n

Their logic presentations (9) are

( 0 )

* * *

Tn (r , . . . , r )1 n 5r r1 1 ∨r r r1 2 2 ∨? ? ?∨r , . . . , r1 n21 n nr r ( 1 )

* * *

Tn (r , . . . , r )1 n 5r r1 1 ∨r r r1 2 2 ∨? ? ?∨r , . . . , r1 n22 nr 21 nr 21∨r , . . . , r .1 n ( 0 ) ( 1 )

* *

It can be proved directly that the operators T₁ 5r₁>r₁ and T₂ 5(r₁>r )₁ <(r₁>

r ) preserve transitivity (i.e. transform TR’s to TR’s). Since all operators (29), (30) are2 ( 0 ) ( 1 )

expressible as some superpositions of T1 and T2 , the fact can be extended to arbitrary

( 1 ) ( 0 ) Tn and Tn .

( 0 ) ( 1 )

*

The operators Tn and Tn are ‘‘transitive analogs’’ of LHLn(13) and ofLn, dual to

a 21

]

s 21 _˜˜ 21

¯

*

r>r (asymmetric), r 5r>r (symmetric), r5r<r (of incomparability). By

( 0 )

means of equivalent transformations, the operators Ln and Tn can be represented as

˜ ˜ ˜ ˜

˜ ˜ ˜ ˜

L 5n r1∨r r1 2∨? ? ?∨r r r1 2 n21 nr , ( 0 ) a s a s s s a Tn 5r1 ∨r r1 2∨? ? ?∨r r . . . r1 2 n21 nr

( 1 )

*

(the accordance betweenLn and Tn is similar). The operators (29), (30) will be called

transitive LH’s (TLH’s). Note, in contrast to TLH, LH do not preserve transitivity. An 21

operator, obtained from a TLH by substituting ri instead of r for some indices i, willi

be called a generalized TLH (GTLH). Similarly to PLH and GPLH (see Section 8), we will define a partial TLH (PTLH) and a generalized PTLH (GPTH). As the relation

21 n

inverse operation ( ) preserve transitivity, all defined operators have type7 →7. The following statement (Sholomov, 1998b) describes explicitly the operators preserving transitivity.

ˆ ˆ ˆ

(a) F[F(7, 7) iff F is equivalent to an operator >1#i#kT , where T , . . . ,T arei 1 k 1

GPTLH’s (PTLH’s, if F[F (7, 7)).

The operators of (a) are in the class F(7, 7) because the operation > preserves transitivity. The proof of the inverse proposition is considerably harder. Let us give its outline.

n

For all preceding types of operators (51) →52, the class51contained asymmetric relations only. It allowed us to describe operators F in terms of functions gF[P3,2(22). If relations of51are not asymmetric (in particular, at5157) it is necessary to use the more general logic form (9).

Let the functionw in (9) be given by a formula in the basish∨, ∧, ∨¯jwith negations applied to elementary variables only. Eliminating the negations in (9) by means of the

]

21 21

¯ * *

equivalent replacements ri5(ri ) and (r )i 5ri , one can transform the formula (9) to the form

21 21 21 21

* * * *

F(r , . . . , r )1 n 5c(r , . . . , r , r1 n 1 , . . . , rn , r , . . . , r , (r1 n 1 ) , . . . , (rn ) ), (31)

21 21

* *

wherec is a formula in the basish∨, ∧j. Call to mind, the symbols r , ri i , r , (ri i ) are considered as Boolean variables which take values 1 or 0 for each concrete (x, y).

The approach, worked out in Section 5 for the study of transitive ordinal relations and

n

based on a special operation of composition, can be adapted to operators7 →7. Let us introduce the suitable notions.

The conjunction K5q . . . q is called an elementary conjunction if each factor q ,1 n i

21 21 21

* *

1#i#n, is of one of following type: (a) r , (b) ri i , (c) r ri i , (d) r , (e) (ri i ) , (f)

21 21 21

* * * *

r (ri i ) , (g) r r , (h) ri i i (ri ) , (j) 1 ( qi51 means the absence of the suitable factor). The factors of types (a)–(j) are called elementary factors.

A disjunction K1∨? ? ?∨K of elementary conjunctions is called a disjunctive normals form (DNF). Any functionc in the form (21) can be presented by means of a DNF. Let

us define a binary operation F+G of operators composition. As in Section 5, the

operation will be described for elementary factors at first, for elementary conjunction after that, and for operators (31) in the form of DNF at last.

1

2

all factors having types (a) or (g), the class Q is formed by all factors of types (b) or

ˆ

(h), and all factors of types (d )2( f ), (j) produce the class Q. The classes do not contain

21 ₉

the factors r ri i (of type (c)) only. The result of operation qi+q on elementary factors qi i

ˆ ˆ

9 9 9

and q is determined in the following way. If qi i[Q or qi[Q then qi+qi51. Now, let

1 2

ˆ

9 9

q , qi i [⁄ Q. In this case, if q and q are contained in the same class Qi i or Q then

1 2

9 9 9

qi+qi 5qi∧q . If they belong to different classes Qi and Q then qi+qi 51. If some

21 _ˆ

9 9 9

of the factors q ,q coincides with r ri i i i (and q , qi i[⁄ Q ) then the result of qi+q isi

equal to the other of the factors.

9 9

For conjunctions K5q , . . . , q , K1 n 9 5q , . . . , q1 n and DNF’s c 5K1∨? ? ?∨K ,s

9 9

c9 5K1∨? ? ?∨K , the result of operation Kt +K9 and c+c9are defined by (14) and (15). For the operators F5c and G5c9, we set F+G5c+c9(it can be proved that the result of composition operation F+G does not depend on chosen DNF’sc andc9). Any operator F satisfying the condition F+F5F is referred to as closed (with respect

to composition).

The role of the composition operation is clarified by the following assertion:

F[F(7, 7) iff F is closed. In this way, the explicit description of all operators

n

7 →7, mentioned in the statement (a), can be found.

12. Summary table

Recall previously introduced notations:

J J

ˆ ˆ ˆ

L(L), L9(L 9), L(L ) are a linear hierarchy (generalized), a partial LH (general-ized), a LH on the set J (general(general-ized), respectively—(Section 8);

ˆ

T(T ) is a partial transitive LH (generalized)—(Section 11); ˆ

Q(Q) is a weighted majority operator (generalized) with right of veto—(Section 10);

1

G(G ) is any (monotone) operator;

a 21

r (a[h1, 21j) is r ata 51 and is r at a 5 21.

n

Now, we shall present the obtained results on explicit forms of operators (51) →52

for all classes 51, 52 of (3) provided51#52 in Table 3. Rows and columns of this table are associated with classes51 and52, respectively. If condition51#52 is false the corresponding entry (the square) contains the line.

In the case of monotone operators F, it is necessary to all entries of the table (except

Table 3

Domain Range

+ 0 6 ( 3 7 !

]

a a a a a_i a_i a_i a

+ ri ri ri ri >iri >ir ,i >is dri ri>G

J _*

ˆ ˆ ˆ ˆ ˆ ˆ ˆ

0 – L 9 L 9 L 9 >iLi >iLi,>is dLi L>G

a a a_i a_{i ˆ}

6 – – ri ri >iri >iri Q>G

a a_i a_i a

( – – – ri >iri >iri ri>G

a_i a_i a

3 – – – – >iri >iri ri>G

ˆ

7 – – – – – >iTi –

a

a_i a

for the ones mentioned later) the following replacements: ri and ri are to be replaced

J

ˆ ˆ ˆ ˆ ˆ ˆ

by r ; instead of generalized operatorsi L,Li,L 9,Li, T ,i Qone has to use the suitable

J

operatorsL,L_i,L9,L_i, T ,_i Q; an arbitrary operator G must be replaced by a monotone

1 n n

operator G . Exceptions are operators+ →7 of the second type, operators0 →3,

]

n 21

operators0 →7 of the first type. First of them is equivalent to the operator >iri ,

9

and other two are equivalent to >i Li.

13. Aggregation operators with given relation of voters power

The idea to reduce problems for aggregation operators to similar problems for ordinal

n

relations on R turns out to be useful also for design of specific operators in practical social choice problems. In this section we consider an example of such a problem.

Let N5h1, . . . , njbe the set of voters, and R be an irreflexive binary relation on N. The notation iRj is interpreted as ‘the voter i is stronger than j’, and R is called the relation of voters power. Subsets M#N will be referred to as coalitions. A coalition M1

is stronger than M , denoted by M RM , if there exists an injection2 1 2 j: M2→M such1

that j(i )Ri for all i[M . With a profile r2 5(r , . . . , r ) and two alternatives x, y1 n [A

we associate the coalition V (r)xy 5hiuxr yji . An operator F(r) is said to be compatible with the relation R of voters power if V (r)RV (r)xy yx ⇒xF(r)y for all x, y, r. We will

suppose that the operators also satisfy the normative constraints of binariness (B), neutrality to alternatives (N), non-imposition (nI), and some rationality condition

n

(51) →52.

We will consider the case of 5150, 5253, R[3. The operator, satisfying

chosen conditions and compatible with R, is nonunique. The intersection of all such operators will be denoted by F . It is easy to see that F is the least (with respect toR R

inclusion) operator which is compatible with R and satisfies the constraints B, N, nI, and

n

condition 0 →3. Moreover, FR automatically satisfies the condition M of

mono-tonicity. The problem is to find the explicit form of F .R

LetwRbe the function (9) for the operator F ,R rRbe the ordinal relation corresponding

to F . The theorem from Section 7 givesR rR[3. By analogy with the operator,

compatible with the relation R of voters power, one can define an ordinal relation, compatible with a given relation R of criteria power (Sholomov, 1996). ThenrR is the least relation which belongs to 3 and is compatible with R. The explicit form of the relation rR obtained in Sholomov (1996) gives the following explicit form of F :R

*

F (r , . . . , r )R 1 n 5

S

<

ri

D

>

>

S

ri <

<

rj

D

21 1#i#n 1#i#n j[R (i )

21

where R (i )5hj[NujRij.

To clarify the result, we will consider the special case of R[0. Let M , . . . , M be1 k

the partition of the voters set N into equivalence classes such that iRj⇔u,v where i[M , ju [M . To reformulate the preceding general result for the considered specialv

*

*

r ,i P 5M PM>SM (the Pareto operator). Then the expression of F for RR [0 can be transformed to the form

FR5PM₁<(PM₁>PM₂)<(PM₁<M₂>PM₃)< ? ? ?<(PM₁<???<M_k21>PM_k). In particular cases, if R[+ (i.e. all M are one-element) then F is the total lineari R

hierarchy LN; if R55 (i.e. all voters are contained in single class M15N ) then FR

coincides with the Pareto operator PN. In other cases, inclusions PN#FR#LN hold. Note that in Sholomov (1996) the problem was solved for the more general case of arbitrary R and 5257.

Acknowledgements

The author is greatly indebted to Prof. Fuad Aleskerov for the idea to write the paper and the kind assistance, as well to the anonymous referee for his very skilled work and suggestions on improvement of the paper quality.

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