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Arrowian characterizations of latticial federation consensus functions
,Mathematical Social Sciences 20 (1990) 5 l-7 I North-Holland
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ARROWIAN CHARACTERIZATIONS OF LATTICIAL FEDERATION CONSENSUS FUNCTIONS B. MONJARDET UniversitP Paris V and C.A ..M.S., 54 bd Raspail 75270, Paris Cedex 06, France Communicated by F. McMorris Received 2.5 May 1989 Revised 7 March 1990
In this paper we begin the construction of an abstract axiomatic theory of consensus functions in order to account for several classes of similar concrete results in the domain of the axiomatic approach to consensus. Such results, which belong to various fields, are mentioned in our introduction and conclusion. Among the consensus functions defined on a semilattice f.. i.e. the functions f.” -+ L. we study the federation consensus functions associated with cohereditary families of subsets of { 1, . . ..n} called ‘federations’ (or simple games). We obtain several axiomatic characterizations of all federation consensus functions, or of significant subclasses of such functions, and especially of the meet projection (‘oligarchic’) consensus functions. These characterizations depend on the structural properties of the considered semilattice, which are the key to understanding the concrete results mentioned above and allow us to obtain new results. We end by pointing out some further lines of research. Key words: Consensus function; semilattice; federation.
1. Introduction Arrow’s theorem (1951) has fostered much research in social choice theory (see, for instance, Kelly, 1978). But as early as 1952, Guilbaud emphasized the fact that the logical problems of aggregation arise in many different fields. He gave a weak form of Arrow’s theorem, useful in a general context, with a proof based on an implicit use of the notion of ultrafilter (see Monjardet, 1983). Nevertheless, it was only in 1975 that Mirkin gave the first Arrowian result in a quite different field of social choice theory: cluster analysis. Mirkin’s result bears on the aggregation of several partitions into one partition. More recently, the problems of obtaining a consensus of taxonomic structures have been investigated by several authors (see the special issue on consensus in the Journal of Ciassijkation, 3, 1986). For instance, Leclerc (1984) obtained an improved version of Mirkin’s theorem (also given by Fishburn and Rubinstein, 1986) as a corollary of a general result on the consensus of ultrametrics (dendrograms). More generally, in data analysis one needs consensus methods to summarize relational (qualitative or ordinal) data. For an overview of this topic, see Day (1988), 0165-4896/90/63.50
0 1990-Elsevier
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6. Monjardet / Federation consensus functions
Faith (1988), and Leclerc (1988). For instance, paired comparison methods used for the expression of individual preferences of several subjects lead to problems of aggregation of tournaments (complete and antisymmetric relations). Several results on the axiomatic theory of aggregation of tournaments obtained by Monjardet (1978) were generalized to other types of relations by Mirkin (1981). For the aggregation of arbitrary relations one can also mention the work of Arkhipoff (1980). Another example concerns the field of individual choice functions where Aizerman and his colleagues (see Aizerman, 1985; Aizerman and Aleskerov, 1986) have begun a systematic study of ‘spaces’ of choice functions along with the problem of aggregation of individual choice functions. In all these fields the axiomatic approach - inspired by Arrow’s work - is the same. One chooses axioms that the consensus function should satisfy and one determines the functions (if any) satisfying the axioms. Among the huge quantity of results obtained by this approach one can distinguish two main cases. In the first case the axioms imposed on the consensus function lead to a very restricted class of such functions, especially ‘oligarchic’ or ‘dictatorial’ functions. A typical example is the case, already quoted above, where the domain of the consensus function F is the set of all possible n-tuples 17= (R, .-.Ri-*-R,,) of equivalence relations on a fixed set X and its codomain the set of all equivalence relations on X (notice this formulation already contains ‘universality’ and ‘functional’ axioms). Then, if F must also satisfy an ‘independence’ axiom and a ‘unanimity’ (Pareto) axiom, it must be ‘oligarchic’, i.e. F(Z7) = nisIi ,,,.., ipl Ri, where {it, . . . . ip} is a fixed subset of the set { 1, . . . , n} (Mirkin, 1975; Leclerc, 1984). When p= 1, F is called a dictatorial consensus function. In the second case, one demands many more properties for the consensus function, for instance ‘neutrality’ and ‘monotonicity’ axioms, but one obtains a much broader class of consensus functions that generalizes the classical majority rule. Then, this last consensus function can be characterized by adding other axioms (especially symmetry). A typical example of this situation is the case of taxonomic n-trees (Margush and McMorris, 1981; McMorris and Neumann, 1983). The aim of this paper is to begin the construction of an ‘abstract’ theory in order to account for these two kinds of results. The crucial observation is that in all the above concrete cases one works with a set of complex objects (i.e. objects formed from simple objects, such as a binary relation is formed of ordered pairs), and that the set of all complex objects (of a given type), endowed with a natural order, has a lattice or semilattice structure. For instance, the sets of all binary relations, of all tournaments, of all equivalences, and of all choice functions are lattices; whereas the sets of all partial orders, of all complete preorders, and of all n-trees are semilattices. Moreover, the consensus functions obtained in the results mentioned above can very often be expressed by means of the lattice operations. For instance, an oligarchic function on equivalences is nothing more than a meet operation in the lattice of equivalences, while the majority rule expresses itself like a lattice polynomial function.
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So in our abstract theory of aggregation the set of complex objects will be replaced by an abstract (finite) semilattice L, which for convenience we choose to be a meet semilattice. Then the simple objects are the join-irreducible elements of L, each element of L being a join of such elements. The consensus functions are the functions with domain L”, the set of all n-uples of elements of L, and codomain L. The abstractsetN={l,..., i,..., n} is the set of voters, criteria, variables, etc. in the concrete examples. In Section 2 we recall some basic notions of lattice and ordered set theory, and give a characterization of a lower distributive meet semilattice by means of a binary relation /3 defined on the set of join irreducible elements of this semilattice. This relation will play a significant role in our work. Section 3 is devoted to consensus functions defined on a meet semilattice. In Subsection 3.1 we introduce the important class of federation consensus functions. Such functions are associated with cohereditary families of subsets of N, here called federations (sometimes called simple games). Well-known examples of federation consensus functions are the dictatorial, oligarchic and majoritary functions. In Subsection 3.2 we study the relations between several axioms defined for arbitrary consensus functions. Section 4 contains the main results of this work which are axiomatic characterizations of classes of federation consensus functions, and especially of oligarchic functions. These results, like those in Subsection 3.2, depend heavily on the structure of the meet semilattice being considered and of the properties of the binary relation /?. In the conclusion, we summarize the main results by distinguishing three different structural situations inducing different characterization results, and we point several concrete results in the field of consensus that are obtained from our results by identifying the relevant situation. Two structural situations (3 and 1) correspond exactly with the two cases mentioned at the beginning of this Introduction, whereas situation 2 represents an intermediate case. Finally, we mention some further lines of research. It should be clear from the above summary of the content of this paper that it is essentially devoted to abstract theory. The use of this theory to obtain old or new results is mentioned but not developed since future papers are planned that will do so. Some preliminary results of this study were published without proofs in Barthelemy, Leclerc and Monjardet (1984a,b, 1986). The last of these papers contains an overview on the use of the ordinal abstract approach in problems of comparison and consensus of classifications. The interest in such an approach has been also made clear in works like those of Day (1983, 1986), Janowitz (1978, 1988), McMorris (1985), Neumann and Norton (1986a,b), or Barthelemy and Janowitz (1991). This last paper contains also an ordinal axiomatic approach to consensus problems, but for multivalued consensus functions. Finally, notice that another abstract approach to consensus problems has been recently tackled by Rubinstein and Fishburn (1986). They use a linear algebra approach and it would be interesting to compare the ordinal and algebraic approaches. N.B. Theorem 3.4 means Theorem 3 in Section 4.
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2. Preliminaries
First we recall some basic notions in lattice and ordered set theory. For a general background and definitions not given here, see, for example, Aigner (1979), Barbut and Monjardet (1970), or Birkhoff (1967). A meet (join) semilattice is an ordered set P such that for all x, y in P, there exists a greatest lower bound (least upper bound) called the meet (join) of x and y and denoted x~y (xvy). A lattice is an ordered set that is both a meet and a join semilattice. In this paper L will denote a finite meet semilattice that can also be a lattice. All results obtained on meet semilattices can be extended to join semilattices by duality and we will use this well-known fact in Section 5. Each subset S of L has a meet that is denoted /\ S. In particular, AL is the least element of L, denoted OL or simply 0. On the other hand, S has a join V S if and only if the elements of S have a common upper bound. Each time that we will use the notation VS it means that such a join exists in L (notice that V 0 =O). Note that L is a lattice if and only if L has a greatest element denoted 1, , or simply 1, and in this case vL=l=I\@. We denote by (x] the set of lower bounds of x, i.e. (x] = {ye L: ylx}. Notice that (x] is always a lattice. An element x of L is join irreducible if and only if x= V S implies XE S. Equivalently, x is join irreducible if and only if x covers a unique element y of L (i.e. y
VX,
implies there exists an i such that s I
Xi.
one has:
for all s + s’ in J and for every x in L, s <
It is not difficult to prove that property
s’Vx
implies s
(P) is a characteristic
property of lower
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distributivity for a semilattice. We give a related characterization based on a binary relation /3 defined on J. The relation p will play a significant role in Section 3. Define a relation p on J by s/Is’ e s #s’ and there exists x in L such that s, s’%x and scs’vx. Proposition 1.2. L is a lower distributive meet semilattice if and only if s/Is’ e s < s’, for all s,s’ in J. Proof. Let L be a meet semilattice. If s
A meet semilattice L is a meet tree semilattice if and only if for all x, y in L, xvy exists, implies x5 y or yrx. So if L is a meet semilattice that is not a tree, there exists x, y in L, x incomparable to y, and xVy exists. Moreover, it is clear that x and y can be taken as join irreducibles. A meet tree semilattice can be also defined as a meet semilattice L such that for every x in L, (x] is a chain, i.e. a subset of L in which any two elements are comparable. So such a semilattice is lower distributive.
3. Consensus functions on a meet semilattice 3. I. Definitions and examples Letl={l,..., i,j,k ,..., n} be an n-set and L a meet semilattice. We denote by 17 an n-tuple of elements of L, indexed by N: n=(xr, . . ..x., . . ..x.) (n is sometimes called a profile). Then A;=, Xi exists and will be generally denoted by Axi. We adopt special notations for some n-tuples that will occur frequently. Let A, B,CbethreesubsetsofNsuchthatAUBUC=NandAnB=BnC=CnA=[a; I7= (A :x; B : y; C: z) is the n-tuple for which for every i in A (respectively in B, C), xi = x (respectively y, z). Similarly, with A = 1v- A, 27= (A : x; A : y) is the n-
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tuple for which for every i in A (respectively, in A), X,=X (respectively u). Finally (N: x) is the n-tuple with ,u,=x for each i in R;. We denote by L” the set of all n-tuples of elements of N. A consensus function on L is a function, denoted by F, with domain L” and range L. Thus, to every ntuple Z7 in L”, F associates its image s= F(n) EL. Notice that from a mathematical point of view a consensus function on L is nothing more than an n-ary operation on L. 3. I. I. Examples (1) If for every I7 in L”, F(I7) equals a fixed element x0 of L, consensus function. We denote by F” the constant function where least element of L. If L has a greatest element 1, we denote by F’ ing constant consensus function. (2) Let A be a fixed subset of N. We denote by FA the consensus ed by: F(l7) = A xi for every n in L”.
F is a constant x0 equals 0, the the correspondfunction defin-
ierl
FA is also called the A-meet projection consensus function, and any is called a meet projection consensus function. In particular, Fii), I$, is the usual projection, and F, is the n-ary operation defined operation. (3) Let L be a median semilattice, and INI = 2p+ 1. We denote consensus function defined by: F(n) =
v !A rp+1
such function denoted also by the meet by FfP+l) the
for each Z7 in L”.
A well-known result (see, for example, Bandelt and Barthelemy, 1984) says that this formula defines an n-ary operation in a median semilattice, called the (n-ary) median. There it is called the median (or majoritary) consensus function. We now define a large class of consensus functions, which we call the class of federation consensus functions. Such functions are associated with families g of subsets of N. By definition, a family 9 on N is a set of subsets of N. If g is a family, gd = {A s N: A = (N-A) $ g} is its dual family. Three pertinent conditions that a family .cYcan satisfy are: (1) [AC.!% B>A] = [Beg]. (2) [AES, BEG] r=$[AfIB#0]. (3) [AE~] a [Af$g]. A family on N is called a federation if it satisfies condition (1); transversal if it satisfies condition (2); and ipsodual if it satisfies condition (3) (indeed in this case g =.@). A federation consensus function is a consensus function F for which there exists a federation .!J such that
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=
V
A
AEd ( ieA
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Xi
>
consemu
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for each f7 in L”.
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(IO)
Such a federation consensus function will also be denoted by F$, and F,-(n) will be sometimes denoted by V,$ (AA). Notice that the federation consensus functions contain most of the above examples. Indeed, if 9 is the empty family @, FG equals F”; if g= {B c N: B 2 A}, F3 equals FA. If n=2p+l and g={(AcN: JAl?p+l}, fF=&+t). If L has a greatest element 1, F2n = F, = F’ (where 2” denotes the family of all the subsets of N). One must emphasize the fact that if L is an arbitrary meet semilattice and 9 an arbitrary federation, there is not necessarily a federation consensus function associated with $. Indeed, the above formula (cp) can be defined only for some ntuples of L, so that it defines a partial n-ary operation and not a consensus function. The following facts are obvious or well known. If L is a lower distributive meet semilattice, a consensus function is associated by formula (cp) with each transversal federation. For instance, in this case with k > L&2] we define the k-majoritary consensus function by: Q)(n)
= ,& > ;?, *; ( >
If L is a lattice, a consensus function is associated Moreover, if L is a distributive lattice we have
with each federation
$?
In this last case with n = 2p + 1, the median consensus function has two expressions (notice that {A : IA / rp + I} is an ipsodual family): 4,+1,(n)
= ,A ,v,+,
(Ax’)
=/A/$+1
(iyAxi)’
3.1.2. Remarks (1) Let s be an arbitrary family of subsets of N, grn the family of minimal sets of .E S+ the family of ‘supersets’ of S (B is a superset if it contains A in 9). When the above formula (v) is defined for one of these three families, it is defined for the two others and it gives the same result for the three families. So, if the consensus function Fg exists, all the families between grn and P give this same consensus function. It is more convenient to work only with ‘representatives’ of these classes of equivalent families. Here we choose to work with families satisfying (l), the so-called federations. Another possible choice would have been the ‘Sperner’ families, or ‘clutters’, i.e. the families S for which two subsets in 9 are always incomparable, which is the case for sm. (2) An ipsodual federation is also called a proper and strong simple game. For the consensus functions associated with such families in the case of tournaments, see Monjardet (1978).
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(3) The term ‘federation’ is taken from Mirkin (1981). It seems to us more convenient here than other equivalent terms such as simple game, cohereditary or hereditary family, etc. Aizerman (1985) uses the term ‘federation’ in a related but different sense. We end this section by characterizing the federation consensus functions F” for which ZZ is ipsodual. First notice that the image of a profile Z7= (A : y; A : y) by a federation consensus functions belongs to {x, y, xl\y, xvy} . A consensus function F is bi-idempotent if and only if for all x, y in L and for each BcN,F(B:x; B:y)~(x,y). Definition.
The notation xlly used below means that x and y are two incomparable
elements.
Proposition 1.3. Let FS be a federation consensus function. (1) [VA s N, A $9 implies A E +cIT]if and only if [for all x, y EL with XIIy, and for each BcN, F,(B:x; B:y)>xl\y]. (2) [VAc N, AE~!T impliesA$9] ifand only [for all x,y~L with x/y andfor each BGN, F,(B:x; 8:y)xAy. If Be.95 then F,(B : x; B : y)~y>xr\y. Conversely, if there exists B (; N such that Beg and Beg, then for every AE9, AflB#B and ACTB#@. Thus, for x[y, F,(B : x; B : y) =xAy, a contradiction. (2) Let x, y be in L, with xlly. We have only to consider the case where xVy exists. If BGNwith B$S, then for each AE.‘~, AnB#@ so that F,(B:x; 8:y)
B : y) =xvy, a contradiction. (3) Since F3, being bi-idempotent, is equivalent to Fg satisfying F,(B : x; B : y) @ {xAy, xVy} for all x, y in L with xlly and for each B c N, (3) is the consequence of (1) and (2). 3.2. Axioms for consensus functions Let L be a meet semilattice and J the set of all its join-irreducible elements. We need the following notation. For 17=(x1, . . . . xi, . . . . x,,) in L”, and s in J, set N,(n) = (in N: s I Xi}. First we define five ‘Arrowian’ properties that can be satisfied by a consensus function F on L. J-Decisivity (J-D): For every SE J and for all Z7,Z~‘E L” [N,(D)
= N,(W)]
I)
[s s F(l7)
w
s s F(l7’)].
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(J-N): For all S.S’E J and for every l7~ L” [N,(n) = N&7)]
=
[s I F(n)
F,
s’ I F(n)].
J-Decisive Neutrality (J-DN): For all S,S’E J and for all l7,l7’~ L” [N,(n) = N,,(W)] J-Decisive Monotonicity
=
o
s’ I F(I7’)].
(J-DM): For every s E J, and for all J7,17’~ L”
[N,(n) c N,(l7’)] J-Neutral Monotonicity
[s I F(l7)
=
[s 5 F(n)
=
s 5 F(W)].
(J-NM): For every S,S’E J, and for all I7,17’~ L”
[N,(n) c N,Q7’)]
=> [s s F(Z7)
=
s’ I FW’)].
F is called J-decisive, respectively J-neutral, J-decisive neutral, J-decisive monotonic, J-neutral monotonic if it satisfies the property J-D, respectively J-N, J-DN, J-DM and J-NM. The obvious implications between the above properties are summarized below: J-NM u J-DM
a
J-DN u =z. J-D
Other possible properties
=
J-N
for F are the following:
Symmetry (S): For every permutation CTof N and for every l7 in L”, F(oI7) = F(17) (here, oJ7 is equal to (x0(]), . . . ,x0(;), . . . ,x,(,,)). Isotony (I): For all Z7,I7’ in L”, IlsZ7’ F(17) 5 F(l7’). J-Unanimity [s i F(n)].
(i.e. x,5x;
for every i in N) implies
(J-U): For every s in J and every I7 in L”, [N,(n) =N] implies
L-Unanimity (L-U): For every x in L and every I7 in L”, [{ieN: plies [xc F(n)].
xcxi)
=N] im-
J-Autonomy
(J-,4): For every s in J, there exists 17 in L” such that s~F(l7).
Idempotence
(ID): For every x in L, F(N: x) =x.
It is obvious that
L-U
e,
J-U
=
J-A.
We are going to state less obvious implications between the above properties, most of them depending on additional conditions on the set of all join irreducibles. In
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all the following propositions F is a consensus function on a meet semilattice L and J is the set of all join irreducible elements of L. Our first result is true without any condition on J. Proposition
(1) (2) (3) (4)
F F F F
2.3.
is J-decisive monotonic e F is J-decisive and isotone. is J-neutral monotonic e F is J-decisive neutral and J-decisive monotonic. is J-neutral monotonic and Fjt FQ imply F is J-unanimous. is J-decisive monotonic and J-autonomous imply F is J-unanimous.
We first prove (1). Let F be J-decisive monotonic, so that F is J-decisive, and let l7,Z7’ be in L” with 17~ J7’. Then for every s in J,N#7) = {i E N : SI Xi} GN,(l7’)=(iEN:s~x~}. Hence, srF(J7) implies slF(JT’), so that J(F(l7))s; J(F(l7’)) and F(n) I F(W). Conversely, let F be J-decisive and isotone, Z7,J7’, be in L”, and s be in J such that A = N,(n) C_B = N,(l7’) and ss F(Z7). Define J7, = (A : s; A : 0), Q’= (B : s; B : 0). Then n, 5ni’ and the isotonicity of F imply F(l7,) I F(&‘), so that ss F(l7,) (by the J-decisivity of F for J7 and n,) s F(l7;). 3 By the J-decisivity of F (for Z7’ and J7,‘), s~F(l7’). In order to prove (2), we have only to prove that when F is J-decisive neutral and J-decisive monotonic it is J-neutral monotonic. Let F be such a consensus function, l7, J7’, in L”, s, s’ in J such that A = N,(J7) G N,.(17’) = B and SC F(n). We consider the profile J7” = (B : s; L?I: 0); F being J-decisive monotonic N,(n) =A C_N#7”) implies srF(17”). By the J-decisive neutrality of F, NS,(17’)=NS(17”)= B implies S’S F(l7’). 0 We now prove (3). Let F be a J-neutral monotonic consensus function not equal to 9. There exists 17, in L” such that O< F(l7,) =x and there exists s1 in J such that S~IX. We set N,,(J7,)=A (rz,cAcN) and we consider 17’=(N:s,). By Jmonotonicity, st sF(n’). Now let Z7 be in L” and s in J such that N,(J7)=N= Z N,,(n). By J-decisive neutrality, slF(Z7). Finally, let us prove (4). Let s be in J and 17 in L” such that N,(n) = N. There exists 17’ in L” such that s~F(17’). Let us consider l7”= (N: s). By J-decisive monotonicity, JVS(J7”)= N ;! N,(Z7’) and s s F(l7’) imply s I F(W). By J-decisivity, q N,(n) = N,(P) and s I F(T) imply s I F(n). Proof.
Other relations between the above properties depend on the structures of the set J of join-irreducibles and of the semilattice L. We shall use two binary relations defined on J and associated notions of connectivity. First, we define the symmetric relation a on J. For all s,s’ in J, sas’ if and only if sf S’ and SVS’ exists. We say that J is a connected if and only if for all s, s’ in J, there exists a sequence se = s, sl, . . . , sir . . . , Sk= s’ of join irreducibles elements such that for all i=O ,..., k-l,s;crsi+t. Notice that if L is a lattice J is always a connected.’ Notice also that this condition is very weak. Indeed, it is equivalent to the condition L-(O) is a connected poset (i.e.
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for all x, y in L-(O), there exists a sequence x0=x,x1, . . . ,x;, . . . ,xp =y such that for all i=O,..., p-l, xi#O and x;.U,, i). However, a meet tree semilattice may not satisfy this condition. Now we recall the definition of the binary relation /3 on J (Section 2). For all s,s’ in J,spS’ if and only if s#s’ and there exists x in L such that s,s’$x and s
Then F is J-decisive
Proof. By proposition 2.3(2), we have only to show that when F is J-decisive neutral it is J-decisive monotonic. Let n, n’ be in L” such that A =N,(LT)C N,(lT’)=B and ssF(J7). If A = B, s~F(l7’) by J-decisive neutrality. Assume A C B. Since p is non-empty one can find either s,,s2 in J with s1
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If A =N, then s’lF(Z7’) by J-unanimity. If A CN, and s =s’, then s’cF(Z7’) by J-decisivity. If A C N and sfs’, by strong connectivity of p, there exists a sequence of elements of J, se=s’,s I,..., s; ,..., sk=s, and a sequence of elements of L,Xr, . . ..x. such that for i=O,...,k-l,s;pS;+i, i.e. Si
if and only if for each
(E): F is extensive if and only if for each x in L, XI F(N: x).
The last property is defined only if L has a greatest element 1 (i.e. if L is a lattice). I-Idempotence
(l-ID): F is I-idempotent
if and only if F(N : 1) = 1.
We have the following obvious result. Lemma 6.3. F is meet compatible 1-idempotent.
or F is idempotent
3 F is extensive
=. F is
Proposition 7.3. (1) F is J-unanimous
o F is meet compatible. (2) Zf F is isotone or J-decisive, then F is J-unanimous o F is extensive. (3) Zf L is a lattice and F is J-decisive, then F is J-unanimous o F is I-idempotent. (4) Zf F is J-decisive monotonic, then F is J-unanimous e F is J-autonomous.
Proof.
(1) If SC AXi, one has N#Z)=N and by J-unanimity sz~F(Z7). Then /\Xi= F(Z7). Conversely, N,(ZZ) = N implies ssxi for each i. Thus, SI l\Xil F(Z7). (2) From (1) and Lemma 6.3. one has only to prove that if F satisfies one of the stated hypotheses, and is extensive, it is J-unanimous or (equivalently) meetcompatible. Assume F is J-decisive. Let s be in J, Z7 in L” such that N,(ZZ) = N and Z7’= (N: s). By extensivity, slF(Z7’) and by J-decisivity ssF(Z7). Assume F is isotone and let Z7= (xi, . . . *Xi,. . . ,x,). Consider Z7’= (N: Axi> 1l7.
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By extensivity and isotony, l\XilF(17’)sF(17). (3) By (1) and Lemma 6.3, if F is J-unanimous, it is meet compatible and thus, if L is a lattice, I-idempotent. If F is I-idempotent, let n such that N,(n) =N; then N,(n) = N&V : 1) = N; by J-decisivity, s I F(N : 1) = 1 implies s I F(D). (4) This is a consequence of assertion (4) in Proposition 2.3. 0 Before giving the final result of this section we notice that s/Is’ implies that szs’ and SVS’ exists, i.e. sas’. Then we have Lemma 8.3. J is /3 strongly connected imply J is CTconnected and J is non-empty. By using Propositions 2.3, 3.3, and 4.3, Theorem 5.3, Proposition 7.3, and Lemma 8.3 one easily obtains the following theorem that summarizes the results of this section. Theorem 9.3. Let L be a meet semilattice such that the set J of ah join irreducible elements of L is p strongly connected. The six following properties are equivalent. Moreover, if L is a lattice they are equivalent to the last property: (1) F satisfies J-NM and F+ F”.
(2) (3) (4) (5) (6) (7)
F F F F F F
satisfies satisfies satisfies satisfies satisfies satisfies
Remark. In connectivity let us define s
J-D and J-U. J-D and MC. J-D and E. J-DM and J-A. J-D, J-N, I, and Ff F". J-D and l-ID.
view of Theorem 9.3 it is important to have conditions for of J. The following condition is obvious but useful. For spJs’ if and only if sfs’ and there exists s” in J such that p, is strongly connected, let us say that J is pJ strongly pJ strongly connected J is /3 strongly connected.
the p strong all s,s’ in J, S,S’YZS”and connected.
4. Characterizations of federation consensus functions In this section we study the consensus functions satisfying properties defined in Subsection 3.2. Especially we characterize the federation consensus functions defined in Subsection 3.1. Depending on the structural properties of the semilattice L, such characterizations can use a quite different system of axioms. Special cases of these results provide characterizations of projection consensus functions and of median consensus functions. The following notion is basic in the proofs of these results.
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Definition.
Let F be a consensus function on the meet semilattice L and s in the set J of all join irreducible elements of L. We say that A C N is an F s-decisive set, or simply an s-decisive sef if there exists f7 in L” such that N,(n) =A and SI F(f7). Recall that if s,s’ are in J, sos if and only if
SVS’
exists.
Proposition
1.4. Let F be a consensus function on the meet semilattice L. If F is J-decisive, for each s in J there exists a family 9” on N such that, for every I7 in L”, F(J7) = V {s E J : N,(n) E SS}. Moreover, if F is isotone, then gS is always a federation and if F is J-neutral, SS = S-s, as soon as s and s’ are in the same connected class of the relation a. Proof. Let F be a J-decisive consensus function, s in J and A an s-decisive set. Since F is J-decisive we get that A is s-decisive if and only if for every J7 in L” such that N,(n) = A,s< F(I7). For each s in J, we set S,= {A EN: A is s decisive}. Thus, for every J7 in L”, ss F(l7) if and only if N,(n) E 9”. Notice one can have .5&= the empty family @, which means s s F(IZ) for every J7 in L”. Then F(t7) = V {s E J : N,(n) E gS}. If F is J-decisive and isotone it is J-decisive monotonic (Proposition 2.3(l)). Let s be in J such that SS#@, A in SS with B3A. We set l7=(A : s; A : 0) and ff’=(B:s, B:O). Then srF(J7) and by J-DM, ssF(Z7’). Thus, BE$$,, which proves that ,9$ is a federation. Notice that Q, is also a federation. Let F be a J-decisive and J-neutral consensus function; let s,s’ be in J such that s#s’, svs exists and gS+@; let J7 be in L” such that N,(Z7)=A ES, and J7’= (A : s’; A : 0). Define J7” = (A : sVs’; A : 0). By J-D for J7 and J7”, SC F(IIn). By J-N for J7”, s’sF(I7”). By J-D for J7” and J7’, S’S F(l7’). Thus, A ~3,,, and s,rs,,. One shows similarly that S,, G 9”. Thus, ,c&= SS, . Now let s and s’ be in a connected component of the relation a, with gS# @. Then there exists a sequence of join irreducible elements so=s,sI, . . . . Q,s~+~, . . . . sp=s’, such that for k=O, . . . . p-l, s,Vs,+, exists. Then by the above result S, = gSk+, for k = 0, . . . ,p - 1, so 5$ = s,..
cl
Proposition
2.4. A J-neutral monotonic
sensus function: in L”,
there exists a federation
F(n) =Av_
(
consensus function F is a federation conS C 2N such that F = F,,i. e. for every l7
i?, xi >
Moreover, if L is not a lattice, g is a transversal federation. Proof.
Let F be a J-neutral monotonic consensus function. It is J-decisive and isotone; so by Proposition 1.4, for every s in J there exists a federation SS such that F(J~)=V{SEJ: N,(JI)ES~}. If F=F ‘, for every s in J S”= Q, and we have F=F,.
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If not, there exists s in J such that .?Fs#@. Let A be in gs and s’#:s in J. By the definition of s-decisive sets there exists 17 in t” such that N,(n) =A and srF(J7). We set J7’= (,4 : s’; A : 0). By J-DN we have s’sF(J7’) so that A ES’,, and gXC .&. Since ++ @, we obtain similarly & G gs, and thus gs2,=gs,. Thus, Z&is the same federation $, for every s in J, and we have for every f7 in L”, F(R) = V{SE J: N,(n) ES}. Let I7 be in L”; we are going to show that F’(J7) = v Ae9
A x; ( iaA
>
exists and equals F(J7). Let A be in .!+Fand s in J such that slAieA xi; then N,(n) 2 A, and since 9 is a federation, N,(n) E .F; so ssF(II), and thus AiEA xir F(n); so finally F’(R) exists and is less than or equal to F(H); let s be in J such that SC F(n); then N,(J7) =A E 9 implies s5 AieA xi5 F’(Z7); then F(l7)~ F’(l7) and finally F(l7) = F’(n). Let us assume that L is not a lattice; there exist X,y in L such that xVy does not exist. Let A be in $ and assume A E g; we set J7= (A : x; A : y); for every SIX, N,(J7) 2 A; since 9 is a federation, one gets s~F(l7), so x~F(l7); similarly, one gets yl F(n), a contradiction. Thus, S satisfies the property: A E 5 implies A e R since $ is a federation, one obtains that if A, BE .9, A n B # 0, i.e. 9r is a transversal federation. 0 Theorem 3.4. Let L be a non-lower distributive meet semilattice and F a consensus function on L. Consider the following conditions: (1) F is J-neutral monotonic and FZ F’. (2) F is J-decisive, J-neutral, isotone and F # F’. (3) There exists A, 0 CA c N such that for every I7 in L”, F(I7) = l\isz Xi (i.e. F= FA). (4) There exists A, or c A G N such that for every r/ in L”, F(I7) = l\ieA Xi* IfL is not a lattice, then conditions (1) and (3) are equivalent. If L is a lattice, then conditions (l), (2) and (4) are equivalent. Moreover, A = N (or Q in the lattice case) if and om’y if F is symmetric, and /A[ = 1 if and only if F is bi-idempotent.
Suppose L is not a lattice. (1) = (3). By Proposition 2.4, a J-neutral monotonic consensus function F is a federation consensus function FS with &@a transversal federation on N and B # @ (since F#p). Condition (3) is equivalent to saying that 9 has an unique minimal set A. Let us assume that $ has two different minimal sets, A and B, so that A tl B is not in S. Since L is not lower distributive, there exist s,s’ in J and x in L such that sss’, s%x and SCS’VX (see Proposition 1.2). Consider the following profile n:(A-AnB:s’; B-AflB:x; AnB:s’vx; N-AUB:s’). N,,(l7)=AU (A U B) is in S so s’~F(l7). Since for each s in J such that SIX, N,(n) a B is in 9 one gets x5 F(l7). Then S
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(3) * (1). We consider a meet projection consensus function FA . Let n, l7’ be in L”,s,s in J, N,(II)=BGN,,(II’)=C and assume s~F~(I7)=l\;~~ xi. Then AGiv#7)=BZ;C, so /\. rcc X; I AieA x,! = FA(I7’). Then N,,(n’) = C implies S’SA\ipcXI sFA(I7’). If L is a lattice, conditions (1) and (2) are equivalent by Proposition 4.3. The same proof as above shows that (1) is equivalent to (4), but in this case, A can be empty since F, = F’. If A = N (or A = 0 in the lattice case) FA is symmetric. If 0 CA c N, let us consider n=(A : s; A : 0) with s>O; FA(17) =s. For any permutation o that exchanges an element of A and A, FA(o17) =O
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6.4. Let L be a lower distributive meet semilattice and Fa consensusfunction on L. Consider the following conditions: (1) F is J-neutral monotonic. (2) F is J-decisive neutral and J-decisive monotonic. (3) F is J-decisive, J-neutral and isotone. (4) F is a federation consensus function F,- with 9 a transversal federation. (5) F is a federation consensus function F9. If L is not a lattice, then conditions (I), (2) and (4) are equivalent. If L is a (distributive) lattice, then conditions (l), (2) (3) and (5) are equivalent. Theorem
In all cases (1) and (2) are equivalent conditions by Proposition 2.3. Let L be a lower distributive meet semilattice that is not a lattice. By Proposition 2.4, if F satisfies (l), F= FS with F a transversal federation. Conversely, let F” be such a consensus function and let IT, J7’ be in Ln, s,s’ in J with N,(Z7) G N,,(J7’) and s~F,(l7) = V, (AA). By lower distributivity, there exists A in S such that s-(/\ieA xi. Thus, A GN,(J~)I;N,,(J~‘) and s’_(l\isA x~sF(I~‘), SO that F” is Jneutral monotonic. If L is a distributive lattice, conditions (1) and (3) are equivalent by Proposition 4.3. By Proposition 2.4, if F satisfies (l), F is a federation consensus function, and 0 one proves the converse implication as above. Proof.
Remarks.
(1) In the above theorem, F can be the constant function F” in the general case or F’ in the lattice case. One can exclude these possibilities by adding axioms, such as the idempotence property: F(N : x) =x. (2) The comparison of Theorems 3.4 and 6.4 shows that essentially the same axiom of J-neutral monotonicity characterizes all the federation consensus functions in the distributive case, but only the meet projection consensus functions in the nondistributive case. Indeed, one can also characterize the meet projection consensus functions in the distributive case by a much stronger neutral monotonicity axiom: the L-NM axiom, obtained by replacing J by L in the definition of the J-NM axiom. Nevertheless, L must not be a meet tree semilattice (see Barthelemy, Leclerc and Monjardet, 1984 b). Corollary 7.4. Let L be a median semilattice and Fa consensus function on L. Consider the following conditions: (1) F is J-neutral monotonic, symmetric and bi-idempotent. (2) F is J-decisive neutral, J-decisive monotonic, symmetric and bi-idempotent. (3) F is J-decisive, isotone, symmetric and bi-idempotent. (4) F is the median (majoritary) consensus function. If L is not a lattice, then conditions (l), (2) and (4) are equivalent. If L is a (distributive) lattice, then all four conditions are equivalent.
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Proof. Let L be a median semilattice that is not a lattice. By Theorem 4.4, conditions (1) and (2) are equivalent and imply that F=F,- with 3 a transversal family. The symmetry of F,- implies that 9 = {A c N: IA 1r k}. By Proposition 1.3, the biidempotence of FJT implies that .9 is ipsodual, i.e. 3 =Bd = {B G N: Beg} = {BsN: lB/>n-k}. Then n=2k-1 and F,(17)=V,Aizk (/jieA Xi) is the median consensus function. Conversely, it is clear that such a consensus function satisfies (1). Assume now that L is a distributive lattice. Then (1) implies (3) by Theorem 4.4. In order to prove the converse implication, it suffices to prove that when F is Jdecisive and bi-idempotent F must be J-neutral, since for a lattice, J-NM is equivalent to J-D, J-N and I (Proposition 4.3). Let s,s’ be in J and I7 in L” such that N,(JT)=N,(JT)=A and s~F(l7). One has A#@, since if A=QI, 0~s~ F(N: O)=O, a contradiction. Thus, there exists Xi in I7 such that s and S’lXi, and then SVS’ exists. Set J7’=(A : svs’; A : 0). Since N,(Z’J)=A, one has by J-decisivity O
5. Conclusion
It is possible to summarize the main results of this paper by distinguishing three different structural situations for the meet semilattice L on which the consensus functions are defined. Situation 1: L is lower distributive. Then, in this case, the axiom of J-Neutral Monotonicity characterizes any federation consensus function (Theorem 6.4). Situation 2: L is non-lower distributive. Then, if F+F’, the axiom of J-Neutral Monotonicity characterizes only the meet projection consensus functions (Theorem 3.4). Situation 3: L is non-lower distributive, and J is p strongly connected. Then the axiom of J-Neutral Monotonicity is equivalent with (for instance) the J-decisive and J-unanimity axioms (Theorem 4.4). In terms of social choice theory, in this situation the independence and Pareto axioms characterize the oligarchic consensus functions. Moreover, since all the results of this paper can be easily dualized, we have the three dual situations l*, 2*, and 3* where L is a join semilattice either upper distributive, or not upper distributive, or not upper distributive and with the set M p* strongly connected. Here, M is the set of all meet irreducible elements of L, and the binary relation p* on M is defined by: sp*s’ 1) s#s’ and there exists x in L such that S,S’ZX and S’>SAX. Notice that in the dual case, meet projection consensus functions become join projection consensus functions (sometimes called cooligarchic functions): F(Z7) = Vip xi. We list now some ‘concrete’ results that are special instances of the ‘abstract’ results proved in the above situations. Notice that to determine the relevant situation is not always trivial, especially for situation 3 or 3*. A
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Situation 1. Characterization of ‘majoritary opinion functions’ and of the median operation for tournaments (Monjardet, 1978). Characterization of ‘social welfare functions’ for binary relations (Mirkin, 1981). Characterization of ‘decisive’ consensus functions and of the majoritary consensus for n-trees (Margush and McMorris, 1981; McMorris and Neumann, 1983). Characterization of ‘local operators’ for the set of all choice functions or the set of all choice functions satisfying the ‘heritage’ property (see Aizerman, 1985). Situation 2. Characterization of local operators for the sets of all choice functions satisfying the ‘heritage’ and ‘concordance’ or ‘outcast variants’ properties (see Aizerman, 1985). Situation 3. Characterization of oligarchic consensus functions for the set of all equivalence relations (Mirkin, 1975; Leclerc, 1984; Fishburn and Rubinstein, 1986). Characterization of oligarchic social choice functions for the set of all partial orders relations (Brown, 1975). Situation 3*. Characterization of co-oligarchic consensus functions for the set of all equivalence relations (Neumann and Norton, 1986). It is also easy to see how the abstract results allow us to obtain new results. We have first to assume that the set of complex objects we wish to aggregate can be naturally endowed with a semilattice structure (a frequent situation). Then we have to identify the join or (and) meet irreducibles of this semilattice and to study the properties of relations fi or (and) /3*. According to these properties we are in one of the six structural situations 1, 2, 3, I*, 2*, or 3* and it suffices to apply the corresponding results. Such a procedure, which could for instance be applied to weak orders, partial orders contained in a linear order, partial orders containing a given partial order, ultrametrics, ‘fuzzy’ relations, ordered trees and Buneman trees (see BarthClemy, Leclerc and Monjardet, 1986), will be not made more precise here. Notice that in some cases it can raise non-trivial problems. Notice also that Leclerc (1990) has rather thoroughly investigated the case of the aggregation of ‘fuzzy preferences’, i.e. equivalently the case of consensus functions defined on the lattice of all residuated mappings defined between two lattices. We end this conclusion by pointing out further directions of research. In Theorems 3.4 and 6.4 we have obtained axiomatic characterizations of projections (i.e. ‘dictatorial’) functions by using the bi-idempotence axiom. It would be interesting to obtain other characterizations of this very special case. Indeed, this case (for instance, Arrow’s theorem for linear orders) could be more clearly understood within the study of more general lattice consensus functions. Such functions would be functions F=D” ---rA, where D and A would be subsets of one or even two semilattices. Finally, a study of the relation p in semilattices and lattices would be useful. In this direction recall that in a distributive lattice the relation p is exactly
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the order relation < defined on the set J of all join irreducible elements of L. In the case of a geometric lattice L (see, for instance, Aigner, 1979), one easily see that the relation p is exactly the ‘perspectivity relation’ defined on the set J of all atoms of L. Then p is strongly connected if and only if L is ‘indecomposable’, which is for instance the case of the (geometric) lattice of all partitions of a set.
Acknowledgements The author is grateful to V. Duquenne and B. Leclerc for helpful discussions concerning the relation p defined in this paper, and to F.R. McMorris and anonymous referees for significant improvements of style.
References M. Aigner, Combinatorial Theory (Springer-Verlag, Berlin, 1979). M.A. Aizerman, New problems in the general choice theory, Social Choice and Welfare 2 (1985) 235-282. M.A. Aizerman and F.T. Aleskerov, Voting operators in the space of choice functions, Math. Sot. Sci. 11 (1986) 201-242. 0. Arkhipoff, An introduction to the axiomatics of procedure of aggregation, Math. Sot. Sci. 1 (1980) 69-83. K.J. Arrow, Social Choice and Individual Values (Wiley, New York, 1951). H.J. Bandelt and J.P. Barthelemy, Medians in median graphs, Discrete Appl. Math. 8 (1984) 131-142. M. Barbut, MCdiane. distributivit&, Cloignements (1961); reprinted in MathCmatiques et Sciences Humaines 70 (1980) 5-31. M. Barbut and B. Monjardet, Ordre et classification, Algebre et Combinatoire (Hachette, Paris, 1970). J.P. BarthClemy and M.F. Janowitz, A formal theory of consensus, preprint, SIAM J. Disc. Math. (1991). J.P. Barthtlemy, B. Leclerc and B. Monjardet, Ensembles ordonnCs et taxonomie mathimatique, in: M. Pouzet and D. Richard, eds., Orders: Descriptions and Roles, Annals of Discrete Mathematics 23 (North-Holland, Amsterdam. 1984a) 523-548. J.P. BarthClemy, B. Leclerc and B. Monjardet, Quelques aspects du consensus en classification, in: E. Diday et al., eds., Data Analysis and Informatics III (North-Holland, Amsterdam, 1984b) 307-316. J.P. Barthblemy, B. Leclerc and B. Monjardet, On the use of ordered sets in problems of comparison and consensus of classifications, J. Classification 3 (1986) 187-224. G. Birkhoff, Lattice Theory, 3rd edn. (American Mathematical Society, Providence, 1967). D.J. Brown, Aggregation of preferences, Quart. J. Econom. 89 (1975) 456-469. W.H.E. Day, The role of complexity in comparing classifications, Math. Biosciences 66 (1983) 97-114. W.H.E. Day, Consensus methods as tools for data analysis, in: H.H. Bock, ed.. Classification and Related Methods of Data Analysis (North Holland, Amsterdam, 1988) 333-340. W.H.E. Day and D.P. Faith, A model in partial orders for comparing objects by dualistic measures, Math. Biosciences 8 (1986) 179-192. D.P. Faith, Consensus applications in the biological sciences, in: H.H. Bock, ed., Classification and Related Methods of Data Analysis (North Holland, Amsterdam, 1988) 325-332. P.C. Fishburn and A. Rubinstein, Aggregation of equivalence relations, J. Classification 3 (1) (1986) 61-65. G. Th. Guilbaud, Les theories de I’intCret g&&al et le probleme logique de l’agregation, Economic appliquee 5 (1952) 501-551; reprinted in ElCments de la ThCorie des Jeux (Dunod, Paris. 1968).
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M.F. Janowitz. An order theoretic model for cluster analysis, SIAM J. Appl. Math. 34 (1978) 55-72. M.F. Janowitz, Induced social welfare functions, Math. Sot. Sci. 15 (3) (1988) 261-276. J.S. Kelly, Arrow Impossibility Theorems (Academic Press, London, 1978). B. Leclerc, Efficient and binary consensus functions on transitively valued relations, Math. Sot. Sci. 8 (1984) 45-61. B. Leclerc, Consensus applications in the social sciences, in: H.H. Bock, ed., Classification and Related Methods of Data Analysis (North Holland, Amsterdam, 1988) 333-340. B. Leclerc, Agregation of fuzzy preferences: a theoretic Arrow-like approach, Fuzzy Sets and Systems (1990) to appear. T. Margush and F.R. McMorris, Consensus n-trees, Bull. ,Math. Biology 43 (1981) 239-244. F.R. McMorris, Axioms for consensus functions on undirected phylogenetic trees, Math. Biosciences 74 (1985) 17-21. F.R. McMorris and D.A. Neumann, Consensus functions on trees, Math. Sot. Sci. 4 (1983) 131-136. B.C. Mirkin, On the problem of reconciling partitions, in: Quantitative Sociology, International Perspectives on Mathematical and Statistical Modelling (Academic Press, New York, 1975) 441-449. B.C. Mirkin. Federations and transitive group choice, Math. Sot. Sci. 2 (1981) 35-38. B. Monjardet, An axiomatic theory of tournament aggregation, Math. Op. Res. 3 (4) (1978) 334-351. B. Monjardet, On the use of ultrafilters in social choice theory, in: P.K. Pattanaik and M. Salles, eds., Social Choice and Welfare (North-Holland, Amsterdam, 1983) 73-78. D.A. Neumann and Jr. V.T. Norton, On lattice consensus methods, J. Classification 3 (1986a) 225-25.5. D.A. Neumann and Jr. V.T. Norton, Clustering and isolation in the consensus problems for partitions, J. Classification 3 (1986b) 281-297. A. Rubinstein and P.C. Fishburn, Algebraic aggregation theory, J. Econom. Theory 38 (1) (1986) 63-77. M. Sholander, Medians, lattices and trees, Proc. Amer. Math. Sot. 5 (1954) 808-812.
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ARROWIAN CHARACTERIZATIONS OF LATTICIAL FEDERATION CONSENSUS FUNCTIONS B. MONJARDET UniversitP Paris V and C.A ..M.S., 54 bd Raspail 75270, Paris Cedex 06, France Communicated by F. McMorris Received 2.5 May 1989 Revised 7 March 1990
In this paper we begin the construction of an abstract axiomatic theory of consensus functions in order to account for several classes of similar concrete results in the domain of the axiomatic approach to consensus. Such results, which belong to various fields, are mentioned in our introduction and conclusion. Among the consensus functions defined on a semilattice f.. i.e. the functions f.” -+ L. we study the federation consensus functions associated with cohereditary families of subsets of { 1, . . ..n} called ‘federations’ (or simple games). We obtain several axiomatic characterizations of all federation consensus functions, or of significant subclasses of such functions, and especially of the meet projection (‘oligarchic’) consensus functions. These characterizations depend on the structural properties of the considered semilattice, which are the key to understanding the concrete results mentioned above and allow us to obtain new results. We end by pointing out some further lines of research. Key words: Consensus function; semilattice; federation.
1. Introduction Arrow’s theorem (1951) has fostered much research in social choice theory (see, for instance, Kelly, 1978). But as early as 1952, Guilbaud emphasized the fact that the logical problems of aggregation arise in many different fields. He gave a weak form of Arrow’s theorem, useful in a general context, with a proof based on an implicit use of the notion of ultrafilter (see Monjardet, 1983). Nevertheless, it was only in 1975 that Mirkin gave the first Arrowian result in a quite different field of social choice theory: cluster analysis. Mirkin’s result bears on the aggregation of several partitions into one partition. More recently, the problems of obtaining a consensus of taxonomic structures have been investigated by several authors (see the special issue on consensus in the Journal of Ciassijkation, 3, 1986). For instance, Leclerc (1984) obtained an improved version of Mirkin’s theorem (also given by Fishburn and Rubinstein, 1986) as a corollary of a general result on the consensus of ultrametrics (dendrograms). More generally, in data analysis one needs consensus methods to summarize relational (qualitative or ordinal) data. For an overview of this topic, see Day (1988), 0165-4896/90/63.50
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Faith (1988), and Leclerc (1988). For instance, paired comparison methods used for the expression of individual preferences of several subjects lead to problems of aggregation of tournaments (complete and antisymmetric relations). Several results on the axiomatic theory of aggregation of tournaments obtained by Monjardet (1978) were generalized to other types of relations by Mirkin (1981). For the aggregation of arbitrary relations one can also mention the work of Arkhipoff (1980). Another example concerns the field of individual choice functions where Aizerman and his colleagues (see Aizerman, 1985; Aizerman and Aleskerov, 1986) have begun a systematic study of ‘spaces’ of choice functions along with the problem of aggregation of individual choice functions. In all these fields the axiomatic approach - inspired by Arrow’s work - is the same. One chooses axioms that the consensus function should satisfy and one determines the functions (if any) satisfying the axioms. Among the huge quantity of results obtained by this approach one can distinguish two main cases. In the first case the axioms imposed on the consensus function lead to a very restricted class of such functions, especially ‘oligarchic’ or ‘dictatorial’ functions. A typical example is the case, already quoted above, where the domain of the consensus function F is the set of all possible n-tuples 17= (R, .-.Ri-*-R,,) of equivalence relations on a fixed set X and its codomain the set of all equivalence relations on X (notice this formulation already contains ‘universality’ and ‘functional’ axioms). Then, if F must also satisfy an ‘independence’ axiom and a ‘unanimity’ (Pareto) axiom, it must be ‘oligarchic’, i.e. F(Z7) = nisIi ,,,.., ipl Ri, where {it, . . . . ip} is a fixed subset of the set { 1, . . . , n} (Mirkin, 1975; Leclerc, 1984). When p= 1, F is called a dictatorial consensus function. In the second case, one demands many more properties for the consensus function, for instance ‘neutrality’ and ‘monotonicity’ axioms, but one obtains a much broader class of consensus functions that generalizes the classical majority rule. Then, this last consensus function can be characterized by adding other axioms (especially symmetry). A typical example of this situation is the case of taxonomic n-trees (Margush and McMorris, 1981; McMorris and Neumann, 1983). The aim of this paper is to begin the construction of an ‘abstract’ theory in order to account for these two kinds of results. The crucial observation is that in all the above concrete cases one works with a set of complex objects (i.e. objects formed from simple objects, such as a binary relation is formed of ordered pairs), and that the set of all complex objects (of a given type), endowed with a natural order, has a lattice or semilattice structure. For instance, the sets of all binary relations, of all tournaments, of all equivalences, and of all choice functions are lattices; whereas the sets of all partial orders, of all complete preorders, and of all n-trees are semilattices. Moreover, the consensus functions obtained in the results mentioned above can very often be expressed by means of the lattice operations. For instance, an oligarchic function on equivalences is nothing more than a meet operation in the lattice of equivalences, while the majority rule expresses itself like a lattice polynomial function.
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So in our abstract theory of aggregation the set of complex objects will be replaced by an abstract (finite) semilattice L, which for convenience we choose to be a meet semilattice. Then the simple objects are the join-irreducible elements of L, each element of L being a join of such elements. The consensus functions are the functions with domain L”, the set of all n-uples of elements of L, and codomain L. The abstractsetN={l,..., i,..., n} is the set of voters, criteria, variables, etc. in the concrete examples. In Section 2 we recall some basic notions of lattice and ordered set theory, and give a characterization of a lower distributive meet semilattice by means of a binary relation /3 defined on the set of join irreducible elements of this semilattice. This relation will play a significant role in our work. Section 3 is devoted to consensus functions defined on a meet semilattice. In Subsection 3.1 we introduce the important class of federation consensus functions. Such functions are associated with cohereditary families of subsets of N, here called federations (sometimes called simple games). Well-known examples of federation consensus functions are the dictatorial, oligarchic and majoritary functions. In Subsection 3.2 we study the relations between several axioms defined for arbitrary consensus functions. Section 4 contains the main results of this work which are axiomatic characterizations of classes of federation consensus functions, and especially of oligarchic functions. These results, like those in Subsection 3.2, depend heavily on the structure of the meet semilattice being considered and of the properties of the binary relation /?. In the conclusion, we summarize the main results by distinguishing three different structural situations inducing different characterization results, and we point several concrete results in the field of consensus that are obtained from our results by identifying the relevant situation. Two structural situations (3 and 1) correspond exactly with the two cases mentioned at the beginning of this Introduction, whereas situation 2 represents an intermediate case. Finally, we mention some further lines of research. It should be clear from the above summary of the content of this paper that it is essentially devoted to abstract theory. The use of this theory to obtain old or new results is mentioned but not developed since future papers are planned that will do so. Some preliminary results of this study were published without proofs in Barthelemy, Leclerc and Monjardet (1984a,b, 1986). The last of these papers contains an overview on the use of the ordinal abstract approach in problems of comparison and consensus of classifications. The interest in such an approach has been also made clear in works like those of Day (1983, 1986), Janowitz (1978, 1988), McMorris (1985), Neumann and Norton (1986a,b), or Barthelemy and Janowitz (1991). This last paper contains also an ordinal axiomatic approach to consensus problems, but for multivalued consensus functions. Finally, notice that another abstract approach to consensus problems has been recently tackled by Rubinstein and Fishburn (1986). They use a linear algebra approach and it would be interesting to compare the ordinal and algebraic approaches. N.B. Theorem 3.4 means Theorem 3 in Section 4.
B. Monjardet
54
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2. Preliminaries
First we recall some basic notions in lattice and ordered set theory. For a general background and definitions not given here, see, for example, Aigner (1979), Barbut and Monjardet (1970), or Birkhoff (1967). A meet (join) semilattice is an ordered set P such that for all x, y in P, there exists a greatest lower bound (least upper bound) called the meet (join) of x and y and denoted x~y (xvy). A lattice is an ordered set that is both a meet and a join semilattice. In this paper L will denote a finite meet semilattice that can also be a lattice. All results obtained on meet semilattices can be extended to join semilattices by duality and we will use this well-known fact in Section 5. Each subset S of L has a meet that is denoted /\ S. In particular, AL is the least element of L, denoted OL or simply 0. On the other hand, S has a join V S if and only if the elements of S have a common upper bound. Each time that we will use the notation VS it means that such a join exists in L (notice that V 0 =O). Note that L is a lattice if and only if L has a greatest element denoted 1, , or simply 1, and in this case vL=l=I\@. We denote by (x] the set of lower bounds of x, i.e. (x] = {ye L: ylx}. Notice that (x] is always a lattice. An element x of L is join irreducible if and only if x= V S implies XE S. Equivalently, x is join irreducible if and only if x covers a unique element y of L (i.e. y
VX,
implies there exists an i such that s I
Xi.
one has:
for all s + s’ in J and for every x in L, s <
It is not difficult to prove that property
s’Vx
implies s
(P) is a characteristic
property of lower
B. Monjardet / Federation consensus functions
55
distributivity for a semilattice. We give a related characterization based on a binary relation /3 defined on J. The relation p will play a significant role in Section 3. Define a relation p on J by s/Is’ e s #s’ and there exists x in L such that s, s’%x and scs’vx. Proposition 1.2. L is a lower distributive meet semilattice if and only if s/Is’ e s < s’, for all s,s’ in J. Proof. Let L be a meet semilattice. If s
A meet semilattice L is a meet tree semilattice if and only if for all x, y in L, xvy exists, implies x5 y or yrx. So if L is a meet semilattice that is not a tree, there exists x, y in L, x incomparable to y, and xVy exists. Moreover, it is clear that x and y can be taken as join irreducibles. A meet tree semilattice can be also defined as a meet semilattice L such that for every x in L, (x] is a chain, i.e. a subset of L in which any two elements are comparable. So such a semilattice is lower distributive.
3. Consensus functions on a meet semilattice 3. I. Definitions and examples Letl={l,..., i,j,k ,..., n} be an n-set and L a meet semilattice. We denote by 17 an n-tuple of elements of L, indexed by N: n=(xr, . . ..x., . . ..x.) (n is sometimes called a profile). Then A;=, Xi exists and will be generally denoted by Axi. We adopt special notations for some n-tuples that will occur frequently. Let A, B,CbethreesubsetsofNsuchthatAUBUC=NandAnB=BnC=CnA=[a; I7= (A :x; B : y; C: z) is the n-tuple for which for every i in A (respectively in B, C), xi = x (respectively y, z). Similarly, with A = 1v- A, 27= (A : x; A : y) is the n-
56
B. Monjarder
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tuple for which for every i in A (respectively, in A), X,=X (respectively u). Finally (N: x) is the n-tuple with ,u,=x for each i in R;. We denote by L” the set of all n-tuples of elements of N. A consensus function on L is a function, denoted by F, with domain L” and range L. Thus, to every ntuple Z7 in L”, F associates its image s= F(n) EL. Notice that from a mathematical point of view a consensus function on L is nothing more than an n-ary operation on L. 3. I. I. Examples (1) If for every I7 in L”, F(I7) equals a fixed element x0 of L, consensus function. We denote by F” the constant function where least element of L. If L has a greatest element 1, we denote by F’ ing constant consensus function. (2) Let A be a fixed subset of N. We denote by FA the consensus ed by: F(l7) = A xi for every n in L”.
F is a constant x0 equals 0, the the correspondfunction defin-
ierl
FA is also called the A-meet projection consensus function, and any is called a meet projection consensus function. In particular, Fii), I$, is the usual projection, and F, is the n-ary operation defined operation. (3) Let L be a median semilattice, and INI = 2p+ 1. We denote consensus function defined by: F(n) =
v !A rp+1
such function denoted also by the meet by FfP+l) the
for each Z7 in L”.
A well-known result (see, for example, Bandelt and Barthelemy, 1984) says that this formula defines an n-ary operation in a median semilattice, called the (n-ary) median. There it is called the median (or majoritary) consensus function. We now define a large class of consensus functions, which we call the class of federation consensus functions. Such functions are associated with families g of subsets of N. By definition, a family 9 on N is a set of subsets of N. If g is a family, gd = {A s N: A = (N-A) $ g} is its dual family. Three pertinent conditions that a family .cYcan satisfy are: (1) [AC.!% B>A] = [Beg]. (2) [AES, BEG] r=$[AfIB#0]. (3) [AE~] a [Af$g]. A family on N is called a federation if it satisfies condition (1); transversal if it satisfies condition (2); and ipsodual if it satisfies condition (3) (indeed in this case g =.@). A federation consensus function is a consensus function F for which there exists a federation .!J such that
B. .Vonjarder
F(n)
=
V
A
AEd ( ieA
/ Federation
Xi
>
consemu
functions
for each f7 in L”.
57
(IO)
Such a federation consensus function will also be denoted by F$, and F,-(n) will be sometimes denoted by V,$ (AA). Notice that the federation consensus functions contain most of the above examples. Indeed, if 9 is the empty family @, FG equals F”; if g= {B c N: B 2 A}, F3 equals FA. If n=2p+l and g={(AcN: JAl?p+l}, fF=&+t). If L has a greatest element 1, F2n = F, = F’ (where 2” denotes the family of all the subsets of N). One must emphasize the fact that if L is an arbitrary meet semilattice and 9 an arbitrary federation, there is not necessarily a federation consensus function associated with $. Indeed, the above formula (cp) can be defined only for some ntuples of L, so that it defines a partial n-ary operation and not a consensus function. The following facts are obvious or well known. If L is a lower distributive meet semilattice, a consensus function is associated by formula (cp) with each transversal federation. For instance, in this case with k > L&2] we define the k-majoritary consensus function by: Q)(n)
= ,& > ;?, *; ( >
If L is a lattice, a consensus function is associated Moreover, if L is a distributive lattice we have
with each federation
$?
In this last case with n = 2p + 1, the median consensus function has two expressions (notice that {A : IA / rp + I} is an ipsodual family): 4,+1,(n)
= ,A ,v,+,
(Ax’)
=/A/$+1
(iyAxi)’
3.1.2. Remarks (1) Let s be an arbitrary family of subsets of N, grn the family of minimal sets of .E S+ the family of ‘supersets’ of S (B is a superset if it contains A in 9). When the above formula (v) is defined for one of these three families, it is defined for the two others and it gives the same result for the three families. So, if the consensus function Fg exists, all the families between grn and P give this same consensus function. It is more convenient to work only with ‘representatives’ of these classes of equivalent families. Here we choose to work with families satisfying (l), the so-called federations. Another possible choice would have been the ‘Sperner’ families, or ‘clutters’, i.e. the families S for which two subsets in 9 are always incomparable, which is the case for sm. (2) An ipsodual federation is also called a proper and strong simple game. For the consensus functions associated with such families in the case of tournaments, see Monjardet (1978).
B. Monjardet / Federation consensus functions
58
(3) The term ‘federation’ is taken from Mirkin (1981). It seems to us more convenient here than other equivalent terms such as simple game, cohereditary or hereditary family, etc. Aizerman (1985) uses the term ‘federation’ in a related but different sense. We end this section by characterizing the federation consensus functions F” for which ZZ is ipsodual. First notice that the image of a profile Z7= (A : y; A : y) by a federation consensus functions belongs to {x, y, xl\y, xvy} . A consensus function F is bi-idempotent if and only if for all x, y in L and for each BcN,F(B:x; B:y)~(x,y). Definition.
The notation xlly used below means that x and y are two incomparable
elements.
Proposition 1.3. Let FS be a federation consensus function. (1) [VA s N, A $9 implies A E +cIT]if and only if [for all x, y EL with XIIy, and for each BcN, F,(B:x; B:y)>xl\y]. (2) [VAc N, AE~!T impliesA$9] ifand only [for all x,y~L with x/y andfor each BGN, F,(B:x; 8:y)
B : y) =xvy, a contradiction. (3) Since F3, being bi-idempotent, is equivalent to Fg satisfying F,(B : x; B : y) @ {xAy, xVy} for all x, y in L with xlly and for each B c N, (3) is the consequence of (1) and (2). 3.2. Axioms for consensus functions Let L be a meet semilattice and J the set of all its join-irreducible elements. We need the following notation. For 17=(x1, . . . . xi, . . . . x,,) in L”, and s in J, set N,(n) = (in N: s I Xi}. First we define five ‘Arrowian’ properties that can be satisfied by a consensus function F on L. J-Decisivity (J-D): For every SE J and for all Z7,Z~‘E L” [N,(D)
= N,(W)]
I)
[s s F(l7)
w
s s F(l7’)].
B. Monjarder
J-Neutrality
/ Federation
consensus
functions
(J-N): For all S.S’E J and for every l7~ L” [N,(n) = N&7)]
=
[s I F(n)
F,
s’ I F(n)].
J-Decisive Neutrality (J-DN): For all S,S’E J and for all l7,l7’~ L” [N,(n) = N,,(W)] J-Decisive Monotonicity
=
o
s’ I F(I7’)].
(J-DM): For every s E J, and for all J7,17’~ L”
[N,(n) c N,(l7’)] J-Neutral Monotonicity
[s I F(l7)
=
[s 5 F(n)
=
s 5 F(W)].
(J-NM): For every S,S’E J, and for all I7,17’~ L”
[N,(n) c N,Q7’)]
=> [s s F(Z7)
=
s’ I FW’)].
F is called J-decisive, respectively J-neutral, J-decisive neutral, J-decisive monotonic, J-neutral monotonic if it satisfies the property J-D, respectively J-N, J-DN, J-DM and J-NM. The obvious implications between the above properties are summarized below: J-NM u J-DM
a
J-DN u =z. J-D
Other possible properties
=
J-N
for F are the following:
Symmetry (S): For every permutation CTof N and for every l7 in L”, F(oI7) = F(17) (here, oJ7 is equal to (x0(]), . . . ,x0(;), . . . ,x,(,,)). Isotony (I): For all Z7,I7’ in L”, IlsZ7’ F(17) 5 F(l7’). J-Unanimity [s i F(n)].
(i.e. x,5x;
for every i in N) implies
(J-U): For every s in J and every I7 in L”, [N,(n) =N] implies
L-Unanimity (L-U): For every x in L and every I7 in L”, [{ieN: plies [xc F(n)].
xcxi)
=N] im-
J-Autonomy
(J-,4): For every s in J, there exists 17 in L” such that s~F(l7).
Idempotence
(ID): For every x in L, F(N: x) =x.
It is obvious that
L-U
e,
J-U
=
J-A.
We are going to state less obvious implications between the above properties, most of them depending on additional conditions on the set of all join irreducibles. In
60
B. Monjardet / Federation consensus functions
all the following propositions F is a consensus function on a meet semilattice L and J is the set of all join irreducible elements of L. Our first result is true without any condition on J. Proposition
(1) (2) (3) (4)
F F F F
2.3.
is J-decisive monotonic e F is J-decisive and isotone. is J-neutral monotonic e F is J-decisive neutral and J-decisive monotonic. is J-neutral monotonic and Fjt FQ imply F is J-unanimous. is J-decisive monotonic and J-autonomous imply F is J-unanimous.
We first prove (1). Let F be J-decisive monotonic, so that F is J-decisive, and let l7,Z7’ be in L” with 17~ J7’. Then for every s in J,N#7) = {i E N : SI Xi} GN,(l7’)=(iEN:s~x~}. Hence, srF(J7) implies slF(JT’), so that J(F(l7))s; J(F(l7’)) and F(n) I F(W). Conversely, let F be J-decisive and isotone, Z7,J7’, be in L”, and s be in J such that A = N,(n) C_B = N,(l7’) and ss F(Z7). Define J7, = (A : s; A : 0), Q’= (B : s; B : 0). Then n, 5ni’ and the isotonicity of F imply F(l7,) I F(&‘), so that ss F(l7,) (by the J-decisivity of F for J7 and n,) s F(l7;). 3 By the J-decisivity of F (for Z7’ and J7,‘), s~F(l7’). In order to prove (2), we have only to prove that when F is J-decisive neutral and J-decisive monotonic it is J-neutral monotonic. Let F be such a consensus function, l7, J7’, in L”, s, s’ in J such that A = N,(J7) G N,.(17’) = B and SC F(n). We consider the profile J7” = (B : s; L?I: 0); F being J-decisive monotonic N,(n) =A C_N#7”) implies srF(17”). By the J-decisive neutrality of F, NS,(17’)=NS(17”)= B implies S’S F(l7’). 0 We now prove (3). Let F be a J-neutral monotonic consensus function not equal to 9. There exists 17, in L” such that O< F(l7,) =x and there exists s1 in J such that S~IX. We set N,,(J7,)=A (rz,cAcN) and we consider 17’=(N:s,). By Jmonotonicity, st sF(n’). Now let Z7 be in L” and s in J such that N,(J7)=N= Z N,,(n). By J-decisive neutrality, slF(Z7). Finally, let us prove (4). Let s be in J and 17 in L” such that N,(n) = N. There exists 17’ in L” such that s~F(17’). Let us consider l7”= (N: s). By J-decisive monotonicity, JVS(J7”)= N ;! N,(Z7’) and s s F(l7’) imply s I F(W). By J-decisivity, q N,(n) = N,(P) and s I F(T) imply s I F(n). Proof.
Other relations between the above properties depend on the structures of the set J of join-irreducibles and of the semilattice L. We shall use two binary relations defined on J and associated notions of connectivity. First, we define the symmetric relation a on J. For all s,s’ in J, sas’ if and only if sf S’ and SVS’ exists. We say that J is a connected if and only if for all s, s’ in J, there exists a sequence se = s, sl, . . . , sir . . . , Sk= s’ of join irreducibles elements such that for all i=O ,..., k-l,s;crsi+t. Notice that if L is a lattice J is always a connected.’ Notice also that this condition is very weak. Indeed, it is equivalent to the condition L-(O) is a connected poset (i.e.
B. Monjardet / Federation consensus funcrions
61
for all x, y in L-(O), there exists a sequence x0=x,x1, . . . ,x;, . . . ,xp =y such that for all i=O,..., p-l, xi#O and x;
Then F is J-decisive
Proof. By proposition 2.3(2), we have only to show that when F is J-decisive neutral it is J-decisive monotonic. Let n, n’ be in L” such that A =N,(LT)C N,(lT’)=B and ssF(J7). If A = B, s~F(l7’) by J-decisive neutrality. Assume A C B. Since p is non-empty one can find either s,,s2 in J with s1
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If A =N, then s’lF(Z7’) by J-unanimity. If A CN, and s =s’, then s’cF(Z7’) by J-decisivity. If A C N and sfs’, by strong connectivity of p, there exists a sequence of elements of J, se=s’,s I,..., s; ,..., sk=s, and a sequence of elements of L,Xr, . . ..x. such that for i=O,...,k-l,s;pS;+i, i.e. Si
if and only if for each
(E): F is extensive if and only if for each x in L, XI F(N: x).
The last property is defined only if L has a greatest element 1 (i.e. if L is a lattice). I-Idempotence
(l-ID): F is I-idempotent
if and only if F(N : 1) = 1.
We have the following obvious result. Lemma 6.3. F is meet compatible 1-idempotent.
or F is idempotent
3 F is extensive
=. F is
Proposition 7.3. (1) F is J-unanimous
o F is meet compatible. (2) Zf F is isotone or J-decisive, then F is J-unanimous o F is extensive. (3) Zf L is a lattice and F is J-decisive, then F is J-unanimous o F is I-idempotent. (4) Zf F is J-decisive monotonic, then F is J-unanimous e F is J-autonomous.
Proof.
(1) If SC AXi, one has N#Z)=N and by J-unanimity sz~F(Z7). Then /\Xi= F(Z7). Conversely, N,(ZZ) = N implies ssxi for each i. Thus, SI l\Xil F(Z7). (2) From (1) and Lemma 6.3. one has only to prove that if F satisfies one of the stated hypotheses, and is extensive, it is J-unanimous or (equivalently) meetcompatible. Assume F is J-decisive. Let s be in J, Z7 in L” such that N,(ZZ) = N and Z7’= (N: s). By extensivity, slF(Z7’) and by J-decisivity ssF(Z7). Assume F is isotone and let Z7= (xi, . . . *Xi,. . . ,x,). Consider Z7’= (N: Axi> 1l7.
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63
By extensivity and isotony, l\XilF(17’)sF(17). (3) By (1) and Lemma 6.3, if F is J-unanimous, it is meet compatible and thus, if L is a lattice, I-idempotent. If F is I-idempotent, let n such that N,(n) =N; then N,(n) = N&V : 1) = N; by J-decisivity, s I F(N : 1) = 1 implies s I F(D). (4) This is a consequence of assertion (4) in Proposition 2.3. 0 Before giving the final result of this section we notice that s/Is’ implies that szs’ and SVS’ exists, i.e. sas’. Then we have Lemma 8.3. J is /3 strongly connected imply J is CTconnected and J is non-empty. By using Propositions 2.3, 3.3, and 4.3, Theorem 5.3, Proposition 7.3, and Lemma 8.3 one easily obtains the following theorem that summarizes the results of this section. Theorem 9.3. Let L be a meet semilattice such that the set J of ah join irreducible elements of L is p strongly connected. The six following properties are equivalent. Moreover, if L is a lattice they are equivalent to the last property: (1) F satisfies J-NM and F+ F”.
(2) (3) (4) (5) (6) (7)
F F F F F F
satisfies satisfies satisfies satisfies satisfies satisfies
Remark. In connectivity let us define s
J-D and J-U. J-D and MC. J-D and E. J-DM and J-A. J-D, J-N, I, and Ff F". J-D and l-ID.
view of Theorem 9.3 it is important to have conditions for of J. The following condition is obvious but useful. For spJs’ if and only if sfs’ and there exists s” in J such that p, is strongly connected, let us say that J is pJ strongly pJ strongly connected J is /3 strongly connected.
the p strong all s,s’ in J, S,S’YZS”and connected.
4. Characterizations of federation consensus functions In this section we study the consensus functions satisfying properties defined in Subsection 3.2. Especially we characterize the federation consensus functions defined in Subsection 3.1. Depending on the structural properties of the semilattice L, such characterizations can use a quite different system of axioms. Special cases of these results provide characterizations of projection consensus functions and of median consensus functions. The following notion is basic in the proofs of these results.
B. Monjardet
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Definition.
Let F be a consensus function on the meet semilattice L and s in the set J of all join irreducible elements of L. We say that A C N is an F s-decisive set, or simply an s-decisive sef if there exists f7 in L” such that N,(n) =A and SI F(f7). Recall that if s,s’ are in J, sos if and only if
SVS’
exists.
Proposition
1.4. Let F be a consensus function on the meet semilattice L. If F is J-decisive, for each s in J there exists a family 9” on N such that, for every I7 in L”, F(J7) = V {s E J : N,(n) E SS}. Moreover, if F is isotone, then gS is always a federation and if F is J-neutral, SS = S-s, as soon as s and s’ are in the same connected class of the relation a. Proof. Let F be a J-decisive consensus function, s in J and A an s-decisive set. Since F is J-decisive we get that A is s-decisive if and only if for every J7 in L” such that N,(n) = A,s< F(I7). For each s in J, we set S,= {A EN: A is s decisive}. Thus, for every J7 in L”, ss F(l7) if and only if N,(n) E 9”. Notice one can have .5&= the empty family @, which means s s F(IZ) for every J7 in L”. Then F(t7) = V {s E J : N,(n) E gS}. If F is J-decisive and isotone it is J-decisive monotonic (Proposition 2.3(l)). Let s be in J such that SS#@, A in SS with B3A. We set l7=(A : s; A : 0) and ff’=(B:s, B:O). Then srF(J7) and by J-DM, ssF(Z7’). Thus, BE$$,, which proves that ,9$ is a federation. Notice that Q, is also a federation. Let F be a J-decisive and J-neutral consensus function; let s,s’ be in J such that s#s’, svs exists and gS+@; let J7 be in L” such that N,(Z7)=A ES, and J7’= (A : s’; A : 0). Define J7” = (A : sVs’; A : 0). By J-D for J7 and J7”, SC F(IIn). By J-N for J7”, s’sF(I7”). By J-D for J7” and J7’, S’S F(l7’). Thus, A ~3,,, and s,rs,,. One shows similarly that S,, G 9”. Thus, ,c&= SS, . Now let s and s’ be in a connected component of the relation a, with gS# @. Then there exists a sequence of join irreducible elements so=s,sI, . . . . Q,s~+~, . . . . sp=s’, such that for k=O, . . . . p-l, s,Vs,+, exists. Then by the above result S, = gSk+, for k = 0, . . . ,p - 1, so 5$ = s,..
cl
Proposition
2.4. A J-neutral monotonic
sensus function: in L”,
there exists a federation
F(n) =Av_
(
consensus function F is a federation conS C 2N such that F = F,,i. e. for every l7
i?, xi >
Moreover, if L is not a lattice, g is a transversal federation. Proof.
Let F be a J-neutral monotonic consensus function. It is J-decisive and isotone; so by Proposition 1.4, for every s in J there exists a federation SS such that F(J~)=V{SEJ: N,(JI)ES~}. If F=F ‘, for every s in J S”= Q, and we have F=F,.
B. Monjardet / Federation consensus functions
65
If not, there exists s in J such that .?Fs#@. Let A be in gs and s’#:s in J. By the definition of s-decisive sets there exists 17 in t” such that N,(n) =A and srF(J7). We set J7’= (,4 : s’; A : 0). By J-DN we have s’sF(J7’) so that A ES’,, and gXC .&. Since ++ @, we obtain similarly & G gs, and thus gs2,=gs,. Thus, Z&is the same federation $, for every s in J, and we have for every f7 in L”, F(R) = V{SE J: N,(n) ES}. Let I7 be in L”; we are going to show that F’(J7) = v Ae9
A x; ( iaA
>
exists and equals F(J7). Let A be in .!+Fand s in J such that slAieA xi; then N,(n) 2 A, and since 9 is a federation, N,(n) E .F; so ssF(II), and thus AiEA xir F(n); so finally F’(R) exists and is less than or equal to F(H); let s be in J such that SC F(n); then N,(J7) =A E 9 implies s5 AieA xi5 F’(Z7); then F(l7)~ F’(l7) and finally F(l7) = F’(n). Let us assume that L is not a lattice; there exist X,y in L such that xVy does not exist. Let A be in $ and assume A E g; we set J7= (A : x; A : y); for every SIX, N,(J7) 2 A; since 9 is a federation, one gets s~F(l7), so x~F(l7); similarly, one gets yl F(n), a contradiction. Thus, S satisfies the property: A E 5 implies A e R since $ is a federation, one obtains that if A, BE .9, A n B # 0, i.e. 9r is a transversal federation. 0 Theorem 3.4. Let L be a non-lower distributive meet semilattice and F a consensus function on L. Consider the following conditions: (1) F is J-neutral monotonic and FZ F’. (2) F is J-decisive, J-neutral, isotone and F # F’. (3) There exists A, 0 CA c N such that for every I7 in L”, F(I7) = l\isz Xi (i.e. F= FA). (4) There exists A, or c A G N such that for every r/ in L”, F(I7) = l\ieA Xi* IfL is not a lattice, then conditions (1) and (3) are equivalent. If L is a lattice, then conditions (l), (2) and (4) are equivalent. Moreover, A = N (or Q in the lattice case) if and om’y if F is symmetric, and /A[ = 1 if and only if F is bi-idempotent.
Suppose L is not a lattice. (1) = (3). By Proposition 2.4, a J-neutral monotonic consensus function F is a federation consensus function FS with &@a transversal federation on N and B # @ (since F#p). Condition (3) is equivalent to saying that 9 has an unique minimal set A. Let us assume that $ has two different minimal sets, A and B, so that A tl B is not in S. Since L is not lower distributive, there exist s,s’ in J and x in L such that sss’, s%x and SCS’VX (see Proposition 1.2). Consider the following profile n:(A-AnB:s’; B-AflB:x; AnB:s’vx; N-AUB:s’). N,,(l7)=AU (A U B) is in S so s’~F(l7). Since for each s in J such that SIX, N,(n) a B is in 9 one gets x5 F(l7). Then S
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(3) * (1). We consider a meet projection consensus function FA . Let n, l7’ be in L”,s,s in J, N,(II)=BGN,,(II’)=C and assume s~F~(I7)=l\;~~ xi. Then AGiv#7)=BZ;C, so /\. rcc X; I AieA x,! = FA(I7’). Then N,,(n’) = C implies S’SA\ipcXI sFA(I7’). If L is a lattice, conditions (1) and (2) are equivalent by Proposition 4.3. The same proof as above shows that (1) is equivalent to (4), but in this case, A can be empty since F, = F’. If A = N (or A = 0 in the lattice case) FA is symmetric. If 0 CA c N, let us consider n=(A : s; A : 0) with s>O; FA(17) =s. For any permutation o that exchanges an element of A and A, FA(o17) =O
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6.4. Let L be a lower distributive meet semilattice and Fa consensusfunction on L. Consider the following conditions: (1) F is J-neutral monotonic. (2) F is J-decisive neutral and J-decisive monotonic. (3) F is J-decisive, J-neutral and isotone. (4) F is a federation consensus function F,- with 9 a transversal federation. (5) F is a federation consensus function F9. If L is not a lattice, then conditions (I), (2) and (4) are equivalent. If L is a (distributive) lattice, then conditions (l), (2) (3) and (5) are equivalent. Theorem
In all cases (1) and (2) are equivalent conditions by Proposition 2.3. Let L be a lower distributive meet semilattice that is not a lattice. By Proposition 2.4, if F satisfies (l), F= FS with F a transversal federation. Conversely, let F” be such a consensus function and let IT, J7’ be in Ln, s,s’ in J with N,(Z7) G N,,(J7’) and s~F,(l7) = V, (AA). By lower distributivity, there exists A in S such that s-(/\ieA xi. Thus, A GN,(J~)I;N,,(J~‘) and s’_(l\isA x~sF(I~‘), SO that F” is Jneutral monotonic. If L is a distributive lattice, conditions (1) and (3) are equivalent by Proposition 4.3. By Proposition 2.4, if F satisfies (l), F is a federation consensus function, and 0 one proves the converse implication as above. Proof.
Remarks.
(1) In the above theorem, F can be the constant function F” in the general case or F’ in the lattice case. One can exclude these possibilities by adding axioms, such as the idempotence property: F(N : x) =x. (2) The comparison of Theorems 3.4 and 6.4 shows that essentially the same axiom of J-neutral monotonicity characterizes all the federation consensus functions in the distributive case, but only the meet projection consensus functions in the nondistributive case. Indeed, one can also characterize the meet projection consensus functions in the distributive case by a much stronger neutral monotonicity axiom: the L-NM axiom, obtained by replacing J by L in the definition of the J-NM axiom. Nevertheless, L must not be a meet tree semilattice (see Barthelemy, Leclerc and Monjardet, 1984 b). Corollary 7.4. Let L be a median semilattice and Fa consensus function on L. Consider the following conditions: (1) F is J-neutral monotonic, symmetric and bi-idempotent. (2) F is J-decisive neutral, J-decisive monotonic, symmetric and bi-idempotent. (3) F is J-decisive, isotone, symmetric and bi-idempotent. (4) F is the median (majoritary) consensus function. If L is not a lattice, then conditions (l), (2) and (4) are equivalent. If L is a (distributive) lattice, then all four conditions are equivalent.
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Proof. Let L be a median semilattice that is not a lattice. By Theorem 4.4, conditions (1) and (2) are equivalent and imply that F=F,- with 3 a transversal family. The symmetry of F,- implies that 9 = {A c N: IA 1r k}. By Proposition 1.3, the biidempotence of FJT implies that .9 is ipsodual, i.e. 3 =Bd = {B G N: Beg} = {BsN: lB/>n-k}. Then n=2k-1 and F,(17)=V,Aizk (/jieA Xi) is the median consensus function. Conversely, it is clear that such a consensus function satisfies (1). Assume now that L is a distributive lattice. Then (1) implies (3) by Theorem 4.4. In order to prove the converse implication, it suffices to prove that when F is Jdecisive and bi-idempotent F must be J-neutral, since for a lattice, J-NM is equivalent to J-D, J-N and I (Proposition 4.3). Let s,s’ be in J and I7 in L” such that N,(JT)=N,(JT)=A and s~F(l7). One has A#@, since if A=QI, 0~s~ F(N: O)=O, a contradiction. Thus, there exists Xi in I7 such that s and S’lXi, and then SVS’ exists. Set J7’=(A : svs’; A : 0). Since N,(Z’J)=A, one has by J-decisivity O
5. Conclusion
It is possible to summarize the main results of this paper by distinguishing three different structural situations for the meet semilattice L on which the consensus functions are defined. Situation 1: L is lower distributive. Then, in this case, the axiom of J-Neutral Monotonicity characterizes any federation consensus function (Theorem 6.4). Situation 2: L is non-lower distributive. Then, if F+F’, the axiom of J-Neutral Monotonicity characterizes only the meet projection consensus functions (Theorem 3.4). Situation 3: L is non-lower distributive, and J is p strongly connected. Then the axiom of J-Neutral Monotonicity is equivalent with (for instance) the J-decisive and J-unanimity axioms (Theorem 4.4). In terms of social choice theory, in this situation the independence and Pareto axioms characterize the oligarchic consensus functions. Moreover, since all the results of this paper can be easily dualized, we have the three dual situations l*, 2*, and 3* where L is a join semilattice either upper distributive, or not upper distributive, or not upper distributive and with the set M p* strongly connected. Here, M is the set of all meet irreducible elements of L, and the binary relation p* on M is defined by: sp*s’ 1) s#s’ and there exists x in L such that S,S’ZX and S’>SAX. Notice that in the dual case, meet projection consensus functions become join projection consensus functions (sometimes called cooligarchic functions): F(Z7) = Vip xi. We list now some ‘concrete’ results that are special instances of the ‘abstract’ results proved in the above situations. Notice that to determine the relevant situation is not always trivial, especially for situation 3 or 3*. A
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Situation 1. Characterization of ‘majoritary opinion functions’ and of the median operation for tournaments (Monjardet, 1978). Characterization of ‘social welfare functions’ for binary relations (Mirkin, 1981). Characterization of ‘decisive’ consensus functions and of the majoritary consensus for n-trees (Margush and McMorris, 1981; McMorris and Neumann, 1983). Characterization of ‘local operators’ for the set of all choice functions or the set of all choice functions satisfying the ‘heritage’ property (see Aizerman, 1985). Situation 2. Characterization of local operators for the sets of all choice functions satisfying the ‘heritage’ and ‘concordance’ or ‘outcast variants’ properties (see Aizerman, 1985). Situation 3. Characterization of oligarchic consensus functions for the set of all equivalence relations (Mirkin, 1975; Leclerc, 1984; Fishburn and Rubinstein, 1986). Characterization of oligarchic social choice functions for the set of all partial orders relations (Brown, 1975). Situation 3*. Characterization of co-oligarchic consensus functions for the set of all equivalence relations (Neumann and Norton, 1986). It is also easy to see how the abstract results allow us to obtain new results. We have first to assume that the set of complex objects we wish to aggregate can be naturally endowed with a semilattice structure (a frequent situation). Then we have to identify the join or (and) meet irreducibles of this semilattice and to study the properties of relations fi or (and) /3*. According to these properties we are in one of the six structural situations 1, 2, 3, I*, 2*, or 3* and it suffices to apply the corresponding results. Such a procedure, which could for instance be applied to weak orders, partial orders contained in a linear order, partial orders containing a given partial order, ultrametrics, ‘fuzzy’ relations, ordered trees and Buneman trees (see BarthClemy, Leclerc and Monjardet, 1986), will be not made more precise here. Notice that in some cases it can raise non-trivial problems. Notice also that Leclerc (1990) has rather thoroughly investigated the case of the aggregation of ‘fuzzy preferences’, i.e. equivalently the case of consensus functions defined on the lattice of all residuated mappings defined between two lattices. We end this conclusion by pointing out further directions of research. In Theorems 3.4 and 6.4 we have obtained axiomatic characterizations of projections (i.e. ‘dictatorial’) functions by using the bi-idempotence axiom. It would be interesting to obtain other characterizations of this very special case. Indeed, this case (for instance, Arrow’s theorem for linear orders) could be more clearly understood within the study of more general lattice consensus functions. Such functions would be functions F=D” ---rA, where D and A would be subsets of one or even two semilattices. Finally, a study of the relation p in semilattices and lattices would be useful. In this direction recall that in a distributive lattice the relation p is exactly
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the order relation < defined on the set J of all join irreducible elements of L. In the case of a geometric lattice L (see, for instance, Aigner, 1979), one easily see that the relation p is exactly the ‘perspectivity relation’ defined on the set J of all atoms of L. Then p is strongly connected if and only if L is ‘indecomposable’, which is for instance the case of the (geometric) lattice of all partitions of a set.
Acknowledgements The author is grateful to V. Duquenne and B. Leclerc for helpful discussions concerning the relation p defined in this paper, and to F.R. McMorris and anonymous referees for significant improvements of style.
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