Abstract
A value of a sequence of elements of a finite metric space is an element for which is minimum. The –function with domain the set of all finite sequences on and defined by is a value of is called the –function on . The and functions are the well-studied median and mean functions, respectively. In this note, simple characterizations of the –functions on the -cube are given. In addition, the center function (using the minimax criterion) is characterized as well as new results proved for the median and antimedian functions.
1. Introduction
A consensus function (aka location function) on a finite connected graph is a mapping , where denotes the set of all subsets of , and with . The elements of are called profiles and a generic one of length is denoted by . Let denote the usual geodesic distance, where is the length of a minimum length path joining vertices and . Suppose the graph represents the totality of possible locations. Then a profile is formed where represents the best location from the point-of-view of client (voter, customer, and user) . A typical approach in location theory is to find those vertices (locations) in that are “closest” to the profile . There has been much work in this area of research, ranging from practical computational methods to more theoretical aspects. Since Holzman’s paper in 1990 [1], there have been many axiomatic studies of the procedures themselves which resulted in a much better understanding of the process of location (for a small sample, see [2–4] and references within). Now suppose the vertex set is the set of all linear orders (preference ranking) on a given set of alternatives. In this consensus situation, a profile could represent the collection of ballots of the voters labeled by the set ; that is, is the preferred ranking of alternatives by voter . Here a closest vertex to would represent the entire group’s preferred consensus ranking. Many references for this classical situation can be found in [5] and other books on voting theory. Another classic situation, and one pertinent to our study, is the process of selecting a committee from a slate of candidates. Here each of voters is to nominate a subset of candidates, so a ballot is simply a profile where each is a subset of the candidates [6, 7]. The vertices of the graph are the subsets of candidates and the committee consensus function will return one or more subsets closest to the profile.
Four popular measures of the closeness, or remoteness, of a vertex to a profile are as follows:(1)The eccentricity of , (2)The status of , (3)The square status of , (4)The status of ,
The consensus functions based on the these measures of remoteness have been defined as follows:(a)The center function, denoted by Cen, is defined by (b)The median function, denoted by Med, is defined by (c)The mean function, denoted by Mean, is defined by (d)The -function, denoted by , is defined by The median and mean functions are special cases of the –function, but earlier work [8–10] shows a striking difference between the case of and .
In this paper we focus on consensus functions on the n-dimensional hypercube whose vertex set is . Of course the natural realization of is the set of all subsets of an -element set. Recall that, for and vertices in , is an edge of if and only if . We set if and only if for all . Let be the usual Hamming distance, where , so that is an edge if and only if . Let denote the addition modulo 2, and define . For a profile and let . Let = and = . Note that for all . Also it is easy to see that, for and vertices in , . We set to be the vertex with ’s everywhere except in the coordinate. So, for example, in
Let denote the subgraph induced by the vertices comprising . Note that is isomorphic to for all , and so intuitively is simply a “translation” of to another position within . Our goal is to use the particular structure of to present a very simple unifying approach to give axiomatic characterizations of the consensus functions Cen, Med, and on these graphs. Mulder and Novick [10, 11] have given an elegant set of axioms characterizing the function on all median graphs (of which is a special case) whereas our axioms are essentially straightforward properties that follow from the definitions. At present the most general graph for which characterizations exist for Cen, Mean, and is a tree [9, 12–14]. An interesting weighted version of on is studied in [6].
We mention that the following results can be framed in the more abstract context of finite Boolean algebras, as it is done in [15–17]. We prefer to work in the more specific situation of the -cube where properties become quite easy to visualize, and yet we are working without loss of generality because every finite Boolean algebra is isomorphic to an -cube.
2. The Axioms and Characterizations of Cen, Med, and -Function
In this section we give two very simple properties that will allow us to establish a general result that can be used to give a new way to view , and defined on . Let be a consensus function on . Our key axiom for a consensus function is the following.
Translation (T). For any profile and vertices and of ,
Note that this is equivalent to if and only if .
Now let and be consensus functions on and let be a vertex. We say and agree at if for any profile
Theorem 1. If the consensus functions and on both satisfy (T) and agree at a vertex , then .
Proof. Let be a profile and . Then there exists such that . Since satisfies (T), we haveBecause and agree at ,Since satisfies (T),Hence if and only if .
Theorem 1 implies that if and are consensus functions on and both satisfy (); then if the conditions placing 0 in are the same as the conditions placing 0 in .
As observed before, for , and vertices in . Using this and the definitions it is easy to see that Cen, Med, and all satisfy (). Therefore, characterizations will follow once the conditions are obtained for when , , and . We present these results in a series of lemmas and corollaries.
Let and set , that is, the number of ones that appear in the representation as a vertex of . Let be a profile on . Then is defined to be
Lemma 2. Let be the center function on and a profile. Then
Proof. The result is clear because in , and for any profile .
Corollary 3. Let be a consensus function on . Then if and only satisfies (T) and for every profile and
Mulder and Novick [10] give an elegant characterization of on , which was extended to all median graphs in [11]. We will give another characterization using the approach given by Theorem 1. For a profile let . The next result has been noted in [10].
Lemma 4. Let be the median function on and a profile. Then
Corollary 5. Let be a consensus function on . Then if and only satisfies (T) and for any profile ,
For the function it is easy to see from the definitions that, for any profile and in ,
As in [17] we consider the -characteristic of a profile to be the numberLemma 3.12 in [17] gives the following result.
Lemma 6. Consider the function on , and let be a profile. Then
Corollary 7. Let be a consensus function on . Then if and only satisfies (T) and for any profile ,
Here are three other examples of consensus functions that satisfy the Translation property. However it is clear that these functions would not be useful in committee elections or as location functions, for instance.
Example 1. Let be the consensus function on defined by for any profile . That is, is a standard projection function. Then clearly satisfies ().
Example 2. Let be the consensus function on defined by for all profiles . That is, is the constant function with ouput being the entire vertex set . Then satisfies (), and moreover it can be easily shown that it is the only constant function that satisfies ().
Example 3. Let be the consensus function on defined by for all where is the set of vertices appearing in the profile . Then clearly satisfies ().
The function allows us to see some of the implications of imposing (). First we need to recall one of the crucial axioms for the characterization of the consensus function [10, 11, 18].
Consistency (C). The consensus function satisfies () if, for profiles and ,
Proposition 8. A consensus function on satisfies (T), (C), andif and only if .
Proof. Clearly satisfies the conditions, so now let be a consensus function that satisfies (), (), and the intersection condition. Let for all . Then since satisfies () we have for all . Now let be an arbitrary vertex. Then and thus . So if is any vertex in , and since satisfies () we haveTherefore , which means that for all profiles of length 1. Using () and induction we conclude that for all profiles , that is, .
3. Alternative Characterizations of the Median and Antimedian Functions on
For any profile such thatfor , let be the vertex in such thatfor . We will say that a location function satisfies the condition iffor any profile . We have previously noted that the median function satisfies () and we will show below that, as expected, satisfies (Maj). However, there are other location functions that satisfy these two conditions, such as , for example. But, arguably is not a very reasonable method of consensus or location. So our next step is to invoke a condition that restricts the range of a location function.
For any profile such thatfor define the Condorcet score of to beObserve that if the profile length is odd, then . A location function satisfies Restricted Range (RR) iffor any profile .
We can now give a completely different characterization of from that found in [10].
Theorem 9. Let be a location function on . Then if and only if satisfies (T), (Maj), and (RR).
Proof. Assume . We already know that satisfies (), so we only need to show that satisfies (Maj) and (RR).
We will follow the notation given above. Let be a profile such thatfor and let . Now let be such that for some . First note that, for every , because and are equal for at least of the ’s,Sincewe haveTherefore and satisfies (Maj).
Let be the vertex in such thatfor . For any vertex such that and for any such that we get that , , and of course . Observe thatSince whenever it follows that and so . Moreover, if is vertex in such that for some where , thenIn this case, and so . It now follows thatwhereTherefore, and hence satisfies (RR).
For the converse, assume that satisfies (), (Maj), and (RR). We will show that . Let be a profile. Then, using Theorem 9,where for . Observe that , and since satisfies it follows that . Since satisfies () we getIt now follows that for any profile . Therefore,for any profile . We know that and, by (RR), that for any profile . Hence for any profile and we are done.
The three consensus functions we have considered all minimize a criterion in order to produce vertices that are close to a given profile of vertices, and as such are useful in location theory. When finding locations to place noxious entities, it is more appropriate to maximize rather than minimize these objective functions, and the resulting “anti”-functions have also been well-studied. Because we have proved Theorem 9 about the median function, we mention the antimedian function, denoted by AM, defined by
has been characterized on in [19], but we will give an alternate characterization as a corollary to Theorem 9. As before is a profile such thatfor . Let be the vertex in such thatfor . We will say that a location function satisfies condition (Min) iffor any profile . Corollary 3 now follows from the proof of Theorem 9 in the obvious way by reversing the inequalities.
Corollary 10. Let be a location function on . Then if and only if satisfies (T), (Min), and (RR).
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.