AN AXIOMATIC APPROACH TO DISTANCES BETWEEN CERTAIN DISCRETE STRUCTURES T. Margush A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August 1980 il ABSTRACT The need for a quantitative tool to measure distances between various kinds of discrete structures is apparent from the literature. Distance concepts can also be used to judge the "goodness" of a candidate for the consensus of a collection of structures. Several axiomatic characterizations of distance functions on collections of binary relations and sets of subsets are presented. Applications to tree structures, preference relations, hierarchical classifications, cladistic characters, and partitions and other structures resulting from non-hierarchical classification techniques are discussed. The median of a collection, with respect to an appropriate distance function, is proposed as a candidate for consensus and methods for determining the median in the various contexts are presented. Ill ACKNOWLEDGMENT I would like to express my appreciation to Dr. F. R. McMorris for his continued interest in the progress of my research and his many suggestions and assistance in the writing of this dissertation. I would also like to thank Linda Shellenbarger for her expertise in typing. IV tS&e TABLE OF CONTENTS Page CHAPTER 0 INTRODUCTION............................................................. 1 CHAPTER 1 DISTANCES BETWEEN BINARY RELATIONS .... 4 CHAPTER 2 DISTANCES BETWEEN PARTITIONS AND OTHER SETS OF SUBSETS......................................................................37 CHAPTER 3 A CONSENSUS FOR DISCRETE STRUCTURES. ... 65 BIBLIOGRAPHY 78 CHAPTER 0 INTRODUCTION In 1951, K. Arrow [2] listed several "reasonable" axioms which should be satisfied by social welfare functions. The axioms proved to be inconsistent, that is, there was no social welfare function that satisfied all of the axioms. Much work has since been done, revising Arrow's axioms in order to obtain characterizations of various social welfare functions. In 1962, Kemeny and Snell [18] gave another approach to the problem of determining "good" social y welfare functions. They gave an axiomatic characterization of a distance function on the collection of choices and then examined properties of medians with respect to the resulting distance function. K. Bogart [5,6] generalized the work of Kemeny and Snell and showed that in many instances, majority rule resulted in a median for certain collections. In numerical taxonomy, and other areas of science, there is considerable interest in the comparison of hierarchical classifications of collections of objects. Several types of "distance functions" have been proposed and their relative advantages discussed [10,13,15,20, 27,28,30]. An axiomatic development of both distance concepts and consensus techniques on the various structures involved is an -1- 2 important step in the attempt to quantify some of the subjective analysis which takes place in these areas. Similar questions also arise in the area of non-hierarchical classification techniques, notably in cases concerning partitions of sets [8,12,22]. Chapter 1 begins with a summary of a portion of K. Bogart's results found in [5], which is an extension of that of Kemeny and Snell [18]. These results are then generalized to arbitrary collec­ tions of binary relations on a finite set J. The seven axioms listed by Bogart are eventually replaced by a pair of axioms which are equivalent to Bogart's on the collections of binary relations considered by him. Chapter 2 deals with similar concepts for sets of subsets of j/. It first parallels the development in Chapter 1, then, following along the lines of a paper by E. L. Perry [22], examines distance concepts for collections of partitions that generalize to a new distance function on sets of subsets. Several representations of sets of subsets are introduced and their relative advantages discussed in light of particular applications. The last chapter examines the question of consensus for the various discrete structures of Chapters 1 and 2. First, Bogart's results for collections of preference relations [5,6] are extended to arbitrary collections of binary relations and sets of subsets. Majority rule is shown to be a median in each case (with respect to the distance functions characterized in Chapters 1 and 2) and questions of existence and uniqueness are examined. 3 For sets of subsets and the extension of Perry's distance function, a different consensus is defined. With the proper representation, it is shown that this may also be viewed as a type of majority rule. Questions of existence and uniqueness are then examined in this context. CHAPTER 1 DISTANCES BETWEEN BINARY RELATIONS The use of binary relations to model preferences on a collection of objects has been widely accepted in the behavioral sciences [5, 26,33,34]. If ji is a finite collection of objects and a and b are elements of j/, then we may represent the preference of a to b by the ordered pair, (b,a). An individual's "preferences" among the elements of if can then be represented by a binary relation, R, on jf, where (b,a) £ R means the individual prefers a to b. Not all binary relations on if can be reasonably interpreted as someone's preferences. For example, {(a,a)} is a binary relation on jf = {a,b}. The definition of "preference" tacitly assumes the comparison of at least two distinct objects, and therefore, this relation could not be the representation of an individual's preferences in if. If a binary relation, R, on if, is to represent preferences, the pair (a, a) should not be an element of R for any a in ft. That is, R should be an irreftexive relation on if. There are other commonly accepted properties of the relation "is preferred to" that can be easily interpreted in the language of binary relations. If a,b € / and a is preferred to b, then b cannot also be preferred to a. Thus, if (b,a) € R, then (a,b) £ R or R is an -4- 5 antisymmetric relation. Preference is usually thought to be transitive: if a is preferred to b, and b to c, then a is preferred to c. In ordered pair notation: if (b,a) € R and (c,b) € R, then (c,a) £ R or, R is a transitive relation. The set j/ will always represent a finite collection of objects. DEFINITION 1.1. A preference relation on J is a binary relation, R (R£ that is irreflexive and transitive. If R is a preference relation on xf, and (a,b) € R, then (b,a) <£ R. Otherwise, since R is transitive, both (a,a) and (b,b) £ R. But R is irreflexive. Preference relations therefore, are antisymmetric, which the previous discussion indicated was a desirable property. In some applications it is useful to consider binary relations which are irreflexive and antisymmetric. Bogart calls these intransitive preference relations or antisymmetric orderings [6]. Associated with each preference relation on J is a partial order and its associated Hasse diagram. If R is a preference relation on J, for each a,b € xf, define a < b if and only if (a,b) £ R. Since R is transitive and antisymmetric, ,<) is a partially ordered set. Since R is irreflexive, a / a for all a € xf and hence '<' is what is called a strict partial order on xf. Example 1.1 shows how a preference relation can be represented in an order theoretic sense by the Hasse diagram of its associated partially ordered set. 6 EXAMPLE 1.1. Let J = {1,2,3,4}, P = {(1,2),(1,3),(1,4),(3,4)}, and Q = {(1,2),(1,3),(1,4)}. P and Q are preference relations on j<f. Their associated Hasse diagrams are: 3 In an attempt to quantify the concept of "agreement between preferences", several "distance functions" on the collection of preference relations on J have been proposed and their relative advantages widely discussed [9,23,24]. if will always denote a collection of binary relations. The underlying set will be clear from the context. DEFINITION 1.2 , A distance function on g is a real valued function, d: £ * £ -> R, satisfying the following properties: DI: If A,B € £ then d(A,B) 0. Equality holds if and only if A = B D2: If A,B € £ then d(A,B) = d(B,A). D3: If A,B,C €$ then d(A,C) < d(A,B) + d(B,C). A distance function on £ is usually called a metric and (£,d) a metric space. K. Bogart, generalizing the work of Kemeny and Snell [18], lists properties DI, D2, and D3, along with four other "reasonable" properties, as a collection of axioms that any measure of distance on collections of preference relations should satisfy [5]. 7 Before listing these additional axioms I will define some geometrical concepts in collections of binary relations. These concepts are generalizations of Bogart's definitions. If we consider ~P, the collection of all preference relations on , there is a natural relationship that sometimes occurs among three elements of "P, P, Q, and R. The preferences represented by Q are said to be "between" those of P and R if Q prefers b to a whenever both P and R do and only if at least one does. This relationship is made more precise by the following definition. DEFINITION 1.3. Let P,Q,R € £. Q is between P and R (denoted B(P,Q,R)) if and only if PDRCQCPUR. If in addition Q / P and Q / R, then Q is strictly between P and R (B*(P,Q,R)). DEFINITION 1.4. Let P,Q £ £. A line segment from P to Q is a sequence, P = P ,P2,...,Pn = Q, of relations from Ç which satisfies the following conditions: 1.