Andrew Young School of Policy Studies Research Paper Series Working Paper 06-57 May 2006 Department of Economics Ordinal Distance, Dominance, and the Measurement of Diversity Prasanta K. Pattanaik University of California, Riverside Yongsheng Xu Georgia State University This paper can be downloaded at: http://aysps.gsu.edu/publications/2006/index.htm The Social Science Research Network Electronic Paper Collection: http://ssrn.com/abstract=905912 ANDREW YOUNG SCHOOL OF POLICY STUDIES Ordinal Distance, Dominance, and the Measurement of Diversity Prasanta K. Pattanaik Department of Economics, University of California, Riverside, CA 92521, U.S.A. ppat@ucrac1.ucr.edu Yongsheng Xu Department of Economics, Andrew Young School of Policy Studies, Georgia State University, Atlanta, GA 30303, U.S.A textttyxu3@gsu.edu May 2006 Abstract. In discussing issues relating to, for example, biodiversity of different ecosystems and cultural diversities of various communities, the measurement of diversity is inevitable. This paper uses an ordinal concept of distance between objects to provide a characterization of all diversity-based rankings of sets of objects that satisfy a plausible property of dominance; we also characterize a specific member of this class. JEL Classification Numbers: D00, Q00, Z00 Keywords: Diversity, ordinal distance, dominance, revealed distance function 1 Introduction The purpose of this paper is to consider a class of rules for comparing sets of objects1 in terms of the degrees of diversity that they offer. Such comparisons of sets are important for many purposes. For example, in discussing biodiversity of different ecosystems, one is interested in knowing whether or not one ecosystem is more diverse than another. Similarly, when discussing issues relating to cultural diversities of various communities, one may be interested in knowing how these communities compare with each other in terms of cultural diversity. In the economics literature, there have been several contributions to the measurement of diversity. Weitzman [6,7,8] develops a measure of diversity based on cardinal distances between objects. Among other things, Nehring and Puppe [3] provide a conceptual foundation for cardinal distances in Weitzman’s framework. Weikard [5] discusses an alternative measure of diversity; Weikard’s measure is based on the sum of cardinal distances between all objects contained in a set. Underlying much of our everyday discussion of diversity, we have some intuition regarding the extent to which objects are dis-similar2 , though, in its coarsest form, this intuition may distinguish between only two degrees of similarity by declaring that two objects are either similar or dissimilar. It is difficult to see how one can compare the diversity of one group of objects with that of another without some notion, however coarse, of the distances or the degrees of dis-similarity between different objects in these sets. The requirement that distances be cardinally measurable does, however, seem rather restrictive in many contexts. Thus, in considering linguistic diversity, we may not be able to compare the extent to which the (linguistic) dis-similarity between an English person and a Chinese person exceeds the difference between a Chinese person and a Hindi-speaking person on the one hand and the extent to which the difference between an Italian person and a Hindi-speaking person exceeds the dis-similarity between a Spanishspeaking person and a Hindi-speaking person on the other. This is not to claim that cardinal distance functions never have sound intuitive foundations. In general, however, the requirement of a cardinal distance function for the measurement of diversity seems quite strong. It is, therefore, of considerable 1 We use the term ‘objects’ rather broadly so that people and animals can also be objects. 2 The basis for assessing dis-similarity or similarity of two objects will, of course, vary depending on the specific notion of diversity in which one is interested. 2 interest to see how far one can proceed in measuring diversity on the basis of an ordinal distance function. The existing literature has some contributions on the measurement of diversity based on ordinal distance. Pattanaik and Xu [4] introduce a coarse ordinal distance function to measure the diversity of different sets: under their distance function, any two objects are either similar or dis-similar. Bossert, Pattanaik and Xu [2] elaborate further on ordinal distances between objects. Bervoets and Gravel [1] also use ordinal distances to provide some rules for ranking sets of objects in terms of diversity; these rules focus on the objects that are most dis-similar in a set. In this paper, we use an ordinal distance function to develop a notion of domination between sets, to characterize the class of all rules for ranking sets in terms of diversity that satisfy the property of dominance, and to characterize a specific ranking rule belonging to this class. The paper is organized as follows. In Section 2, we present the basic notation and definitions of our analysis. Section 3 introduces our central concepts of weak domination and domination between sets. In Section 4, we provide a characterization of the class of all rankings of sets of objects, which satisfy the property of dominance. In this section, we also present a characterization of a distinguished member of this class. Section 5 considers some related analytical frameworks. Section 6 contains some brief concluding remarks. 2 Notation and definitions Let X be the universal set of objects; X is assumed to contain a finite number of elements. It is clear that the interpretation of the objects in X will depend on the specific context in which one is interested in the notion of diversity. Thus, if one is interested in changes in biodiversity over time in a given region of China, then X can be the set of all animals that have been known to exist in that region of China at different points of time over that period (note that, for the purpose of this interpretation, we would consider, say, two different Chinese tigers as two distinct animals in the universal set). If one is interested in comparing the degrees of linguistic diversity in different countries, then X may be the set of all people living in all these countries (again, two different persons speaking exactly the same language will be considered to be two distinct elements of the universal set). The formal structure of the problem 3 of measuring diversity as we consider it in this paper does not, however, depend on the specific type of diversity in which we may be interested or on any specific interpretation of the universal set. Let K be the class of all non-empty subsets of X and let K2 be the class of all sets A in K such that #A ≤ 2. Our problem is one of ranking the different subsets, A, B, . . ., of K in terms of the degrees of diversity that they offer. For example, does a forest with 24 tigers, 22 bears, 2003 deer, and 5000 rabbits offer more biodiversity than a forest with 14 tigers, 40 bears, 4500 deer, and no rabbits? To analyze this type of questions, let º be a binary relation over K; for all A, B ∈ K, A º B means that the set A is at least as diverse as the set B. The symmetric and asymmetric parts of º are denoted, respectively, by ∼ and ≻. º is assumed to be a quasi-ordering, i.e., it is assumed to be reflexive and transitive, but not necessarily connected. Much of this paper will be concerned with the structure of the binary relation º . 3 Ordinal distance function, indistinguishable objects, and dominance In analyzing the structure of º, our central primitive concept will be that of an ordinal distance function. Suppose we are interested in biodiversity so that our universal set is a set of animals, and suppose, on the basis of some criteria, we believe that an elephant and a panther are more dissimilar than a panther and a leopard. The notion of an ordinal distance function is intended to capture the comparison of the dis-similarity or distance between any two such elements in the set under consideration with the dis-similarity or distance between two other elements in the set. More formally, an ordinal distance function is a function d : X × X → [0, ∞) such that (3.1) (x, x) = 0 for all x ∈ X; (3.2) d(x, y) = d(y, x) for all x, y ∈ X. The function d has the following interpretation: for all x, y, z, w ∈ X, d(x, y) > d(z, w) denotes that the degree of dis-similarity between x and y is greater than the degree of dis-similarity between z and w, and d(x, y) = d(z, w) denotes that the degree of dis-similarity between x and y is the same as the degree of dis-similarity between z and w. As the name ‘ordinal distance function’ suggests, we do not attach any meaning to the comparison 4 of d(x, y) − d(z, w) and d(x′ , y ′ ) − d(z ′ , w′ ) for any x, y, z, w, x′ , y ′ , z ′ , w′ ∈ X. (3.1) says that the degree of dissimilarity between an object and itself is 0; this is simply a convention. (3.2) requires that the degree of dissimilarity between an object x and an object y is the same as the degree of dissimilarity between y and x (so that the distance function is symmetric). Several points may be noted about our ordinal distance function d. First, d is a primitive concept in our framework. We do not enquire into the criteria that constitute the basis of d, but it is clear that the criteria underlying d will be very different depending on the type of diversity (e.g., biodiversity, cultural diversity, linguistic diversity, and so on) under consideration. Secondly, the definition of the ordinal distance function d implies an implicit assumtion, namely, that the dis-similarity or distance between any two objects can be compared with the dis-similarity or distance between any two other objects. This may be a strong assumption. It is possible that, sometimes in practice, we may not have such universal comparability of the dis-similarities involved. In Section 5 below, we shall indicate how one can relax the assumption of universal comparability of distances. Lastly, our definition of d permits d(x, y) to be 0 for distinct objects x and y in X. Let I be a binary relation (“indistinguishable from”) defined over X as follows: for all x, y ∈ X, xIy iff for all z ∈ X, d(x, z) = d(y, z). I is clearly an equivalence relation. We say that x and y are distinguishable iff not(xIy). Note that, if xIy, then d(y, x) = d(x, x) = 0. In the absence of further restrictions on d, d(x, y) = 0 does not necessarily imply xIy. Consider, however, the following rather mild restriction on d : (3.3) For all x, y ∈ X, if d(x, y) = 0 then d(x, z) = d(y, z) for all z ∈ X. It is easy to check that, if d satisfies (3.3), then, for all x, y ∈ X, [d(x, y) = 0] is equivalent to xIy. Though we believe that (3.3) is a relatively mild and ‘natural’ property of an ordinal distance function, we do not need it for our results and we do not impose it on d. An object x0 ∈ X will be said to be a null object iff d(x0 , x) = 0 for all x ∈ X . For all A ∈ K, the set of all null objects in A will be denoted by A0 .We say that a set A is heterogeneous iff [A does not contain any null object, and, for all distinct x, y ∈ A, d(x, y) 6= 0]. For all A ∈ K with #A ≥ 2, if A = A0 , then let e(A) ≡ {a} for some a ∈ A; otherwise, partition A − A0 into I−equivalence classes, A1 , A2 , ..., Am(A) , and let e(A) denote {a1 , a2 , ..., am(A) }, where for all i ∈ {1, 2, ..., m(A)}, ai is some object 5 (arbitrarily) chosen from Ai . Our task is to use the information contained in the ordinal distance function d to rank various subsets of X in terms of diversity. This is a rather complex exercise, to say the least. To make the maximum possible use of our initial intuition, we shall start with the notion of ‘weak domination’. Definition 3.1. For all A, B ∈ K , we say that A weakly dominates B iff (3.4) if #e(A) ≥ #e(B) and there exist a subset e′ (A) of e(A) and a one-toone correspondence f between e′ (A) and e(B) such that [for all x, y ∈ e′ (A), d(x, y) ≥ d(f (x), f (y))]. For all A, B ∈ K , we say that A dominates B iff A weakly dominates B, but B does not weakly dominate A. Several features of our notions of weak domination and domination may be noted. First, when d(x, y) = 0, but x and y are distinguishable, {x, y} dominates {z} for all z ∈ X: a set of two objects that have 0 distance from each other but are distinguishable dominates every singleton set. Second, if [d(x, y) = d(y, z) = d(x, z) = 0 but x, y, z are pairwise distinguishable] and [ d(a, b) > 0], there does not exist any weak domination relation either way between {x, y, z} and {a, b}. Third, for all x, y, z, w ∈ X, {x, y} weakly dominates {z, w} or {z, w} weakly dominates {x, y} (note that this is true irrespective of whether x = y or z = w). 4 Ranking rules satisfying the property of dominance Definition 4.1. We say that º satisfies dominance (D) iff, for all A, B ∈ K, [if A weakly dominates B, then A º B] and [if A dominates B, then A ≻ B]. Note that D, by itself, does not identify a unique quasi-ordering over K; instead, we have a class of quasi-orderings over K, each of which satisfies D. It is easy to check that every quasi-ordering º over K , which satisfies D, satisfies the following intuitively appealing properties: (4.1) for all x, y ∈ X, {x} ∼ {y}; (4.2) for all x, y, x′ , y ′ ∈ X, if x and y are distinguishable, and x′ and y ′ are distinguishable, then d(x, y) > d(x′ , y ′ ) ⇒ {x, y} ≻ {x′ , y ′ }; 6 (4.3) for all A ∈ K and all x ∈ X, [if x is indistinguishable from some a ∈ A or x is a null object, then A ∪ {x} ∼ A], and [if for every a ∈ A, x is distinguishable from a and x is not a null object, then A ∪ {x} ≻ A]; and (4.4) for all A, B ∈ K with #A = #B, for all x ∈ X\A and all y ∈ X\B such that both A∪{x} and B∪{y} are heterogeneous, if there exists a one-toone mapping f from A to B such that d(x, a) ≥ d(y, f (a)) for all a ∈ A, then [A º B ⇒ A ∪ {x} º B ∪ {y}] and [A ≻ B ⇒ A ∪ {x} ≻ B ∪ {y}]. (4.1) states that every singleton set {x} is as diverse as every other singleton set {y}. This property has been used by many writers implicitly or explicitly in measuring diversity. (4.2) requires that the diversity of every doubleton set containing two distinguishable objects depends exclusively on the ordinal distance between the two objects. (4.3) stipulates that, the addition of an object x to a set A will not change the degree of diversity already offered by A when x is indistinguishable from some existing object in A or x is a null object, while the addition of x to A will increase the degree of diversity offered by A when x is distinguishable from every object in A and x is not a null object. This property needs careful interpretation. In real life, people sometimes make the remark that an increase in the population of, say, giant pandas will increase biodiversity. Such remarks seem to go counter to the intuition of (4.3). The remark, however, seems to be based on the belief that there is a critital level of present population below which giant pandas have no reasonable chance of surviving in the future, that the current population level of giant pandas is below this critical level, and that, as a consequence, an increase in the population of giant pandas now will increase biodiversity by ensuring the survival of giant pandas. (4.3) seems to be a reasonable property when such intertemporal issues are considered in a framework analogous to the standard analytical framework in economics where we consider the same physically identifiable commodity available at two different points of time as two different commodities. Suppose, we have only one species, giant pandas, and two periods, ‘the present’ (0) and ‘the future’ (1). The number of giant pandas in the present is denoted by g0 , and the number of giant pandas in the future is denoted by g1 . Let the minimum number of giant pandas required in the present for its survival in the future be 100. Suppose g0 is 60. This cannot ensure the survival of giant pandas in the future. Hence we have a set of animals consisting of 60 giant pandas 7 in the present and 0 giant pandas in the future. If g0 increases to 80, then we shall have a set of animals consisting of 80 giant pandas in the present and none in the future. On the other hand, if g0 increases to 100, then that will ensure the survival of giant pandas in the future and we would have a set of animals consisting of 100 giant pandas in the present and, say, 40 giant pandas in the future. If a giant panda in the future is considered to be ‘different’ from a giant panda in the present, while two giant pandas in the present are considered ‘exactly similar’, then it is not unintuitive to say that, biodiversity will not increase in the first case but will increase in the second case. This is consistent with (4.3). Finally, (4.4) is a type of independence property. Theorem 4.2. º satisfies D if and only if it satisfies (4.1), (4.2), (4.3) and (4.4). Proof: The proof is given in the appendix. As we noted earlier, property D does not determine a unique rule for ranking sets in terms of diversity. Instead, it identifies a (non-singleton) class rules. The following definition introduces a rule, denoted by º∗ , which belongs to this class and is of some interest. Definition 4.3. For all A, B ∈ K, A º∗ B if and only if A weakly dominates B. It is clear that º∗ satisfies D. The feature of º∗ that distiguishes it from other rules satisfying D is that, for every pairs of sets in K, if neither of the two sets weakly dominates the other, then they are declared non-comparable by º∗ . Now, consider the following properties of a ranking º over K. (4.5) For all A, B ∈ K with #A > 1 and #B > 1, if A º B, then A \ {a∗ } º B \ {b∗ } for some a∗ ∈ A and some b∗ ∈ B. (4.6) For all A, B ∈ K , if both A and B are heterogeneous, #A = #B ≥ 2, A º B, and A \ {a′ } º B \ {b′ } for some a′ ∈ A and some b′ ∈ B, then, there exist a∗ ∈ A and b∗ ∈ B such that A \ {a∗ } º B \ {b∗ } and for some one-to-one correspondence f from A \ {a∗ } to B \ {b∗ }, d(a∗ , a) ≥ d(b∗ , f (a)) for all a ∈ A \ {a∗ }. 8 (4.7) For all A, B ∈ K, if both A and B are heterogeneous, A º B, and #A > #B, then there exists a proper subset A′ of A such that A′ º B. Theorem 4.4. º=º∗ if and only if º satisfies (4.1) through (4.7). Proof: The proof is given in the appendix. 5 Revealed ordinal distance functions and the comparability of distances between objects In this section, we indicate some alternatives to the approach that we have adopted in the earlier sections.. So far, we have treated the ordinal diastance function d as a primitive concept in our framework and based our ranking of the sets in K on this exogenously given d. One can, however, follow an approach where the quasiordering º over K is the primitive concept and the ordinal distance function is defined in terms of º . Let º (“at least as diverse as”) be a given quasiordering over K. Let º2 be a binary relation over K2 such that for all A, B ∈ K, A º2 B iff A º B. Assume that (5.1) [ for all x, y ∈ X, {x, y} º {x} ∼ {y}], (5.2) [for all x, y, z, w ∈ X, {x, y} º {z, w} or {z, w} º {x, y}]. Since º is assumed to be a quasi-ordering over K (i.e., º is reflexive and trasitive over K], (5.2) implies that º2 is an ordering over K2 (i.e., º2 satisfies reflexivity, connectedness and transitivity over K2 ). Define a function d′ :X 2 ⇒ R+ such that (5.3) for all (x, y), (z, w) ∈ X 2 , d′ (x, y) ≥ d′ (z, w) iff {x, y} º {z, w} (5.4) for all x, y ∈ X, d(x, y) = d(y, x) and d′ (x, x) = 0. It can be easily checked that, given the finiteness of X and given (5.1), such a real valued function d′ can be found. One can then treat this function d′ as an ordinal distance function which is ‘revealed’ by º. Our concepts of weak domination and domination of sets and the property of dominance can now be developed in terms of this ‘revealed ordinal distance function, d′ , and the property of dominance for º, in its turn, can be defined in terms of these newly defined relations of weak domination and domination between 9 sets. One can then prove the counterparts of Theorems 4.2 and 4.4 in this framework, the proofs being exactly analogous to the proofs of Theorems 4.2 and 4.4, respectively. It may be worth noting some possible extensions of Theorems 4.2 and 4.4. The ordinal distance function introduced in Section 3 implicitly assumes that, for all x, y, z, w ∈ X, the extent of dis-similarity between x and y can be compared with the extent of dis-similarity between z and w. This intuitive assumption regarding the comparability of dis-similarities between objects is also inherent in all aproaches based on cardinal distance functions. The assumption of universal comparability of dis-similarities may, however, be considered rather strong in some contexts. It is, therefore, of interest to note that results analogous to our Theorems 4.2 and 4.4 can be proved without this intuitive assumption. To do this, we can start with a given reflexive and transitive, but not necessarily connected, binary relation D defined over X 2 such that, for all x, y ∈ X, (x, y) D (y, x) D (x, x). For all x, y, z, w ∈ X, (x, y) D (z, w) means that the distance between x and y is at least as great as the distance between z and w. It is possible to develop the counterparts of Theorems 4.2 and 4.4 using the binary relation D instead of the real-valued ordinal distance function d, but we do not undertake the exercise here since the reasoning involved is very similar to the reasoning in Section 4. 6 Concluding remarks In this paper, we have used an ordinal concept of distance between objects to provide a characterization of all diversity-based rankings of sets of objects that satisfy a plausible property of dominance; we have also characterized a specific member of this class, which declares two sets of objects to be non-comparable if they are not comparable in terms of the relation of weak domination. We have discussed a parallel aproach where we rely on the ‘revealed ordinal distances’ rather on an exogenously given ordinal distance function. Our results constitute only the first step in the exploration of diversity-based rankings of sets of objects, using ordinal distance functions. The class of diversity-based rankings that satisfy dominance is a very wide class. Can we narrow down this class by imposing other reasonable properties in addition to dominance, but requiring only information about ordinal distances? This ‘natural’ extension of our analysis requires a separate study. 10 Acknowledgements For helpful comments, we are grateful to Nick Baigent, Rajat Deb, Mark Freurbaey, Peter Hammond, Kotaro Suzumura, and other participants in the Conference on Rational Choice, Individual Rights, and Non-welfaristic Normative Economics held in Tokyo in March 2006. References 1. Bervoets S and Gravel N (2003), Appraising diversity with an ordinal notion of similarity: an axiomatic approach. Mimeo, Université de la Méditerranée 2. Bossert W, Pattanaik PK, and Xu Y (2003), Similarity of objects and the measurement of diversity. Journal of Theoretical Politics 15: 405421. 3. Nehring K and Puppe C (2002), A theory of diversity. Econometrica 70: 1155-1190. 4. Pattanaik PK and Xu Y (2000), On diversity and freedom of choice. Mathematical Social Sciences 40: 123-130. 5. Weikard HP (2002), Diversity functions and the value of biodiversity. Land Economics 78: 20-27. 6. Weitzman ML (1992), On diversity. Quarterly Journal of Economics 107: 363-406. 7. Weitzman ML (1993), What to preserve? an application of diversity theory to crane conservation. Quarterly Journal of Economics 108: 157-183. 8. Weitzman ML (1998), The Noah’s ark problem. Econometrica 66: 1279-1298. 11 Appendix Proof of Theorem 4.2 Suppose that º satisfies (4.1), (4.2), (4.3), and (4.4). We show that º is a dominance-based rule. Let º satisfy (4.1), (4.2), (4.3) and (4.4). By the repeated use of (4.3) and transitivity of º, it is straightforward to show that A ∼ e(A) for every A ∈ K,. We first show that, for all A, B ∈ K, if #e(A) = #e(B) and there is a one-to-one correspondence f between e(A) and e(B) such that [for all x, y ∈ e(A), d(x, y) = d(f (x), f (y))], then A ∼ B. Let A, B ∈ K be such that #e(A) = #e(B) and, for some one-to-one correspondence f between e(A) and e(B), we have [for all x, y ∈ e(A), d(x, y) = d(f (x), f (y))]. Let e(A) = {a1 , . . . , am } and e(B) = {b1 , . . . , bm } be such that f (ai ) = bi for i = 1, . . . , m. Then, d(ai , aj ) = d(bi , bj ) for all i, j = 1, . . . , m. By (4.1), {a1 } ∼ {b1 }. By (4..4) and noting that d(a1 , a2 ) = d(b1 , b2 ), it follows that {a1 , a2 } ∼ {b1 , b2 }. By (4.4) and noting that d(a3 , a2 ) = d(b3 , b2 ) and d(a3 , a1 ) = d(b3 , b1 ), we obtain, {a1 , a2 , a3 } ∼ {b1 , b2 , b3 }. By the repeated use of (4.4) if necessary, and noting that d(ai , aj ) = d(bi , bj ) for all i, j = 1, . . . , m, we have {a1 , . . . , am } ∼ {b1 , . . . , bm }. That is, e(A) ∼ e(B). Noting [A ∼ e(A) and B ∼ e(B)], the transitivity of º then gives us A ∼ B. Next, we show that, for all A, B ∈ K, if #e(A) = #e(B) and there is a one-to-one correspondence f between e(A) and e(B) such that [for all x, y ∈ e(A), d(x, y) ≥ d(f (x), f (y))] and [for some x, y ∈ e(A), d(x, y) > d(f (x), f (y))], then A ≻ B. Let A, B ∈ K be such that #e(A) = #e(B) and, for some one-to-one correspondence f between e(A) and e(B), we have [for all x, y ∈ e(A), d(x, y) ≥ d(f (x), f (y))] and [for some x, y ∈ e(A), d(x, y) > d(f (x), f (y))]. Again, let e(A) = {a1 , . . . , am } and e(B) = {b1 , . . . , bm } be such that f (ai ) = bi for i = 1, . . . , m. Then, d(ai , aj ) ≥ d(bi , bj ) for all i, j = 1, . . . , m, and for some h, k = 1, . . . , m, d(ah , ak ) > d(bh , bk ). Without loss of generality, let h = 1 and k = 2. Then, d(a1 , a2 ) > d(b1 , b2 ). By (4.2), noting that d(a1 , a2 ) > d(b1 , b2 ), {a1 , a2 } ≻ {b1 , b2 } follows immediately. By (4.4) and noting that d(a3 , a1 ) ≥ d(b3 , b1 ), d(a3 , a2 ) ≥ d(b3 , b2 ), it follows that {a1 , a2 , a3 } ≻ {b1 , b2 , b3 }. By the repeated use of (4.4) if necessary, and noting that d(ai , aj ) ≥ d(bi , bj ) for all i, j = 1, . . . , m, we obtain {a1 , . . . , am } ≻ {b1 , . . . , bm }. That is, e(A) ≻ e(B). By the transitivity of º, A ≻ B follows from e(A) ∼ A and e(B) ∼ B. Finally, let A, B ∈ K, let #e(A) > #e(B), and let there be a subset e′ (A) 12 of e(A) and a one-to-one correspondence f between e′ (A) and e(B) such that [for all x, y ∈ e′ (A), d(x, y) ≥ d(f (x), f (y))]. We show that A ≻ B. Let e′ (A) = {a1 , . . . , am }, e(A) = {a1 , . . . , am , am+1 , . . . am+n } and e(B) = {b1 , . . . , bm } be such that f (ai ) = bi for i = 1, . . . , m, and n ≥ 1. From the above analysis, we have e′ (A) º e(B). By (4.3), it follows that e′ (A) ∪ {am+1 } ≻ e′ (A), e′ (A) ∪ {am+1 } ∪ {am+2 } ≻ e′ (A) ∪ {am+1 }, . . . , e(A) = e′ (A) ∪ {am+1 } ∪ . . . ∪ {am+n−1 } ∪ {am+n } ≻ e′ (A) ∪ {am+1 } ∪ . . . ∪ {am+n−1 }. By the transitivity of º, it follows that e(A) ≻ e′ (A). Another application of transitivity of º yields e(A) ≻ e(B). Now, A ≻ B follows from the transitivity of º by noting that A ∼ e(A) and B ∼ e(B). To complete the proof of Theorem 4.2, we note that it is straightforward to check that every º that satisfies D must satisfy (4.1), (4.2), (4.3), and (4.4). ⋄ Proof of Theorem 4.4 It can be verified that º∗ satisfies (4.1) through (4.7). In what follows, we show that, if º satisfies (4.1) through (4.7), then º=º∗ . Let º satisfy (4.1) through (4.7). We first note that, by Theorem 4.2, A ∼ e(A) for all A ∈ K,. Consider any A, B ∈ K. If either of the two sets, A and B, weakly dominates the other, then, by Theorem 4.2 and the definition of º∗ , it is clear that A º B iff A º∗ B and B º A iff B º∗ A. To complete the proof, therefore, we need only to show that, if neither of the two sets, A and B, weakly dominates the other, then A and B are noncomparable. Assume that neither A weakly dominates B nor B weakly dominates A. Given this, it can be checked that #e(A) ≥ 2 and #e(B) ≥ 2. There are several cases that need to be considered. First, we note that, if #e(A) < #e(B), then we must have not[e(A) º e(B)]. This is because, if e(A) º e(B), by (4.5), we obtain e(A) \ {a1 } º B \ {b1 } for some a1 ∈ e(A) and some b1 ∈ e(B). If e(A) \ {a1 } contains one object, then there is an immediate contradiction with e(B) \ {b1 } ≻ e(A) \ {a1 } given Theorem 4.2. If e(A) \ {a1 } contains more than one element, by repeated application of (4.5), we obtain a similar contradiction. Therefore, when #e(A) < #e(B), it must be true that not[e(A) º e(B)]. By transitivity of º, it follows that, if #e(A) < #e(B), then not[A º B] holds. Next, we consider A and B such that (i) #e(A) = #e(B); and (ii) for every one-to-one correspondence f from e(A) to e(B), there exist x, y, z, w ∈ 13 e(A) such that d(x, y) > d(f (x), f (y)) and d(z, w) < d(f (z), f (w)). We need to show that e(A) and e(B) are not comparable. Suppose to the contrary that they are comparable. If e(A) º e(B), by (4.5), there exist a1 ∈ e(A) and b1 ∈ e(B) such that e(A)\{a1 } º e(B)\{b1 }. Then, by (4.6), there exists a∗ ∈ e(A) and b∗ ∈ e(B) such that e(A) \ {a∗ } º e(B) \ {b∗ } and for some one-toone correspondence g from e(A)\{a∗ } to e(B)\{b∗ }, d(a∗ , a) ≥ d(b∗ , g(a)) for all a ∈ e(A) \ {a∗ }. If e(A) \ {a∗ } contains two objects, say a and a′ , then, by Theorem 4.2, it must be true that d(a, a′ ) ≥ d(g(a), g(a′ )). Consider the oneto-one correspondence g ′ from e(A) = {a∗ , a, a′ } to e(B) = {b∗ , g(a), g(a′ )} defined as: g ′ (a∗ ) = b∗ , g ′ (a) = g(a) and g ′ (a′ ) = g(a′ ). Then, for the correspondence g ′ from e(A) to e(B), we have d(u, v) ≥ d(g ′ (u), g ′ (v)) for all u, v ∈ e(A), which contradicts the fact that there exist x, y, z, w ∈ e(A) such that d(x, y) > d(f (x), f (y)) and d(z, w) < d(f (z), f (w)). Therefore, in this case, it cannot be true that e(A) º e(B). If e(A) \ {a∗ } contains more than two objects, then by the repeated use of (4.5) and (4.6), a similar contradiction can be derived. Therefore, e(A) º e(B) does not hold. Similarly, it can be shown that e(B) º e(A) cannot hold. Consequently, we must have that e(A) and e(B) are non-comparable. By transitivity of º, it follows that A and B are not comparable. Finally, we consider A and B such that (i) #e(A) > #e(B); and (ii) for every subset e′ (A) of e(A) with #e′ (A) = #e(B), and every one-toone correspondence f from e′ (A) to e(B), there exist x, y, z, w ∈ e′ (A) such that d(x, y) > d(f (x), f (y)) and d(z, w) < d(f (z), f (w)). Suppose e(A) º e(B). Then, by (4.7), there exists a proper subset C of e′ (A) such that C º e(B). We can assume that #C = #e(B) since (i) #C ≥ #e(B) and (ii) if #C > #e(B), by possibly several applications of (4.7), we can reduce the cardinality of C to #e(B). Note that, C is a subset of e(A) and that #C = #e(B). Then, for every one-to-one correspondence f from C to e(B), there exist x, y, z, w ∈ e′ (A) such that d(x, y) > d(f (x), f (y)) and d(z, w) < d(f (z), f (w)). It then follows that C and e(B) are noncomparable, a contradiction. Therefore, e(A) º e(B) cannot be true. Since #e(B) < #e(A), it must be true that not[e(B) º e(A)]. Therefore, e(A) and e(B) are not comparable. The transitivity of º now implies that A and B are not comparable. This completesthe proof of Theorem 4.4. ⋄ 14