Measuring one-dimensional diversity

ABSTRACT

A large number of diversity measures have been proposed in the literature, almost all of which are designed for sets with elements that differ in multiple aspects. However, these types of measures are not appropriate for sets with elements that differ in a single aspect only, such as duration, value or probability. In this essay I present a new measure of diversity, designed specifically for single aspect diversity. The measure captures the intuitive idea that single aspect diversity is affected by the maximal dissimilarity between the elements, the number of different elements, and the even spacing of the elements’ aspect values, when represented as points in one dimension. I show that the measure satisfies six plausible conditions for a single aspect diversity measure, which concern scale independence, minimal diversity, maximal dissimilarity, restricted function growth, strict monotonicity, and even spacing. The measure is also characterised.

KEYWORDS:

• One-dimensional diversity
• diversity measure
• dissimilarity

Diversity is an important concept in many scientific areas, such as statistics, biology and economics. Considering its importance, it is hardly surprising that there are a large number of diversity measures in use. It is surprising, however, that none of these measures seem to work for the diversity of sets with elements which differ in just one quantitative aspect. While extant measures work for the diversity of people with respect to age and income, for example, they do not work for the diversity of people with respect to age or income only. This is a disadvantage.

In this essay I will present a new real valued measure of diversity, designed for sets with elements that differ in a single quantitative aspect. The measure can, for example, be used for the diversity of populations with respect to well-being, or for the diversity of choice sets with respect to price. Since single aspect differences can be represented by one-dimensional distances, this type of diversity can be called ‘one-dimensional diversity’.

The diversity of a set plausibly depends on the dissimilarities between its elements. The more dissimilar the elements are overall, the more diverse the set. For one-dimensional diversity, this intuitively involves a larger maximal dissimilarity and a larger number of dissimilar elements. It also intuitively involves aspect values being closer to evenly spaced. However, no previously proposed measure captures all of these intuitions. If we simply aggregate all dissimilarities, as some have proposed, we get the result that the set of aspect values (20, 21, 60) is as diverse as the set (20, 100), which intuitively underestimates the importance of a larger maximal dissimilarity. But if we instead consider only average dissimilarity, as others have proposed, we get the result that the set (20, 100) is more diverse than the set (20, 21, 100), which intuitively underestimates the importance of additional elements. Also, both these measures have the result that the set (20, 21, 100) is as diverse as the set (20, 60, 100), which ignores the intuition that an even spacing of values matters as well.

All these intuitions can be captured, however, if we measure single aspect diversity by aggregating the absolute differences between adjacent aspect values by a strictly concave root function. This is the diversity measure that I shall propose here.

The measure rewards a larger maximal dissimilarity, more elements, and an even spacing of single aspect values. More precisely, it satisfies the following plausible conditions for a single aspect diversity measure:

• A ratio scale independence condition, stating that ratio relations between degrees of diversity should be preserved when the underlying dissimilarity measure is multiplied by a positive number.

• A minimal diversity condition, stating that the empty set and singleton sets should be classified as offering no diversity.

• A diameter condition, stating that a set with a larger maximal dissimilarity should be ranked above any set with a smaller maximal dissimilarity, in comparisons between sets that have the same number of elements and whose elements’ single aspect values are evenly spaced.

• A restricted growth condition stating that the addition of an element y to a set A should not increase diversity by more than the diversity assigned to the set (x, z), where d(x, z) is the largest dissimilarity between the elements of A ∪ (y).

• A strict monotonicity condition, stating that the addition of an element that is dissimilar to all the other elements of a set, should result in more diversity.

• An even spacing condition, stating that a set of elements with single aspect values that are evenly spaced should be ranked above any set of elements with single aspect values that are not evenly spaced, in comparisons between sets with the same number of elements and the same maximal dissimilarity.

The essay will be structured as follows: in Section 1 I motivate the need for a measure of one-dimensional diversity; in Section 2 I discuss the concept of diversity and how the conception used in this essay differs from other conceptions; in Section 3 I make some necessary assumptions; in Section 4 I propose conditions that a measure of diversity should satisfy; in Section 5 I present some previously proposed measures and show how none of them satisfies all of the proposed conditions; in Section 6 I present my own measure, which satisfies the conditions, in Section 7 I explain how the measure is unique, and in Section 8 I make some concluding remarks. An appendix with a uniqueness proof follows.

1. Motivation

Diversity is discussed in a wide variety of scientific areas, including applied mathematics, statistics, information science, computer science, linguistics, chemistry, biology, earth science, geography, anthropology, psychology, sociology, and economics. In philosophy, diversity is discussed in (for example) epistemology, philosophy of science, ethics, political philosophy, and decision theory.¹ These discussions have in part concerned the concept of diversity and its measurement, and as a result, a large number of diversity measures have been proposed. One may thus wonder why we need another one. The straightforward answer is that there are no measures for one-dimensional diversity, based on the assumption that the diversity of a set is a function of single aspect ratio scale dissimilarities between its elements, and which also accommodate the intuitions that the largest dissimilarity, the number of dissimilar elements, and the even spacing of the elements’ aspect values determine the diversity of a set. In fact, there are no one-dimensional diversity measures at all, designed specifically for single aspect ratio scale dissimilarities – at least not if we distinguish between diversity and dispersion, which I will argue below that we should. (Ratio scale dissimilarities are dissimilarities measurable on a ratio scale, allowing representation of ratio relations between dissimilarities, as well as representation of equality, order and interval relations. Ratio scales are characterised by having non-arbitrary zero points and arbitrary units.)

The lack of one-dimensional diversity measures would not be a problem if there were no use for one-dimensional measures in any of these areas. But this is hardly the case. Single aspect properties are of interest in all sciences, and whenever the objects of some set differ in only one aspect, it could be useful to measure the diversity of the set with regard to that aspect. Among one-dimensional properties of scientific interest we have: data size, probability, confirmation, time, length, speed, mass, luminosity, pitch, pressure, temperature, humidity, acidity, sweetness, enjoyment, willingness, credence, price, inflation, income, equality, age, and birth rate. Additional properties of philosophical interest include properties such as cogency, goodness, and well-being. A one-dimensional diversity measure could also be used to compare sets of elements which include different numbers of some kind of element, such as the diversity of a set of countries which include different numbers of people.

The lack of one-dimensional diversity measures, dependent on ratio scale dissimilarities, would also not be a problem if measures designed for other purposes could be used instead. However, there are at least two reasons why they cannot. First, most diversity measures cannot be adapted to register ratio scale dissimilarities at all, whether one-dimensional or multi-dimensional. Second, even measures that are designed for multi-dimensional ratio scale dissimilarities are not suitable for one-dimensional ratio scale dissimilarities, since there is a specific feature desirable for multi-dimensional measures which is not desirable for one-dimensional measures. Let us look at these two problems in turn.

The first problem with finding a measure suitable for one-dimensional diversity is that most measures are designed for models that do not register dissimilarity relations at all. There are many measures that only register number of elements, attributes, or classes, for example, and these cannot be used to measure diversity regarded as a function of single aspect dissimilarities.² A count of elements does not register the dissimilarities between the elements at all (other than them being distinct elements); a count of attributes does not register that elements can differ within a single aspect; and a count of classes does not register that elements can differ within a single class. These types of measures are thus disqualified from the start. Trying to redesign these types of measures for single aspect dissimilarities is not going to work either, because even if we could treat specific values of some single aspect as elements, attributes, or classes, we could not capture the dissimilarity relations between them in this way.

The second problem is that even measures that are designed for diversity regarded as a function of dissimilarity relations are not suitable for one-dimensional ratio scale dissimilarity relations. Some measures will not work well because they are designed only for ordinal relations (relevant information is thus lost if they are applied).³ Other measures will not work well because they are designed for multiple aspect dissimilarities.⁴ This last point is perhaps surprising because one might think that multiple aspect measures should be capable of handling single aspect diversity as well. There is a reason, however, to think that they are not. Relative to multiple aspect diversity, it is reasonable to think that elements that are equally dissimilar to one another are more diverse than elements that are not – everything else being equal (number of elements and largest dissimilarity).⁵ For example: if the elements are represented as points in Euclidean space and their dissimilarities are represented as distances, a set appears more diverse if the elements are represented as positioned in the corners of a simplex (everything else being equal). This way the elements are as dissimilar to one another as possible, relative to some fixed number of elements and a fixed largest dissimilarity.⁶ As an extension of this ideal, it is also reasonable to think that sets with elements that are closer to being equally dissimilar to one another are more diverse than sets with elements that are further away from being equally dissimilar, everything else being equal. This equidistance ideal, however, is hardly reasonable for single aspect diversity, not just because only two elements could be equally dissimilar to all other elements with respect to a single aspect, but also because the ideal conflicts with the intuition that objects are more diverse when their aspect values are closer to being evenly spaced in one-dimensional space, everything else being equal. This conflict might seem surprising, but an example can illustrate the point: Let us assume that closeness to equidistance can be measured by a sum of differences between the dissimilarities between the elements of a set, where a smaller sum represents larger closeness and 0 represent equidistance. Let us next compare the sets of aspect values: A = (0, 53, 106, 159, 212, 265) and B = (0, 70, 105, 140, 210, 280) with the same total sum of dissimilarities. Here, the values in A are distributed at even intervals on a real line, but the values in B are closer to being equidistant, relative to the dissimilarities between all pairs of values. (The sum of differences between the distances between the values in B is smaller than the sum of differences between the distances between the values in A.) A single aspect diversity measure should rank A over B in diversity, but a multiple aspect diversity measure would rather rank B over A, in keeping with the ideal of equidistance. For this reason, a multiple aspect diversity measure will not be adequate for single aspect diversity. A specific one-dimensional diversity measure based on ratio scale dissimilarities is thus needed.

2. Diversity

The term ‘diversity’ does not express a single, precise concept; rather, it is used to express a variety of different diversity conceptions. Among these, there is a specific conception of one-dimensional diversity which will be the focus of this paper. Let us see how this conception differs from some of its alternatives.

In general, diversity is a quantitative property of sets (which could be populations or assortments or something else). While the elements of a set can be more or less dissimilar to one another, only sets can be more or less diverse. However, a set, A, is more diverse than another set, B, if and only if the elements of A are more dissimilar to one another than what the elements of B are. Diversity is thus the same as collective dissimilarity.⁷

According to the single aspect diversity conception which will be the focus here, the diversity of a set is dependent on dissimilarity relations between elements and is affected in particular by the largest dissimilarity, the number of dissimilar elements, and the even spacing of the elements’ aspect values, when represented as points in one-dimensional space. Since this is not the only conception of diversity in use, let us contrast it with others that occur in the literature. (Conceptions will be individuated rather coarsely here because if we were to regard each measure as the expression of a distinct conception, the presentation would be much too long.)

All conceptions of diversity seem somehow to involve the idea that the diversity of a set is dependent on dissimilarities. However, the relevant dissimilarities are not always considered to be dissimilarities between the elements of the set, and the dependence on dissimilarities is not always considered to be a direct one. There seems to be at least three views where dissimilarities only play an indirect role. Let us consider these views first.

One indirect view is that the diversity of a set is the same thing as the number of elements of the set.⁸ Since the individual elements of a set are separated by being dissimilar, dissimilarities play an indirect role at least for individuating elements. To identify diversity with number of elements is a rather blunt view, however. It does not capture that the diversity of a set could be affected by the elements being dissimilar in different aspects. Neither does it capture that the diversity of a set could be affected by degrees of dissimilarities between its elements.

Another indirect view is that the diversity of a set is the same thing as the number of attributes that the elements of the set exemplify.⁹ Since attributes are separated by being dissimilar, dissimilarities play an indirect role also in this case, although for individuating attributes. This view is rather blunt as well, since it does not capture that a set of elements that differ in just one aspect could be diverse. Thus, this view is at most appropriate for multiple aspect diversity.

A third indirect view is that the diversity of a set is dependent (somehow) on the classes that the elements of the set could be sorted into – either the number of classes (richness), the number of individuals per class (abundance), the dissimilarities between classes, how evenly the elements can be sorted into classes (with maximal evenness being that each class has an equal number of members), or some combination of these four features.¹⁰ Since classes are separated by being dissimilar, dissimilarities play an indirect role even for these cases. However, these views cannot capture all aspects of diversity either, since they cannot capture that a set of elements belonging to just one class can be diverse.

Other views on diversity consider the diversity of a set to be more directly dependent on dissimilarities. Let us consider such views next.

One direct view is that the diversity of a set is a function of dissimilarities, but not just a function of the dissimilarities between the elements of the set, but a function of the dissimilarities between the elements of the set and elements outside the set: either all the elements of the universal set or some element representing the mean of the universal set (which may or may not belong to the set).¹¹ This conception is not very plausible, however, since it represents diversity as an extrinsic property of sets, rather than as an intrinsic one. An implausible implication of this view is that it is not possible to rank sets in terms of diversity, without knowledge of the elements of the universal set. Another implausible implication is that a set of two elements which are very similar could be ranked as more diverse than a set of two elements which are very dissimilar, due to their different relations to the universal set.

A closely related view is that the diversity of a set is a function of the dissimilarities between the elements of a set and some element representing the mean of the set (which also may or may not belong to the set).¹² This view equates diversity with dispersion, which can be measured by variance, standard deviation, or mean deviation.¹³ One problem with these types of measures is that the addition of a mean element to a set would not increase the diversity of the set, which it intuitively should.

Another direct view is that the diversity of a set is just the totality of the dissimilarities between the elements of a set.¹⁴ This view is not sensitive to the distribution of the totality of dissimilarities among individual dissimilarities, however. It is not sensitive to whether the dissimilarities are equal or whether the aspect values of the elements are evenly spaced.

Other direct views regard the diversity of a set as some aggregate of some selected dissimilarities between the elements of a set.¹⁵ Many different proposals have been made here, but none of them is sensitive to all three properties mentioned above (magnitude, number and even spacing). We will look at some of these proposals later.

The direct view that is most similar to the one that I will consider is that the diversity of a set is a function of the dissimilarities between the elements in such a way that it is affected by the magnitude of dissimilarities, the number of dissimilar elements and the equal size of the dissimilarities.¹⁶ However, as we have just seen, this view seems appropriate only for multiple aspect diversity as it cannot accommodate the view that even spacing between one-dimensional values matters for diversity.

3. Assumptions

Before we can consider proposals for measures of one-dimensional diversity, some assumptions regarding diversity, dissimilarity and single quantitative aspects need to be made. The purpose of a diversity measure is to assign numbers to sets so that quantitative relations between degrees of diversity are accurately represented by numerical relations. Since the diversity of a set is intrinsically dependent on the dissimilarities between the elements of a set, a diversity measure should be a function of a dissimilarity measure. And since dissimilarity, in turn, is intrinsically dependent on some aspect of the elements, a dissimilarity measure should, in turn, be a function of a measure of some quantitative aspect. Thus, both diversity and dissimilarity measures are derived measures of a measure of some quantitative aspect.¹⁷

Let us first assume that there is a finite universal set of objects, X. Next, let us assume that there is a measure q of some single aspect Q. The measure q assigns real numbers to all objects in a way that represents the degree to which they are Q and also represents the quantitative relations between degrees of Q by numerical relations. I will assume that the measure q is ratio scale, which means that the measure represents ratio relations between degrees of Q, as well as difference relations, ordinal relations and equality relations. A ratio scale measure of a quantitative property can more precisely be described as follows: First, let X be the set of relevant objects and let φ₁,  … φ_n be n-place relations that hold between the elements x_i in X, and which compare them in degrees of Q; S = 〈X, φ₁,  … φ_n〉 is then a relational system. Next, let R be a set of real positive numbers and let ρ₁,  … ρ_n be n-place relations that hold between the numbers in R; T = 〈R, ρ₁,  … ρ_n〉 is then another relational system. The property Q is measureable by real positive numbers when the system S is represented by the system T through a function f: X → R such that for each sequence of elements x_i ∈ X and each relation φ_i, it is the case that φ_i(x₁ … x_n) if and only if ρ_i(f(x₁) … f(x_n)).¹⁸ And Q is measureable on a ratio scale when S is also represented by T through any function g, such that g = αf, where α > 0.¹⁹ The function f and each function g are measures of Q.

Next, let us assume that the dissimilarity d between two objects, x and y, with respect to Q, can be measured by the absolute difference between q(x) and q(y), thus:

Dissimilarity as absolute difference: d(x, y) = |q(x) – q(y)|.²⁰

Dissimilarity as absolute difference can be regarded as a distance in a one-dimensional Euclidean space.²¹ Such a space can be regarded as a set (here the set X) combined with a function d that assigns a distance to all ordered pairs of elements (x, y) from X. The distance d(x, y) represents how different x and y are to one another in an aspect Q. Since the relevant space here is a one-dimensional Euclidean space, there are specific requirements on the distance function d. For all x, y, z ∈ X, the distance function d satisfies:

1. Non-negativity: d(x, y) ≥ 0.

2. Identity of Indiscernibles: d(x, y) = 0 if and only if x = y.

3. Symmetry: d(x, y) = d(y, x).

4. Triangle Equality: d(x, y) = d(x, z) + d(z, y).

The Identity of Indiscernibles implies that objects that have exactly the same degree of Q are identical (this is a natural view to take since the objects are identical with respect to Q). It also implies that sets cannot be expanded by adding elements that are exactly alike elements that are already included in the set.

The one-dimensional Euclidean space 〈X, d〉 can be associated with a vector d_X, where the distances are indexed with positive integers and then ordered. To construct this vector, the elements in X are first indexed from 1 to n, as x₁, x₂ … x_n−1, x_n in such a way that q(x₁) ≥ q(x₂) ≥ … q(x_n−1) ≥ q(x_n). Then the distances are indexed so that d₁ = d(x₁, x₂), d₂= d(x₂, x₃) etc. For each subspace 〈A, d〉 of 〈X, d〉 there is a vector d_ASP which contains the distances d₁ to d_n−1, which are the shortest path distances connecting all the elements of A. This vector is particularly important since all distances between the elements of a set A can be derived from the shortest path distances, using the Triangle Equality. To give an example: for the set of lengths (10 m, 20 m, 25 m) the associated shortest path vector is [10 m, 5 m], which can be used to derive the vector of all distances [10 m, 5 m, 15 m].

The maximal dissimilarity (or distance) between two elements in a set A will be called the diameter of A, and will be denoted by Ø(A) (it is also known as the range of A). The number of elements in A will be denoted by n or #A.

With these assumptions and notations in place, we can continue to consider conditions for a measure of diversity.

4. Conditions for a one-dimensional diversity measure

There are an infinite number of ways in which a measure could be a function of dissimilarities. The idea that a diversity measure should be a function of a dissimilarity measure is thus insufficient for identifying a diversity measure and we must rely on further ideas about diversity. Here I will use the ideas that the diversity of a set is affected by the magnitude of the dissimilarities between the elements, the number of dissimilar elements, and the even spacing of the elements’ aspect values when represented as points on the real line. These three ideas will be used to formulate necessary conditions for a measure of diversity. First, however, I will consider a more general condition.

4.1 Ratio scale independence

The first consideration is important for any derived measure: a derived measure should be independent of scale. That is: ratio relations between different degrees of whatever quantity that is measured should be preserved even if an underlying ratio measure on which the derived measure depends changes scale. In the case of diversity, ratio relations between degrees of diversity should be preserved even if the underlying dissimilarity (and aspect) measure changes scale. It should not matter for the assessment of the diversity of a set of incomes whether incomes are measured in dollars or cents, for example. A diversity measure should thus satisfy the following condition:

The Ratio Scale Independence Condition: For a diversity measure D that is a function of a dissimilarity function d, it holds that for all numbers K > 0, there is a number L = L(K) > 0, where L is an increasing function of K such that if d is multiplied by K, then D is multiplied by L.²²

Interestingly, there are rather many diversity measures that do not satisfy the Ratio Scale Independence Condition. Measures based on logarithmic and natural logarithmic functions do not, for example.²³ Unless these measures are based on measures of absolute scales, such as probability measures, they are thus problematic. (Absolute scales do not admit of scale changes as they have non-arbitrary units and non-arbitrary zero points.)

4.2 Minimal diversity

A second condition concerns the minimal limit of diversity. Since diversity is a function of dissimilarity, a set whose members are not dissimilar could not be diverse. This is the case both for the empty set and for singleton sets: the empty set because it does not contain any elements that can be dissimilar at all and singleton sets because they contain only one element, which cannot be dissimilar to itself. A measure of diversity should thus satisfy the following condition:

The Minimal Diversity Condition: For a diversity measure D and any set A, if #A ≤ 1, then A is not diverse and thus D(A) = 0, or D(A) is undefined.²⁴

All diversity measures that are derived measures of dissimilarity measures satisfy this condition. However, there are some other kinds of diversity measures, mentioned earlier, that do not, such as count of elements, attributes or classes. Even though they give the empty set a value of 0 or no value at all, they give singleton sets a value of at least 1.

4.3 Magnitude

The smallest sets that can be diverse are thus sets of two elements. Between these two elements, there is only one dissimilarity relation (assuming that d(x, y) and d(y, x) are regarded as the same relation). If diversity is a function of dissimilarities, the diversity of this kind of set should plausibly depend on the magnitude of this one dissimilarity relation. A higher degree of dissimilarity should imply a higher degree of diversity. Thus the following condition should hold:

The Limited Diameter Condition: For a diversity measure D and any sets A and B and any elements x, y, w, z, such that A = (x, y) and B = (w, z), if d(x, y) > d(w, z), then A is strictly more diverse than B and thus D(A) > D(B).²⁵

The Limited Diameter Condition shows how closely diversity and dissimilarity are connected when there is just one dissimilarity relation to consider. It could be used to rank the set A = (20, 60) above the set B = (20, 40), for example. Later on I will use it to characterise the proposed measure.

For sets of two elements, diversity is a function of the magnitude of a dissimilarity relation. But, obviously, also for larger sets, diversity is a function of dissimilarity relations, and for them too, the magnitude of the dissimilarity relations should matter. Here, there is a difference between multiple aspect diversity and single aspect diversity, however.

For multiple aspect diversity, it seems important that each dissimilarity relation be as large as possible in order for the elements of a set to be as dissimilar to one another as possible. The reason for this is that for multiple aspect cases, represented in several dimensions, each dissimilarity relation can be of the same maximal degree. For example, for a set of three elements, three dissimilarity relations could have the same maximal degree. (If the elements were represented as points and the dissimilarity relations were represented as straight lines, such a set would be represented as a maximised triangle in two dimensions.) For single aspect cases, however, it cannot be important that each dissimilarity relation be as large as possible in order for the elements of a set to be as dissimilar to one another as possible. Firstly, all dissimilarity relations cannot be of the same maximal degree, if a set has cardinality larger than two. Secondly, even if it is favourable that one dissimilarity relation is maximised (the diameter), it is not favourable that the other relations are. This can be shown with some examples. The set A = (30, 70) seems more diverse than the set B = (30, 50) due to the larger diameter of A. However, the set C = (30, 69, 70) seems less diverse than the set D = (30, 50, 70), even though C has a larger second largest dissimilarity. An even spacing of values is intuitively more important here than the size of the second largest dissimilarity.

The diameter thus plays a special role for single aspect diversity since it is the only dissimilarity relation that should be maximised. We can use the diameter to formulate a condition for a single aspect diversity measure. Obviously, it will not do to formulate a condition that says that any set with a larger diameter is more diverse than any set with a smaller one. Such a condition would be reasonable only if the diameter would always be more important than either the number of elements or an even spacing of elements, and this does not seem to be the case. The set A = (25, 70) is intuitively less diverse than the set B = (25, 40, 55, 69), and the set C = (25, 26, 70) is intuitively less diverse than the set D = (25, 47, 69), for example. A larger diameter cannot compensate for fewer elements or an uneven spacing of values in these cases. The ideal of a larger diameter must thus be combined with the ideal of a larger number of elements and an even spacing of values. Nevertheless, if we keep numbers and spacing fixed, and only vary the diameter of sets, the set with the larger diameter should be more diverse. Thus, the following condition should hold:

The Diameter Condition: For a diversity measure D and any sets A and B, such that #A = #B and all elements of A and all elements of B have values that are evenly spaced in one dimension, if Ø(A) > Ø(B) then A is strictly more diverse than B and thus D(A) > D(B).²⁶

We could use the Diameter Condition to rank, for example, the set A = (20, 40, 60) above the set B = (20, 30, 40), in terms of diversity. The condition implies the Limited Diameter Condition, mentioned previously.

4.4 Cardinality

I have proposed that diversity is dependent on the number of dissimilar elements. Not everyone agrees with this, however. There is a tradition of separating diversity from number, claiming that diversity is independent of number.²⁷ This idea can be captured for example by measuring diversity by average dissimilarity. Intuitively, however, this seems wrong. The average dissimilarity is larger in the set (20, 100) than in the set (20, 50, 100). Yet, the second set seems more diverse since it has an additional value exemplified.

There is thus reason to think that diversity is affected by number. But the question is what number: number of elements or number of dissimilarities?

Since diversity is dependent on dissimilarities, it might seem that diversity should be affected by the number of dissimilarities. One proposal is that the diversity of a set is proportional to the number of dissimilarities. This view is problematic, however. When the number of elements is n, the number of dissimilarities is n²/2. Of these, all but n dissimilarities (those between elements and themselves) are non-zero dissimilarities that might contribute to the diversity of a set. For each added element, the number of elements grows with 1, but the number of non-zero dissimilarities grows linearly with the size of the set. If a set has 100 elements, the addition of a single element results in 100 additional dissimilarities. If all of these contributed proportionally to the diversity of a set, diversity would grow excessively with each added element.²⁸ This excessive growth has several implausible implications. One implausible implication is that for large sets, the more numerous set is more likely to be more diverse – regardless of its diameter or its spacing of values. For example, the set A = (20, 21, 59, 60) might counterintuitively be considered more diverse than the set B = (20, 40, 60), although the elements of B seem more collectively dissimilar. Another implausible implication is that in the choice of adding an element to either a small set or a large set in order to maximise diversity, one should add the element to the larger set – even in those cases where there is already a very similar element in the larger set. For example, in the choice of adding an object with a value of 21 to either of the two sets C = (40, 60) and D = (20, 40, 60) in order to maximise diversity, the recommendation would be to add the object to the second set. Both these implications are implausible. A larger set is not generally more diverse. And the contribution of an element to a set should not depend on the size of the set in this way.

Due to the implausible implications of excessive growth, the diversity of a set cannot be proportional to the number of dissimilarities (or anything else that leads to excessive growth). It should at most be proportional to the number of elements. This restriction will prevent the implication that the contribution of an additional element to a set increases with the size of the set. We can capture this idea by formulating a condition that restricts the growth of a diversity measure. One way to do this is to propose that the addition of an element to a set should never add more to the diversity of the set than whatever value the diversity measure gives to the set that contains the two elements that are most dissimilar in the larger set – the diameter of the larger set, in other words. More formally this could be expressed as follows:

The Restricted Growth Condition: For a diversity measure D and any set A and any element y ∉ A, it holds that D((A ∪ (y))) ≤ D(A) + D((x, z)), where x, z ∈ A ∪ (y) and x and z are chosen so that the dissimilarity d(x, z) = Ø(A ∪ (y)).²⁹

The Restricted Growth Condition assures that the contribution to diversity by an element in a set is independent of the size of the set. It could hold even if the number of elements would not matter at all for diversity. However, it seems as if the number of elements does matter, which the example given in the beginning of this section illustrates. The question is just how.

One proposal is that if diameter and even spacing are kept fixed, and only numbers are varied, the set with the larger number of elements is more diverse. However, such an idea would be controversial. According to this idea, the set A = (20, 40, 60, 80) is more diverse than the set B = (20, 50, 80). But this does not seem obvious. Perhaps the aspect value of 50 in B compensates for the absence of the aspect values of 40 and 60 in A in terms of diversity. After all: 50 is more dissimilar to both 20 and 80 than either of 40 and 60 is dissimilar to 20 and 80. Nevertheless, cardinality seems to matter at least if we compare sets with their own subsets. If we start with the set A = (20, 60) and then add a new aspect value to A, for example, y = 50, we would judge A ∪ (y) as more diverse than A because it contains an additional (and qualitatively different) element. Thus, satisfying the following condition should be required of any diversity measure:

The Strict Monotonicity Condition: For a diversity measure D and any elements x, y and a set A such that #A ≥ 1, it holds that: If an element y is dissimilar to all elements x ∈ A and y ∉A then A ∪ (y) is strictly more diverse than A and thus D(A ∪ (y)) > D(A).³⁰

The Strict Monotonicity Condition is reasonable since all the dissimilarities that hold between the elements in A also hold between the elements in A ∪ (y), but there are some dissimilarities that hold between the elements in A ∪ (y) that do not hold between the elements in A. Therefore, A ∪ (y) seems more diverse than A. Because the addition of an element might result in the elements being less evenly spaced, the Strict Monotonicity Condition shows a situation in which number of elements is more important than even spacing.

There are some potential objections to the Strict Monotonicity Condition, however. One possible objection is that A ∪ (y) would not be more diverse than A if the element y is very similar to an element in A. For example: if A = (20, 60) and y = 20.1, then A ∪ (y) might not be more diverse than A.

This reasoning seems to presuppose that it is types of elements that affect diversity, rather than individual elements. Two very similar elements would not affect diversity separately, since they belong to the same type. Such an idea is problematic, however. For example, let us define some types of Q so that each aspect value would belong to one and only one type. We could, for example, define the types of Q as half-open intervals of q, as follows: [0, 0.2], [0.2, 0.4], [0.4, 0.6] … etc. In this case we would get the result that y = 59.9 increases the diversity of A = (20, 60) whereas y = 20.1 does not, since the values 59.9 and 60 are of different types whereas the values 20 and 20.1 are not. This is not really a plausible solution since both y:s differ from an element of A by the same difference of 0.1. However, if we instead defined types so that all aspect values within (for example) 0.1 of each other always would count as belonging to the same type, we would get an infinite number of types and any set (apart from the empty set) would contain elements of an infinite number of types. The set A would, for example, exemplify elements of the types [19.9, 20.1], [19.99, 20.11], [19.999, 20.111] and so on. This is not a plausible solution either.

Perhaps there is a way to formulate this idea without implausible consequences, but there is really no need to. It is much easier to stick with the Strict Monotonicity Condition. If an element is added to A which is very similar to an element already in A, we should not say that diversity does not increase, but rather that it increases just slightly.

Another possible objection to the Strict Monotonicity Condition is the following: A ∪ (y) cannot be more diverse than A because A ∪ (y) must be less diverse than A ∪ (y). We can come to this conclusion by assuming, first that diversity is the opposite of unity and second, that unity is an increasing function of similarities (just as diversity is an increasing function of dissimilarities). If this is the case we can say that A ∪ (y) must be more unitary than A since all the similarities that hold between the elements in A also hold between the elements in A ∪ (y), but there are some similarities that hold between the elements in A ∪ (y) that do not hold between the elements in A. We can also say that since diversity and unity are opposites, A ∪ (y) must be less diverse than A.

This reasoning presupposes a certain relationship between unity and diversity which is problematic, however. If unity depends on similarities and diversity depends on dissimilarities, and if each element is both similar and dissimilar to every other element, then unity and diversity would both increase when a set increases. Thus, they cannot be opposites in the sense that if one increases the other decreases. So this objection will not work either.

4.5 Even spacing

The last property that I have proposed as affecting the diversity of a set is the even spacing of the elements’ aspect values in one dimension.³¹ When the elements of a set have aspect values that are evenly spaced, they seem to be more collectively dissimilar. The set A = (20, 40, 60) seems more diverse than the set B = (20, 21, 60), for example.

The importance of even spacing is unique for single aspect diversity. For multiple aspect diversity, it is rather equal dissimilarities that matter. This is because in multiple aspect cases all elements in a set could have the same dissimilarity to all other elements, which they cannot have in single aspect cases, if the set has more than two elements. (For n elements, there must be at least n – 1 dimensions in order for all the elements to have the same distance to all the other elements.)

For the property of even spacing, we might again propose an everything-else-being-equal type of condition, where spacing is varied and the other two factors are held fixed. Such a condition would claim that if two sets A and B have the same number of elements and the same diameter, then A is strictly more diverse than B, if the elements of A are closer to being evenly spaced than are the elements of B. The problem with such a condition would be that it is not clear exactly what ‘closer to being evenly spaced’ means.

A more precise understanding of ‘closer to being evenly spaced’ requires a measure of even spacing, and considering the many possibilities for such a measure, any particular choice would be controversial. So rather than provide an interpretation of ‘closer to being evenly spaced’, it is better to adopt a different condition. I propose that we adopt a condition that only concerns comparisons of elements that are completely evenly spaced with elements that are not. In order to formulate such a condition, we must first define the expression ‘completely evenly spaced’. To do this, we first index the elements of each set with positive integers from 1 to n according to the degree of q that they exemplify, so that the elements x₁ to x_n are ordered such that q(x₁) < q(x₂) < … < q(x_n-1) < q(x_n). We then have:

A set A with n elements x_i ∈ A, indexed and ordered from 1 to n, has elements that are completely evenly spaced with respect to q, if and only if with respect to their dissimilarities in q, d(x₁, x₂) = d(x₂, x₃) … = d(x _n-2, x_n-1) = d(x_n-1, x_n).

Next we can formulate a limited even spacing condition that only concerns comparisons involving elements that are completely evenly spaced with elements that are not completely evenly spaced and, in addition, only concerns comparisons between sets that differ with respect to a single exemplified aspect value:

The Limited Even Spacing Condition: For a diversity measure D, and for all sets A, B such that #A = #B and Ø(A) = Ø(B), if there is a bijection from the elements x ∈ A to the elements y ∈ B such that for each pair of elements (x_i, y_i) it is that case that q(x_i) = q(y_i), except for one case (x_j, y_j) which is such that q(x_j) ≠ q(y_j) and where the consecutive elements x_j-1, x_j, x_j+1 are completely evenly spaced, whereas the consecutive elements y_j-1, y_j, y_j+1 are not completely evenly spaced, then A is strictly more diverse than B, and thus D(A) > D(B).

The Limited Even Spacing Condition will be used to characterise the measure that I will propose later. It is not very general in its immediate application, but if we assume that a diversity ordering is transitive (as we should), then we can derive an even spacing condition that is more general and is of a similar everything-else-being-equal type as the Diameter Condition. We then have:

The Even Spacing Condition: For a diversity measure D, and for all sets A, B such that #A = #B and Ø(A) = Ø(B), if the elements x ∈ A have values that are completely evenly spaced and the elements y ∈ B have not, then A is strictly more diverse than B, and thus D(A) > D(B).

The Even Spacing Condition can be used to order for example the set A = (20, 30, 40) over B = (20, 39, 40) in terms of diversity.

4.6 Summary of conditions

Three properties were proposed to affect the one-dimensional diversity of a set: the largest dissimilarity between the elements, the number of dissimilar elements and the even spacing of the elements’ aspect values. The importance of these three properties for diversity was captured in several conditions for a measure of diversity: for diameter, a Diameter Condition was accepted (implying a Limited Diameter Condition); for number, a Strict Monotonicity Condition; and for even spacing, an Even Spacing Condition (implied by a Limited Even Spacing Condition, which will be used later). The Diameter Condition requires that a set with a larger diameter is ranked above a set with a smaller diameter, for comparisons between sets with the same number of elements and the same even spacing; the Strict Monotonicity Condition requires that a set is ranked above its proper subsets; and the Even Spacing Condition requires that a set with evenly spaced elements is ranked above a set without evenly spaced elements, for comparisons between sets with the same diameter and the same number of elements. In addition, a Ratio Scale Independence, a Minimal Diversity and a Restricted Growth Condition were accepted. The Scale Independence Condition requires that the measure should preserve ratio relations between degrees of diversity when the underlying dissimilarity measure changes scale; the Minimal Diversity Condition requires that the empty set and singleton sets are classified as offering no diversity and the Restricted Growth Condition requires that the diversity measure does not present diversity as growing excessively with the addition of an element to a set.

All six conditions are independent of one another, mostly because they do not concern the same situations. The Ratio Scale Independence Condition is the only condition that concerns how a diversity measure should respond to a change of scale of the underlying dissimilarity measure. The Diameter Condition is the only condition that concerns comparisons between sets where both cardinality and even spacing are the same but diameter differs. The Even Spacing Condition is the only condition that concerns comparisons between sets where both cardinality and diameter are the same but the distribution of exemplified aspect values differs. The other three conditions concern comparisons between sets of different cardinality. However, the Minimal Diversity Condition concerns only sets of one or fewer members. It does not imply the Strict Monotonicity Condition since that condition applies to sets with a larger cardinality than 1. Neither is it implied by the Strict Monotonicity Condition since that condition does not concern the empty set. The Minimal Diversity Condition does not imply the Restricted Growth Condition since that condition applies to sets of all sizes. Neither is it implied by the Restricted Growth Condition since that condition does not exclude that the empty set or singleton sets could have a diversity value above 0. Moreover, the Restricted Growth Condition does not imply the Strict Monotonicity Condition since it is compatible with additional elements not adding additional value. Neither does the Strict Monotonicity Condition imply the Restricted Growth Condition since it does not exclude the possibility that D((x, y)) could be 0. All of these conditions are fulfilled by the proposed measure, which we will consider in the sixth section. First, however, we shall look at some previously proposed measures which do not satisfy all of the conditions. These measures are not designed specifically for one-dimensional ratio scale dissimilarity relations, but they are still relevant to consider as they are designed for diversity regarded as a function of dissimilarities.

5. Previous measures

The presentation of previously proposed measures will be brief, as I will only present a list of measures and a table that shows how they fare in relation to the proposed conditions. All the proposals are designed for diversity as a function of dissimilarities (sometimes represented by metric space distances). The measures in question are:

• The Total Dissimilarity Measure, which measures diversity by the total sum of all dissimilarities between all elements of a set.³²

• The Average Dissimilarity Measure, which measures diversity by the average dissimilarity between all elements of a set.³³

• The Range Measure, which measures diversity by the diameter of a set.³⁴

• The Lexical Measure, which measures diversity (ordinally) by comparing sets with respect to their largest dissimilarity, their second largest dissimilarity and so on.³⁵

• The Average Aggregation Measure, which measures diversity by aggregating the average dissimilarity from each element to all the other elements of the set (disregarding the dissimilarity between an element and itself).³⁶

• The Maximal Aggregation Measure, which measures diversity by aggregating the maximal dissimilarity from each element to the other elements of the set.³⁷

• The Minimal Aggregation Measure, which measures diversity by aggregating the minimal dissimilarity from each element to the other elements of the set (disregarding the dissimilarity between an element and itself).³⁸

• The Elimination Measure, which measures diversity by aggregating dissimilarities selected by an iterative procedure where elements are eliminated one by one.³⁹

• The Minimal Path Measure, which measures diversity by aggregating those dissimilarities which connect all elements of a set by a minimal path.⁴⁰

These measures are described and characterised in more detail in the literature where they occur. Here we shall only look at how they fare with regard to the proposed conditions. This is captured in the following table:

Diversity measures and conditions

Measures/Conditions	Scale Independence	Minimal Diversity	Diameter	Restricted Growth	Strict Monotonicity	Even Spacing
Total Dissimilarity	YES	YES	YES	NO	YES	NO
Average Dissimilarity	YES	YES	YES	YES	NO	NO
Range	YES	YES	YES	YES	NO	NO
Lexical	YES	YES	YES	N/A	YES	NO
Average Aggregation	YES	YES	YES	YES	NO	NO
Maximal Aggregation	YES	YES	YES	YES	YES	NO
Minimal Aggregation	YES	YES	YES	YES	NO	YES
Elimination	YES	YES	YES	YES	YES	NO
Minimal Path	YES	YES	YES	YES	NO	NO

The measures fare the worst with respect to the Even Spacing Condition, since this condition is satisfied only by the Minimal Aggregation Measure. This can be shown with some examples. Let us first consider the set A = (20, 30, 40) which has values that are evenly spaced. This set should be ranked above B = (20, 21, 40) which does not have values that are evenly spaced. However, the Total Dissimilarity Measure ranks A as equally diverse as B and so does the Average Dissimilarity Measure, the Range Measure, the Average Aggregation Measure and the Minimal Path Measure. The Lexical Measure and the Maximal Aggregation Measure even rank B over A. Although the Elimination Measure ranks A over B, as the condition requires, it ranks (10, 17.5, 25, 40) above (10, 20, 30, 40), and so it does not fulfil the condition either.

6. The shortest path root measure

We have finally come to the measure proposed in this essay.  

To construct this measure we first index and order all the n elements x in the set A that we are to assess, in accordance with their degree of Q, measured by q. (Note again that elements that have the same degree of Q are considered to be identical elements.) The elements are indexed from 1 to n, as x₁ to x_n, in such a way that q(x₁) < q(x₂) … q(x_n-1) < q(x_n). The dissimilarity d between two elements, x_i and x_j with respect to Q is then measured by their absolute difference in q. Last we define the diversity measure as follows:

D(A, d) = $\alpha \sum _{i=2}^{n-1}{d}_i{(x_i,x_{i-1})}^r$, where D is a measure of diversity, A is a set, d is a measure of dissimilarity, as defined above, n is the number of elements in A, x_i are elements in A, indexed and ordered from 1 to n, as described above, and 0 < r < 1 and α > 0.

Since neither r nor α are precisely defined, the measure is actually a class of measures. Because the measure applies a root function on the dissimilarities (regarded as one-dimensional distances) that constitute the shortest path between the elements of A, it can be called the Shortest Path Root Measure.  A simple measure in this class is the Shortest Path Square Root Measure, using r = ½, and defined as follows:

D(A, d) = $\sum _{i=2}^n\sqrt{d(x_i,x_{i-1})}$

Let us see how this last measure works by using an example. Suppose we are to compare the set A = (20, 30, 40) to the set B = (20, 21, 40) in terms of diversity. First, we index the elements of A, so that q(x₁) = 20, q(x₂) = 30 and q(x₃) = 40. Then we calculate D(A)  = $\sqrt{|30-20|}$ + $\sqrt{|40-30|}$ = $2\sqrt{10}$ ≈ 6.3. Next we use the same technique for B and get the result that D(B) = $\sqrt{|21-20|}$ + $\sqrt{|40-21|}$ ≈ 4.4. Thus, we conclude that A is more diverse than B.

The Shortest Path Root Measure satisfies all of the six conditions. The measure satisfies the Scale Independence Condition, since when all the dissimilarities between the elements of A are multiplied by K, D(A) is multiplied by K^r . The measure satisfies the Minimal Diversity Condition, since it does not assign any value at all to the empty set and singleton sets. The measure satisfies the Diameter Condition, since it is a strictly increasing function of dissimilarity values. The measure satisfies the Restricted Growth Condition, since D(A) cannot grow by more than D((x, y)) when y is added to A (in fact, unless A is a singleton set, it always grows by less). The measure satisfies the Even Spacing Condition, since it is an additively separable strictly concave function of dissimilarities between adjacent aspect values.⁴¹ Lastly, the measure satisfies the Strict Monotonicity Condition, for two reasons: (i) when an element y is added that expands the diameter of a set A, the measure satisfies the condition because it is a strictly increasing function of dissimilarity values; (ii) when an element y is added that does not expand the diameter of A (because it is added in between two other elements), the measure satisfies the condition because it is an additively separable strictly concave function. It thus has the property of strict subadditivity, a property which is defined as follows:

A function f is strictly subadditive if and only if f(d₁ + d₂) < f(d₁) + f(d₂) holds for all d_i > 0.

For a diversity measure D, based on dissimilarities, this means that for a fixed total of dissimilarities, it is possible to obtain a greater value for D by splitting the total between more individual dissimilarity values. This is what happens when an element y is added that has a value that lies between the values of two other elements x and z, since the dissimilarity value d(x, z) is then replaced by the two dissimilarity values d(x, y) and d(y, z) and d(x, z) = d(x, y) + d(y, z). This allows the measure to satisfy the Strict Monotonicity Condition also in this case, and thus the measure satisfies all six conditions for a one-dimensional diversity measure.

7. The uniqueness of the shortest path root measure

The Shortest Path Root Measure satisfies the six conditions, but one may wonder how unique the measure is in this regard. As it turns out, it is possible to show that the Shortest Path Root Measure is unique among continuous analytic functions in satisfying all of the conditions in combination with a Symmetry Condition. More precisely, it is possible to show that the Shortest Path Root Measure is the only continuous analytic function that jointly satisfies the Ratio Scale Independence Condition, the Limited Diameter Condition, and the Limited Even Spacing Condition, together with a Symmetry Condition. A proof of this can found in the Appendix.

The restriction to continuous analytical functions is standard for characterisation proofs of various measures. Symmetry assumptions are standard too, but worth discussing. Since the Shortest Path Root Measure is a function of the shortest path distances, the symmetry condition that the measure satisfies concerns only these distances. We can thus formulate a relevant symmetry condition as follows:

The Symmetry Condition: For a diversity measure D, and for all sets A, B such that #A = #B if there is a bijection from the elements x ∈ A to the elements y ∈ B so that for each pair of dissimilarities between adjacent elements (d(x_i, x_i+1), d(y_j, y_j+1)), it is that case that d(x_i, x_i+1) = d(y_j, y_j+1), then A is equally diverse as B, and thus D(A) = D(B).⁴²

According to the Symmetry Condition it does not matter in which order the distances of a shortest path vector are arranged. The distance vector [1, 2, 3] is as diverse as the distance vector [1, 3, 2] or the distance vectors [2, 1, 3], [2, 3, 1], [3, 1, 2], and [3, 2, 1], for example. Symmetry conditions are common in the literature on diversity measures and are often taken for granted.⁴³ However, if one does not want to take them for granted, one could argue for them in various ways. One possible argument is that the Symmetry Condition fits perfectly with the Limited Even Spacing Condition. This condition implies that if two sets A and B have the same number of elements and the same diameter, diversity is maximised if all the elements are evenly spaced, for example if the shortest path vector has the form [α₁, α₂, … , α_n-1, α_n], where α > 0. If we accept this, we could also assume that the degrees of diversity of sets with the same number of elements n and the same diameter are wholly determined by how close their shortest path vectors are to the ideal evenly spaced vector [α₁, α₂, … , α_n-1, α_n]. And if we accept this, we could also assume that a measure of closeness to even spacing should be symmetric since the closeness between [b₁, b₂, … , b_n-1, b_n] and [α, α, … , α, α] seems no different than the closeness between, for example, [b₂, b₁, … , b_n-1, b_n] and [α, α, … , α, α]. All of this suggests that the diversity measure should be a symmetrical measure.

Against the view that a one-dimensional diversity measure should be symmetric, one could propose the following counterexample: Consider the following two sets of objects: A = (1, 2, 11, 12) and B = (1, 2, 3, 12). The vectors of shortest path distances are d_ASP = [1, 9, 1] and d_BSP = [1, 1, 9]. According to the Symmetry Condition, the two sets are equally diverse. However, A is intuitively more diverse than B. This shows that the sizes of the dissimilarities are not the only things that matter, but something else matters as well. One proposal is that what additionally matters is the shortest path dissimilarities between the shortest path dissimilarities, and the shortest path dissimilarities between the shortest path dissimilarities between the shortest path dissimilarities a.s.o. (The vector of shortest path dissimilarities between the shortest path dissimilarities is [8, 8] for A, and [0, 8] for B.)

If we wish to accommodate the above intuition, we cannot use a symmetric measure on vectors of shortest path distances but must use an asymmetric measure. A potential method would be to use a lexical measure which first compares aggregate dissimilarities, modified by a root function, and if equal, compares aggregate dissimilarities between dissimilarities, modified by a root function and, if equal, continues this process until there are no values left to aggregate. I will not discuss the virtues or problems of such a complex measure here, however, but will just note that one could argue both for and against symmetrical one-dimensional diversity measures.

8. Conclusion

In this essay I have introduced a new kind of diversity measure, designed for sets with elements that differ only in a single aspect, which I called the Shortest Path Root Measure. The measure is responsive to the magnitude of the diameter, the number of dissimilar elements and the even spacing of the elements’ aspect values. More precisely, it satisfies six necessary conditions for a single aspect diversity measure: Ratio Scale Independence, Minimal Diversity, Diameter, Restricted Growth, Strict Monotonicity, and Even Spacing. By satisfying these conditions the measure improves on previous proposals, which do not satisfy all of them. In addition, it can be shown that the Shortest Path Root Measure is the only one, within the class of continuous analytic functions, which jointly satisfies four conditions: the Ratio Scale Independence Condition, listed above, a Limited Diameter Condition (implied by the Diameter Condition), a Limited Even Spacing Condition (implying the Even Spacing Condition, assuming transitivity) and a Symmetry Condition. If these conditions are necessary for a one-dimensional diversity measure, then the Shortest Path Root Measure is necessary for measuring one-dimensional diversity as well.⁴⁴

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Philosophical overviews can be found in for example Page 2007 (epistemology), Gravel 2009 (philosophy of science), Audi 2007 (ethics), Crowder 2007 (political philosophy), and Bossert, Pattanaik, and Xu 2003 (decision theory).

2 For number of elements diversity measures, see Van Hees (2004) and Gravel (2009); for number of attributes diversity measures, see Walker et al (1999), Nehring and Puppe (2002), and Shi and Eberhart (2008); and for number of classes diversity measures, see Connell and Orias (1964) and Lloyd and Ghelardi (1964).

3 For ordinal diversity measures, see Nehring and Puppe (2002), Bervoets and Gravel (2007), and Pattanaik and Xu (2008).

4 For multiple aspect diversity measures, see Rao (1982), Wen, Wang, and Cheng (1998), Barker and Martin (2000), Ursem (2002), Bossert, Pattanaik, and Xu (2003), Van Hees (2004), Lacevic, Konjicija, and Avdagic (2007), Lacevic and Arnaldi (2010), and Enflo (2012).

5 Cf. Dowden (2011, 6).

6 See Enflo (2012).

7 Compare Junge who defines homogeneity as similarity among more than two objects (1978, 114).

8 For diversity as cardinality, see Van Hees (2004) and Gravel (2009).

9 For diversity as number of attributes, see Walker, Kinzig, and Langridge (1999) and Nehring and Puppe (2002).

10 For diversity as number of classes, see Connell and Orias (1964) and Lloyd and Ghelardi (1964); for diversity as dissimilarities between classes, see Vane-Wright, Humphries, and Williams (1991), Eiswerth and Haney (1992), Faith (1992), Weitzman (1992), Walker, Kinzig, and Langridge (1999), and Izsák and Papp (2000); for diversity as number and evenness, see Shannon and Weaver (1949), Simpson (1949), McIntosh (1967), Junge (1994), and Justus (2011); for diversity as number and dissimilarities, see Helmus et al. (2007), Chao, Chiu, and Jost (2010) and Dowden (2011); for diversity as abundance and dissimilarities, see Barker (2002); for diversity as evenness and dissimilarities, see Barker (2002), Ricotta (2004); for diversity as number, evenness and dissimilarities, see Warwick and Clarke (1995), Ganeshaiah, Chandrashekara, and Kumar (1997), Shimatani (2001), Mason et al. (2003), Ricotta and Avena (2003), Weikard, Punt, and Wesseler (2006), Helmus et al. (2007), Stirling (2007), and Allen, Kon, and Bar-Yam (2009).

11 For diversity as dissimilarities to elements in the universal set, see Gravel (2009) and Eiswerth and Haney (1992). Cf. Gustafsson (2010). For diversity as dissimilarities to a mean element of the universal set, see Ursem (2002).

12 For diversity as dissimilarities to the mean of a set, see Crocker (1980), Morrison and De Jong (2002), and Shi and Eberhart (2008).

13 For variance, see Fisher (1919); for standard deviation, see Pearson (1894); and for mean deviation, see Eddington (1914).

14 For diversity as totality of dissimilarities, see Barker and Martin (2000).

15 For diversity as selected dissimilarities, see Rao (1982), Wen, Wang, and Cheng (1998), Ursem (2002), Bossert, Pattanaik, and Xu (2003), Van Hees (2004), and Bervoets and Gravel (2007).

16 For diversity as number, equality and dissimilarities, see Enflo (2012). Cf. Dowden (2011).

17 See Suppes and Zinnes (1963, 17–19).

18 See Suppes and Zinnes (1963, 5, 7, 11).

19 See Stevens (1946, 678).

20 See Russell (1903, 179).

21 An early example of dissimilarity represented by distance can be found in Russell (1903, 171). A later example can be found in Shepard (1958, 510). For the original presentation of metric spaces, see Fréchet (1910). Discussion of these and other dissimilarity measures can be found in Enflo (2020).

22 Similar scale independence conditions have previously been proposed by Solow and Polasky (1994, 97–98), Hubálek (2000, 255), Mason et al. (2003, 573), and Dowden (2011, 4).

23 Shannon’s diversity (and entropy) index (1949) and Rényi’s diversity (and entropy) index (1961) are examples of logarithmic measures.

24 See Dowden (2011, 3).

25 See Bossert, Pattanaik, and Xu (2003, 418), Bervoets and Gravel (2007, 264), and Gravel (2009, 39).

26 This diameter condition is implied by Weitzman’s Monotonicity in Distances condition (Weitzman 1992, 392) and Nehring and Puppe’s Monotonicity in Dissimilarity condition (2002, 1182).

27 For the view that diversity is independent of number, see for example Patil and Taillie (1982, 548).

28 Cf. Page (2010, 59).

29 This restricted growth condition is rather similar to a condition proposed by Weitzman (but for species). See Weitzman (1992, 392).

30 Similar monotonicity conditions have been proposed previously by Solow, Polasky, and Broadus (1993, 62), Bossert, Pattanaik, and Xu (2003, 413), Van Hees (2004, 258), Pattanaik and Xu (2008, 264), Gravel (2009, 20), and Dowden (2011, 3). Monotonicity conditions for species have been proposed by Weitzman (1992, 376), Daly, Baetens, and De Baets (2018, 8).

31 Cf. Nehring and Puppe (2003, 177).

32 See Barker and Martin (2000, 1005), Wineberg and Oppacher (2003, 1494), and Hashemi and Endriss (2014, 424).

33 See Rao (1982, 3) and Wen, Wang, and Cheng (1998, 773).

34 See Gravel (2009, 43). Cf. Shi and Eberhart (2008, 1064).

35 See Bervoets and Gravel (2007, 264–265).

36 See Van Hees (2004, 263).

37 See Van Hees (2004, 261).

38 See Weitzman (1992, 367) and Van Hees (2004:, 260).

39 See Weitzman (1992, 375).

40 See Faith (1996, 1286), Nehring and Puppe (2002, 1158), and Lacevic and Arnaldi (2010, 5). Cf. Petchey and Gaston (2002, 403).

41 According to both Kempton (1979) and Rao (1982), concavity is a necessary feature of a diversity measure. Concave measures have also been used for measuring other kinds of properties in areas were even spacing, evenness or equality of some aspect values are considered important. There are, for example, concave egalitarian social welfare measures (in Weirich 1983), concave risk averse expected utility measures (in Pratt 1964) and concave information measures (in Hartley 1928).

42 Symmetry conditions have been proposed for multiple aspect dissimilarity diversity by, for example, Laxton (1978, 53), Jost (2008, 927), Page (2010, 67), Leinster and Cobbold (2012, 9), Hashemi and Endriss (2014, 426), and Daly et al. (2018, 126).

43 That diversity measures should be symmetrical is taken for granted by, for example, Laxton (1978, 53), Kreutz-Delgado and Rao (1999, 1082), Jost (2008, 927), Leinster and Cobbold (2012, 9), and Hashemi and Endriss (2014, 426).

44 I am grateful to Per Enflo, Victor Moberger, and several referees for their help in improving this paper.

45 See Glaeser (1963, 205–206).

Appendix

The uniqueness of the shortest path root measure

Uniqueness theorem: If a diversity measure D is a continuous analytic function of a vector of positive real numbers, representing the dissimilarities between the elements of a set with respect to the elements’ values of some single aspect q, and if D satisfies the four conditions of Limited Diameter, Limited Even Spacing, Symmetry and Ratio Scale Independence, then D is a Shortest Path Root Measure.

*  *  *

The conditions and the measure:

The Limited Diameter Condition: For a diversity measure D and any sets A and B and any elements x, y, w, z, such that A = (x, y) and B = (w, z), if d(x, y) > d(w, z), then A is strictly more diverse than B and thus D(A) > D(B).

The Limited Even Spacing Condition: For a diversity measure D, and for all sets A, B such that #A = #B and Ø(A) = Ø(B), if there is a bijection from the elements x ∈ A to the elements y ∈ B such that for each pair of elements (x_i, y_i) it is that case that q(x_i) = q(y_i), except for one case (x_j, y_j) which is such that q(x_j) ≠ q(y_j) and where the consecutive elements x_j-1, x_j, x_j+1 are completely evenly spaced so that d(x_j-1, x_j) = d(x_j, x_j+1), whereas the consecutive elements y_j-1, y_j, y_j+1 are not completely evenly spaced so that d(y_j-1, y_j) ≠ d(y_j, y_j+1), then A is strictly more diverse than B, and thus D(A) > D(B).

The Symmetry Condition: For a diversity measure D, and for all sets A, B such that #A = #B if there is a bijection from the elements x ∈ A to the elements y ∈ B so that for each pair of dissimilarities between adjacent elements (d(x_i, x_i+1), d(y_j, y_j+1)), it is that case that d(x_i, x_i+1) = d(y_j, y_j+1), then A is equally diverse as B, and thus D(A) = D(B).

The Ratio Scale Independence Condition: For a diversity measure D that is a function of a dissimilarity function d, it holds that for all numbers K > 0, there is a number L = L(K) > 0, where L is an increasing function of K such that if d is multiplied by K, then D is multiplied by L.

The Shortest Path Root Measure:

D(A, d) = α$\sum _{i=2}^{n-1}d{(x_i,x_{i-1})}^r$, where D is a measure of diversity, A is a set, d is a measure of dissimilarity d(x_i, x_j) = |q(x_i) – q(x_j)|, n is the number of elements in A, and x_i are elements in A, indexed and ordered from 1 to n, as x₁ to x_n, in such a way that q(x₁) < q(x₂) … q(x_n-1) < q(x_n) and where 0 < r < 1 and α > 0.

*  *  *

Assumptions

We assume that the one-dimensional diversity of a set is a function of the single aspect dissimilarities between the elements of a set, which in turn is a function of the elements’ values of some single aspect q. Consequently, the one-dimensional diversity of a set is ultimately a function of the element’s values of q. We also assume that there is a real valued measure q of some single aspect Q and that we can measure the dissimilarity d between two objects, x and y, with respect to Q, by the absolute difference between q(x) and q(y).

We further assume that there is a finite universal set of objects, X. We index all the elements of X from 1 to n, as x₁, x₂ … x_n−1, x_n so that q(x₁) < q(x₂) < … q(x_n−1) < q(x_n). The same principle is used to index the elements of subsets of X. All the shortest path dissimilarities between the elements of X are indexed so that d₁ = d(x₁, x₂), d₂= d(x₂, x₃) etc.

Proof

First, we make the assumption that a one-dimensional diversity measure D is a continuous analytic function of a vector of positive real numbers, representing the dissimilarities between the elements of a set with respect to the elements’ values of some single aspect q.

We will see that under very general assumptions on the diversity measure D, the shortest path dissimilarities d(x_i, x_i+1) = | q(x_i) – q(x_i+1) |, 1 ≤ i ≤ n – 1, play a special role when measuring the one-dimensional diversity of a set. We will prove the following theorem:

Theorem: If the diversity D of a set A is given by a strictly concave, continuous function D of a vector of shortest path dissimilarities between the elements of A with respect to q, and thus of a vector of values of the elements of A with respect to q, and if q(x₁) = a and q(x_n) = b are fixed, then D is maximized when d(x_i, x_i+1) = d(x_j, x_j+1) = (b – a)/(n – 1), for all i and j, 1 ≤ i ≤ n – 1, 1 ≤ j ≤ n – 1.

We first show that D satisfies the Limited Even Spacing Condition. The concavity of the diversity function D gives the following: Assume we have two sets A and B with the same number n of elements, x₁, … , x_n ∈ A and y₁, … ., y_n ∈ B. Assume that q(x_i) = q(y_i), except for one of the i:s, say i = m, 1 < m < n. So q(x_m) ≠ q(y_m). Assume that in A, q(x_m) = 1/2(q(x_m+1) + q(x_m-1)). Then the strict concavity of D gives D(A) > D(B). Thus D satisfies the Limited Even Spacing Condition. … (I).

From (I) we get the Theorem as follows: Assume that for a set B* not all the d(x_i, x_i+1):s are the same. Then there must be one i, for which d(x_i, x_i+1) ≠ d(x_i+1, x_i+2) for B*. Then, by forming a set A* for which q(x_i+1) = 1/2(q(x_i) + q(x_i+2)), we get from (I) that D(A*) > D(B*). Thus D(B*) is not maximal and thus the Theorem is proved.

That D satisfies the Limited Even Spacing Condition trivially implies that D is a strictly concave, continuous function of a vector of shortest path dissimilarities between the elements of A with respect to q.

Next, let the shortest path distances between the elements of A be written as a vector d_ASP = [d₁, d₂, … , d_m] with m = the number of shortest path distances, m = n – 1. We get D(A, d) = D(f(d₁), f(d₂), … , f(d_m)). The Symmetry Condition gives that D is a symmetric function. Since D is an analytic and symmetric function of f(d₁), f(d₂), … , f(d_m), the expansion of a differentiable symmetric function via elementary symmetric functions gives:

D(A, d) = D(d_ASP) = $C(n_1)\left(\sum _{i=1}^m(f(d_i))\right)$ + $C(n_2)\left(\sum _{i=1,i\ne j}^m\sum _{j=1}^m(f(d_i)f(d_j)\right)$ + $C(n_3)\left(\sum _{i=1}^m{(f(d_i))}^2\right)$ + higher order terms in the f(d_i) f(d_j)’s. … (I)⁴⁵

Since D has a non-vanishing derivative at 0 as a function of f(d₁), f(d₂), … , f(d_m), we must have C(n₁) different from 0.

Now consider A = A_j with d₁ = d₂  = … =  d_j = d and d_j+1 = d_j+2  = … =  d_m = 0, 1 ≤ j ≤ m (this is hypothetical since we have assumed that we cannot have 0 distances without identity). Putting this into (I) gives:

D(A_j) = g_j(d) = jC(n₁) f(d) +{C(n₂)J + C(n₃) j}(f(d))² + higher order terms with J = j(j – 1) = the number of (d_i)(d_j)’s different from 0. … . (II)

According to the Ratio Scale Independence Condition multiplying d by a number K will multiply g_j(d) by a number L = L(K). It is a well-known mathematical fact that this gives g_j(d) = α(j)d^r. Since f(d) → 0 as d → 0, the first degree term in f(d) on the right hand side of (II) will dominate as d → 0 and so from (II) get that ${\displaystyle \frac{\alpha (j)d^r}{jC(n_1)f(d)}}$ →1 as d → 0. Since this holds for all j, 1 ≤ j ≤ m, we have that ${\displaystyle \frac{\alpha (j)}{j}}$ is constant, say ${\displaystyle \frac{\alpha (j)}{j}}$ = α and ${\displaystyle \frac{g_j(d)}{j}}$ = g(d) = αd^r. By dividing (II) by j, this gives:

${\displaystyle \frac{g_j(d)}{j}}$ = αd^r= C(n₁) f(d) +{C(n₂)${\displaystyle \frac{J}{j}}$ + C(n₃) }(f(d))² + higher order terms … (III)

Since this holds for all j, 1 ≤ j ≤ m and ${\displaystyle \frac{J}{j}}$ = j – 1, we must have C(n₂) = 0 in (III) and so C(n₂) = 0 in (I). Thus putting d = d_i in (III) and putting that back into (I), we get D(A, d) = α$\sum _{i=1}^m(d_i{)}^r$. The Limited Even Spacing Condition gives that this function should be strictly concave and the Limited Diameter Condition gives that this function should be strictly increasing so 0 < r < 1, α > 0. Thus we have: D(A, d) = α$\sum _{i=1}^n(d_i{)}^r$, where 0 < r < 1, α > 0.

We thus see that the acceptance of the four conditions necessitates a Shortest Path Root Measure.

Q.E.D.