Unidimensional Development Ranking and Fuzzy Lorenz Dominance

Abstract

This chapter is devoted to an attempt at extending the unidimensional theory of development ranking so as to reduce the possibility of ranking failures. For this purpose, we borrow from the theory of fuzzy sets in mathematics and use the notion of fuzzy Lorenz dominance relation. The approach of modelling Lorenz dominance by a fuzzy binary relation was proposed in the 1980s. The idea does not seem to have been followed up actively in the subsequent literature. The fuzzy dominance relation proposed in the chapter is a follow-up on this line of research. It is, however, different from the earlier suggestions. Moreover, if the idea of Lorenz dominance is “fuzzified”, it would be natural to fuzzify the notion of development ranking itself. Indeed, this is what we do in this chapter. It is seen that such fuzzy development rankings can also be used to induce crisp (i.e. non-fuzzy) development rankings. The ranking methods developed in this chapter seem to be able to reduce the preponderance of the problem of ranking failures that arises frequently under the crisp (i.e. non-fuzzy) approach. In particular, it is shown that if two economies have the same per capita income, we shall now always be able to rank them in terms of development. While that is not the case when per capita incomes differ, completeness of the ranking is achieved under weaker conditions than in crisp theory.

Appendix: Fuzzy Sets and Relations

Since the notion of fuzzy relations may not be familiar territory to all readers, some of the basic ideas behind the notion have been sought to be summarised in this Appendix. This is not meant to be an exhaustive review all aspects of the theory.

Sets and relations

Start with the notion of a set. When we say that A is a set, we implicitly assume that in the background, there is a universal set X (say) of which the set A under consideration is a subset. (For instance, if A is a set of some particular countries, we may consider X to be the set of all countries in the world). A set can be formally described with the help of a function. For any set A, consider the function m_A (say) on the domain X into the real line constructed in the following way. For every member x of X, check whether x is in A or not. If it is, then choose the value of the function to be 1; if it is not, then choose it to be zero. Thus, the range of the function is the set {0, 1} of the two numbers 0 and 1. We can, of course, take any other pair of numbers instead of 0 and 1. As long as the numbers are different, they would serve our purpose. Taking 0 and 1 is, however, the convention. m_A is called the membership function of A.

Note that a set A and its membership function are logically equivalent. Once we specify a set (i.e. we specify which members of the universal set belong to A and which ones do not), we can construct the membership function of A. Conversely, if we specify the membership function, then we know the set A (i.e. we know which members of X are in set A and which are not). Thus, for any set, there is function to which it is logically equivalent. In this sense, we say that a set is a function.

Next, consider the notion of binary relation on a set. This notion has already been introduced and discussed in Chap. 1. A binary relation means a relation between two things. Since this is the only type of relation that is used in this book, we drop the prefix “binary” and talk about relations. Recall that a relation R on a set is the description of a relationship between pairs of members of the set. For example, if X is the set of all countries of the world, R may stand for the phrase “is at least as developed as”. For any two members x and y of X x R y would mean that x is at least as developed as y i.e. either x is more developed than y or x and y are equally developed. Similarly, if X is the set all possible income vectors, R may stand for the statement “weakly Lorenz dominates” (i.e. the Lorenz curve of x does not lie below that of y at any point). For any two income vectors x and y, x R y would then mean that x weakly Lorenz dominates y. In all these cases what is important is that, for any x and y in X, we must be able to say whether each of the two statements x R y and y R x is true or not. If x R y is true, we say that the truth value of the statement that x R y is 1. If it is false, then we say that the truth value of x R y is 0.

The set of all ordered pairs (x, y) such that both x and y are in X is denoted by X × X or X². Mathematicians point out that a relation R on a set X can be considered to be a subset of X² (and, therefore, we can write $R \subseteq {\mathbf{X}}^{2}$) if we specify that a pair (x, y) is in the subset R of X² if and only if x R y.

As we have already seen above, a set can be interpreted to be a function. Now we see that a relation is a set. It follows that a relation is also a function. We only have to identify the membership function of the set with the truth value function of the relation. We may use the symbol R itself as the membership or the truth value function of R. But now the domain of the function is X² (rather than X), and, again, 0 and 1 are the only possible values of the function. For any x and y in X, i.e. for any ordered pair (x, y) in X², we put R(x, y) = 1 if it is true that x R y; otherwise, R(x, y) = 0.

Fuzzy relations: The cardinal approach

We come now to the matter of fuzzy sets and relations. The notion of fuzzy sets was introduced in mathematics independently by Zadeh (1965) and Klaua (1965). (Klaua’s contribution was published in German. A recent analysis of the paper is by Gottwald (2010)) There were, however, several earlier thinkers, especially in the fields of logic and mathematics, who had anticipated some of the basic ideas. (The curious reader is referred to the first two papers in Dubois and Prade (2015)).

The notion of a fuzzy set is a generalisation of the notion at of a set considered above. In many cases, it happens that the question whether a member x of the universal set X belongs to a set A in X does not have a clear yes/no answer. For instance, if X is the set of all countries and A is the set of developed countries, A would not be an unambiguously defined set unless we arbitrarily define exactly what development means, i.e. exactly how developed a country has to be in order to be considered developed. Usually, only the extreme cases are unambiguous. For example, we may agree that if the per capita income of a country is zero, then the country is not developed. There may also be a consensus that if a country’s per capita income is the highest in the world, then it is developed. The intermediates cases, however, will be ambiguous. (In reality, the task of classification will be even more complex if we say that per capita income is not the only criterion of development. For the sake of simplicity, however, forget this additional complexity).

One way of handling such ambiguous cases is to generalise the notion of the membership function of a set by expanding its range. A fuzzy set A in the universal set X is a set whose membership function again has the domain X but whose range is the closed unit interval [0, 1] rather than just the set {0, 1} of the two numbers 0 and 1. Thus, for any x in X, m_A(x) can now take intermediate values between 0 and 1. If m_A(x) = 0, it would mean that x is definitely not in A; if m_A(x) = 1, it would mean that x is definitely in A. If m_A(x) is, say 0.7, it would mean that the “extent to which x is in A is 0.7” (on a scale from 0 to 1). The type of set discussed earlier is now called a crisp or an exact set. A crisp set is a special case of a fuzzy set in which the range of the membership function is the subset {0, 1} of the closed interval [0, 1].

It is important to note that the type of ambiguity discussed in fuzzy set theory has nothing to do with the notion of probability. The statement that the extent to which a particular member x of the universal set X belongs to the set A is 0.7 is quite different in meaning from the statement that the probability of the event that x is in A is 0.7. The latter statement means (under certain simplifying assumptions which we need not go into here) that if it was repeatedly observed whether x is in A or not, then 70% of the times it will be observed to be in A. In each single observation, however, x is either in A or it is not. There is no ambiguity in that respect. Fuzzy set theory, on the other hand, is concerned with ambiguity (in any given observation) regarding whether x is in A. 0.7 is here the degree or the extent to which it is true that x is in A.

The notion of fuzzy relations is, similarly, a generalisation of that of relations as discussed above. A fuzzy relation on a universal set X is a function from X² into the closed unit interval [0, 1]. Start, again, by stating that, for any x and y in X, x R y means that x and y stand in a certain relationship to each other. (For instance, again, it may mean that country x is at least as developed as y or that the income vector x weakly Lorenz dominates the income vector y). For any ordered pair (x, y), we now put R(x, y) = 1 if it is definitely true that x R y. We put R(x, y) = 0 if x R y is definitely not true. But now, we permit R(x, y) to take intermediate values. R(x, y) = 0.6 would mean that the degree or the extent to which it is true that x R y is 0.6. It is easily seen that a relation of the type described earlier, called a crisp (or an exact) relation, is a special case of a fuzzy relation in which, for all x and y in X, R(x, y) is either 0 or 1. Moreover, just as a crisp relation can be interpreted as a crisp set, a fuzzy relation is a fuzzy set.

Next, recall the notion of a crisp ordering. Recall that if R is a crisp relation on a set X, it called reflexive if, for all x in X, x R x. Both the examples of a crisp relation mentioned above (the relation “is at least as developed as” on a set of countries and the relation “weakly Lorenz dominates” on a set of income vectors) are instances of reflexive crisp relations. R is called complete if, for all x and y in X, either x R y or y R x (or both) must be true. Weak Lorenz dominance is an example of a relation which may fail to be complete (since income vectors x and y may be such that their Lorenz curves cross). Whether “is at least as developed as” is a complete relation on a set of countries depends on how development is measured. If the state of development of a country is measured by its per capita income, then this relation is complete since either the per capita income of x is greater than or equal to that of y or that of y is greater than or equal to that of x. (If per capita incomes in x and y are the same, then both of these are true). However, if development is measured by some other criterion (e.g., if distributional equity considerations are also a part of the criterion) then completeness is not guaranteed. R is called transitive if, for all x, y and z in X, [x R y and y R z] implies x R z. It is easily checked that weak Lorenz dominance is a transitive relation. If development of a country is judged by per capita income, then so is the relation “at least as developed as”. If a crisp relation is reflexive, complete and transitive, it is called an ordering.

The properties of reflexivity, completeness and transitivity have been sought to be formulated for fuzzy relations also. For any x and y, we now have to make use of the numerical values of R(x, y) and R(y, x). It is a simple matter to extend the definition of reflexivity to the fuzzy case. Comparing with the crisp case, it is seen that reflexivity would require that for any x in X, R(x, x) = 1. It is easily seen that this is equivalent to saying that R is reflexive if, for all x and y in X, if x and y are the same member of X, then the statement that x R y must be definitely true. A fuzzy relation R is defined to be complete if, for all x and y in X, R(x, y) + R(y, x) ≥ 1. This is easily seen to be a generalisation of the definition of completeness from the crisp to the fuzzy context. In particular, in the special case where R happens to be crisp, completeness requires that, for all x and y in X, either x R y or y R x (or both) must be definitely true. In other words, either R(x, y) = 1 or R(y, x) = 1 or both of these two equations must be true. Obviously, if any one of these two equations are valid, then R(x, y) + R(y, x) cannot be less than 1 (because both R(x, y) and R(y, x) are in the closed interval [0, 1], i.e. neither of them can be negative). Hence, R(x, y) + R(y, x) ≥ 1.

It is more difficult to formulate a transparent definition of transitivity in the fuzzy case. What is clear is that any proposed definition must be consistent with crisp transitivity. Thus, for all x, y and z in X, if R(x, y) = 1 and R(y, z) = 1, then we should require R(x, z) = 1. R(x, y) = 1 means that it is definitely true that x R y and R(y, z) = 1 means that y R z is definitely true. Obviously, therefore, any concept of transitivity should require that it is definitely true that x R z, i.e. R(x, z) = 1. This weak notion of transitivity was called “E-transitivity” in Basu (1987).

Beyond this, there is no generally accepted definition of transitivity. Different authors have put forward different proposals. Zadeh (1965) proposed a definition which is now called “max-min” transitivity in order to distinguish it from other definitions. R is max-min transitive if, for all x, y and z in X, R(x, z) ≥ min[R(x, y), R(y, z)]. (Dutta (1987) calls it T₁ transitivity.) It is easy to check that max-min transitivity satisfies the minimal requirement of consistency with the definition of transitivity in the crisp case: if R(x, y) = 1 and R(y, z) = 1, then min[R(x, y), R(y, z)] = 1 so that max-min transitivity would require R(x, z) ≥ 1. However, R(x, z) has to be in the closed interval [0, 1]. In other words, R(x, z) ≤ 1. Therefore, R(x, z) must be exactly 1. The subsequent literature contains many other suggestions all of which are consistent with transitivity in the crisp case. We desist from providing a detailed survey of all of these proposals since these have not have been used in this book. For the sake of completeness of the present discussion, however, we mention a broad concept of transitivity in the fuzzy case (called “max-star” transitivity). This is a broad class of transitivity definitions rather than one specific definition. This class of definitions was proposed by Ovchinnikov (1984). Stated in slightly different words from those used by Ovchinnikov, max-star transitivity requires that, for all x, y and z in X, R(x, z) ≥ f(R(x, y), R(y, z)) where f is any function with the set [0, 1] × [0, 1] as its domain into the set [0, 1] and where it is required that the function f satisfies a number of properties. The following are the desired properties of f.

(a) Monotonicity: f(p, q) ≤ f(r, s) for all p, q, r and s in [0, 1] such that p ≤ r and q ≤ s; (b) Commutativity: for all p and q in [0, 1], f(p, q) = f(q, p); (c) Associativity: for all p, q and r in [0, 1], f(f(p, q), r) = f(p, f(q, r)) and (d) Boundary Condition: for all p in [0, 1], f(p, 1) = p.

Any specific f function satisfying the above-mentioned properties would yield a specific definition of a transitive fuzzy relation. It is easily checked that max-min (or T₁) transitivity mentioned above is a member of the max-star class of definitions of transitivity: for all p and q in [0, 1], just put f(p, q) = min(p, q), remembering that, for any fuzzy relation R on X and for any x and y in X, R(x, y) is in the interval [0, 1]. Another member is T₂-transitivity examined in Dutta (1987). It requires that, for all x, y and z in X, R(x, z) ≥ R(x, y) + R(y, z) − 1. To see that this is also a type of max-star transitivity, put f(p, q) = p + q − 1 for all p and q in [0, 1]. For an analysis of some of the properties of T₁ and T₂ transitivities, see Dutta (1987).

The fact that there are a whole class of transitivity definitions for fuzzy relations is somewhat perplexing for the applied researcher since it leaves one wondering as to which particular definition to adopt. It is possible that results obtained by applying one of these specific definitions in the analysis of an economic (or any other) problem may not be valid if a different definition were adopted. Moreover, there have been further extensions of the theory. For instance, why not make transitivity itself a fuzzy concept? In other words, instead of making transitivity an all-or-nothing proposition (i.e. instead of assuming that a given fuzzy relation is either transitive or not), should we not talk about the degree or the extent to which it is transitive? It turns out that many new and interesting results can be obtained by following this line of thought. In the first few sections of the text of this chapter, we have followed Basu (1987) in avoiding all of these complexities by adopting E-transitivity, i.e. in insisting only on the minimal requirement that transitivity in the fuzzy case must be consistent with transitivity in the crisp case.

Another question that arises in the theory of fuzzy relations is how to decompose a weak relation into its asymmetric and symmetric components. The types of relation that we have so far been talking about (“at least as developed as”, “weakly Lorenz dominates”, etc.) are examples of weak relations. If country x is at least as developed as y, x is weakly preferred to y in the sense that it is not necessarily more developed than y. It is either more developed than y or at the same level of development as y. Similarly, if the income vector x weakly Lorenz dominates the vector y, x is not necessarily a strictly more equal distribution of income than y. Rather, it is at least as equal a distribution as y, i.e. it is either more equal than y or is at the same level of equality as y. An obvious requirement of a weak relation is that is reflexive.

Given a crisp weak relation R on X, we know how to decompose it into its asymmetric and symmetric parts. Define P to be the relation on X such that, for all x and y in X, x P y if and only if [x R y and not y R x] and define I to be the relation on X such that, for all x and y in X, x I y if and only if [x R y and y R x]. A relation S on X is called asymmetric if, for all x and y in X, if x S y, then we cannot have y S x. It is called symmetric if, for all such x and y, if x S y, then it is also true that y S x. It is easily seen that if, from a given weak relation R, we derive the relations P and I in the way described above, then P is an asymmetric relation, and I is a symmetric one. Moreover, if R is a complete relation on X, then it is easily checked that if x R y, then either x P y or x I y. We cannot have x P y and x I y at the same time [since x P y means (x R y and not y R x) and x I y means (x R y and y R x)]. On the other hand, it cannot be that neither x P y nor x I y since completeness of R implies that then the only possibility is that y P x, i.e. (y R x and not x R y). but we have assumed that x R y. Thus, y P x is ruled out. Using the fact that a relation is a set, we can use the set theoretic notations of union and intersection. We say that $R = P \cup I$ and $P \cap I = \emptyset$ where $\emptyset$ is the null set. In other words, if R is reflexive and complete, then P and I are mutually exclusive and collectively exhaustive subsets of R.

Coming to fuzzy weak relations, however, it is seen that there seems to be no unique way of decomposing a given fuzzy weak relation R on X into its asymmetric and symmetric components. There is a broad agreement over what symmetry of a fuzzy relation means. If S is a fuzzy relation on X, it is symmetric if, for all x and y in X, S(x, y) = S(y, x). It is less clear what the definition of an asymmetric relation should be in the fuzzy case. What is clear is that there should be consistency with the crisp case. Thus, if S is to be asymmetric, S(x, x) must be zero for all x. Similarly, if, for any x and y, S(x, y) = 1, then S(y, x) = 0. However, beyond such obvious requirements, there is little agreement regarding the precise characteristics of a asymmetric fuzzy relation. For instance, Dutta (1987) proposed that P, the asymmetric component of R, should be what he called an antisymmetric relation, meaning that whenever P(x, y) > 0, P(y, x) should be zero. Some, however, seem to be of the opinion that we should not rule out cases where both P(x, y) and P(y, x) are positive.

As a result, no unique decomposition rule has emerged in the fuzzy case. In the text, we have followed the decomposition rule due to Barrett and Pattanaik (1989), according to which, for all x and y in X, P(x, y) = 1 − R(y, x) and I(x, y) = R(x, y) + R(y, x) − 1. However, the literature contains other proposals. For instance, Dutta (1987), Banerjee (1994) and Richardson (1998) agree on how I should be defined: for all x and y in X, I(x, y) = min{R(x, y), R(y, x)}. However, their definitions of P differ from each other. Dutta defined P to be such that for all such x and y, P(x, y) = R(x, y) if R(x, y) > R(y, x), and it is 0 otherwise. Banerjee concurred with Barrett and Pattanaik and required P(x, y) to be 1 − R(y, x). Richardson’s definition of P(x, y) is the maximum of the two quantities R(x, y) − R(y, x) and 0. Dasgupta and Deb (2001) also adopted this definition. We desist from attempting an exhaustive survey of all the different proposals in this regard.

It is because of this lack of unanimity among mathematicians regarding how to derive the asymmetric component P of a given fuzzy relation R that some economists have avoided this controversy by working directly with a given asymmetric fuzzy relation P instead of deriving P from R. Basu’s (1987) work on fuzzy Lorenz dominance discussed in the text is of this type. In the broader context of measuring development, however, it would be more natural to start from the weak relation R. That is what we have done in the text. Whenever we have needed to derive P and I from R, we have adopted the Barrett and Pattanaik (1989) procedure mentioned in the previous paragraph. While it is true that which of the different procedures would be suitable in a specific context would depend on the context, we have found that in the context of development ranking which is our concern, this particular procedure seems to be appropriate.

Fuzzy relations: An ordinal approach

The type of fuzzy relations that we have so far been talking about in this Appendix is known as cardinal fuzzy relations. Under this approach, in order to specify a fuzzy relation on a set X, we specify a numerical value (between 0 and 1) for R(x, y) for all x and y in X. In this framework, it is meaningful to say, for instance, that for some x, y, z and w in X, R(x, y) is twice R(z, w). Needless to say, the statement does not have to be true. But the statement is meaningful in the sense that its veracity can be checked by comparing the numerical values of R(x, y) and R(z, w).

From an intuitive viewpoint, however, the cardinal approach to fuzzy relations is sometimes criticised for not being in conformity with the basic ideas of the fuzzy sets approach. If the basic point is that some statements are inherently ambiguous, then it is somewhat self-contradictory to specify the degree of ambiguity by giving to it a precise numerical value. For this reason, a somewhat different type of approach to fuzzy relations has been proposed. It is known as the theory of ordinally fuzzy relations. There are different versions of the theory. In the version that we adopted for our purposes in the text, we still assume, for convenience, that, for any x and y in X, R(x, y) is numerically specified: for any x and y in X, R(x, y) is still a real number. However, we do not perform arithmetic operations such as additions or multiplication on these numbers. For any x, y, z and w in X, all we care about is whether or not R(x, y) ≥ R(z, w). In other words, all we need to use is the natural order of real numbers given by the (crisp) relation ≥ on the real line.

R now is assumed to be a mapping from X² into a bounded subset A of the real line with the usual order relation ≥ on the real line. Since A is bounded, it will have a supremum (a*, say) and an infimum (a_*, say).

It may be noted in passing that the notion of an ordinal fuzzy relation formulated here is an example of what are called “L-fuzzy binary relations” in mathematics. An L-fuzzy binary relation S on a set B is a mapping from B × B into a lattice L. A lattice is any partially ordered set (not necessarily a set of real numbers) in which every pair of members has a least upper bound and a greatest lower bound with respect to the specified partial order relation (T, say). Salii (1965) (in Russian) contained an early exploration of the idea. Recent contributions in this area are developments based on Goguen (1967). It should be noted that in our framework an ordinal fuzzy relation is, trivially, a complete relation if completeness is defined to mean that, for any x and y in X, R(x, y) ≥ R(y, x) or R(y, x) ≥ R(x, y). An L-fuzzy relation S, however, would be complete if and only if S(x, y) T S(y, x) or S(y, x) T S(x, y). It would be a non-trivial restriction. An arbitrary L-fuzzy relation is not necessarily complete. In this book, we do not work in the more general L-fuzzy framework and confine ourselves to the notion of an ordinal fuzzy relation as formulated in the previous paragraph. For an application of the Goguen framework (in the context of a problem in social choice theory), see Barrett, Pattanaik and Salles (1992).

To continue with our formulation, an ordinal fuzzy relation R on X is called reflexive if, for any x in X, R(x, x) = a*. As noted above R is, by definition, complete in the sense that for any admissible x and y, at least one of the two inequalities R(x, y) ≥ R(y, x) and R(y, x) ≥ R(x, y) must be true.

What makes R a transitive relation is, again, a question on which there is no general agreement. It may be noted that some of the definitions of transitivity suggested in the cardinal context would make sense in the ordinal framework also. Notably, the notion of max-min (or T₁) transitivity mentioned above remains applicable since it is stated by using the order relation ≥ on real numbers and does not involve any arithmetic operation. The same is true of some of the other members of the max-star class of definitions. However, it is not true of all members of that class. One member of the class that would no longer be applicable in the ordinal framework is the notion of T₂-transitivity which was also mentioned above because it involves arithmetic operations such as additions.

In the text of this chapter, we did not use max-min transitivity for the ordinal framework. We proposed (and used) the following notion of ordinal transitivity. An ordinal fuzzy relation R on X is transitive if, for all x, y and z in X, [R(x, y) ≥ R(y, x) and R(y, z) ≥ R(z, y)] implies [R(x, z) ≥ R(z, x)] and if, whenever at least one of the first two inequalities is strict, so is the last. This condition (which we called fuzzy transitivity in the text) is in spirit essentially similar to a condition introduced in Banerjee (1993) and also to what was called “strong transitivity” in Kolodziejczyk (1986).

In the ordinal context, again, one can raise the question how one would decompose a given ordinal weak fuzzy relation into its asymmetric and symmetric components. For our purposes in this book, however, we do not need to enter into this question since the discussion is carried out in terms of the weak relation itself.