Stochastic dominance on unidimensional grids: theory and applications Satya, R. Chakravarty∗ Indian Statistical Institute Kolkata, India

Claudio Zoli† Department of Economics University of Verona, and CHILD, Italy

March 2010

Abstract The objective of this paper is to derive some integer-majorization results for variable-sum comparisons. We use an axiomatic framework to establish equivalence between several intuitively reasonable conditions. Relationship of our findings with the existing results along this line is clearly demonstrated. It is also shown that the results developed in the paper have a wide range of applications. Keywords: Stochastic dominance, stochastic orders, grids, measures. JEL Classification Numbers: D63, D81.

1

Introduction

Stochastic dominance analysis is employed in various areas of economics for ranking alternative social states (see Levy, 2006, for a survey). Traditional bases of stochastic dominance analysis are iterates of cumulative distribution functions and classes of utility functions defined on the continuum. Fishburn and Lavalle (1995) (henceforth F&L) considered stochastic dominance analysis for probability distributions on a finite grid. By a grid we mean a finite set of evenly spaced points. Their first and second order stochastic dominance results can be regarded as unidimensional grid counterparts to the traditional ones. The central idea underlying the F&L analysis goes back to the Muirhead (1903) integer-majorization result with a constant total. A related contribution to this literature, which compares profiles of opportunity sets using set-inclusion filtral preorders, was recently made by Savaglio and Vannucci ∗

Economic Research Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata - 700 108; W.B, INDIA (e-mail: [email protected]). † Claudio Zoli, Dipartimento di Scienze Economiche, Università degli Studi di Verona, Palazzina 32, Viale dell’Università n. 3, 37129 Verona, ITALY. (e-mail: [email protected]) and Centre for Household, Income, Labour and Demographics, Turin, ITALY.

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(2007). They considered the height functions (of non-negative integer value) of each individual opportunity set with respect to some opportunity set thresholds and the majorization result holds for a fixed sum of heights for each opportunity profile. In each of the above contributions the vectors under comparison have a fixed sum. F&L compared probability distributions so that the sums are always equal to one. The main objective of our paper is to develop integer-majorization analysis for variable-sum comparisons. An attractive feature of our contribution is demonstration of equivalence between several seemingly unrelated conditions in an axiomatic structure. Our results are quite general in the sense that variability of the population size is also permissible in the framework considered. Alternative interpretations of our results at various applications can be provided. For instance, under appropriate interpretation, for a fixed mean income, the Atkinson (1970) result on the equivalence between Lorenz dominance and the principle of transfers, and the Rothschild-Stiglitz (1970) mean preserving spreads can be related to our result. Shorrocks (1983) showed that if u and v are two income distributions over a given population size, then u generalized Lorenz dominates v if and only if u is regarded at least as good as v by all increasing and S-concave social welfare functions (see also Kolm, 1969). Generalized Lorenz domination of u over v is also equivalent to the condition that the former is obtained from the latter by successive applications of a finite number of T-transformations, assuming that the incomes are ordered nondecreasingly (Marshall and Olkin, 1979, p.108). Our main theorem can be regarded as the integer-majorization counterpart to the Shorrocks (1983)-Marshall and Olkin (1979) result on generalized Lorenz ordering. The proof of this theorem clearly identifies the minimal number of monotonic and equalizing transformations that take us from a socially inferior distribution to a better one. Other applications of our result include measurement of health inequality, inequality of opportunity sets, literacy and social exclusion. Moreover, our results are also related to (integer) stochastic orders induced by the class of linear rank dependent evaluation functionals (Yaari, 1987 and Weymark, 1981, 2003). Analogies with F&L results for third and higher orders of integer stochastic dominance are derived in the integer majorization framework by focussing on third order inverse stochastic dominance results (see Muliere and Scarsini, 1989). The paper is organized as follows. After presenting the preliminaries and notation in Section 2, we state the dominance results in Section 3. Section 4 provides a comparison of our results with the F & L results. In Section 5 we discuss some applications of our results. Finally, Section 6 concludes.

2

Preliminaries and notation

Let N (N0 ) be the set of positive (non-negative) integers and R (R+ ) the (nonnegative) real line. For any society with a population size of n ∈ N, there is a finite non-empty set F of characteristics or functionings relevant for social integration. A functioning can be an attribute that a person may value doing, having or being. The valued functionings may vary from simple ones like adequate nourishment, life expectancy, literacy, housing, income to complex individual characteristics like having 2

self-respect and communing with friends. We assume that F is the same for all societies under consideration so that cross population comparisons of our dominance results becomes possible in terms of the elements of F. The characteristics profile of individual i is represented by the m−dimensional vector xi,. = (xi,1 , xi,2 , ..., xi,j , ..., xi,m ), where xi,j ∈ {0, 1} . We say that individual i possesses functioning or characteristic j if xi,j = 1, otherwise xi,j = 0. Each Pindividual’s functioning attainment is represented by the functioning score xi := m j=1 xi,j . That is, the personal level of functioning attainment of individual i is given by the sum of his/her functioning scores across all characteristics. Thus, in calculating the level of functioning score of a person we assume, for simplicity, that all characteristics are equally important.1 The whole population attainment profile is the vector x = (x1 , x2 , ..., xi , ..., xn ). Dn is the set of attainment profiles for any n−person society, while D := ∪n∈N Dn is the set of all possible attainment profiles. Note that by construction xi ∈ {0, 1, 2, ..., m} , that is, the maximum level of individual functioning score is m, hence Dn = ⊗ni=1 {0, 1, 2, ..., m}i . A measure of functioning evaluation is a function E : D → R, with E n : Dn → R representing the restriction of the measure over the set of distributions of the same population size n. For any n ∈ N, x ∈ Dn , E n (x) is an indicator of the degree of functioning attainment enjoyed by the persons in the society. We write xˆ to denote the non-decreasingly ordered permutation of x, that is, xˆ1 ≤ xˆ2 ≤ ... ≤ xˆi ≤ ... ≤ xˆn−1 ≤ xˆn . The following properties, specify the behavior of the functioning evaluation indicators E n (.). Axiom 1 ((ANY) Anonymity) For all n ∈ N, x ∈ Dn , E n (x) = E n (xP ), where P is an n × n permutation matrix. Axiom 2 ((MON) Monotonicity) For all n ∈ N, x ∈ Dn , all i ∈ {1, 2, ..., n} such that xi < m E n (x1 , x2 , ..., xi + 1, ..., xn ) ≥ E n (x1 , x2 , ..., xi , ..., xn ). Axiom 3 ((NIME) Non-Increasingness of Marginal Evaluation) For all n ∈ N, x ∈ Dn , all i, j ∈ {1, 2, ..., n} , if m > xi > xj then E n (x1 , x2 , ..., xj + 1, ..., xi , ..., xn ) ≥ E n (x1 , x2 , ..., xj , ..., xi + 1, ..., xn ). To make comparisons between distributions of functioning evaluation indicators with different population sizes we consider a replication of the original population. Let xr denote the r−times replica of the vector x, that is, xr := (x, x, x, ..., x, x) with x repeated r times for r ∈ N. Axiom 4 ((POP) Population Principle) For all n, r ∈ N, x ∈ Dn , E rn (xr ) = E n (x). 1

Some characteristics may be more important than others, in this case different characteristic attainments will get different weights. Our analysis with the same weight (= 1) for different characteristics can be extended to the unequal weight case.

3

This set of axioms identifies the impact on the functioning evaluation of some transformations of the vectors x. We now make these transformations explicit. Axiom ANY requires that social evaluation does not depend on the identities of the individuals but only on their values of the attainment profiles. Therefore, permuting the identities of the individuals does not affect the functioning evaluation. We introduce first the following transformation. Definition 1 (TA transformation) Let x, y ∈ Dn . x is obtained from y through a “TA transformation”, that is, x = TA (y) if and only if: there exists a permutation function π : {1, 2, ..., n} → {1, 2, ..., n} such that xπ(i) = yi for all i ∈ {1, 2, ..., n} . Let TA denote the set of all TA transformations associated with all permutation functions π. Therefore, ANY is equivalent to stating that for all TA ∈ TA , if x = TA (y), then E n (x) = E n (y). The following transformation is associated with the MON axiom, it requires that x is obtained from y through an increase in the realization of one individual in one characteristic. Definition 2 (TM transformation) Let x, y ∈ Dn . x is obtained from y through a “TM transformation”, that is, x = TM (y) if and only if: there exist i ∈ {1, 2, ..., n} such that xi = yi + 1 ≤ m, and xh = yh for all h ∈ {1, 2, ..., n} \i. MON says that under TM transformation the final distribution cannot show lower level of functioning evaluation than the original one. Let TM denote the set of all TM transformations associated with all i ∈ {1, 2, ..., n} , then MON is equivalent to stating that for all TM ∈ TM , if x = TM (y), then E n (x) ≥ E n (y). The next transformation requires that an increase in an individual’s functioning attainment score has an higher impact on the social evaluation the lower is the individual’s functioning score. Definition 3 (TNIME transformation) Let x, y ∈ Dn . x is obtained from y through a “TNIME transformation”, that is, x = TNIME (y) if and only if: there exist i, j ∈ {1, 2, ..., n} such that m > yi − 1 > yj , xh = yh for all h 6= i, j, xi = yi − 1 and xj = yj + 1. Let TNIME denote the set of all TNIME transformations associated with all i, j ∈ {1, 2, ..., n} such that xi > xj + 1.2 Axiom NIME is, therefore, equivalent to stating that for all TNIME ∈ TNIME , if x = TNIME (y), then E n (x) ≥ E n (y).3 Clearly, NIME captures the relativity aspect of functioning evaluation. 2

Note that TN IM E transformations are rank preserving. That is, if x ˆ is obtained from yˆ through a TN IM E transformation (where xi > xj + 1) , then the transformation does not affect the ranking of the individuals involved. Thus, x ˆj − yˆj = 1 and x ˆi − yˆi = −1, where j < i and x ˆh − yˆh = 0 for all h 6= i, j. 3 This implication can be obtained as follows: note that the NIME axiom posits that E n (x1 , ..., xj + 1, ..., xi , ..., xn ) − E n (x1 , ..., xj , ..., xi , ..., xn ) ≥ E n (x1 , ..., xj , ..., xi + 1, ..., xn ) − E n (x1 , ..., xj , ..., xi , ..., xn ),

4

Finally, we need a transformation that will allow us to link distributions with different population sizes. Definition 4 (TP R transformation) Let x, y ∈ Dn . x is obtained from y through a “TP R transformation”, that is, x = TP R (y) if and only if: there exist r ∈ N such that x = yr . Let TP R denote the set of all TP R transformations associated with all r ∈ N. Axiom POP is, therefore, equivalent to stating that for all TP R ∈ TP R , if x = TP R (y) then E rn (x) = E n (y).

3

The results

The following theorem identifies the set of partial orderings defined over distributions in Dn consistent with the measures satisfying the previous axioms. A rank-dependent class of linear social evaluation indices is also presented (see Weymark, 1981 and Yaari, 1987). It can be considered equivalent to the generalized Gini social evaluation function for profile of opportunity sets characterized in Weymark (2003) for a domain analogous to Dn . According to condition 1 of Theorem 1, the weighted sum of individual functionings levels in profile xˆ is at least as high as that in the profile yˆ, where the non-negative weights are arranged non-increasingly. Given that xˆ and yˆ are arranged in non-decreasing order, condition 2 says that the cumulative sum of the functioning scores of the first k persons in xˆ is at least as large as that in yˆ, where k = 1, 2, ..., n, that is, x dominates y by the social evaluation criterion. Theorem 1 shows that these two conditions are equivalent to the requirement that y does not have more social functioning evaluation than x for all social evaluation indices that fulfil ANY, MON and NIME. n Theorem x, y ∈ . The following statements are equivalent: Pn 1 Let PD n n n (1) i=1 vi · xˆi ≥ i=1 vin · yˆi for all vin ≥ vi+1 ≥ 0. Pk Pk (2) i=1 xˆi ≥ i=1 yˆi for all k = 1, 2, ..., n. (3) x can be obtained from y through a finite sequence of TA , TM and TNIME transformations. (4) E n (x) ≥ E n (y) for all indices E n (.) satisfying ANY, MON and NIME.

Proof of Theorem 1. If xˆ = yˆ the implications are immediate. We consider the case xˆ 6= yˆ. P n (1) =⇒ (2) : Let δ i := xˆi − yˆi . We prove that ni=1 vin · δ i ≥ 0 for all vin ≥ vi+1 ≥0 Pk implies i=1 δ i ≥ 0 for all k = 1, 2, ..., n. Without loss of generality we define while if x = TN IM E (y), then the condition E n (x) ≥ E n (y) is

E n (y1 , ..., yj + 1, ..., yi − 1, ..., yn ) ≥ E n (y1 , ..., yj , ..., yi , ..., yn ). Subtracting E n (y1 , ..., yj , ..., yi − 1, ..., yn ) from both sides of the inequality, we obtain the NIME axiom requirement if yi − 1 = xi and yk = xk for all k 6= i.

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P n vin = nk=i αnk . If αnk ≥ 0 for all k = 1, 2, ..., n, then vin ≥ vi+1 ≥ 0. As a result we can rewrite Xn Xn Xn Xn Xk vin · δ i = αnk · δ i = αnk δi. i=1 i=1 k=i i=1 i=1 ³P ´ P k n x ˆ − y ˆ Condition (1) is, therefore, equivalent to ni=1 αnk i ≥ 0 for all αk ≥ 0. i=1 i P∗ P∗ Condition (2) is sufficient as well as necessary for (1). Suppose that ki=1 xˆi < ki=1 yˆi for a given k∗ , letting αnk = 0 for all k 6= k∗ and αnk∗ > 0, then condition (1) is violated. (2) =⇒ (3) : Note that because of TA transformations we can restrict attention to non-decreasingly ordered vectors xˆ, yˆ. We show that if (2) is satisfied it is possible to decompose the vector of elements δ i into a finite sequence of changes associated with TM and/or TNIME transformations. The decomposition process is divided Pn into ∗ two First, we identify the TM transformations δ i such that i=1 xˆi = Pn parts. (A) ∗ (ˆ y + δ ) , that is, starting from yˆ we get a distribution with the same total i i i=1 functioning score as xˆ. (B) Then we compare the two distributions xˆ and yˆ + δ ∗ with same total scores identifying the set of TNIME transformations leading to xˆ starting from yˆ + δ ∗ , where δ ∗ is the vector of δ ∗i ’s. Part (A): Let δ i := xˆiP − yˆi and ∆i := yˆi+1P− yˆi ≥ 0 for i ∈ {1, 2, ..., n}, where yˆn+1 := m. k n Denote Xk = ˆi and X = ˆi , with analogous notation Yk and Y for i=1 x i=1 x distribution y. Suppose that X > Y. Let i∗ := max {i : δi > 0, ∆i > 0} . Then we identify the (t) sequence of TM transformations making use of a sequence of increases δ i ∈ {1, ..., m}, where t denotes the index of the element in the sequence and i is the position of the individual experiencing the increase in functioning score. Each of these increases (t) corresponds to δi many TM -type transformations and will lead to a sequence of distributions yˆ(t) starting from yˆ(0) := yˆ. (1) Consider now δ i∗ := min {δ i∗ , ∆i∗ , X − Y } , that is, the first element of the (1) sequence. By construction, the new distribution yˆ(1) is obtained letting yˆi := yˆi (1) (1) (1) for all i 6= i∗ and yˆi∗ := yˆi∗ + δ i∗ . According to the definition of δ i∗ , the ranking (1) in yˆ(1) is preserved (because ∆i∗ > 0 is considered in the definition of δ i∗ ). Further(1) (1) more yˆi∗ < yˆi∗ ≤ xˆi∗ (given that δ i∗ > 0 is considered in the definition of δ i∗ ) and (1) X ≥ Y (1) > Y (because X − Y > 0 is considered in the definition of δ i∗ ). We prove that: Claim (i): if xˆ 6= yˆ, then i∗ exists; P P P (1) Claim (ii): ki=1 xˆi ≥ ki=1 yˆi ≥ ki=1 yˆi for all k ∈ {1, 2, ..., n}. Proof of Claim (i). Consider xˆ 6= yˆ and suppose that i∗ does not exist. Then, recalling that, by definition ∆i ≥ 0, it follows that for any i ∈P{1, 2, ..., n} either [δ i ≤ 0 and ∆i > 0] or [δ i ≥ 0 and ∆i = 0]. Recalling again that ki=1 δ i ≥ 0 for all i and xˆ 6= yˆ, it follows that there exists at least a position i0 such that δ i0 = xˆi0 −ˆ yi0 > 0. Thus, we necessarily have ∆i0 = 0 implying yˆi0 = yˆi0 +1 . Suppose that i0 6= n, then, since the vector xˆ is ranked in non-decreasing order we also have xˆi0 +1 ≥ xˆi0 leading to xˆi0 +1 ≥ xˆi0 > yˆi0 = yˆi0 +1 . Therefore, xˆi0 +1 − yˆi0 +1 > 0, thereby, contradicting our initial assumption, unless ∆i0 +1 = 0. By repeating the same argument, we get to the case where i0 + 1 = n. In view of the initial assumption, then xˆn − yˆn > 0 6

and ∆n = 0 have to hold. However, by construction ∆n = 0 ↔ m = yˆn+1 = yˆn . Thus, since yˆn = m reaches the maximum functioning score, it is impossible that δ n = xˆn − yˆn > 0. This final consideration contradicts the initial hypothesis that i∗ does not exist, thereby, proving Claim (i). P P (1) Proof of Claim (ii). By construction ki=1 yˆi ≥ ki=1 yˆi for all k. We need to P P (1) (1) prove that Xk = ki=1 xˆi ≥ ki=1 yˆi = Yk for all k ∈ {1, 2, ..., n}. (1) (1) Condition Xk ≥ Yk holds by construction for k ∈ {1, 2, ..., i∗ } because Yk = Yk (1) (1) for k ∈ {1, 2, ..., i∗ −1} and consequently, since δ i∗ ≥ δ i∗ > 0, we have also Xi∗ −Yi∗ = (1) (1) (1) Xi∗ −1 − Yi∗ −1 + (δ i∗ − δ i∗ ). Recalling that Xi∗ −1 − Yi∗ −1 = Xi∗ −1 − Yi∗ −1 ≥ 0 and (1) (1) that δ i∗ − δ i∗ ≥ 0, we obtain Xi∗ ≥ Yi∗ . (1) We consider now the case where k ∈ {i∗ + 1, ..., n}. Condition Xk ≥ Yk does not hold if there exist k0 ∈ {i∗ + 1, ..., n} (not necessarily unique) such that Xk0 − Yk0 = (1) mink∈{i∗ +1,...,n} {Xk − Yk } and Xk0 − Yk0 < δ i∗ ≤ Xi∗ − Yi∗ . Note that if k0 6= n, then there exists at least one i0 > k0 such that δ i0 > 0. However, recalling that k0 ≥ i∗ + 1 we have i0 > i∗ [with δ i0 > 0], which leads to a contradiction in view of what we proved for Claim (i). If k0 = n, then Xk0 − Yk0 = X − Y > 0. In this case by construction we have (1) argued that X − Y ≥ δ i∗ , thereby, establishing the final contradiction that proves Claim (ii). Having derived the first stage of the algorithm, we can move to its general speci(t) (t−1) fication. The decomposition algorithm can be constructed letting yˆi := yˆi for all (t) (t−1) (t) (t−1) ∗ (0) and i 6= i and yˆi∗ := yˆi∗ + δ i∗ , where yˆ := yˆ. We then denote δ i,t := xˆi − yˆi Pn (t) (t−1) (t−1) (t) ∆i,t := yˆi+1 − yˆi for i ∈ {1, 2, ..., n}, where yˆn+1,t := m and Y = i=1 yˆi . The algorithm is obtained by identifying at each stage t i∗ := max {i : δ i,t > 0, ∆i,t > 0} and letting

© ª (t) δ i∗ := min δi∗ ,t ; ∆i∗ ,t ; X − Y (t−1) .

(1) (2) (T )

The procedure followed by the algorithm stops after T stages when δ i∗ = X −Y (T −1) . Given that after each stage the difference X − Y (t−1) is reduced by at least one unit, T equals at most X − Y, which is finite. As a result the final distribution yˆ(T ) is P (t) obtained as yˆ + δ ∗ , where the vector δ ∗ is such that δ∗i = Tt=1 δ i for all i associated (t) with at least one positive δ i and 0 for all the other elements of δ ∗ . Part (B): We consider now the case X = Y, where the distribution yˆ can also be considered as obtained through the set of transformations in part A. We then apply a sequence of TNIME transformations to yˆ in order to obtain xˆ. Recall that by construction xˆ is obtained from yˆ through a rank preserving TN IME transformation if xˆj − yˆj = 1 and xˆi − yˆi = −1, where j < i and xˆh − yˆh = 0 for all h 6= i, j, where these transformations do not affect the ranking of the agents involved. Note that any TNIME transformation, as defined for distributions that are not necessarily ranked, can be considered as rank preserving when xi > xj + 1. However, if xˆ is obtained from yˆ through many TNIME transformations then it might not be immediate to decompose the transition from 7

yˆ to xˆ applying rank preserving TNIME transformations. This is the scope of our algorithm. (t) (t−1) (t) (t−1) The algorithm is constructed letting yˆi := yˆi for all i 6= i− , i+ , yˆh := yˆh + (t) − + (0) δ h for h = i and h = i , where yˆ := yˆ. We then define δ i,t and ∆i,t as in part A. We let i− := max {i : δi,t < 0, ∆i−1,t > 0} , (3) and i+ := max {i : δ i,t > 0, ∆i,t > 0} .

(4)

Claim (iii): If xˆ 6= yˆ then i− and i+ exist, moreover i− > i+ . Proof of Claim (iii): For expositional purposes, without loss of generality, we let t = 1 and suppress the subscript t in the notation of this proof. The claim of existence of i+ is thus analogous to Claim (i) in part A concerning i∗ . − + In order to prove that i− exists Pthat Xk ≥ Yk for all k Pk and i > i , we note with P X = Y, or equivalently, i=1 δi ≥ 0 for all k and ni=1 δ i = 0, which imply that ni=k δ i ≤ 0 for all k ∈ {1, 2, ..., n − 1}. Thus, the last δ i element which is different from 0, is negative. Suppose that i− does not exist. Then there exists j such that δ j < 0 and δ i = 0 for i ∈ {j + 1, ..., n} with ∆j−1 = 0. Note that δ j < 0 ↔ xˆj < yˆj and ∆j−1 = 0 ↔ yˆj = yˆj−1 . Recalling that xˆj−1 ≤ xˆj , it follows that xˆj−1 ≤ xˆj < yˆj = yˆj−1 , thus, δ j−1 < 0. Then if ∆j−2 > 0 it follows that i− exists and i− > i+ . Otherwise, if ∆j−2 = 0, we can repeat the previous argument. However, notice that in order to guarantee that Xk ≥ Yk for all k, there must exist i such that δ i > 0 and, therefore, there is at least one agent < n such that δ ≥ 0 while δ +1 < 0. Hence, it is necessarily true that ∆ > 0, otherwise, if ∆ = 0, we obtain xˆ ≥ yˆ = yˆ +1 > xˆ +1 , which contradicts xˆ ≤ xˆ +1 . Combining ∆ > 0 with δ +1 < 0, we obtain that i− = + 1. Moreover, i+ ≤ , thus i− > i+ , which completes the proof of Claim (iii). Then we let (t)

(t)

−δi− = δ i+ = min {δ i+ ,t ; −δi− ,t ; ∆i+ ,t ; ∆i− −1,t } ,

(5)

which completes the algorithm for deriving the set of rank preserving TNIME trans(t) (t) formations. Note that the transformations −δ i− = δ i+ > 0, where i+ < i− , coincides (t) with δ i+ many TNIME transformations. Furthermore, these transformations are rank (t) (t) preserving given that by construction δi+ ≤ ∆i+ ,t and 0 > δ i− ≥ −∆i− −1,t . P Note also that, by construction, after each transformation we reduce ni=1 |δ i | Pn by 2 units. Thus, T = i=1 |δ i | /2 is the (finite) number of TN IME transformations implementing xˆ from yˆ. In order to complete the we show Pk Pproof Pk that: P (t) (t−1) k Claim (iv): x ˆ ≥ y ˆ ≥ ˆi ≥ ki=1 yˆi for all k ∈ {1, 2, ..., n}, i=1 i i=1 i i=1 y t ∈ {1, 2, ..., T }. P P (t) (t−1) Proof of Claim (iv): Each TNIME transformations implies that ki=1 yˆi ≥ ki=1 yˆi Pk (1) for all k ∈ {1, 2, ..., n}, where t ∈ {1, 2, ..., T }. For t = 1 we have yˆi ≥ Pk (0) Pk Pk Pk (T ) Pk i=1 (T −1) ˆi = i=1 yˆi , while for t = T we have i=1 xˆi = i=1 yˆi ≥ i=1 yˆi . Bei=1 y Pk cause of transitivity of the dominance relation, it is always guaranteed that i=1 xˆi ≥ 8

Pk

(t)

yˆi for t ∈ {1, 2, ..., T − 1}. Thus, dominance holds at each stage of the algorithm. (3) =⇒ (4) : By definition. (4) =⇒ (1) : We prove that the class P of social evaluation measures in (1) is a n special case of those in (4). Let Ev (x) := ni=1 vin · xˆi . The index Evn satisfies ANY since it is defined P in terms ¡of ranked vectors xˆ. Evn satisfies MON if and only if ¢ Evn (x0 ) − Evn (x) = nj=1 vjn · xˆ0j − xˆj ≥ 0 if x0 = TM (x). Letting xˆj − xˆ0j = 0 for all j 6= i and xˆ0i − xˆi = 1, then Evn (x0 ) − Evn (x) ≥ 0 implies that vin ≥ 0. Given that we can choose any i ∈ {1, 2, ..., n}, it follows that a necessary condition for Evn to satisfy MON is vin ≥ 0 for all i. This condition is also sufficient. Evn satisfies NIME if and only if Evn (x0 ) − Evn (x) ≥ 0, where x0 = TNIME (x). Suppose that x is obtained from x0 through a TNIME transformation involving individuals n n i and i + 1 for i ≤ n − 1. Then Evn (x0 ) − Evn (x) = vin − vi+1 , that is, vin ≥ vi+1 for all n i ∈ {1, 2, ..., n − 1} is necessary for Ev to satisfy NIME. It is also sufficient. Condition (2) in Theorem 1 parallels weak supermajorization of y over x, and majorization for equal mean scores comparisons (see Marshall and Olkin, 1979). Of interest is the algorithm adopted for the decomposition of the integer weak supermajorization condition in two sequences of TM and TNIME transformations. Both sequences of transformations are rank preserving. In this respect the algorithm in part B of our proof differs from the one used to prove integer majorization by Muirhead (1903) [and also the one in Lemma B.B.1 in Marshall and Olkin, 1979]. Moreover, our algorithm in part A differs from the one adopted in Shorrocks (1983) (t) for the same purposes because the latter does not necessarily guarantee that yˆi ≤ xˆi at each stage t, if individual i is affected by a TM transformation. Our algorithm allows to derive the minimal number of TM and TNIME transformations leading from yˆ to xˆ. P P Remark 1 Consider x, y ∈ Dn such that ki=1 xˆi ≥ ki=1 yˆi for all k = 1, 2, ..., n, then the minimal number P of TM and TNIME transformations leading from yˆ to xˆ are xi − yˆi − δ ∗i | /2, where δ ∗i is derived in part A of the respectively (X − Y ) and ni=1 |ˆ proof of Theorem 1. we PnBy construction Pncan express them in more compact form ∗ using the fact that i=1 |ˆ xi − yˆi − δ i | = i=1 |ˆ x − yˆi | − (X − Y ). Pk Pki n That is, for x, y ∈ D where i=1 xˆi ≥ i=1 yˆi for all k, we can consider the minimal number of TM and TNIME transformations leading from y to x, denoted n n respectively by #TM (y, x) and #TNIME (y, x), as bidimensional measures of "distance" between the two distributions, with Pn |ˆ xi − yˆi | − (X − Y ) n n #TM (y, x) = X − Y ; #TNIME (y, x) = i=1 . (6) 2 P n If X = Y, then #TNIME (y, x) = ni=1 |ˆ xi − yˆi | /2. When X > Y the algorithm in part A of the proof of Theorem 1 allows us to obtain a distribution yˆ − δ ∗ , which is "as close as possible" to xˆ in the metric of the number of TNIME transformations. This is evident because any operation is rank preserving (t−1) (t) } ≥ yˆi ≥ and at each stage t ∈ {1, 2, ..., T } of the algorithm we have max{ˆ xi , yˆi (t−1) (T ) (0) min{ˆ xi , yˆi } for any i ∈ {1, 2, ..., n}, where yˆi = xˆi and yˆi = yˆi . The following examples may clarify our claim. i=1

9

P P Example 1 Let xˆ = (2, 4, 4) and yˆ = (1, 2, 3). Note that ki=1 xˆi ≥ ki=1 yˆi for all k ∈ {1, 2, 3} and X = 10 > 6 = Y. Moreover, xˆ rank dominates yˆ, that is, xˆi ≥ yˆi for all i ∈ {1, 2, 3}. If m > 7 an immediate option to modify yˆ with TM transformations is to increase the higher functioning score by X −Y = 4 getting yˆ0 = (1, 2, 7). The integer majorization condition for yˆ0 over xˆ is then satisfied. Then 3 TNIME transformations are necessary to lead to xˆ starting from yˆ0 . They are: yˆ0 = (1, 2, 7) → (1, 3, 6) → (1, 4, 5) → (2, 4, 4) = xˆ. If instead we use the algorithm in part A of the proof of Theorem 1, we construct yˆ00 making sure to preserve rank dominance of xˆ over yˆ. That is, the 4 TM transformations are: yˆ = (1, 2, 3) → (1, 2, 4) → (1, 3, 4) → (1, 4, 4) → (2, 4, 4) = yˆ00 = xˆ. In this case no TNIME transformation is necessary! The result in the previous example is based on the fact that xˆ rank dominates yˆ. Now, we provide an example where this condition does not hold. Pk P ˆi Example 2 Let xˆ = (2, 3, 3, 5) and yˆ = (1, 2, 4, 4). Note that ki=1 xˆi ≥ i=1 y for all k ∈ {1, 2, 3, 4} and X = 13 > 11 = Y. If m > 6 an immediate option to modify yˆ with TM transformations is to increase the higher functioning score by X − Y = 2 getting yˆ0 = (1, 2, 4, 6). The integer majorization condition for yˆ0 over xˆ is then satisfied. Then 2 TNIME transformations are necessary to lead to xˆ starting from yˆ0 . They are: yˆ0 = (1, 2, 4, 6) → (1, 3, 4, 5) → (2, 3, 3, 5) = xˆ. If instead we use the algorithm in part A of the proof of Theorem 1, we obtain yˆ00 through 2 TM transformations: yˆ = (1, 2, 4, 4) → (1, 2, 4, 5) → (1, 3, 4, 5) = yˆ00 . Note that now it is necessary only 1 TNIME transformation to obtain xˆ from yˆ00 . We now extend the previous results in order to compare distributions with different population sizes. Let Q+ denote the set of non-negative rational numbers. We construct the Discrete Generalized Lorenz curve DGLx (p) of distribution x ∈ Dn for p ∈ [0, 1] in the following way:4 1 Xk k DGLx ( ) = xˆi for k = 1, 2, ..., n, DGLx (0) = 0 i=1 n n ¡ ¤ ¡ k k+1 ¢ ¢£ k ) − DGL ( ) for all p ∈ , n , and DGLx (p) = DGLx ( nk ) + p − nk DGLx ( k+1 x n n n for k = 0, 1, 2, ..., n − 1. Theorem 2 allows us to extend comparisons of functioning scores distributions to variable population distributions making use of DLG P curves and the specification of the rank dependent linear evaluation functional ni=1 vin · xˆi satisfying POP (see 4

DGL is the generalized Lorenz curve for integer-variable distributions. Even though the definition of the curve coincides with the one defined for real variables, the fact that x ∈ Dn may affect some properties of the associated dominance conditions. For instance, for comparisons of distributions with fixed total functioning score X, the distribution with the higher curve may not be obtained by assigning the same score to each individual. Let h ∈ N0 be the smaller value such that (X − h)/n = µ ∈ N0 , then the distribution with higher DGL contains n − h individuals with functioning score µ and h individuals with functioning score µ + 1.

10

Theorem 4 in Donaldson and Weymark, 1980), where the weights are vin = V ( ni ) − V ( i−1 ), with V : Q+ ∩ [0, 1] → R being non-decreasing and concave.5 n Theorem 2 hLet x ∈ Dnx , y ∈i Dny . Then the following statements are equivalent: i Pnx Pny h i i−1 i i−1 (1) i=1 V ( nx ) − V ( nx ) · xˆi ≥ i=1 V ( ny ) − V ( ny ) · yˆi for all non-decreasing and concave V : Q+ ∩ [0, 1] → R . (2) DLGx (p) ≥ DLGy (p) for all p ∈ [0, 1]. (3) There exist x0 and y 0 which can be obtained, respectively from x and y, through TP R transformations such that x0 can be obtained from y 0 through a finite sequence of TA, TM and TNIME transformations. (4) E(x) ≥ E(y) for all indices E(.) satisfying MON, ANY, NIME and POP. Proof of Theorem 2. Let x0 = xr and y 0 = y s such that rnx = sny = n. Given that the population sizes of x0 and y 0 are the same, all the conditions in Theorem 1 apply. ¤ P £ Note that EV (x) = ni=1 V ( ni ) − V ( i−1 ) · xˆi satisfies POP, moreover for a given n P same conditions specified in part (1) n, letting V ( ni ) := ij=1 vjn , we get precisely Pn the n n of Theorem 1 with EV (x) = Ev (x) := i=1 vi · xˆi . It follows that Evnx (x) = Evn (x0 ) ≥ n Evn (y 0 ) = Ev y (y). Furthermore, note that DGL(p) satisfies POP, therefore, DGLx (p) = DGLx0 (p) ≥ DGLy0 (p) = DGLy (p). Similarly, for all indices E(.) satisfying POP, we get E(x) = E(x0 ) ≥ E(y 0 ) = E(y). Thus, we obtain the desired results.

4

Comparison with results on stochastic dominance on unidimensional grids

In this section we highlight the analogies between our results and F&L results based on the stochastic orders induced by additively decomposable functionals and existing results on stochastic dominance for real valued outcomes. For expositional purposes we will focus here on second and third degrees of stochastic dominance. Let Fx (t) denote the cumulative distribution function on R+ of a distribution x with bounded support [0, z¯] and finite mean µx . Moreover, let Fx−1 (p) := inf {t ∈ R+ : Fx (t) ≥ p} ,

0 ≤ p ≤ 1,

be the left continuous inverse distribution function showing the realization at the p quantile of the distribution. If x ∈ D, then the associated probability distribution is fx (t) with t ∈ {0, 1, 2, ..., m}. 5

We define V : Q+ ∩ [0, 1] → R as concave if and only if the usual concavity condition holds for all points in Q+ ∩[0, 1] or, equivalently, if and only if we have V (p + ρ) − V (p) ≥ V (q + ρ) − V (q) for all p, q, ρ ∈ Q+ ∩ [0, 1] with p < q such that p + ρ, q + ρ ∈ Q+ ∩ [0, 1]. See Marshall and Olkin (1979), pp. 453-454.

11

Rt Letting D1 (t) := Fy (t) − Fx (t) and deriving recursively Dk (t) := 0 Dk−1 (s)ds for k ∈ {2, 3} the Stochastic Dominance (SD) condition of order k for x over y, denoted by x Uk y if and only if Wu (x) ≥ Wu (y) for all u ∈ Uk . It is well known that x >U2 y is equivalent to x <2 y (see Rothschild and Stiglitz, 1970) and x >U3 y is equivalent to x <3 y and µx ≥ µy (see Whitmore, 1970). For ISD analogous results concern the set V1 of bounded and non-negative weighting functions v(p) with continuous first and second derivatives, and the nested subsets respectively V2 of all v ∈ V1 such that v 0 ≤ 0 and V3 of all v ∈ V2 such that v00 ≥ 0. For k ∈ {2, 3} we can thus specify the following stochastic orderings: x >Vk y if and only if Wv (x) ≥ Wv (y) for all v ∈ Vk . It is also known that x >V2 y is equivalent to −1 x <−1 2 y (see Yaari, 1987) and x >V3 y is equivalent to x <3 y and µx ≥ µy (see Zoli, 1999). Condition x <−1 y coincides with generalized Lorenz dominance and 2 with Lorenz dominance for equal means comparisons. The results in Theorem 1 focus on fixed population comparisons over unidimensional grids. Within this framework, that is, for a distribution x ∈ P Dn , the two related n families of evaluation functions can be specified as Eu (x) := n m t=0 fx (t) · u(t) = Pn P n n n ˆi . i=1 u(xi ) and Ev (x) := i=1 vi · x The related restrictions on u and v can be derived in analogy with F&L. Let the grid of the outcome realizations be G := {0, 1, 2, ..., m}. Now, consider the function h(t) for t ∈ G and let ∆h(t) := h(t) − h(t + 1) denote the difference applied to h(t) for t ∈ {0, 1, 2, ..., m − 1}. As in F&L, the set U1G includes bounded functions u(t) such that ∆u(t) ≥ 0. While U2G contains all u ∈ U1G such that ∆u(t) ≥ ∆u(t + 1) for all t ∈ {0, 1, 2, ..., m − 2}, U3G considers all u ∈ U2G such that ∆[∆u(t)] ≥ ∆[∆u(t + 1)] for all t ∈ {0, 1, 2, ..., m − 3}. For the dual functionals, we need to consider the grid N := {1, 2, ..., n}. Then the set V1N considers bounded weighting functions such that vin ≥ 0 for i ∈ N , and the n nested subsets are respectively V2N of all vin ∈ V1N such that ∆[vin ] = vin − vi+1 ≥ 0 for N n N n n n all i ∈ {1, 2, ..., n−1} and V3 of all vi ∈ V2 such that ∆[∆vi ] = ∆[vi ]−∆[vi+1 ] ≥ 0 for all i ∈ {1, 2, ..., n − 2}. For k ∈ {2, 3} the associated stochastic orders are respectively: x >U G y if and k

12

only if Eun (x) ≥ Eun (y) for all u ∈ UkG and x >VkN y if and only if Evn (x) ≥ Evn (y) for all v ∈ VkN . The dominance tests for additively decomposable evaluation functionals on G can be constructed making use of the following indicators: let d0t :=P fx (t) − fy (t) t k−1 for t ∈ {0, 1, 2, ..., m} and define recursively, on grid points, dkt := for j=0 dt k ∈ {1, 2, 3}. Analogously, it is possible to construct the relevant indicators for the 1 dual approach on N . Let δP ˆi − yˆi for i ∈ {1, 2, ..., n} and define recursively, on i := x individual positions, δ ki := ij=1 δ k−1 for k ∈ {2, 3}. j according to the direct approach the test PmNote that P Pninvolves comparisons Pn where m f (t) = f (t), while in the dual approach x ˆ = X and ˆi = Y t=0 x t=0 y i=1 i i=1 y may differ. For k = 2 the following results hold: Proposition 1 For x, y ∈ Dn the following statements are equivalent: (i) d2t ≤ 0 2 for t ∈ {0, 1, 2, ..., m − 1}; (ii) x >U2G y; (iii) x <2 y; (iv) x <−1 2 y; (v) δ i ≥ 0 for i ∈ {1, 2, ..., n}; (vi) x >V2N y. Proof. (i) ↔ (ii): See Theorem 2 in F&L. (ii) ↔ (iii) : See Theorem 6 in F&L. (iii) ↔ (iv) : It is the well known equivalence between second degree stochastic dominance and generalize Lorenz dominance, for equal means comparisons, see Atkinson (1970). For a general result see, among others, Levy (2006) Ch. 4.2.b. (iv) ↔ (v) : Follows from piecewise linearity of ∆2 (p) and the fact that ∆2 (i/n) = δ 2i /n. (v) ↔ (vi) : Conditions (v) and (vi) coincides respectively with conditions (2) and (1) in Theorem 1. For fixed population comparisons over unidimensional grids when k = 3, the above equivalences do not hold anymore. It is well known that third degree SD and ISD do not coincide (Muliere and Scarsini, 1989) and, as shown in F&L, x >U3G y implies x <3 y but the reverse may not be true. We formalize here the following facts concerning inverse stochastic dominance relations. Proposition 2 (i) For x, y ∈ Dn , x >V3N y ⇐⇒ δ 3i ≥ 0 for i ∈ {1, 2, ..., n − 1} and δ 2n ≥ 0; n (ii) x >V3N y ⇒ x <−1 3 y and µx ≥ µy , where x, y ∈ D ; (iii) there exist n ≥ 3, and x, y ∈ Dn such that x <−1 3 y and µx ≥ µy but not x >V3N y. P Proof. (i) The result is equivalent to ni=1 vin · δ 1i ≥ 0 for all vin ∈ V3N . Applying P P Pi Abel’s formula for summation by parts, that is, ni=1 ai · bi = n−1 i=1 ( j=1 aj ) · (bi −

13

bi+1 ) +

³P n

n X i=1

´ a · bn , we obtain: j j=1

vin · δ 1i = = =

n−1 X i=1 n−2 X i=1 n−2 X i=1

δ 2i · ∆[vin ] +

à n X i=1

δ 1i

!

· vnn

à n−1 ! X ¡ ¢ n n δ 3i · ∆[vin ] − ∆[vi+1 ] + δ 2i · ∆[vn−1 ] + δ2n · vnn i=1

n δ 3i · ∆[∆vin ] + δ 3n−1 · ∆[vn−1 ] + δ 2n · vnn .

Given that vin ∈ V3N , we have ∆[∆vin ] ≥ 0, ∆[vin ] ≥ 0 and vnnP≥ 0. Then the conditions δ 3i ≥ 0 for i ∈ {1, 2, ..., n − 1} and δ 2n ≥ 0 are sufficient for ni=1 vin · δ 1i ≥ 0. They are also necessary because the values of ∆[∆vin ], ∆[vin ] and vnn are independent. Thus, if either δ 3i < 0 for some iP ∈ {1, 2, ..., n − 1} or, δ 2n < 0, then it is possible to construct sequences vin such that ni=1 vin · δ 1i < 0. ³ ´ R i/n P P (ii) Let Evn (x) := ni=1 vin · xˆi = ni=1 i−1/n v(p)dp · xˆi and note that the set of R i/n functions i−1/n v(p)dp for v ∈ V3 are a subset of vin ∈ V3N . Thus, x >V3N y ⇒ x >V3 y ⇐⇒ x <−1 3 y and µx ≥ µy . (iii) Consider the following example with n = 3, where x = (2, 2, 16) and y = (0, 7, 7), so that xˆ − yˆ = δ 1 = (2, −5, 9). Note that µx > µy and ∆3 (p) > 0 for p ∈ (0, 7/3) ∪ (7/3, 1], while ∆3 (7/3) = 0. Thus, x >V3 y. Through recursive summations we obtain δ 2 = (2, −3, 6) and δ 3 = (2, −1, 5). Hence the test δ 3i ≥ 0 for i ∈ {1, 2} is violated, therefore, x >V3N y does not hold. Moreover, for x, y ∈ Dn it can be shown that >U3G 6=>V3N . If X = Y, it is well known that <3 6=<−1 3 , however, we have shown that >U3G and >V3N respectively imply but do not coincide with <3 and <−1 3 . Thus, we cannot rely on the known results to prove the following fact. Proposition 3 There exist x, y ∈ Dn such that (i) x >V3N y but not x >U3G y, (ii) x >U3G y but not x >V3N y. Proof. Consider x, y ∈ Dn such that X = Y. We prove the proposition making use of two examples. (i) Consider n = 3, m = 5, x = (1, 1, 5) and y = (0, 3, 4). It follows that δ 1 = xˆ − yˆ = (1, −2, 1) leading to δ3 = (1, 0, 0), thus, x >V3N y, while we have d0 = fx −fy = 1/3 · (−1, 2, 0, −1, −1, 1) leading to d3 = 1/3 · (−1, −1, 0, 1, 1, 1). Thus, the condition d3 (t) ≤ 0 for t ∈ {0, 1, ..., m − 2} is violated, hence x >U3G y does not hold. (ii) Consider n = 4, m = 4, x0 = (1, 1, 3, 4) and y 0 = (0, 3, 3, 3). It follows that d00 = fx0 − fy0 = 1/4 · (−1, 2, 0, −2, 1) leading to d03 = 1/4 · (−1, −1, 0, 0, 0). Thus, the condition d3 (t) ≤ 0 for t ∈ {0, 1, ..., m − 2} holds. Hence x0 >U3G y 0 . Note that δ 01 = xˆ0 − yˆ0 = (1, −2, 0, 1) leading to δ03 = (1, 0, −1, −1). Thus, δ 03 (i) ≥ 0 for all i ∈ {1, ..., n − 1} does not hold, hence we do not have x0 >V3N y 0 . Clearly, the results presented in the section complement the F&L results with appropriate majorization and inverse stochastic dominance. 14

5

Interpretations

In this section we present some interpretations of our general results developed in the previous sections. Social Exclusion. Social exclusion refers to inability of individuals to participate in basic economic and social activities related to different functionings of the society in which they live.6 In this case we say that individual i is excluded in terms of characteristics j if eij = 1, otherwise eij = 0, where eij := 1−xij . Consequently, person i’s P level of exclusion is given by the deprivation (exclusion) score ei := m e j=1 i,j = m−xi . The function E, defined over distributions e := (e1 , e2 , ..., en ), then becomes a measure of social exclusion. Since social exclusion is a bad, the inequality in Axiom 4 should now be reversed and we refer to it as Non-Decreasingness of Marginal Social Exclusion (NMS) (see Chakravarty and D’Ambrosio, 2006). This parallels an argument put forward by Sen (1976) in the context of poverty measurement. He argued that in the construction of the poverty index higher deprivation should be assigned higher weight, where deprivation of a poor person is given by his income shortfall from the poverty line representing the income necessary to maintain a subsistence standard of living. Take the vector e¯ whose elements are ranked order, that is e¯i ≥ Pk in non-incresing k 1 k e¯i+1 . Let us define the plot of SEe ( n ) = n i=1 e¯i against n , where k ∈ {1, 2, ..., n}, SEe (0) = 0, as the social exclusion curve associated with e. This curve is directly related to the ‘absolute poverty gap profile curve’ introduced, among others, by Jenkins and Lambert (1997) and Shorrocks (1998). Our Theorem 2 shows that of two exclusion profiles e and e0 , e0 is not more excluded that e socially by all exclusion indices satisfying MON, ANY, NMS and POP if and only if the social exclusion curve of e dominates that of e0 , that is, the social exclusion curve of e is nowhere below that of e0 . This is equivalent to the condition that the weighted sum of individual exclusions in e is at least as high as that in e0 , where the rank dependent weights are vin := V ( ni ) − V ( i−1 ), with V : n Q+ ∩[0, 1] → R being non-decreasing and concave. Note that we do not need equality of the population sizes and total exclusions for this general result to hold. From Theorem 2 it that under¤the conditions stated in part (1) of the theorem, we Pemerges n £ i can regard i=1 V ( n ) − V ( i−1 ) · e¯i as a social exclusion index. If we assume that n i i β V ( n ) = ( n ) , where 0 < β ≤ 1, then the resulting index becomes the social exclusion counterpart to the Donaldson and Weymark (1980) generalized Gini inequality index. For e, e0 ∈ Dn , the results in Proposition 2 involve social exclusion comparisons based on comparisons of partial sums in δ 3 obtained from δ 1 = e¯ − e¯0 . Social Welfare. We begin by specifying a result on generalized Lorenz domination. Let u and v be two non-decreasingly ordered income distributions over a given population Shorrocks (1983) showed that u generalized Lorenz dominates v P size n. P (that is, ki=1 ui ≥ ki=1 vi , 1 ≤ k ≤ n) if and only if W (u) ≥ W (v) for all S-concave social welfare functions W , where S-concavity of W demands that a rank preserving 6

See, among others, Akerlof (1997), Atkinson (1998), Klasen (1998), Chakravarty and D’Ambrosio (2006) and Bossert et al. (2007), for alternative interpretations of social exclusion.

15

transfer of income from a person to anyone with a lower income does not decrease welfare. It may be noted that all S-concave functions satisfy anonymity. Marshall and Olkin’s (1979, p. 108) Proposition A.2 shows that generalized Lorenz domination of u over v is equivalent to the condition that u can be obtained from v by applications of a finite number of T transformations of the form T (z1 , z2, ..., αzi , zi+1, ..., zn ), where α > 1. If we interpret xi as income class realization of person i and E as a social welfare function, then equivalence between conditions (2), (3) and (4) of Theorem 1 can be regarded as integer analogue to the Marshall and Olkin (1979)-Shorrocks (1983) results. Moreover, it generalizes Muirhead 1903 result (see Marshall and Olkin, 1979) linking majorization and T transformations for distributions whose components are non-negative integers with fixed equal sums. For a fixed total of functioning scores, equivalence between conditions (2) and (3) of Theorem 1 becomes tantamount to the Atkinson (1970) theorem on equivalence between Lorenz ordering and the principle of transfers (see also Dasgupta, Sen and Starrett, 1973 and Rothschild and Stiglitz, 1973). Individual Utility-Well Being Ranking. E is a measure of individual utility over n indivisible goods and xi represents the person’s consumption level of good i in discrete numbers {0, 1, ..., mi }, for example, TV, refrigerator, electric bulb and so on. Then E n (x) ≥ E n (y) simply means utility ranking of the consumption baskets x, y ∈ Dn . More generally, the logic can be extended to goods/attributes whose realizations xi are finite and isomorphic to the set {0, 1, ..., mi }. Moreover, for n alternative equiprobable states i with realizations xi ∈ {0, 1, ..., m}, our result can be applied to derive individual rankings under risk. Further interpretations can be derived exploiting analogies with applications to ranking of multisets in Sertel and Slinko (2007). Opportunity Inequality. The desirability of reducing inequality in the distribution of opportunities among individuals in a society is a well-accepted principle of distributive justice. In Savaglio and Vannucci (2007) an opportunity-threshold has been introduced so that any opportunity set that does not meet this threshold is taken as a null set. The height function associated with each opportunity set, which assumes a non-negative integral value, ‘provides a total extension of the. . . . resulting higher than relation’ (op. cit., p. 480). Transformations of the opportunity profiles, in order to lead to the dominance conditions, were devised so that the equalizing opportunity transfers do not destroy total aggregate height. If we regard E as a measure of opportunity equality and xi as height, then our Theorem 1 can be treated as an extension of the Savaglio —Vannucci result to the variable total case. The rank dependent class of indices given by condition (1) of Theorem 1 resembles the generalized Gini social evaluation function for profiles of opportunity sets characterized in Weymark (2003). Moreover, Remark 1 suggests a meaningful metric, in terms of height transformations, in order to measure differences between opportunity sets. In this case differences between opportunity sets evaluated in terms of number of transformations corresponds to absolute distances in realizations, where comparisons are made between individuals in the same position following the spirit of Roemer (1998) apporach to measure equality of opportunities. 16

Health Inequality. Often self-reported health status data, which represent the people’s opinion about their health status, are presented qualitatively. For instance, the five health status categories ‘ poor’, ‘fair’, ‘good’, ‘very good’ and ‘excellent’ may be assigned positive integral values in increasing order. Allison and Foster (2004) presented an ordinal approach to the measurement of inequality. Their median-based approach regards health inequality as a deviation from the median. In our case we regard E as a health equality measure and xi as person i’s perception of his health. Then our Theorem 1 provides a comparison between two health profiles. Literacy Assessment. The ‘knowledge’ component of the human development index suggested by the UNDP is measured as the weighted average of the adult literacy rate and the combined primary, secondary and tertiary gross enrolment ratio, where the weights are respectively two third and one third. If we define a literacy profile as the vector of enrolments at different levels of education, our Theorem 1 can then be employed to compare two literacy profiles, where xi indicates the enrolment at the education level i. Clearly, i can be primary, secondary or tertiary. Basu and Foster (1998) suggested an index of literacy under the assumption that illiterate person of a household can benefit from the knowledge of a literate person in the household. Following this perspective it is possible to suggest an alternative interpretation of our result. We can consider i as individuals and xi ∈ {0, 1, ..., m} as literacy achievements, where xi = 0 indicates that person i is illiterate otherwise he/she is litterate. A higher value of xi represents a higher level of literacy (possibly, including also proximate literacy). Then Theorem 1 provides a preliminary tool for ranking literacy profiles.

6

Conclusion

Often it becomes necessary to rank social states where each component of a state is represented by an integer. In this paper we have developed some majorization results for ranking states of this type under quite general assumptions like variability of the total and the population size. Our characterizations can be treated as generalizations of the existing results in the literature. Some examples of applications of our results are also provided. Acknowledgments: Chakravarty thanks the Bocconi University, Milan, Italy, for support. Financial support from the Italian Ministero dell’Istruzione, dell’Università e della Ricerca (Prin 2007) is gratefully acknowledged by Claudio Zoli.

References [1] Akerlof, G. A. (1997): Social distance and social decision, Econometrica, 65, 1005-1027. [2] Allison, R. A. and J. E. Foster (2004): Measuring health inequality using qualitative data, Journal of Health Economics, 23, 505-524.

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[3] Atkinson, A. B. (1970): On the measurement of inequality, Journal of Economic Theory, 2, 244-263. [4] Atkinson, A. B. (1998): Social exclusion, poverty and unemployment, CASE/4, Centre for Analysis of Social Exclusion, London School of Economics, 1-20. [5] Basu, K. and J. E. Foster (1998): On measuring literacy, Economic Journal, 108, 1733-1749. [6] Bossert, W. D’Ambrosio, C. and V. Peragine (2007): Deprivation and social exclusion, Economica, 74, 773-803. [7] Chakravarty, S. R. and C. D’Ambrosio (2006): The measurement of social exclusion, Review of Income and Wealth, 52, 377-398. [8] Dasgupta, P., Sen, A. K. and Starrett, D. (1973): Notes on the measurement of inequality, Journal of Economic Theory, 6, 180-187. [9] Donaldson, D. and J. A Weymark (1980) :A single-parameter generalization of the Gini indices of inequality, Journal of Economic Theory, 22, 67-86. [10] Fishburn, P. and Lavalle I. H. (1995): Stochastic dominance on unidimensional grids, Mathematics of Operations Research, 20, 3, 513-525. [11] Jenkins, S. and Lambert, P. J. (1997): Three I’s of poverty curves, with an analysis of UK poverty trends, Oxford Economic Papers, 49, 317-327. [12] Kolm, S. C. (1969): The optimal production of social justice. In Public Economics (Margolis, J. and Gutton, H. eds.), 145-200. London: Mcmillan. [13] Klasen, S. (1998): Social exclusion and children in OECD countries: some conceptual issues, Centre for Educational Research and Innovation, OECD. [14] Levy, H. (2006): Stochastic dominance. Investment decision making under uncertainty, 2nd edition, Springer. [15] Marshall, A. W. and I. Olkin (1979): Inequalities: theory of majorization and its applications. Academic Press. [16] Muirhead, R. F. (1903): Some methods applicable to identities and inequalities of symmetric algebraic functions of letters, Proceedings of the Edinburgh Mathematical Society, 21,144-167. [17] Muliere, P. and Scarsini, M. (1989): A note on stochastic dominance and inequality measures, Journal of Economic Theory, 49, 314-323. [18] Roemer, J. E. (1998): Equality of opportunity. Harvard University Press, Cambridge, MA. [19] Rothschild, M. and Stiglitz, J. E.(1970): Increasing risk: I. A definition, Journal of Economic Theory, 3, 225-243. 18

[20] Rothschild, M. and Stiglitz, J. E. (1973): Some further results on the measurement of inequality, Journal of Economic Theory, 6, 188-204. [21] Savaglio, E. and Vannucci, S. (2007): Filtral preorders and opportunity inequality, Journal of Economic Theory, 132, 474-492. [22] Sen, A. K. (1976): Poverty: an ordinal approach to measurement, Econometrica, 44, 219-231. [23] Shorrocks, A. F. (1983): Ranking income distributions, Economica, 50, 3-17 [24] Shorrocks, A. F. (1998): Deprivation profiles and deprivation indices. In The Distribution of Welfare and Household Production: International Perspectives (Jenkins, S. A., Kaptein, S. A. and van Praag, B. eds.), 250-267. London: Cambridge University Press. [25] Sertel, M. and Slinko, A. (2007): Ranking committees, income streams or multisets, Economic Theory, 30, 265-287. [26] Weymark, J. A. (1981): Generalized Gini inequality indices, Mathematical Social Sciences, 1, 409-430. [27] Weymark, J. A. (2003): Generalized Gini indices of equality of opportunity, Journal of Economic Inequality, 1, 5-24. [28] Yaari, M. E. (1987): The dual theory of choice under risk, Econometrica, 55, 95-115. [29] Zoli, C. (1999): Intersecting generalized Lorenz curves and the Gini index, Social Choice and Welfare, 16, 183-196.

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