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Symmetric measures of segregation, segregation curves, and Blackwell’s criterion
Mathematical Social Sciences 73 (2015) 63–68
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Symmetric measures of segregation, segregation curves, and Blackwell’s criterion Robert Hutchens Cornell University, United States
highlights • • • •
A new approach is proposed for using segregation curves to measure segregation. The resulting incomplete segregation order incorporates a ‘‘symmetry of types’’ property. Using Blackwell’s criterion, the approach is extended to more than two types of people. The new approach results in a more complete segregation order.
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Article history: Received 25 February 2014 Received in revised form 11 October 2014 Accepted 27 November 2014 Available online 8 December 2014
abstract This paper first proposes a new way to use segregation curves to examine whether one distribution of people across groups (e.g., occupations or neighborhoods) is more segregated than another. It then uses Blackwell’s criterion to extend the argument to more than two types of people. The basic idea is that by introducing additional assumptions about the nature of segregation, one obtains a more complete ranking of distributions. The paper demonstrates that the assumption of ‘‘symmetry in types’’ – an assumption that appears frequently in the literature on segregation measurement – has implications for both segregation curves and Blackwell’s criterion. © 2014 Elsevier B.V. All rights reserved.
A large and growing empirical literature seeks to measure segregation in school systems, occupational networks, and city neighborhoods.1 The topic is important because ethnic and gender segregation is intimately tied to issues of access and opportunity— issues that are of fundamental interest to social scientists and the larger society. Standing alongside this empirical literature is a second more theoretical literature that grapples with the problem of how to properly measure segregation. When can we say that one distribution of people across groups (e.g., occupations or neighborhoods) is more segregated than another? This paper contributes to that second literature. In particular, the paper proposes a new way to use segregation curves to examine whether one distribution is more segregated than another. Segregation curves were introduced into the literature in a seminal paper by Otis and Beverly Duncan.2 In their analysis of racial segregation in US cities, the Duncans assumed two types of people (specifically, whites and non-whites) and
E-mail address: [email protected]. 1 For example, Massey and Denton (1988), Flückiger and Silber (1999), Hutchens (2001, 2004), Weeden (2004) and Frankel and Volij (2011). 2 Otis D. Duncan and Beverly Duncan (1955). http://dx.doi.org/10.1016/j.mathsocsci.2014.11.007 0165-4896/© 2014 Elsevier B.V. All rights reserved.
plotted curves that indicate whether segregation in city A is greater than that in city B. The resulting segregation curves are similar to the Lorenz curves used to assess income inequality. Like Lorenz curves, segregation curves have the advantage of resting on a few weak and plausible assumptions about the nature of segregation. Also like Lorenz curves, segregation curves yield an incomplete order of distributions; when segregation curves cross, they yield no information about whether one distribution of people across groups can be ranked as more segregated than another. Whether such an incomplete order is an advantage or disadvantage is in the eye of the beholder. On the one hand it is awkward to announce that because of intersecting segregation curves we do not know whether distribution X is more segregated than Y . There exist several numerical measures of segregation that yield a complete order of distributions; why not use one of them? On the other hand, intersecting segregation curves force us to say, ‘‘it is only by making additional assumptions about the nature of segregations – assumptions that are often neither explicit nor explained – that we can claim that X is more segregated than Y ’’. There can be virtue in such candor. This paper first shows that by introducing an additional plausible assumption about the nature of segregation, one can increase the number of alternatives that can be ranked by segregation
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R. Hutchens / Mathematical Social Sciences 73 (2015) 63–68
curves. It then extends that result to a more general segregation order that is based on Blackwell’s criterion. In the case of segregation curves, the practical implication is as follows: if the segregation curves for X and Y cross, then plot a new segregation curve denoted X T . This new segregation curve is closely related to the segregation curve for X , and is thus called the ‘‘twin’’ of X . If X T lies beneath Y – more precisely, if X T lies at some point below and at no point above Y – then despite the fact that the segregation curves for X and Y cross, distribution X can be declared more segregated than distribution Y , and that increases the number of alternatives that can be ranked. But since that result pertains to segregation curves, it is restricted to situations where one can reasonably assume two types of people. A recent paper by Lasso de la Vega and Volij (2014) applies Blackwell’s criterion (Blackwell, 1951, 1953) to the problem of measuring segregation. They not only obtain an incomplete order that can be used to assess segregation when there are more than two types of people (e.g., white Hispanic, black Hispanic, white non-Hispanic, black non-Hispanic), but they also establish an intimate link between that incomplete order and segregation curves. Their work raises the possibility of extending this paper’s result to the general problem of two or more types of people. The final section of the paper shows that the result does, indeed, generalize. Thus, by making an additional plausible assumption about the nature of segregation, one can increase the number of alternatives that can be ranked by Blackwell’s criterion, and that result applies to two or more types of people. 1. The problem, properties, segregation curves, and previous results Consider I types of people distributed over J occupations. Let xij denote the number of type i people in occupation j (i = 1, . . . , I ; j = 1, . . . , J ), let Ni (X ) = j xij , i = 1, . . . , I denote the total number of type i people over all occupations, and let X be a data matrix of the xij . For example, type 1 could be women, type 2 could be men, and occupation j could be one of J occupations. To insure a meaningful problem, assume that J is greater than one, that the xij are non-negative real numbers,3 and that the number of type i people is positive (Ni (X ) > 0, i = 1, . . . , I). Thus, with I types of people and J occupations, the data matrix takes the form,
x11
.
X = x1 , x2 , . . . , xJ = .. xI1
··· .. . ···
x1J
.. .
∞
J =2 DIJ where DIJ =
I ×J
X ∈ R+
: Ni (X ) > 0, i =
1, . . . , I .
Occupation 1 type 1 type 2
6 5
2 4
4 , 5
3 8
2
1 5
3
4
1 7
4 .
P1. Scale invariance. Let Y be obtained from X by multiplying the number of type i people in every occupation of X by a positive scalar βi . Then Y ∼ X . P2. Symmetry in occupations. Let (j1 , . . . , jJ ) beany permutation of 1, . . . , J, X = x1 , x2 , . . . , xJ and Y = xj1 , xj2 , . . . , xjJ . Then Y ∼ X .5 P3. Organizational equivalence. Let Y be obtained from X by dividing occupation J into two occupations such that for some α ∈ (0, 1), i = 1, . . . , I ; j = 1, . . . , J − 1 i = 1, . . . , I
yiJ +1 = xiJ (1 − α),
i = 1, . . . , I .
Then Y ∼ X . P4. Neighborhood division property. Let Y be obtained from X by dividing occupation J into two occupations such that, yij = xij ,
i = 1, . . . , I ; j = 1, . . . , J − 1 i = 1, . . . , I
and for no α ∈ [0, 1], yiJ = xiJ α,
i = 1, . . . , I
yiJ +1 = xiJ (1 − α), (1)
(2)
this paper seeks to evaluate whether the X distribution is more segregated than Y . To that end, let < denote a segregation order on DI .4 If X ≻ Y then X is ranked as more segregated than Y , and if X v Y then X has the same level of segregation as Y . Of course, the order may be partial; if the order does not rank some X and Y , then the result is an incomplete order. Like any measure of inequality, a ‘‘good’’ measure of segregation should have properties that accord with perceptions of segregation. These properties are ultimately value judgments about what it means to say that one social state is more segregated than another. As such, the properties should be as unrestrictive as possible. Restrictive properties imply strong value judgments, which in turn lead to more controversial conclusions about whether X is more segregated than Y . The literature on measuring segregation advances several properties for a good measure. Four of these properties are particularly important not only because they are unrestrictive, but also because they define a broad class of measures that are linked to segregation curves. While these properties are often stated for two types of people (e.g., men and women), because this paper ultimately addresses the more general problem of I types, it is useful to state the properties in their more general form. Finally, since the four properties are discussed extensively elsewhere, the following is a brief summary.
yiJ + yiJ +1 = xiJ ,
For example, with two types of people and four occupations the data matrix could take the form,
X =
3 4
yiJ = xiJ α,
xIJ
Y =
yij = xij ,
where xj is a column vector of length I that contains data on the types of people in occupation j. Denoting the vector space of all I × J I ×J real matrices with non-negative elements by R+ , the domain of X shall be DI =
Thus, letting type 1 people be women, the distribution in (1) has three women and eight men in the first occupation, and the total number of men and women is respectively 24 and 9 (N1 (X ) = 3 + 1 + 1 + 4 = 9 and N2 (X ) = 8 + 5 + 7 + 4 = 24). Letting Y be another matrix of two types of people over four occupations, e.g.,
i = 1, . . . , I .
Then Y ≻ X .
4
3 Not only does this assumption address the general case, but it is also plausible. Part-time workers could be treated as fractional workers. Irrational numbers are also theoretically admissible (e.g., a full time worker is counted as ‘‘1’’ and a parttime worker as the square-root of 0.3).
4 See Foster (1985, p. 42) for a particularly clear definition of a measure of inequality and an inequality order. 5 The property could be equivalently stated in terms of a permutation matrix. Specifically, let P be a J × J permutation matrix with only one coefficient in each row equal to 1 and only one coefficient in each column equal to 1. If Y = XP then Y ∼ X.
R. Hutchens / Mathematical Social Sciences 73 (2015) 63–68
P1 simply says that if the total number of type i people over all occupations (Ni (X )) is multiplied by a positive scalar (βi ) and the share of type i people in each occupation does not change (where a share is calculated as the number of type i people in an occupation divided by Ni (X )), then segregation does not change. To illustrate, suppose the number of type 1 people (e.g., women) doubles (β1 = 2), and occupations that initially contained 0%, 10%, 20%, etc. of all women continue to do so. Then in accordance with P1 this change in the level of women in the population does not affect segregation. P2 says that occupations can be relabeled without affecting segregation; what is important is numbers of different types of people in an occupation, not whether the occupation is labeled 2, 4, or 10. P3 says that if an occupation is divided in two and the distribution of types within the new occupations stays the same, then measured segregation does not change. For example, suppose that an occupation is comprised of 40% women (type 1) and 60% men (type 2). If that occupation is divided in two and the new occupations are both comprised of 40% women and 60% men, then segregation does not change. This division of an occupation essentially distributes the old segregation over new occupations; the ratio of type 1 to type i people (i = 2, . . . , I) in the new occupations is the same as in the old. The Neighborhood Division Property (P4) is closely related to P3. It arises out of work on residential segregation (see Frankel and Volij, 2011 and Lasso de la Vega and Volij, 2014), hence the reference to neighborhoods. Applied to occupations, this property essentially says that if an occupation is divided into two occupations and the distribution of types in the two new occupations differs from that in the original occupation, then segregation increases. For example, if the original occupation was comprised of 40% women and 60% men, and the division results in two new occupations, one comprised of 20% women and 80% men and the other of 60% women and 40% men, then segregation increases. The Neighborhood Division Property is also closely related to a transfer principle whereby segregation increases when people move between two occupations with particular characteristics. That principle is, in turn, closely related to the Pigou–Dalton transfer principle in the income inequality literature. While the transfer principle works well in situations where there are two types of people (e.g., men and women), it does not generalize to situations where there are multiple types. Thus, this paper is built on the Neighborhood Division Property.6 In situations where there are two types of people, a very useful measure of segregation – a measure that satisfies Properties 1–4 – is the segregation curve. A segregation curve is a plot of the cumulative fraction of type 1 people (on the vertical axis) against the cumulative fraction of type 2 people (on the horizontal axis), with fractions ranked from small to large values of the ratio x1j /x2j . More precisely, let the X matrix be permuted in such a way that x1j /x2j is ordered from smallest to largest, and let xˆ ih = j=1 xij /Ni (X ), i = 1, 2; h = 1, . . . , J. To draw the segregation curve, plot (ˆx1h , xˆ 2h ) h = 1, . . . , J, and use straight lines to connect consecutive points beginning at the origin (0, 0). Fig. 1 illustrates for the X and Y distributions in (1) and (2). Since the segregation curve for Y lies at no point below and at some point above that for X , we say that segregation curve for Y dominates the segregation curve for X . Write this as S(X ) ≻ S(Y ), since X is the more segregated of the two distributions. If the segregation curve for Y is coincident with that of X , then the curves are equivalent and we write S(X ) ∼ S(Y ). Thus,
h
6 Lasso de la Vega and Volij (2013) define the transfer principle and show that a segregation order that satisfies P3 satisfies the transfer principle if and only if it satisfies P4.
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Fig. 1. The segregation curves S(X ) and S(Y ).
Definition 1. Let X and Y be two distributions in D2 . We say that X is at least as segregated as Y according to the segregation curve criterion, denoted X
7 The assumed domain of X had non-negative rational numbers in its second row and non-negative real numbers in its first row. See Hutchens (2001) for additional discussion of the proof. 8 See Proposition 4 in Lasso de la Vega and Volij (2013). That proposition is stated for four properties: ANON, CI, OE, and T. ‘‘ANON’’ is equivalent to P2 above, ‘‘CI’’ is equivalent to P1, ‘‘OE’’ is equivalent to P3, and T is a transfer property. They go on to not only establish what is Theorem 3 in Section 3 of this paper, but also that the Blackwell and segregation curve orders on D2 are the same (see their Corollary 2), which in turn implies Theorem 1.
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R. Hutchens / Mathematical Social Sciences 73 (2015) 63–68
index, the square root index, and the Gini index, denoted Od , Os , and Og respectively. Od (X ) =
J 1 x1j /N1 − x2j /N2 2 j =1
Os (X ) = 1 −
(3)
J (x1j /N1 )0.5 (x2j /N2 )0.5 ,
(4)
j =1
Og (X ) = 1 −
J
x1j /N1
x2j /N2 + 2
j =1
J
x2k /N2
,9
(5)
k=j+1
where the Gini index is computed on data that are arranged such that x11 /x21 < x12 /x22 < · · · < x1J /x2J . All three measures lie between zero and one, where ‘‘zero’’ indicates complete integration and ‘‘one’’ complete segregation. Both the square root and Gini index satisfy P1–P4 and are thereby included in the class of measures encompassed by Theorem 1. Thus, if S(X ) ≻ S(Y ) both the square root index and the Gini index will have numerical values indicating that X is more segregated than Y . The dissimilarity index satisfies three of the four properties, but violates the Neighborhood Division Property (P4). In contrast to segregation curves, such numerical measures possess two related problems. First, they rest on more restrictive – usually implicit and unexplained – assumptions about the nature of segregation. Second, in part because they rest on such assumptions, they can disagree with each other. For example, a dissimilarity index can declare one distribution more segregated than another, while a Gini index comes to the opposite conclusion. An alternative way to proceed is to build measures of segregation on plausible and explicit assumptions and accept the possibility that in some cases we cannot say whether X is more segregated than Y , i.e., accept an incomplete order. The next section examines that alternative.
Fig. 2. The segregation curve S(X ) and its twin S(X T ).
E =
1 0
0 1
and Γ =
0 1
1 0
. In the former case Y is identical
to X , and in the latter case the first row of Y is the second row of X and vice versa. In either case Y ∼ X . This property is not new. It is explicitly noted in James and Taeuber (1985), Hutchens (2001, 2004) and Frankel and Volij (2011). Moreover, most commonly used numerical measures of segregation exhibit this property (e.g., the Gini index, the dissimilarity index, and the square root index). Yet the property’s relevance to segregation curves seems to have been missed. To see this it is useful to introduce the ‘‘twin’’ of a segregation curve. In Fig. 1 the segregation curve for the X distribution (denoted S(X )) assigns women as type 1 and men as type 2. The twin of S(X ) (denoted S(X T )) is just the segregation curve that results when the assignments are switched, i.e., men are assigned type 1 and 3 8
women type 2. For example, if X =
8 3
5 1
7 1
4 4
1 5
1 7
4 4
then X T =
and the resulting segregation curves are S(X ) and
S(X ) in Fig. 2. Alternatively stated, S(X T ) = S(Γ X ), where Γ is a T
2. An additional property and its implication for segregation curves This section seeks a middle ground between the restrictive properties underlying numerical measures and the less restrictive properties underlying segregation curves. The paper introduces an additional property for a measure of segregation and then uses that property along with P1–P4 to derive a segregation order. While the result is not the complete segregation order generated by numerical measures, it is more complete than the order generated by the standard application of segregation curves. The additional property is called symmetry in types. According to this property, a measure of segregation should be insensitive to who is labeled type 1 or 2. For example, if men and women stay in the same occupation but exchange labels so that men are type 1 and women are type 2 (or vice versa), then measured inequality should stay the same. For I types of people, this property can be written, P5. Symmetry in types. Let P be some I × I permutation matrix. If PX = Y , then X ∼ Y . For example, if there are two types of people, there are only two possible permutation matrices: the identity matrix
1 2
9 There are several alternative and equivalent expressions for the Gini including,
T T i=1
j=1
(x2i /N2 ) x2j /N2 xx1i2i //NN12 −
Duncan (1955) for yet another.
x1j /N1 x2j /N2
. See Otis D. Duncan and Beverly
permutation matrix of the form, Γ =
0 1
1 0
.
Of course, the twin of a segregation curve is closely linked to P5 above. If a measure of segregation satisfies P5, then a segregation curve and its twin (i.e., S(X ) and S(X T )) indicate the same level of segregation. Thus, despite the intersection of S(Y ) and S(X ), it may still be possible to use segregation curves to determine whether one of the distributions is more segregated than the other. Specifically, plot S(X T ); if S(Y ) dominates S(X T ), then the X distribution is more segregated than Y . Alternatively stated, if either S(X ) or S(X T ) lie at some point below and nowhere above S(Y ), then under Properties 1–5, distribution X is more segregated than distribution Y .10 This idea can be stated in terms of a weak segregation criterion. Thus, Definition 2. Let X and Y be two distributions in D2 . We say that X is at least as segregated as Y according to the weak segregation curve criterion, denoted X
10 It makes no difference whether one compares X T with Y or X with Y T . In the standard application of segregation curves, it makes no difference which type of people is on the ordinate and which is on the abscissa. (If the segregation curve for distribution X dominates that for distribution Y when the cumulative proportion of males is on the abscissa, then X will also dominate Y when females are on the abscissa.) Alternatively stated, it makes no differences whether one compares X with Y or X T with Y T ; the dominance results will be the same. Similarly, it makes no difference whether one compares X T with Y or X with Y T ; the dominance results will be the same.
R. Hutchens / Mathematical Social Sciences 73 (2015) 63–68
Since a 2 × 2 permutation matrix P can take only two forms, E and Γ , this definition simply says that X is at least as segregated as Y if either S(X ) < S(Y ) or S(Γ X ) < S(Y ). Given Definition 2, we can extend Theorem 1 to include symmetry in types (P5):
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Theorem 3. Let < be a segregation order on DI . It satisfies Properties 1–4 if and only if for any X and Y in DI , X ≻I Y ⇒ X ≻ Y X vI Y ⇒ X v Y .
Theorem 2. Let < be a segregation order on D2 . It satisfies Properties 1–5 if and only if for any X and Y in D2 , X ≻wsc Y ⇒ X ≻ Y
(6)
X vwsc Y ⇒ X v Y .
(7)
Proof. See Appendix. Thus, if the segregation order not only satisfies P1–P4 but also symmetry of types (P5), then the segregation curve for X and its twin can be used to determine whether X is more segregated than Y . If the segregation curve for X intersects with the segregation curve for Y , then plot the twin, S(Γ X ). If that lies at some point below and at no point above the segregation curve for Y , then ignore the intersection; X is more segregated than Y . 3. More than two types Since the above analysis applies to two types of people, it is natural to ask whether a similar result obtains with more than two types. Recent work by Lasso de la Vega and Volij (2014) applies Blackwell’s criterion to the problem of measuring segregation when there are additional types, and also establishes a link between the Blackwell partial order and the segregation order produced by segregation curves. This section examines whether their work can be used to extend the above analysis to more than two types of people.11 Blackwell (1951, 1953) developed methods for ordering Markov matrices from least informative to most informative. A Markov matrix has non-negative real numbers as elements and rows that sum to one. Let M be the set of Markov matrices with I rows. Given two matrices AI ×JA , BI ×JB ∈ M , we say that A is at least as informative as B if there is a JA × JB Markov matrix Π such that B = AΠ . Intuitively, since Π is a matrix of weights that lie between zero and one, A is at least as informative as B if the elements of B can be expressed as weighted averages of the elements of A. Lasso de la Vega and Volij (2014) use Blackwell’s idea to order data matrices like those in Section 1 from least segregated to most segregated. To do that, they first transform the data matrix, X , into a Markov matrix. Let M (X ) be an I × J matrix where the element mij is the proportion of type i people in occupation j.12 Thus, mij = xij /Ni (X ). Applying Blackwell’s criterion to the transformed matrices we have, Definition 3. Let X and Y be two data matrices in DI . We say that X is at least as segregated as Y according to Blackwell’s criterion, denoted X
11 I am indebted to an anonymous referee for suggesting this line of argument. 12 Lasso de la Vega and Volij also include notation that accommodates a reordering of the columns of the X matrix. That notation is not essential for present purposes, and thus not included here. 13 See Lasso de la Vega and Volij (2014, Theorem 1).
Note the similarity between Theorems 3 and 1. Both link Properties 1–4 to a segregation order. The difference is that Theorem 1 deals with two types of people, while Theorem 3 deals with two or more types. In that sense Theorem 3 is a more general result than Theorem 1. But more remarkably, Lasso de la Vega and Volij establish that in the case where there are two types of people, the order based on Blackwell’s criterion (
(8)
X vWB Y ⇒ X v Y .
(9)
Proof. See Appendix. This theorem relates to Theorem 2 in much the same way as Theorem 3 relates to Theorem 1. Whereas Theorem 2 dealt with two types of people, Theorem 4 deals with I types. Moreover, since the segregation order based on Blackwell’s criterion (
14 See Lasso de la Vega and Volij (2014, Theorem 2).
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R. Hutchens / Mathematical Social Sciences 73 (2015) 63–68
Moreover, this approach extends to measuring segregation when there are more than two types of people. Blackwell’s criterion provides a segregation order for more than two types—an order that is intimately tied to that provided by segregation curves. By assuming symmetry in types, one can increase the number of alternatives that can be ranked by Blackwell’s criterion. Thus, the assumption of symmetry in types results in a more complete order. Acknowledgments I am indebted to Debi Prasad Mohapatra for suggestions and comments that improved the proofs as well as the broader argument. I am also indebted to two anonymous referees. Of course, I am responsible for all errors that remain. Appendix This appendix presents proofs of Theorems 2 and 4. It is easiest to begin with the following two lemmas. Lemma 1.
that X vWSC Y . Then, by definition, there is a permutation matrix P such that PX ∼SC Y . Since < satisfies Properties 1–4, it follows from Theorem 1 that PX ∼ Y . Since < satisfies Property 5, it follows that X ∼ Y , which establishes (7). Proof of Theorem 4 (Theorem 4). Let < be a segregation order on DI . It satisfies Properties 1–5 if and only if for any X and Y in DI , X ≻WB Y ⇒ X ≻ Y
(8)
X vWB Y ⇒ X v Y .
(9)
Sufficiency: If < satisfies (8) and (9), then < satisfies Properties 1–5. Let X be a data matrix in DI and let Y (i) be formed from X as specified in property i, i = 1, 2, 3, and 5. Since by Lemma 2,
Lemma 2.
(6)
X vWSC Y ⇒ X v Y .
(7)
Sufficiency: If < satisfies (6) and (7), then < satisfies Properties 1–5. Let X be a data matrix in D2 and let Y (i) be formed from X as specified in property i, i = 1, 2, 3, and 5. Since by Lemma 1,
Blackwell, David, 1951. Comparison of experiments. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley, pp. 93–102. Blackwell, David, 1953. Equivalent comparisons of experiments. Ann. Math. Stat. 24 (2), 265–272. Duncan, Otis D., Duncan, Beverly, 1955. A methodological analysis of segregation indices. Amer. Sociol. Rev. 20, 210–217. Flückiger, Yves, Silber, Jacques, 1999. The Measurement of Segregation in the Labor Force. Physica-Verlag, Heidelberg, New York. Foster, James, 1985. Inequality measurement. In: Peyton Young, H. (Ed.), Fair Allocation. American Mathematical Society, Providence, RI, pp. 31–68. Frankel, David M., Volij, Oscar, 2011. Measuring school segregation. J. Econom. Theory 146 (1), 1–38. Hutchens, Robert M., 1991. Segregation curves, Lorenz curves, and inequality in the distribution of people across occupations. Math. Social Sci. 21, 31–51. Hutchens, Robert M., 2001. Numerical measures of segregation: Desirable properties and their implications. Math. Social Sci. 42 (1), 13–29. Hutchens, Robert M., 2004. One measure of segregation. Internat. Econom. Rev. 45 (2), 555–577. James, David R., Taeuber, Karl E., 1985. Measures of segregation. In: Tuma, Nancy Brandon (Ed.), Sociology Methodology, 1985. Jossey-Bass, Inc., San Francisco, London, pp. 1–32. Lasso de la Vega, Casilda, Volij, Oscar, 2013. Segregation, informativenes and Lorenz dominance. Discussion Paper No. 13-12, Monaster Center for Economic Research, Ben-Gurion University of the Negev, November. Lasso de la Vega, Casilda, Volij, Oscar, 2014. Segregation, informativenes and Lorenz dominance. Soc. Choice Welf. 43 (3), 547–564. Massey, Douglas, Denton, Nancy, 1988. The dimensions of residential segregation. Soc. Forces 67, 281–315. Weeden, Kim A., 2004. Profiles of change: sex segregation in the United States, 1910–2000. In: Charles, Maria, Grusky, David B. (Eds.), Occupational Ghettos: The Worldwide Segregation of Women and Men. Stanford University Press, Palo Alto, CA, pp. 131–178.
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Mathematical Social Sciences journal homepage: www.elsevier.com/locate/econbase
Symmetric measures of segregation, segregation curves, and Blackwell’s criterion Robert Hutchens Cornell University, United States
highlights • • • •
A new approach is proposed for using segregation curves to measure segregation. The resulting incomplete segregation order incorporates a ‘‘symmetry of types’’ property. Using Blackwell’s criterion, the approach is extended to more than two types of people. The new approach results in a more complete segregation order.
article
info
Article history: Received 25 February 2014 Received in revised form 11 October 2014 Accepted 27 November 2014 Available online 8 December 2014
abstract This paper first proposes a new way to use segregation curves to examine whether one distribution of people across groups (e.g., occupations or neighborhoods) is more segregated than another. It then uses Blackwell’s criterion to extend the argument to more than two types of people. The basic idea is that by introducing additional assumptions about the nature of segregation, one obtains a more complete ranking of distributions. The paper demonstrates that the assumption of ‘‘symmetry in types’’ – an assumption that appears frequently in the literature on segregation measurement – has implications for both segregation curves and Blackwell’s criterion. © 2014 Elsevier B.V. All rights reserved.
A large and growing empirical literature seeks to measure segregation in school systems, occupational networks, and city neighborhoods.1 The topic is important because ethnic and gender segregation is intimately tied to issues of access and opportunity— issues that are of fundamental interest to social scientists and the larger society. Standing alongside this empirical literature is a second more theoretical literature that grapples with the problem of how to properly measure segregation. When can we say that one distribution of people across groups (e.g., occupations or neighborhoods) is more segregated than another? This paper contributes to that second literature. In particular, the paper proposes a new way to use segregation curves to examine whether one distribution is more segregated than another. Segregation curves were introduced into the literature in a seminal paper by Otis and Beverly Duncan.2 In their analysis of racial segregation in US cities, the Duncans assumed two types of people (specifically, whites and non-whites) and
E-mail address: [email protected]. 1 For example, Massey and Denton (1988), Flückiger and Silber (1999), Hutchens (2001, 2004), Weeden (2004) and Frankel and Volij (2011). 2 Otis D. Duncan and Beverly Duncan (1955). http://dx.doi.org/10.1016/j.mathsocsci.2014.11.007 0165-4896/© 2014 Elsevier B.V. All rights reserved.
plotted curves that indicate whether segregation in city A is greater than that in city B. The resulting segregation curves are similar to the Lorenz curves used to assess income inequality. Like Lorenz curves, segregation curves have the advantage of resting on a few weak and plausible assumptions about the nature of segregation. Also like Lorenz curves, segregation curves yield an incomplete order of distributions; when segregation curves cross, they yield no information about whether one distribution of people across groups can be ranked as more segregated than another. Whether such an incomplete order is an advantage or disadvantage is in the eye of the beholder. On the one hand it is awkward to announce that because of intersecting segregation curves we do not know whether distribution X is more segregated than Y . There exist several numerical measures of segregation that yield a complete order of distributions; why not use one of them? On the other hand, intersecting segregation curves force us to say, ‘‘it is only by making additional assumptions about the nature of segregations – assumptions that are often neither explicit nor explained – that we can claim that X is more segregated than Y ’’. There can be virtue in such candor. This paper first shows that by introducing an additional plausible assumption about the nature of segregation, one can increase the number of alternatives that can be ranked by segregation
64
R. Hutchens / Mathematical Social Sciences 73 (2015) 63–68
curves. It then extends that result to a more general segregation order that is based on Blackwell’s criterion. In the case of segregation curves, the practical implication is as follows: if the segregation curves for X and Y cross, then plot a new segregation curve denoted X T . This new segregation curve is closely related to the segregation curve for X , and is thus called the ‘‘twin’’ of X . If X T lies beneath Y – more precisely, if X T lies at some point below and at no point above Y – then despite the fact that the segregation curves for X and Y cross, distribution X can be declared more segregated than distribution Y , and that increases the number of alternatives that can be ranked. But since that result pertains to segregation curves, it is restricted to situations where one can reasonably assume two types of people. A recent paper by Lasso de la Vega and Volij (2014) applies Blackwell’s criterion (Blackwell, 1951, 1953) to the problem of measuring segregation. They not only obtain an incomplete order that can be used to assess segregation when there are more than two types of people (e.g., white Hispanic, black Hispanic, white non-Hispanic, black non-Hispanic), but they also establish an intimate link between that incomplete order and segregation curves. Their work raises the possibility of extending this paper’s result to the general problem of two or more types of people. The final section of the paper shows that the result does, indeed, generalize. Thus, by making an additional plausible assumption about the nature of segregation, one can increase the number of alternatives that can be ranked by Blackwell’s criterion, and that result applies to two or more types of people. 1. The problem, properties, segregation curves, and previous results Consider I types of people distributed over J occupations. Let xij denote the number of type i people in occupation j (i = 1, . . . , I ; j = 1, . . . , J ), let Ni (X ) = j xij , i = 1, . . . , I denote the total number of type i people over all occupations, and let X be a data matrix of the xij . For example, type 1 could be women, type 2 could be men, and occupation j could be one of J occupations. To insure a meaningful problem, assume that J is greater than one, that the xij are non-negative real numbers,3 and that the number of type i people is positive (Ni (X ) > 0, i = 1, . . . , I). Thus, with I types of people and J occupations, the data matrix takes the form,
x11
.
X = x1 , x2 , . . . , xJ = .. xI1
··· .. . ···
x1J
.. .
∞
J =2 DIJ where DIJ =
I ×J
X ∈ R+
: Ni (X ) > 0, i =
1, . . . , I .
Occupation 1 type 1 type 2
6 5
2 4
4 , 5
3 8
2
1 5
3
4
1 7
4 .
P1. Scale invariance. Let Y be obtained from X by multiplying the number of type i people in every occupation of X by a positive scalar βi . Then Y ∼ X . P2. Symmetry in occupations. Let (j1 , . . . , jJ ) beany permutation of 1, . . . , J, X = x1 , x2 , . . . , xJ and Y = xj1 , xj2 , . . . , xjJ . Then Y ∼ X .5 P3. Organizational equivalence. Let Y be obtained from X by dividing occupation J into two occupations such that for some α ∈ (0, 1), i = 1, . . . , I ; j = 1, . . . , J − 1 i = 1, . . . , I
yiJ +1 = xiJ (1 − α),
i = 1, . . . , I .
Then Y ∼ X . P4. Neighborhood division property. Let Y be obtained from X by dividing occupation J into two occupations such that, yij = xij ,
i = 1, . . . , I ; j = 1, . . . , J − 1 i = 1, . . . , I
and for no α ∈ [0, 1], yiJ = xiJ α,
i = 1, . . . , I
yiJ +1 = xiJ (1 − α), (1)
(2)
this paper seeks to evaluate whether the X distribution is more segregated than Y . To that end, let < denote a segregation order on DI .4 If X ≻ Y then X is ranked as more segregated than Y , and if X v Y then X has the same level of segregation as Y . Of course, the order may be partial; if the order does not rank some X and Y , then the result is an incomplete order. Like any measure of inequality, a ‘‘good’’ measure of segregation should have properties that accord with perceptions of segregation. These properties are ultimately value judgments about what it means to say that one social state is more segregated than another. As such, the properties should be as unrestrictive as possible. Restrictive properties imply strong value judgments, which in turn lead to more controversial conclusions about whether X is more segregated than Y . The literature on measuring segregation advances several properties for a good measure. Four of these properties are particularly important not only because they are unrestrictive, but also because they define a broad class of measures that are linked to segregation curves. While these properties are often stated for two types of people (e.g., men and women), because this paper ultimately addresses the more general problem of I types, it is useful to state the properties in their more general form. Finally, since the four properties are discussed extensively elsewhere, the following is a brief summary.
yiJ + yiJ +1 = xiJ ,
For example, with two types of people and four occupations the data matrix could take the form,
X =
3 4
yiJ = xiJ α,
xIJ
Y =
yij = xij ,
where xj is a column vector of length I that contains data on the types of people in occupation j. Denoting the vector space of all I × J I ×J real matrices with non-negative elements by R+ , the domain of X shall be DI =
Thus, letting type 1 people be women, the distribution in (1) has three women and eight men in the first occupation, and the total number of men and women is respectively 24 and 9 (N1 (X ) = 3 + 1 + 1 + 4 = 9 and N2 (X ) = 8 + 5 + 7 + 4 = 24). Letting Y be another matrix of two types of people over four occupations, e.g.,
i = 1, . . . , I .
Then Y ≻ X .
4
3 Not only does this assumption address the general case, but it is also plausible. Part-time workers could be treated as fractional workers. Irrational numbers are also theoretically admissible (e.g., a full time worker is counted as ‘‘1’’ and a parttime worker as the square-root of 0.3).
4 See Foster (1985, p. 42) for a particularly clear definition of a measure of inequality and an inequality order. 5 The property could be equivalently stated in terms of a permutation matrix. Specifically, let P be a J × J permutation matrix with only one coefficient in each row equal to 1 and only one coefficient in each column equal to 1. If Y = XP then Y ∼ X.
R. Hutchens / Mathematical Social Sciences 73 (2015) 63–68
P1 simply says that if the total number of type i people over all occupations (Ni (X )) is multiplied by a positive scalar (βi ) and the share of type i people in each occupation does not change (where a share is calculated as the number of type i people in an occupation divided by Ni (X )), then segregation does not change. To illustrate, suppose the number of type 1 people (e.g., women) doubles (β1 = 2), and occupations that initially contained 0%, 10%, 20%, etc. of all women continue to do so. Then in accordance with P1 this change in the level of women in the population does not affect segregation. P2 says that occupations can be relabeled without affecting segregation; what is important is numbers of different types of people in an occupation, not whether the occupation is labeled 2, 4, or 10. P3 says that if an occupation is divided in two and the distribution of types within the new occupations stays the same, then measured segregation does not change. For example, suppose that an occupation is comprised of 40% women (type 1) and 60% men (type 2). If that occupation is divided in two and the new occupations are both comprised of 40% women and 60% men, then segregation does not change. This division of an occupation essentially distributes the old segregation over new occupations; the ratio of type 1 to type i people (i = 2, . . . , I) in the new occupations is the same as in the old. The Neighborhood Division Property (P4) is closely related to P3. It arises out of work on residential segregation (see Frankel and Volij, 2011 and Lasso de la Vega and Volij, 2014), hence the reference to neighborhoods. Applied to occupations, this property essentially says that if an occupation is divided into two occupations and the distribution of types in the two new occupations differs from that in the original occupation, then segregation increases. For example, if the original occupation was comprised of 40% women and 60% men, and the division results in two new occupations, one comprised of 20% women and 80% men and the other of 60% women and 40% men, then segregation increases. The Neighborhood Division Property is also closely related to a transfer principle whereby segregation increases when people move between two occupations with particular characteristics. That principle is, in turn, closely related to the Pigou–Dalton transfer principle in the income inequality literature. While the transfer principle works well in situations where there are two types of people (e.g., men and women), it does not generalize to situations where there are multiple types. Thus, this paper is built on the Neighborhood Division Property.6 In situations where there are two types of people, a very useful measure of segregation – a measure that satisfies Properties 1–4 – is the segregation curve. A segregation curve is a plot of the cumulative fraction of type 1 people (on the vertical axis) against the cumulative fraction of type 2 people (on the horizontal axis), with fractions ranked from small to large values of the ratio x1j /x2j . More precisely, let the X matrix be permuted in such a way that x1j /x2j is ordered from smallest to largest, and let xˆ ih = j=1 xij /Ni (X ), i = 1, 2; h = 1, . . . , J. To draw the segregation curve, plot (ˆx1h , xˆ 2h ) h = 1, . . . , J, and use straight lines to connect consecutive points beginning at the origin (0, 0). Fig. 1 illustrates for the X and Y distributions in (1) and (2). Since the segregation curve for Y lies at no point below and at some point above that for X , we say that segregation curve for Y dominates the segregation curve for X . Write this as S(X ) ≻ S(Y ), since X is the more segregated of the two distributions. If the segregation curve for Y is coincident with that of X , then the curves are equivalent and we write S(X ) ∼ S(Y ). Thus,
h
6 Lasso de la Vega and Volij (2013) define the transfer principle and show that a segregation order that satisfies P3 satisfies the transfer principle if and only if it satisfies P4.
65
Fig. 1. The segregation curves S(X ) and S(Y ).
Definition 1. Let X and Y be two distributions in D2 . We say that X is at least as segregated as Y according to the segregation curve criterion, denoted X
7 The assumed domain of X had non-negative rational numbers in its second row and non-negative real numbers in its first row. See Hutchens (2001) for additional discussion of the proof. 8 See Proposition 4 in Lasso de la Vega and Volij (2013). That proposition is stated for four properties: ANON, CI, OE, and T. ‘‘ANON’’ is equivalent to P2 above, ‘‘CI’’ is equivalent to P1, ‘‘OE’’ is equivalent to P3, and T is a transfer property. They go on to not only establish what is Theorem 3 in Section 3 of this paper, but also that the Blackwell and segregation curve orders on D2 are the same (see their Corollary 2), which in turn implies Theorem 1.
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R. Hutchens / Mathematical Social Sciences 73 (2015) 63–68
index, the square root index, and the Gini index, denoted Od , Os , and Og respectively. Od (X ) =
J 1 x1j /N1 − x2j /N2 2 j =1
Os (X ) = 1 −
(3)
J (x1j /N1 )0.5 (x2j /N2 )0.5 ,
(4)
j =1
Og (X ) = 1 −
J
x1j /N1
x2j /N2 + 2
j =1
J
x2k /N2
,9
(5)
k=j+1
where the Gini index is computed on data that are arranged such that x11 /x21 < x12 /x22 < · · · < x1J /x2J . All three measures lie between zero and one, where ‘‘zero’’ indicates complete integration and ‘‘one’’ complete segregation. Both the square root and Gini index satisfy P1–P4 and are thereby included in the class of measures encompassed by Theorem 1. Thus, if S(X ) ≻ S(Y ) both the square root index and the Gini index will have numerical values indicating that X is more segregated than Y . The dissimilarity index satisfies three of the four properties, but violates the Neighborhood Division Property (P4). In contrast to segregation curves, such numerical measures possess two related problems. First, they rest on more restrictive – usually implicit and unexplained – assumptions about the nature of segregation. Second, in part because they rest on such assumptions, they can disagree with each other. For example, a dissimilarity index can declare one distribution more segregated than another, while a Gini index comes to the opposite conclusion. An alternative way to proceed is to build measures of segregation on plausible and explicit assumptions and accept the possibility that in some cases we cannot say whether X is more segregated than Y , i.e., accept an incomplete order. The next section examines that alternative.
Fig. 2. The segregation curve S(X ) and its twin S(X T ).
E =
1 0
0 1
and Γ =
0 1
1 0
. In the former case Y is identical
to X , and in the latter case the first row of Y is the second row of X and vice versa. In either case Y ∼ X . This property is not new. It is explicitly noted in James and Taeuber (1985), Hutchens (2001, 2004) and Frankel and Volij (2011). Moreover, most commonly used numerical measures of segregation exhibit this property (e.g., the Gini index, the dissimilarity index, and the square root index). Yet the property’s relevance to segregation curves seems to have been missed. To see this it is useful to introduce the ‘‘twin’’ of a segregation curve. In Fig. 1 the segregation curve for the X distribution (denoted S(X )) assigns women as type 1 and men as type 2. The twin of S(X ) (denoted S(X T )) is just the segregation curve that results when the assignments are switched, i.e., men are assigned type 1 and 3 8
women type 2. For example, if X =
8 3
5 1
7 1
4 4
1 5
1 7
4 4
then X T =
and the resulting segregation curves are S(X ) and
S(X ) in Fig. 2. Alternatively stated, S(X T ) = S(Γ X ), where Γ is a T
2. An additional property and its implication for segregation curves This section seeks a middle ground between the restrictive properties underlying numerical measures and the less restrictive properties underlying segregation curves. The paper introduces an additional property for a measure of segregation and then uses that property along with P1–P4 to derive a segregation order. While the result is not the complete segregation order generated by numerical measures, it is more complete than the order generated by the standard application of segregation curves. The additional property is called symmetry in types. According to this property, a measure of segregation should be insensitive to who is labeled type 1 or 2. For example, if men and women stay in the same occupation but exchange labels so that men are type 1 and women are type 2 (or vice versa), then measured inequality should stay the same. For I types of people, this property can be written, P5. Symmetry in types. Let P be some I × I permutation matrix. If PX = Y , then X ∼ Y . For example, if there are two types of people, there are only two possible permutation matrices: the identity matrix
1 2
9 There are several alternative and equivalent expressions for the Gini including,
T T i=1
j=1
(x2i /N2 ) x2j /N2 xx1i2i //NN12 −
Duncan (1955) for yet another.
x1j /N1 x2j /N2
. See Otis D. Duncan and Beverly
permutation matrix of the form, Γ =
0 1
1 0
.
Of course, the twin of a segregation curve is closely linked to P5 above. If a measure of segregation satisfies P5, then a segregation curve and its twin (i.e., S(X ) and S(X T )) indicate the same level of segregation. Thus, despite the intersection of S(Y ) and S(X ), it may still be possible to use segregation curves to determine whether one of the distributions is more segregated than the other. Specifically, plot S(X T ); if S(Y ) dominates S(X T ), then the X distribution is more segregated than Y . Alternatively stated, if either S(X ) or S(X T ) lie at some point below and nowhere above S(Y ), then under Properties 1–5, distribution X is more segregated than distribution Y .10 This idea can be stated in terms of a weak segregation criterion. Thus, Definition 2. Let X and Y be two distributions in D2 . We say that X is at least as segregated as Y according to the weak segregation curve criterion, denoted X
10 It makes no difference whether one compares X T with Y or X with Y T . In the standard application of segregation curves, it makes no difference which type of people is on the ordinate and which is on the abscissa. (If the segregation curve for distribution X dominates that for distribution Y when the cumulative proportion of males is on the abscissa, then X will also dominate Y when females are on the abscissa.) Alternatively stated, it makes no differences whether one compares X with Y or X T with Y T ; the dominance results will be the same. Similarly, it makes no difference whether one compares X T with Y or X with Y T ; the dominance results will be the same.
R. Hutchens / Mathematical Social Sciences 73 (2015) 63–68
Since a 2 × 2 permutation matrix P can take only two forms, E and Γ , this definition simply says that X is at least as segregated as Y if either S(X ) < S(Y ) or S(Γ X ) < S(Y ). Given Definition 2, we can extend Theorem 1 to include symmetry in types (P5):
67
Theorem 3. Let < be a segregation order on DI . It satisfies Properties 1–4 if and only if for any X and Y in DI , X ≻I Y ⇒ X ≻ Y X vI Y ⇒ X v Y .
Theorem 2. Let < be a segregation order on D2 . It satisfies Properties 1–5 if and only if for any X and Y in D2 , X ≻wsc Y ⇒ X ≻ Y
(6)
X vwsc Y ⇒ X v Y .
(7)
Proof. See Appendix. Thus, if the segregation order not only satisfies P1–P4 but also symmetry of types (P5), then the segregation curve for X and its twin can be used to determine whether X is more segregated than Y . If the segregation curve for X intersects with the segregation curve for Y , then plot the twin, S(Γ X ). If that lies at some point below and at no point above the segregation curve for Y , then ignore the intersection; X is more segregated than Y . 3. More than two types Since the above analysis applies to two types of people, it is natural to ask whether a similar result obtains with more than two types. Recent work by Lasso de la Vega and Volij (2014) applies Blackwell’s criterion to the problem of measuring segregation when there are additional types, and also establishes a link between the Blackwell partial order and the segregation order produced by segregation curves. This section examines whether their work can be used to extend the above analysis to more than two types of people.11 Blackwell (1951, 1953) developed methods for ordering Markov matrices from least informative to most informative. A Markov matrix has non-negative real numbers as elements and rows that sum to one. Let M be the set of Markov matrices with I rows. Given two matrices AI ×JA , BI ×JB ∈ M , we say that A is at least as informative as B if there is a JA × JB Markov matrix Π such that B = AΠ . Intuitively, since Π is a matrix of weights that lie between zero and one, A is at least as informative as B if the elements of B can be expressed as weighted averages of the elements of A. Lasso de la Vega and Volij (2014) use Blackwell’s idea to order data matrices like those in Section 1 from least segregated to most segregated. To do that, they first transform the data matrix, X , into a Markov matrix. Let M (X ) be an I × J matrix where the element mij is the proportion of type i people in occupation j.12 Thus, mij = xij /Ni (X ). Applying Blackwell’s criterion to the transformed matrices we have, Definition 3. Let X and Y be two data matrices in DI . We say that X is at least as segregated as Y according to Blackwell’s criterion, denoted X
11 I am indebted to an anonymous referee for suggesting this line of argument. 12 Lasso de la Vega and Volij also include notation that accommodates a reordering of the columns of the X matrix. That notation is not essential for present purposes, and thus not included here. 13 See Lasso de la Vega and Volij (2014, Theorem 1).
Note the similarity between Theorems 3 and 1. Both link Properties 1–4 to a segregation order. The difference is that Theorem 1 deals with two types of people, while Theorem 3 deals with two or more types. In that sense Theorem 3 is a more general result than Theorem 1. But more remarkably, Lasso de la Vega and Volij establish that in the case where there are two types of people, the order based on Blackwell’s criterion (
(8)
X vWB Y ⇒ X v Y .
(9)
Proof. See Appendix. This theorem relates to Theorem 2 in much the same way as Theorem 3 relates to Theorem 1. Whereas Theorem 2 dealt with two types of people, Theorem 4 deals with I types. Moreover, since the segregation order based on Blackwell’s criterion (
14 See Lasso de la Vega and Volij (2014, Theorem 2).
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R. Hutchens / Mathematical Social Sciences 73 (2015) 63–68
Moreover, this approach extends to measuring segregation when there are more than two types of people. Blackwell’s criterion provides a segregation order for more than two types—an order that is intimately tied to that provided by segregation curves. By assuming symmetry in types, one can increase the number of alternatives that can be ranked by Blackwell’s criterion. Thus, the assumption of symmetry in types results in a more complete order. Acknowledgments I am indebted to Debi Prasad Mohapatra for suggestions and comments that improved the proofs as well as the broader argument. I am also indebted to two anonymous referees. Of course, I am responsible for all errors that remain. Appendix This appendix presents proofs of Theorems 2 and 4. It is easiest to begin with the following two lemmas. Lemma 1.
that X vWSC Y . Then, by definition, there is a permutation matrix P such that PX ∼SC Y . Since < satisfies Properties 1–4, it follows from Theorem 1 that PX ∼ Y . Since < satisfies Property 5, it follows that X ∼ Y , which establishes (7). Proof of Theorem 4 (Theorem 4). Let < be a segregation order on DI . It satisfies Properties 1–5 if and only if for any X and Y in DI , X ≻WB Y ⇒ X ≻ Y
(8)
X vWB Y ⇒ X v Y .
(9)
Sufficiency: If < satisfies (8) and (9), then < satisfies Properties 1–5. Let X be a data matrix in DI and let Y (i) be formed from X as specified in property i, i = 1, 2, 3, and 5. Since by Lemma 2,
Lemma 2.
(6)
X vWSC Y ⇒ X v Y .
(7)
Sufficiency: If < satisfies (6) and (7), then < satisfies Properties 1–5. Let X be a data matrix in D2 and let Y (i) be formed from X as specified in property i, i = 1, 2, 3, and 5. Since by Lemma 1,
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