Soc Choice Welfare (2008) 30:475–493 DOI 10.1007/s00355-007-0252-1 ORIGINAL PAPER

Second degree Pareto dominance Ronny Aboudi · Dominique Thon

Received: 22 November 2005 / Accepted: 4 May 2007 / Published online: 17 July 2007 © Springer-Verlag 2007

Abstract We present and discuss a binary relation aimed at the study of the re-distribution of income. We characterize, in a number of ways, the set of income allocations that can be reached from an initial allocation through a sequence of pairwise equalizing transfers, where the sequence contains no flow of income from any donor to anyone else who ends up strictly richer than this donor himself ends up at the outcome of the entire sequence. 1 Introduction In the consideration of a voluntary act of altruism between two persons, it is sensible to assume that the benefactor does not intend to end up strictly worse-off than the beneficiary. In the case of a voluntary transfer of income from one person to another, this condition is that the donor does not end up strictly poorer than the recipient does. Such a pair-wise altruistic transfer could be called non-excessive. The set of income distributions that can be reached through a succession of such pair-wise transfers, starting from some original distribution, has been characterized in Thon and Wallace (2004).

Thanks are due to Peter Lambert and James Mirrlees for their comments. Thon was with the Department of Economics, University of Macau, when some of the work on this paper was done. R. Aboudi (B) Department of Management Science, University of Miami, P.O.B. 248237, Coral Gables, FL 33124, USA e-mail: [email protected] D. Thon Bodø Graduate School of Business, 8049 Bodø, Norway e-mail: [email protected]

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Consider now a redistributive scheme, that is, a redistributive black box through which a distribution of income between n persons is transformed into another distribution between the same n persons, keeping the total amount of income constant. Think if you will of the fabled taxation cum social security system at constant total income. Assume now that one agrees on the suitability of non-excessive altruism for a single pair-wise transfer. One would presumably then require from a redistributive scheme that the re-distribution it performs could have been realized through a sequence of such pair-wise non-excessive altruistic transfers. One should note though that it is quite possible for a beggar to receive a sequence of modest alms from a number of non-excessively altruistic persons, and yet to be at the end strictly richer than some of his benefactors end up being themselves. Accordingly, if one wanted to extend the notion of non-excessive altruism to the case of the participation in a redistributive scheme, and consider some initial distribution and a distribution reached through that scheme, then one would naturally require that the latter distribution could have been reached from the former one through a sequence of non-excessively altruistic pairwise transfers, and furthermore, that the sequence contains no flow of income from any donor to anyone who ends up strictly richer than this donor himself ends up at the outcome of the sequence. The characterization of the binary relation that relates two such income distributions is the subject of this paper. This relation appeared in an entirely different context in Aboudi and Thon (2006). The framework of this paper is thus the comparison of allocations of a total income to a set of n persons that is prescribed, that is, the persons are the same in the allocations being compared. What is more, our concern is re-distribution from some initial allocation, rather than distribution considered from an “original position” where a veil of ignorance makes individuals interchangeable [see Thon and Wallace (2004) for the distinction between the two cases]. The binary relation which is most central to the study of income distribution is certainly majorization, which is the cornerstone of the notion of “more equally distributed” in an anonymous context (Lorenz 1905; Dalton 1920; Kolm 1966). In order to apply it, it is sufficient to know the two income vectors to be compared, each one arbitrarily and independently indexed, i.e. there is no need to match every income in one distribution to an income in the other one. In fact, in many cases, it is not even meaningful to try to do so: majorization is applicable, and commonly applied, to a situation where the persons in x and the ones in y are not the same. Think of comparing the income distributions of two countries, after having somehow normalized the sizes of the populations and of the total incomes. Majorization is a symmetric relation in the sense that it is invariant to a permutation of the incomes in either distribution being compared. Likewise, all degrees of stochastic dominance are, by definition, symmetric, and, more generally, so is any binary relation based on some class of expected utility functions. Such relations thus deal with the distribution of income rather than its re-distribution. As far as we know, the only binary relation appearing in the relevant literature which is not symmetric is the partial order analyzed in Thon and Wallace (2004); the one presented in this paper is a smaller binary relation and, likewise, it is non-symmetric (≺ is a smaller binary relation than  if x ≺ y → x  y). Using a non-symmetric binary relation assumes that one is in a context where one is able to index the n persons

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whose incomes appear both in x and y. In other words, if the two allocations of income are viewed as random variables, it is necessary to know their joint distribution. In such a redistributive context, symmetry is certainly inappropriate: one is concerned with the way the incomes of given individuals change. We emphasize that we are by no means concerned with the measurement of income inequality, which is inherently a symmetric concept. Section 2 derives a number of equivalences for the binary relation introduced above. Section 3 discusses the properties of this relation and how it relates to other relations appearing in the literature. Section 4 gives a characterization in terms of a family of social welfare functions. Section 5 sums up and concludes. “Larger”, “smaller”, “richer”, “poorer”, etc. are understood in the weak sense. We n x =
n y when no confusion can arise. By “transfer”, write
x =
y for
i=1 i i=1 i we always mean a strictly positive pair-wise transfer. We let N = {1, 2, . . . , n} and we write x ↑ = (x(1) , x(2) , . . . , x(n) ) for an increasing re-arrangement of x ∈ n . We have as far as feasible harmonized our notation with Thon and Wallace (2004). The proofs are relegated to an Appendix. 2 Equivalences In this section, we provide three quite distinct definitions and then show that they define the same binary relation. We first recall, from Thon and Wallace (2004), the following definition. Definition 2.1 (Asymmetric Dalton transfer) A transfer of income between two persons is an asymmetric Dalton transfer if, after the transfer is performed, the income of the recipient is not strictly larger than the income of the donor. This formalizes what we called in the Introduction a transfer that is not excessively altruistic. We now define the notion of an R-P-in-y transfer (Rich-to-Poor-in-y), which stands for “a transfer from a donor who is richer in y than the recipient is in y”. Definition 2.2 Let x, v, y ∈ n ;
x =
y. We say that v is obtained from x through an R-P-in-y transfer if: v j = x j − ; vk = xk + ;  > 0, yk ≤ y j , and vi = xi for i = j, k. Thus if y is reached from x through a sequence of transfers that are all R-P-in-y transfers, then no one ends up in y poorer than anyone he has donated income to. The property of a redistributive scheme that we introduced in Sect. 1 (that y could be reached from x through a sequence of asymmetric Dalton transfers, none of which is to someone who ends up strictly richer than the donor does) can easily be seen to be equivalent to x, y being related by the following binary relation. Definition 2.3 Let x, y ∈ n . We say that x  ˜ y if y = x or if it is possible to reach y from x through a sequence of transfers, each of which is both an asymmetric Dalton transfer and an R-P-in-y transfer. Note that each transfer in the sequence described in Definition 2.3 satisfies two distinct conditions: the one that it is an asymmetric Dalton transfer (a condition based

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on comparing the incomes of the donor and recipient of this transfer just before it is performed) and the independent condition that it is from a donor who is in y richer than the recipient is in y (a condition based on comparing the size of two incomes in y, something entirely different, and which does not depend on the order in which the transfers are performed, while the first condition does). Our second definition of a binary relation is: Definition 2.4 Let x, y ∈ n . We say that x ◦ y if y = x or if it is possible to reach y from x through a set of transfers that are all asymmetric Dalton transfers, regardless of the order in which they are performed. Note that while Definition 2.3 refers to a sequence of transfers, that is transfers performed in a particular order, Definition 2.4 refers to a set of transfers, as there, the order in which they are performed does not matter. We now provide, following Aboudi and Thon (2006), a third definition of a binary relation. First we need the following notational definition. Definition 2.5 Let x, y ∈ n . We say that the vectors x and y are lexicographically increasing in y if y1 ≤ y2 ≤ · · · ≤ yn and whenever yi = y j for i < j, xi ≤ x j . Essentially, Definition 2.5 means that the persons have been indexed in such a way that the vector y is increasingly ordered, with the provision that x is used as a “secondary key”. Definition 2.6 Let x, y ∈ n . We write x ˜ P2 y if there exists a permutation π of N such that (xπ1 , xπ2 , . . . , xπn ) and (yπ1 , yπ2 , . . . , yπn ) are lexicographically increasing in y, and k
i=1

xπi ≤

k
i=1

yπi , k = 1, . . . , n − 1,

n
i=1

xi =

n


yi .

(2.1)

i=1

We then say that y dominates x by constant-sum second degree Pareto dominance. To illustrate the role of Definition 2.5 in constructing Definition 2.6, consider y = (20, 10, 20) and x = (5, 7, 38). Both x = (7, 5, 38) and x = (7, 38, 5) constitute re-arrangements of x according to the increasing order of y , but only the first one is such that x and y = (10, 20, 20) are lexicographically increasing in y and only this re-arrangement satisfies the inequalities of Definition 2.6. The binary relation x ˜ P2 y derives its name from the fact that it has the same relationship to constant-sum second degree stochastic dominance as the standard Pareto ordering has to first degree stochastic dominance; see Aboudi and Thon (2006, p. 208). Informally, x ˜ P2 y means:
x =
y and The poorest person in y is richer in y than this person is in x. The two poorest persons in y are collectively richer in y than those (2.2) two persons are in x. etc.

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We can now present the following main result. Theorem 2.1 Let x, y ∈ n . The following are equivalent: x ˜ y x ◦ y x ˜ P2 y

(Definition 2.3) (Definition 2.4) (Definition 2.6).

In view of the equivalences provided by Theorem 2.1, we shall henceforth always refer to our one binary relation as constant-sum second degree Pareto dominance, and represent it by ˜ P2 . Note that while Definitions 2.3 and 2.4 are not easy to check, Definition 2.6 provides a simple algorithm to determine whether or not the binary relation holds for a given pair, x, y. 3 Binary relations We now look at the properties of ˜ P2 as a binary relation and at its position in the hierarchy of binary relations that have been used in the study of the distribution and re-distribution of income. The following concept appeared in Thon and Wallace (2004). ˜ A y if y = x or if y can be reached from Definition 3.1 Let x, y ∈ n . We say that x ≺ x through a sequence of asymmetric Dalton transfers. ˜ M y if Definition 3.2 Let x, y ∈ n . We say that x ≺ k
i=1

x(i) ≤

k
i=1

y(i) ; k = 1, . . . , n − 1 and

n
i=1

xi =

n


yi .

(3.1)

i=1

This is the usual definition of “x majorizes y” (see Marshall and Olkin 1979, p. 21). ˜ M y. More generally, we have Comparing (3.1) to (2.1) shows that x ˜ P2 y implies x ≺ the following, which follows directly from the nature of the transfers that define each one of the binary relations, and, for the first implication, from Theorem 2.1: ˜ A y → x≺ ˜ M y. x ˜ P2 y → x ≺

(3.2)

˜ M ” as the set of all For a given x, we define the “better-than-x set according to ≺ M A M ˜ ≺ ˜ y, and call this set S (x). We define similarly the sets S ≺˜ (x) y’s such that x ≺ A M P2 P2 and S ˜ (x). Then, (3.2) is equivalent to: S ˜ (x) ⊆ S ≺˜ (x) ⊆ S ≺˜ (x). Figure 1 A M P2 represents the sets S ˜ (x), S ≺˜ (x) and S ≺˜ (x) for x = (1, 4, 13). For x ∈ n , let x = = (
x/n, . . . ,
x/n). The simplex has been divided into the six cells corresponding to a particular ordering of the three incomes; for example, C213 represents M the set where x2 ≤ x1 ≤ x3 , etc. The construction of S ≺˜ (x) (that is, the closed A set x M N P Q R) is well-known (Kolm 1966). A description of S ≺˜ (x) (the closed set x AC E F G H K ) can be found in Thon and Wallace (2004) where the same example P2 is used. The construction of S ˜ (x) in Fig. 1, that is, the closed set (x AB L x = K ), is

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C123

C 213

A M

x

B K L

C

x= E

R C

N

G F

H

C 231

132

C 312

Q

P

C321

M A P2 Fig. 1 The sets S ≺˜ (x), S ≺˜ (x) and S ˜ (x) for n = 3

not completely straightforward; it is done in Appendix 2. For the geometric principles behind the construction of Fig. 1, see Aboudi and Thon (2006, p. 202) and references within. P2 We establish some elementary properties of S ˜ (x) using the concepts just introduced. We first show that the perfectly equal distribution is a maximum for ˜ P2 . P2

Theorem 3.1 Let x ∈ n . Then x = ∈ S ˜ (x). P2

Figure 1 illustrates that S ˜ (x) is not convex. Our next result shows that it possesses a weak form of convexity. A set S is starshaped with respect to x ∈ S if, for any x ∈ S, any convex combination of x and x belongs to S. (While a set S is convex if it is starshaped with respect to any x ∈ S.) P2

Theorem 3.2 Let x ∈ n . The set S ˜ (x) is not necessarily convex; it is starshaped with respect to x and with respect to x = . P2

˜ A and ≺ ˜ M the property that Thus, while S ˜ (x) is not convex, it still shares with ≺ P2 P2 if x ˜ y, then x ˜ z for any z that is a convex combination of x and y, and x ˜ P2 w for any w that is a convex combination of y and x = . We now turn to the properties of ˜ P2 and related binary relations. The properties we consider for a binary relation R are as follows: Reflexivity: x Rx Transitivity: {x Ry and y Rz} implies x Rz Antisymmetry: {x Ry and y Rx} implies x = y Symmetry: {x Ry and y Rx} if y is a permutation of x.

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Symmetry and antisymmetry preclude each other. We say that a binary relation is a preorder if it is reflexive and transitive, and that it is a partial order if it is also ˜ M is a symmetric preorder. antisymmetric. It is well-known that the binary relation ≺ M M ˜ x hold if and only if y is a permutation of x (Alberti and ˜ y and y ≺ In fact, both x ≺ ˜ A , on the other hand,is a Uhlmann 1982, Lemma 1–10, p. 16). The binary relation ≺ partial order. ˜ A is reflexive, transitive and antisymmetric. Theorem 3.3 The binary relation ≺ ˜ P2 is neither a partial order nor a preorder. The binary relation ≺ ˜ P2 is reflexive and antisymmetric. It is not tranTheorem 3.4 The binary relation ≺ sitive. The fact that ˜ P2 is not transitive is demonstrated by the following example: x = (1, 4, 13); y = (6.9, 4, 7.1); z = (6.9, 5.5, 5.6). It is easy to check that x ˜ P2 y and y ˜ P2 z both hold, while x ˜ P2 z does not. ˜ A and ˜ P2 goes deeper than indicated in (3.2). We have The relationship between ≺ A ˜A ˜ y is not merely a necessary condition for x ˜ P2 y. The relation ≺ in effect that x ≺ P2 is furthermore the transitive closure of ˜ . There are a number of alternative formulations of the notion of the transitive closure of a binary relation; we adopt the one of Fishburn (1973, p. 74). Definition 3.3 The transitive closure t of the binary relation  on X is t =  ∪ ()2 ∪ ()3 ∪ ...., so that xt y ⇔ {x i−1  x i , i = 1, . . . , m, for some x 1 , ....., x m ∈ X , with x 0 = x; x m = y}. ˜ A is the transitive closure of the binary relation Theorem 3.5 The partial order ≺ P2 ˜ . ˜ A y means that y is reachable from x through Thus, it is not simply the case that x ≺ a sequence of asymmetric Dalton transfers (as it is by definition); y is furthermore reachable from x through a sequence of allocations x 0 , x 1 , x 2 , . . . , x m with x = x 0 ; y = x m , where at each step x i+1 can be reached from x i through a sequence of asymmetric Dalton transfers satisfying the further conditions of Definition 2.3, or, equivalently, Definition 2.4. Consider now the standard second degree stochastic dominance binary relation, ˜ M y if
x =
y. which specializes to x ≺ Definition 3.4 Let x, y ∈ n . We represent the second degree dominance binary relation: k
i=1

x(i) ≤

k


y(i) ;

k = 1, . . . , n

(3.3)

i=1

˜ 2 y. by x ≺

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In the terminology of Marshall and Olkin (1979), (3.3) reads: “y is weakly supermajorized by x”; the relation also appears in the literature as “y dominates x by generalized Lorenz dominance”. See Kolm (1966), Shorrocks (1983). Note that, unlike ˜ 2 y implies
x ≤
y, not necessarily
x =
y. ˜ M y, x ≺ x≺ Now, informally, x ˜ P2 y means that: “the s persons who are the poorest in y are collectively richer in y than in x, for s = 1, . . . , n”, with
x =
y. Consider now, still with
x =
y, the condition: “the income allocation in y of the s persons who ˜ 2 -dominate the income allocation of the same persons in are the poorest in y does ≺ x, for s = 1, . . . , n.” We now show that this condition, which is seemingly more demanding than x ˜ P2 y, is in fact merely equivalent to it. Theorem 3.6 Let x, y ∈ n . Then x ˜ P2 y is equivalent to:
x =
y and there exists a permutation π of N such that (xπ1 , xπ2 , . . . , xπn ) and (yπ1 , yπ2 , . . . , yπn ) are lexicographically increasing in y, and such that, with x [s] = (x π1 , xπ2 , . . . , xπs ), y [s] = (y π1 , yπ2 , . . . , yπs ), ˜ 2 y [s] ; s = 1, . . . , n. x [s] ≺

(3.4)

Working out an example will readily show that while (3.4) contains not only the inequalities that define x ˜ P2 y but also further inequalities, the latter set of inequalities is implied by the former. We conclude this section by closing a blind alley. Definition 2.6 of the binary relation x ˜ P2 y is based on the ordering of the incomes in y; the one of the incomes in x is ancillary. This suggests a natural analogue to x ˜ P2 y, where the opposite holds, namely the binary relation informally defined by:
x =
y and The poorest person in x is richer in y than this person is in x. The two poorest persons in x are collectively richer in y than those (3.5) two persons are in x. etc.; compare to (2.2). Here, it is the ordering of x “that matters”. A natural question is whether this relation has a role either as a complement to x ˜ P2 y, or as an interesting alternative to it. We now show that it has no possible role as a complement to x ˜ P2 y, because it is implied by it. We then show that it is of very little interest as a binary relation over income vectors. Formally, (3.5) is defined as follows and written x ˜ y: P2

Definition 3.5 Let x, y ∈ n . We write x ˜ y if there exists a permutation π of N such P2

that (xπ1 , xπ2 , . . . , xπn ) and (−yπ1 , −yπ2 , . . . , −yπn ) are lexicographically increasing in x, (that is, the vector x is increasing but for equal values of x, the vector y is decreasing), and
k i=1

123

xπi ≤


k i=1

yπi , k = 1, . . . , n − 1,


n i=1

xi =


n i=1

yi .

(3.6)

Second degree Pareto dominance

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˜ (x) for n = 3 Fig. 2 The set S P2

x

By inspecting the definitions, it is easy to see that x ˜ P2 y implies x ˜ y. The folP2

lowing example illustrates that x ˜ y is a rather uninteresting binary relation in itself P2

as the set of income allocations that dominates x according to ˜ contains all kinds of P2

allocations, some of which are equalizations of x and some quite the opposite. Example 3.1 Let x = (1, 2, 3, 4), y = (2.5, 2.5, 2.5, 2.5) and z = (10, 0, 0, 0). We have x ˜ y and x ˜ z, as well as y ˜ z. P2

P2

P2

The failure of x ˜ y to exhibit any distributional consistency clearly arises from P2

the uncompromising priority given to the person who happens to have the smallest ˜

income in x. The set S P2 (x) is represented as the shaded area in Fig. 2. If n = 3 and x is increasingly ordered, then x ˜ y boils down to: x1 ≤ y1 and x3 ≥ y3 , with y2 falling P2

˜

wherever it may under the constraint that
x =
y. Thus in Fig. 2, S P2 (x) appears as the unbounded convex polyhedron that is the intersection of the two half-spaces determined by the two inequalities x1 ≤ y1 , x3 ≥ y3 . 4 Social welfare functions In this section, we are concerned with the order-preserving functions of x ˜ P2 y and related binary relations, interpreted as social welfare functions. [A function W (z) is an order-preserving function for the binary relation  if z  v implies W (z) ≤ W (v)]. m u(z ), where z repreWe confine ourselves to the functions of the form W (z) =
i=1 i sents the vector of incomes of a number m of persons. We first recall two well-known results. By Hardy, Littlewood and Polya’s Theorem (see Marshall and Olkin 1979, B1, p. 108), we have:

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˜ M y is equivalent to: Lemma 4.1 Let x, y ∈ n . Then x ≺ n


u(x i ) ≤

i=1

n


u(y i ) f or all continuous and concave u.

(4.1)

i=1

The following result is also well-known. ˜ 2 y is equivalent to: Lemma 4.2 Let x, y ∈ n . Then x ≺ n


u(x i ) ≤

i=1

n


u(y i ) for all continuous, increasing and concave u.

(4.2)

i=1

Note that, while in Lemma 4.2 one has that
x ≤
y, and that the functions are increasing, neither is the case in Lemma 4.1. See Aboudi and Thon (2003, p. 301) for a short bibliography of Lemma 4.2. The following theorem provides a result for x ˜ P2 y which is in the nature of the above two lemmas. It follows readily from Theorem 3.6 and Lemma 4.2. Theorem 4.1 Let x, y ∈ n . Then x ˜ P2 y is equivalent to:
x =
y, and there exists a permutation π of N such that (xπ1 , xπ2 , . . . , xπn ) and (yπ1 , yπ2 , . . . , yπn ) are lexicographically increasing in y, and s
i=1

u(x πi ) ≤

s


u(y πi ); s = 1, .., n,

(4.3)

i=1

for all continuous, increasing and concave u. The meaning of (4.3), and therefore of x ˜ P2 y, is that the welfare of all the subsocieties of the persons who are the s poorest in y is higher in y than in x, for any s, if welfare is evaluated by the unanimous ranking provided by the welfare functions such as in (4.2) applied to those sub-societies. Note that, as it should be, the set of inequalities in (4.3) is not symmetric, i.e. the binary relation they collectively define does not satisfy the Symmetry Axiom. It is well-known that functions of the type (4.2) are particular cases of increasing ˜ 2 . One Schur concave functions, which are by definition the functions that preserve ≺ could easily generalize the result of this section along the line of Schur concavity but we shall not do so here. 5 Conclusion In our concluding comments, we summarize our main results, and then illustrate them with a couple of simple examples. Thon and Wallace (2004) were concerned with characterizing the set of income allocations that result from the actual application of a sequence of non-excessively altruistic transfers. This paper is concerned with characterizing any redistributive scheme

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(which, of course, is not meant to operate through pair-wise transfers) whose outcome could have been obtained through the application of a sequence of such pair-wise transfers, with the further condition that no transfer in the sequence is from a poorer to a richer person when their incomes are measured in the resulting allocation. Such a sequence of transfers is, in a sense, non-excessively altruistic not only in the small but also in the large. This motivates the definition of the binary relation in Definition 2.3, namely that: • y can be reached from x through a sequence of asymmetric Dalton transfers, none of which is a transfer in which the recipient ends (5.1) up, at the outcome of the sequence, strictly richer than the donor does. When characterizing a redistributive scheme by the fact that it that could have been executed through a set of pair-wise transfers satisfying some conditions, the order in which those putative transfers are performed should not be essential. If one retains the notion of non-excessively altruistic transfers but considers a set of transfers that are such regardless of the order in which they are performed, one is led to consider the binary relation of Definition 2.4, namely: • y can be reached from x through a set of transfers that are all asymmetric Dalton transfers regardless of the order in which they (5.2) are performed. Theorem 2.1 shows that (5.1) and (5.2) define the same binary relation. The same theorem also establishes that the binary relation ˜ P2 of Definition 2.6, namely: • the k persons who are the poorest in y are collectively richer in y than in x, for all k’s,

(5.3)

which is a variation on the concept of majorization, is also equivalent to (5.1), (5.2). This relation has the advantage of being easy to check and to compare to other relations that have currency in the literature on income inequality. The relation of Theorem 3.6, namely • the allocation in y of the incomes of the k persons who are the poorest in y dominates by the second degree the allocation in x of the incomes of (5.4) the same person, for all k’s, is shown to be also equivalent to ˜ P2 . The formulation (5.4) paves the way to a further equivalence in terms of social welfare functions, as in Theorem 4.1. We illustrate with a couple of examples. Suppose we have a beggar (B), two middleclass persons (M1 and M2) and a rich person (R) with $ incomes x = (1,60,100,900), respectively. Suppose some taxation/subsidy scheme produces the allocation y = (76,63,77,845). One of the many ways this re-distribution could have been performed through a sequence of pair-wise transfers is as follows: first R gives $ 55 to B, then M2 gives $20 to B, and finally, M2 gives $3 to M1:

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⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 76 76 56 1 ⎜ 63 ⎟ ⎜ 60 ⎟ ⎜ 60 ⎟ ⎜ 60 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ x =⎜ ⎟=y ⎟→⎜ ⎟→⎜ ⎟→⎜ 77 80 100 100 ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ 845 845 845 900 ⎛

(5.5)

Now, this is a sequence of asymmetric Dalton transfers and no transfer is from someone to someone else who ends up strictly richer in y. One can also easily check that if the three transfers were performed in any other order, then each of them would still be an asymmetric Dalton transfer. That x ˜ P2 y holds is also easy to check. This illustrates Theorem 2.1. Consider now a different situation. Let x = (1,60,100,900) and y = (81,60,80,840). There exist sequences of asymmetric Dalton transfers that can produce y from x. One such sequence is: first M2 gives 20 to B, and then R gives 60 to B: ⎞ ⎛ ⎞ ⎛ ⎞ 81 21 1 ⎜ 60 ⎟ ⎜ 60 ⎟ ⎜ 60 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ x =⎜ ⎟ = y. ⎟→⎜ ⎟→⎜ ⎝ 80 ⎠ ⎝ 80 ⎠ ⎝ 100 ⎠ 840 900 900 ⎛

(5.6)

This sequence contains a transfer from M2 to B, who ends up strictly richer than M2. One can also check that if this set of two transfers is performed in the opposite sequence, that is if R makes his donation first, then the second transfer is not an asymmetric Dalton transfer. By Theorem 2.1, we know that this feature is common to all possible sets of asymmetric Dalton transfers that could produce y from x. As a result, there is no way to argue that M2 could have failed to contribute to the enrichment of B, who ends up strictly richer than him. By Theorem 2.1, checking that x ˜ P2 y does not hold would easily establish this fact. The results of this paper are easy to extend to a situation where
x ≤
y, rather than
x =
y, holds. This means dealing with a binary relation whose properties are ˜ 2 are related to the related to the ones of ˜ P2 in the same way as the properties of ≺ M ˜ . ones of ≺ Appendix 1 The proof of Theorem 2.1 requires the following definition. Definition A.1 Let x, y ∈ n . Let Rx = {i ∈ N |xi > yi } and R y = {i ∈ N |yi > xi }. The set Rx denotes all the persons who are strictly richer in x than in y and the set R y denotes all the persons who are strictly richer in y than in x. Proof of Theorem 2.1. (x ◦ y ⇒ x  ˜ y) Since x ◦ y, any transfer from j to k can be the last one to be performed and therefore, after this transfer, the incomes of persons j and k will be y j and yk respectively. Since this is an asymmetric Dalton transfer, we have yk ≤ y j , thus this is an R-P-in-y transfer, so x  ˜ y.

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(x  ˜ y ⇒ x ˜ P2 y) Assume that x  ˜ y. Re-arrange x and y so that they are lexicographically increasing in y. Suppose that x ˜ P2 y does not hold and let r be the first r x > r y . It is clear that x > y . Let now s be the largest occurrence where
i=1
i=1 i i r r index where ys = yr . As xr > yr , the lexicographic ordering implies that xi > yi s x > s y . Note that since n x = n y , for r ≤ i ≤ s and, therefore,
i=1
i=1
i=1 i
i=1 i i i s s s ≤ n − 1. Since
i=1 xi >
i=1 yi , there must be at least one transfer from j to k where j ≤ s and k > s. Thus, by the choice of s we have yk > ys ≥ y j , so this is not an R-P-in-y transfer, contradicting x  ˜ y, therefore x ˜ P2 y holds. (x ˜ P2 y ⇒ x ◦ y) Theorem 4.1 in Aboudi and Thon (2006, p. 210) shows that ˜ M y. The result is obtained by constructing a sequence of R-P-in-y transx ˜ P2 y ⇒ x ≺ fers with the additional property that for each transfer from j to k, j ∈ Rx and k ∈ R y . We will show that this set of transfers can be performed in any order and each one is an asymmetric Dalton transfer. Since all transfers are from Rx to R y , all persons j ∈ Rx donate but never receive income, and all persons k ∈ R y receive but never donate income. As a consequence, if x˜ is a vector obtained from x after the m-th application of a transfer in the above set in any order, x˜ j ≥ y j for all j ∈ Rx and x˜k ≤ yk for all k ∈ R y . Since all transfers are R-P-in-y transfers, whenever there is a transfer from j to k, yk ≤ y j . Assume that the last transfer applied to obtain x˜ was from j to k. As j ∈ Rx and k ∈ R y , the last three inequalities can be combined to obtain x˜ j ≥ y j ≥ yk ≥ x˜k , showing that person k is not strictly richer than person j, and thus the transfer is an asymmetric Dalton transfer. The generality of x˜ implies that all transfers can be performed in any order and all be asymmetric Dalton transfers.   k , . . . , xn ) is increasingly ordered, then i=1 xi ≤ kα Lemma A.1 If the vector (x1 , x2 n xi /n. for k = 1, 2, . . . , n, where α = i=1 Proof By definition of α it is clear that the  inequality holds at equality for k = n. k xi > kα. Since x is increasingly Suppose that for some k, 1 ≤ k ≤ n − 1, i=1 ordered, the strict inequality  implies that xk > α and as a result, we have xi > α for n i = k + 1, .n. . , n. Therefore, i=k+1 xi > (n − k)α. Combining the two partial sums we have i=1 xi > nα contradicting the fact that the latter holds at equality.   n n Proof of Theorem 3.1. It is clear that i=1 xi = i=1 xi= . Let π be a permutation = is a vector of constants, x = is increasingly where x is increasingly ordered. Since x k k xπi ≤ i=1 xπ=i for k = 1, 2, . . . , n−1, ordered as well. By Lemma A.1 we have i=1 P2 =   implying that x ˜ x . P2

Proof of Theorem 3.2. That S ˜ (x) is not necessarily convex is demonstrated by the following example: let x = (1, 4, 13), y = (2.5, 2.5, 13) and y = (7, 4, 7). We have both x ˜ P2 y and x ˜ P2 y , but with z = 0.5y + 0.5y = (4.75, 3.25, 10), x ˜ P2 z does not hold. As regards starshapeness, we first prove the result with respect to x = . Let α = n ˜ P2 (x). Without loss of generality we can assume that y is i=1 x i /n and let y ∈ S  n k n increasingly ordered. Since i=1 xi = i=1 yi , by Lemma A.1, i=1 yi ≤ kα for k = 1, 2, . . . , n. k k xi ≤ i=1 yi ≤ kα = Let z = λx = + (1 − λ)y where 0 ≤ λ ≤ 1. We have i=1 k P2 =  ˜ (x), and i=1 x i for k = 1, 2, . . . , n, where the inequalities hold because y ∈ S

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y is increasingly ordered. It is immediate that k


xi = λ

i=1

k


k k k


xi + (1 − λ) xi ≤ λ xi= + (1 − λ) yi

i=1

=

k


i=1

i=1

(λxi= + (1 − λ)yi ) =

i=1

k


i=1

for k = 1, 2, . . . , n.

zi

i=1

Since y is increasingly ordered and x = is a vector of constants, it is clear that z is increasingly ordered. Combining this observation with the above inequality we have P2 x ˜ P2 z implying that z ∈ S ˜ (x). Now consider the result with respect to x. Let z = λx + (1 − λ)y where 0 ≤ λ ≤ 1. For M ⊆ N , it is clear that
i∈M

xi ≤




yi

if and only if

i∈M




xi ≤

i∈M




zi .

(A.1)

i∈M

P2

Let y ∈ S ˜ (x). In order to simplify the notation, assume, without lossof generk xi ≤ ality that x and y are lexicographically increasing in y. Therefore we have i=1 k k k y for k = 1, 2, . . . , n, implying, by (A.1), that x ≤ z for k = i=1 i i=1 i i=1 i P2 1, 2, . . . , n. Note that one cannot immediately conclude that x ˜ z, as the vector z is not necessarily increasingly ordered. In order to show that x ˜ P2 z, the vectors x, y and z will be permuted iteratively, where in each iteration the following properties will hold: There exists 0 ≤ t ≤ n where z 1 = z (1) , z 2 = z (2) , . . . , z t = z (t) , yt+1 ≤ yt+2 ≤ · · · ≤ yn and k
i=1

xi ≤

k


zi

for k = 1, 2, . . . , n.

(A.2) (A.3) (A.4)

i=1

That is, the first t elements of vector z are the smallest ones and are increasingly ordered, and the remaining n − t elements of y are increasingly ordered. Since x and y are lexicographically increasing in y, (A.4) holds initially, and (A.2) and (A.3) hold for t = 0. It remains to show that the vectors x, y and z can be re-arranged so that (A.2)–(A.4) hold for t = n. This will be done by an inductive type argument. Let s be the largest value of t where (A.2) and (A.3) are satisfied. Note that if (A.2) and (A.3) hold when t = n − 1, they will hold for t = n, therefore, s = n − 1. So either t = s = n, which combined with (A.4) implies that x ˜ P2 z or we have that 0 ≤ s ≤ n − 2. Let z (s+1) = zr . By choice of s it is clear that s + 1 < r ≤ n. By property (A.2) we have zr = z (s+1) ≤ z q for s + 1 ≤ q ≤ r . By definition of z, this is equivalent to λxr + (1 − λ)yr ≤ λxq + (1 − λ)yq

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for s + 1 ≤ q ≤ r,

Second degree Pareto dominance

489

or, λ(xr − xq ) ≤ (1 − λ)(yq − yr ) ≤ 0, for s + 1 ≤ q ≤ r , where the last inequality follows from property (A.3). Thus we have xr ≤ xq for s + 1 ≤ q ≤ r . Note that for q−1 q−1 q q−1 s + 1 ≤ q ≤ r , one has xr + i=1 xi ≤ xq + i=1 xi ≤ i=1 yi ≤ yr + i=1 yi . The first inequality is due to the fact that xr ≤ xq for s + 1 ≤ q ≤ r , the second stems from (A.4) and (A.1), and the third holds since (A.3) implies that yq ≤ yr , for s + 1 ≤ q ≤ r. Thus, by (A.1) we can conclude that xr +

q−1


xi ≤ zr +

i=1

q−1


zi

for s + 1 ≤ q ≤ r.

(A.5)

i=1

Now we will re-arrange the vectors x, y and z to show that (A.2)–(A.4) hold for t = s + 1. This is done by moving up the r -th component to position s + 1 and by moving components s + 1 to r − 1 downward by one position. This permutation guarantees that (A.2) and (A.3) hold for t = s + 1. The inequalities (A.5) guarantee that (A.4) holds as well. Formally, we have x˜s+1 = xr , y˜s+1 = yr

and

z˜ s+1 = zr ,

x˜q+1 = xq , y˜q+1 = yq and z˜ q+1 = z q for q = s + 1, . . . , r − 1, and x˜i = xi , y˜i = yi and z˜ i = z i for i ∈ / {s + 1, s + 2, . . . , r }. By choice of r and by the fact that r is moved to position s + 1, (A.2) and (A.3) hold for x, ˜ y˜ and + 1. By definition ˜ y˜  and z˜ , when k ∈ / {s + 1, s + 2, . . . , r } k of x, kz˜ for t = s k k x˜i = x and z = z ˜ , and by (A.4) we can conwe have i=1 i i=1 i i=1 i i=1 k k k x˜i = clude that i=1 x˜i ≤ i=1 z˜ i . For k ∈ {s + 1, s + 2, . . . , r }, we have i=1 k−1 k−1 k xr + i=1 xi ≤ zr + i=1 z i = i=1 z˜ i , where the inequality stems from (A.5), so (A.4) holds for x, ˜ y˜ and z˜ , so the procedure can be applied again and after each iteration the index t will have a larger value, eventually reaching the value n. As noted   above this implies x ˜ P2 z. ˜ A is reflexive is obvious. The fact that it is Proof of Theorem 3.3. The fact that ≺ transitive follows from its definition. It remains to prove ˜ A x hold. By (3.2), we have x ≺ ˜ My ˜ A y and y ≺ antisymmetry. Assume that both x ≺ M ˜ x, therefore by Alberti and Uhlmann (1982, Lemma 1–10, p. 16) the vector and y ≺ y is a permutation of x. We will show, by contradiction, that x = y. Assume that x = y and construct the vectors x, ¯ y¯ ∈ m by deleting all the elements where xi = yi . It ˜ A y¯ . Let y¯k = min{ y¯1 , y¯2 , . . . , y¯m }. Since is clear that y¯ is a permutation of x¯ and x¯ ≺ y¯ is a permutation of x, ¯ it is clear that x¯i ≥ y¯k for i = 1, 2, . . . , m. By construction of x, ¯ y¯ , x¯k = y¯k and thus x¯k > y¯k . Let x r denote the vector that is obtained after r asymmetric Dalton transfers have been performed. By Definition 3.1 it is clear that the minimum value of x r is at least y¯k for all r . Since x¯k > y¯k , person k must transfer some income and therefore must be involved in at least one transfer. Assume that δ > 0 is transferred from person k to person j during transfer number s, where s is the last + δ > y¯k , transfer that involves person k. After this transfer, xks = y¯k and x sj = x s−1 j so the recipient’s income will exceed the donor’s, violating the definition of an

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˜ A is asymmetric Dalton transfer, a contradiction. Therefore, x = y and hence ≺ antisymmetric.   Proof of Theorem 3.4. The fact that ˜ P2 is reflexive is obvious. That it is not transitive is established in the text following the statement of Theorem 3.4. It remains to prove antisymmetry. Assume that both x ˜ P2 y and y ˜ P2 x. Thus, by (3.2), x ≺ M y and y ≺ M x and therefore x is a permutation of y (Alberti and Uhlmann 1982, Lemma 1–10, p. 16). Without loss of generality assume that x and y are lexicographically increasingly in y. Since x ˜ P2 y, we have x1 ≤ y1 = y(1) . But since x(1) = y(1) = y1 and x1 ≥ x(1) we have x1 ≥ y1 , and thus, combined with the first inequality, x1 = y1 . The second inequality generated by x ˜ P2 y states that x1 + x2 ≤ y1 + y2 = y(1) + y(2) . By deleting x1 and y1 and by observing that x2 ≥ x(2) , one can conclude that x2 = y2 by employing a similar argument. By repeating this procedure n − 1 times we conclude that x = y.   Proof of Theorem 3.5. Denote the transitive closure ˜ P2 by C . Let x, y ∈ n where x C y. By Definition 3.3, there exist x 1 , x 2 , . . . , x m ∈ n where x i−1 ˜ P2 x i for i = 1, 2, . . . , m, where x 0 = x and x m = y. By (3.2) we have that v ˜ P2 w implies ˜ A x i for i = 1, 2, . . . , m. Since the binary relation ≺ ˜ A is transitive, ˜ A w, thus x i−1 ≺ v≺ A i A C ˜ y. ˜ x for i = 1, 2, . . . , m. Thus x  y implies x ≺ x≺ ˜ A y for some y ∈ n , then x C y. To complete the proof, we must show that if x ≺ ˜ A y then y can be reached from x by a sequence of asymmetric By definition, if x ≺ Dalton transfers. We denote by x i the vector of incomes after i transfers have been performed. If the total number of transfers is m then it is clear that y = x m . Note that in order to show that x C y it suffices to show that x i−1 ˜ P2 x i for i = 1, 2, . . . , m. But since x i is obtained from x i−1 by an application of a single asymmetric Dalton transfer, it suffices to show that if z is obtained from w after a single transfer, then w ˜ P2 z. Assume that z is obtained from w by a single asymmetric Dalton transfer where δ units are transferred from person j to person k. By definition we have wk < w j and z k = wk + δ ≤ w j − δ = z j . After choosing a permutation π that arranges the vector z increasingly, we have that πr = k and πs = j with r < s. The latter always holds when wk + δ < w j − δ, but, in the event that wk + δ = w j − δ, one can choose a permutation π so that r < s. Since a single transfer affects only persons k and j, we have: wπi = z πi

for i ∈ N \{r, s},

z πr = wπr + δ

and

z πs = wπs − δ.

These equalities readily imply that k
i=1

i=1

k


k


i=1

implying that w ˜ P2 z.

123

wπi ≤

k


wπi =

z πi

for k = 1, 2, . . . , s − 1 and

z πi

for k = s, . . . , n,

i=1

 

Second degree Pareto dominance

491

˜ 2 y [s] for s = 1, 2, . . . , n. Then, we have Proof of Theorem 3.6. (⇐) Suppose x [s] ≺ [s] [s] s s s s that
i=1 xπi =
i=1 x(i) ≤
i=1 y(i) =
i=1 yπi for s = 1, 2, . . . , n, and since
x =
y, we have x ˜ P2 y. [s] [s] [s] , x(2) , . . . x(k) ) (⇒) Suppose x ˜ P2 y. Let 1 ≤ s ≤ n and 1 ≤ k ≤ s. Since (x(1) [s] are the smallest k values of x [s] = (xπ1 , xπ2 , . . . , xπs ), we have
1k x(i) ≤
1k xπi ≤

[s]
1k yπi =
1k y(i) . The second inequality is due to the fact that x ˜ P2 y and the equality

˜ 2 y [s] for s = 1, 2, . . . , n. The fact stems from the definition of π . Therefore, x [s] ≺ P2   that
x =
y follows from x ˜ y. Proof of Theorem 4.1. If
x =
y and (4.3) hold, then taking u(·) in this expression to be linear increasing gives (2.1), and thus one has x ˜ P2 y. Conversely, if x ˜ P2 y holds, ˜ 2 y [s] holds for s = 1, . . . , n and that
x =
y; one has, by Theorem 3.6, that x [s] ≺ see (3.4). Then (4.3) follows by repeatedly applying Lemma 4.2.   Appendix 2 P2

We construct S ˜ (x) for n = 3, under the assumption that x1 < x2 < x3 , as in Fig. 1. Let x = = (µx , µx , µx ) with µx = (x1 + x2 +x 3 )/3. The simplex can be partitioned into 13 subsets: S1: y1 < y2 < y3 . S2: y1 = y2 < y3 . S3: y2 < y1 < y3 . S4: y2 < y1 = y3 . S5: y2 < y3 < y1 . S6: y2 = y3 < y1 .

S8: y3 < y1 = y2 . S9: y3 < y1 < y2 . S10: y1 = y3 < y2 . S11: y1 < y3 < y2 . S12: y1 < y2 = y3 . S13: y1 = y2 = y3 .

S7: y3 < y2 < y1 . P2

One cannot have y3 < min(y1 , y2 ) if y ∈ S ˜ (x), as the latter would imply x3 ≤ y3 which cannot hold (“the smallest yi cannot be equal to or larger than the largest xi if
x =
y and the xi ’s are distinct”). Likewise, one cannot have y1 > max(y2 , y3 ), P2 if y ∈ S ˜ (x), as the latter would imply y1 ≤ x1 which cannot hold (“the largest yi cannot be equal to or smaller than the smallest xi if
x =
y and the xi ’s are P2 distinct”). Thus if y ∈ S ˜ (x), one must have: y ∈ (S1 ∪ S2 ∪ S3 ∪ S4 ∪ S10 ∪ S11 ∪ S12 ∪ S13). Now, as regards x, there are three possibilities: x2 = µx , x2 < µx and x2 > µx . Let M S ∗ (x) be the subset of S ≺˜ (x) where y1 ≤ y2 ≤ y3 . (A) If x2 = µx , then it is not possible to have y2 < y1 = y3 , as it would imply x2 = (x1 +x 2 + x3 )/3 ≤ y2 , which cannot hold (“the yi which is strictly smaller than the other two cannot be equal to or larger than the average of all three”). It is also not

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Fig. 3 The set S ˜ (x) for n = 3 and x2 = µx

x

P2

Fig. 4 The set S ˜ (x) for n = 3 and x2 > µx

x

possible to have y2 < y1 < y3 for the same reason. It is not possible either to have y1 = y3 < y2 , as this would require y2 ≤ x2 = (x1 +x 2 + x3 )/3 which cannot hold (“the yi which is strictly larger than the other two cannot be equal to or smaller than the average of all three”). The configuration y1 < y3 < y2 is not possible for the same reason. Thus we must have y1 < y2 < y3 , or y1 = y2 < y3 , or y1 < y2 = y3 , or P2 y1 = y2 = y3 , that is: y ∈ (S1 ∪ S2 ∪ S12 ∪ S13). Thus S ˜ (x) ≡ S ∗ (x). See Fig. 3. (B) If x2 < µx then one can have neither y1 = y3 < y2 nor y1 < y3 < y2 by the same argument as under (A). Thus y ∈ (S1 ∪ S2 ∪ S3 ∪ S4 ∪ S12 ∪ S13). Now, P2 if y ∈ (S3 ∪ S4), then we need to have x 2 ≤ y2 . Thus S ˜ (x) is as represented on Fig. 1. It is the non-convex union of two convex sets: S ∗ (x) and the intersection of (S3 ∪ S4) with the half-space defined by x2 ≤ y2 .

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(C) If x2 > µx , then one can have neither y2 < y1 = y3 nor y2 < y1 < y3 by the same argument as under (A). Thus y ∈ (S1 ∪ S2 ∪ S10 ∪ S11 ∪ S13). Now, if P2 y ∈ (S10 ∪ S11), then we need to have x2 ≥ y2 . Thus S ˜ is the non-convex union of two convex sets: S ∗ (x) and the intersection of (S10 ∪ S11) with the half-space P2 defined by x2 ≥ y2 . The set S ˜ for this case is represented on Fig 4. P2 There are thus three possible configurations for S ˜ (x) if n = 3. References Aboudi R, Thon D (2003) Transfer principles and relative inequality aversion; a majorization approach. Math Soc Sci 45:299–311 Aboudi R, Thon D (2006) Refinements of Muirhead’s Lemma and income inequality. Math Soc Sci 51: 201–216 Alberti P, Uhlmann A (1982) Stochasticity and partial order. D. Reidel, Dordrecht Dalton H (1920) The measurement of the inequality of income. Econ J 30:348–361 Fishburn P (1973) The theory of social choice. Princeton University Press, Princeton Kolm S (1966) The optimal production of social justice. In: Guitton H, Margolis J (eds) International economic association conference on public economics, Biarritz (proceedings). Also in Guitton H, Margolis J (eds) Public economics. Macmillan, 1969, London Lorenz M (1905) Methods of measuring the concentration of wealth. J Ame Statist Assoc 9:209–219 Marshall A, Olkin I (1979) Inequalities: Majorization theory and its applications. Academic, New York Shorrocks A (1983) Ranking income distributions. Economica 50:3–18 Thon D, Wallace S (2004) Dalton transfers, inequality and altruism. Soc Choice Welf 22:447–456

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