Deviation from proportionality and Lorenz-domination for claims problems

The Lorenz order is commonly used to compare rules for claims problems. In this paper, we incorporate the average of awards rule, the mean value of the set of awards vectors for a claims problem, to the ranking of the standard rules by proving some properties that are satisfied by this rule. We define a pair of coefficients, inspired by the Gini index, aimed at measuring, for any given claims problem, the discrepancy between the awards assigned by a rule and the proportional division. We generalize the proportionality deviation indices by introducing coefficients that measure the deviation between the awards selected by any two division rules. We show how these deviation indices are related to the Lorenz order.


Introduction
A firm going bankrupt, the division of property among heirs, a government taxing incomes to implement a public project, a rationing problem, the distribution of insufficient supplies such as food or vaccines, or the global carbon budget are just some examples of conflicting claims problems. In all of them, a scarce resource has to be divided or distributed among a group of claimants. The mathematical model used to formally study these problems may look, at first, quite simple: a non-negative real number that represents the endowment, and a finite vector of claims whose coordi-B Estela Sánchez-Rodríguez esanchez@uvigo.es 1 Departamento de Matemáticas, Universidade de Vigo, Vigo 36310, Spain nates add up to more than the total amount available. But, in fact, the model is very rich. The book by Thomson (2019) presents a comprehensive review of the fascinating literature on claims problems.
Aristotle is credited to propose sharing the endowment proportional to claims. O'Neill (1982) describes historical instances of different division procedures found in the Talmud and in several medieval texts. In general, a division rule is a way of associating with each claims problem a division among the claimants of the amount available. Therefore, for each claims problem a rule must select an allocation satisfying three basic requirements: no claimant should be asked to pay, no claimant should be awarded more than his claim, and the sum of the awards should be equal to the endowment. The set of all the allocations that meet these basic properties is the set of awards vectors for the claims problem. The inventory of division rules is now large. We consider in this paper nine of the central rules: the proportional, the constrained equal awards, the constrained equal losses, the constrained egalitarian, the Talmud, Piniles', the minimal overlap, the adjusted proportional, and the random arrival rules. In addition, we study the average of awards rule, introduced by Mirás Calvo et al. (2020), that selects for each claims problem the expected value of the (continuous) uniform distribution over its set of awards vectors.
The axiomatic approach has dominated the study of rules. Properties of rules are formulated that one may want to impose because they have some appeal for a particular situation, or because they cover a theoretical or even an ethical aspect. Then, rules are examined, classified, and characterized according to the properties that they satisfy (or violate). Another important issue when evaluating a rule is how differently it treats larger claimants as compared with smaller claimants. The economist Max Otto Lorenz proposed in 1905 a simple method, now called the Lorenz curve, for visualizing distributions of income or wealth (Lorenz 1905). In the context of claims adjudication the closely related Lorenz order is used as a general criterion to rank rules. Basically, an awards vector Lorenz-dominates another if the cumulative sums of ordered awards are bigger for the first vector. The Lorenz order is a partial order. Using different methods, several authors, among others Schummer and Thomson (1997), Chun et al. (2001), Bosmans and Lauwers (2011), and Thomson (2012), study whether or not the division rules are Lorenz-comparable. As a corollary, we have a complete picture of the ranking of the nine central rules.
Our first goal is to rank the average of awards rule. We rely on the characterizations of Piniles' rule and the minimal overlap rule given by Schummer and Thomson (1997) and Bosmans and Lauwers (2011) respectively. We need to show that the average of awards rule satisfies, besides the basic properties already proven by Mirás Calvo et al. (2020), null claims consistency, order preservation under endowment variations, and order preservation under claims variations. We conclude that the average of awards rule Lorenz-dominates the minimal overlap rule and is Lorenz-dominated by Piniles' rule. Naturally, since the average of awards rule is self-dual it is not Lorenz-comparable with the other self-dual rules: the proportional, the adjusted proportional, the Talmud, and the random arrival rules.
Whether or not the recommendations made by two rules for a claims problem are Lorenz-comparable, the corresponding awards vectors can be similar or they can differ greatly. So, our second objective is to define some coefficients aimed at measuring the discrepancy between the awards vectors selected by two rules. Our basic reference is the Gini index, introduced in 1912 by the statistician and sociologist Corrado Gini as a coefficient intended to measure the degree of income inequality within a population. Ceriani and Verme (2012) provide a historical account of Gini's original formulation. Mathematically, the Gini coefficient is based on the Lorenz curve that represents in the horizontal axis the proportion of the population, from lowest to highest income, and in the vertical axis the cumulative percentage of income or wealth owned. A perfectly equal distribution of wealth would have a Lorenz curve equal to the line y = x. The Gini coefficient measures how far the actual Lorenz curve for a population's income is from the line of equality.
Given a claims problem, if one plots the cumulative percentage of awards with respect to the proportion of claimants, from lowest to highest claims, the line of equality represents the egalitarian division of the endowment. But, in general, the egalitarian division does not select an awards vector for the problem, so it is not a rule. Therefore, instead of the proportion of population, we represent in the horizontal axis the cumulative percentage of claims, ordered from small to large. Then, since the proportional rule shares the endowment in the same proportion as claims, the line y = x is now the line of proportionality. So, we plot the cumulative percentage of the endowment that is assigned by a rule to the cumulative percentage of claims. The monotonically increasing continuous piecewise linear function thus obtained, whose graph lies in the unit square, is called the cumulative claims-awards curve. Naturally, the line of proportionality corresponds to the cumulative claims-awards curve of the proportional rule. We show that the claims-awards curve fully captures the Lorenz ranking of rules. Then, adapting the definition of the Gini index, we introduce a pair of coefficients, the proportionality deviation index, and the signed proportionality deviation index, that measure the deviation of the claims-awards curve from the line of proportionality as the ratio of the area, and the net signed area respectively, that lies between that line and the curve over the total area under the line of proportionality. In this framework, the proportional rule is the rule of reference: given the initial inequality of the vector of claims, the proportionality deviation indices measure the deviation of the distribution of the endowment with respect to this initial inequality.
Certainly, the proportional rule stands out as the best-known rule, and questionnaire studies on claims problems, such as Bosmans and Schokkaert (2009), show that it performs very well in describing the choices of the respondents. Even Thomson (2019) states that "proportionality is often taken as the definition of fairness for claims problems", only to successfully challenge this view. Lately, several authors have analyzed the preservation in gains and losses (the differences between claims and awards) of the inequality in claims. Order preservation is a basic property, met by our ten rules, that requires that a rule should respect the ordering of claims and that the losses should also be ordered as claims are. Now, fix an endowment and take two Lorenz-comparable awards vectors whose coordinates add up to the same amount. Hougaard and Østerdal (2005) propose the requirement that the awards and losses vectors selected by a rule for those two problems are also Lorenz comparable in the same direction. Velez (2010, 2011) show that, when there are more than three claimants, the only rule that satisfies order preservation and claims-inequality preservation in gains and losses is the proportional rule. These results reinforce the role of the proportional rule as the rule of reference to define the deviation indices. The information provided by the pair of proportionality deviation indices of a rule for a given claims problem not only indicates if the rule and the proportional division are Lorenz-comparable but also gives a clear and simple numerical value that quantifies how far from proportionality is the awards vector selected by the rule. We also show that, for a fixed vector of claims, the graph plotting the corresponding index for a given rule as a function of the endowment, the index path, is a good visual instrument that conveys much information about the rule itself.
We choose, for the reasons explained above, the proportional rule as the base rule to define the pair of deviation coefficients. But, each division rule entails different principles of fairness, equity, or justice. In order to make the best decision when solving a particular claims problem, it could be interesting to measure the degree of discrepancy of the awards vectors selected by two arbitrary rules. Fix a rule as the base for comparison. We define the curve representing the vector of cumulative percentages of the awards selected by any given rule against the vector of cumulative percentages of the awards recommended by our base rule. Now, the identity line represents the distribution of resources given by the rule of reference. Then, we introduce the deviation index (or signed deviation index) of a rule with respect to the rule of reference, by measuring, for each claims problem, the deviation (respectively, signed deviation) between the cumulative proportions of the initial endowment assigned by both rules. Therefore, the corresponding deviation indices quantify how far any rule moves away from the reference rule. Of particular interest are the indices with respect to the constrained equal awards and the constrained equal losses rules, since they are Lorenz-maximal and Lorenz-minimal among the order preserving rules, and the indices with respect to the average of awards rule, because it is the mean value of all the awards vectors. Bosmans and Lauwers (2011) and Thomson (2012) explicitly emphasize that the fact that a rule Lorenz-dominates another rule should not be interpreted as a sign that the first rule is superior or inferior to the other. Obviously, the same applies to the claims-awards curves and the deviation indices. Given two rules, the relative position of its curves or the value of their indices just reveal how they are related, how they treat large claims in relation to small claims or how they depart from the proportional division or from any other rule of reference. Of course, it is up to the decision maker to use this information depending on the specific real-world context of the claims problem.
In Sect. 2 we introduce the basic definitions, notations, rules and properties and recall the Lorenz-ranking of the basic rules. We compare in Sect. 3 the average of awards and the other rules. We introduce, in Sect. 4, the cumulative claims-awards curve, the proportionality deviation indices, and the index path, three alternative tools to compare rules with the proportional division. Section 5 generalizes the indices to compare any two given rules.

Preliminaries
Let N be the set of all finite non-empty subsets of the natural numbers N. Given N ∈ N , x ∈ R N , and S ∈ 2 N let |N | be the number of elements of N and x(S) = i∈S x i . Given x, y ∈ R N , the notation x ≤ y means that A claims problem with set of claimants N ∈ N is a pair (E, d) where E ≥ 0 is the endowment to be divided and d ∈ R N is the vector of claims satisfying d i ≥ 0 for all i ∈ N and d(N ) ≥ E. We denote the class of claims problems with set of players N by C N .
For Let R n ≤ be the set of nonnegative n-dimensional vectors x = (x 1 , . . . , x n ) with coordinates ordered from small to large, i.e., 0 ≤ x 1 ≤ . . . ≤ x n . For simplicity, given (E, d) ∈ C N with |N | = n, we will assume throughout the paper that N = {1, . . . , n} and that d ∈ R n ≤ . As a consequence of such an arrangement of the claims we have Fig. 1, either d n ≤ D −n or D −n ≤ d n , but in both cases 1 2 d(N ) is the middle point of the line segment with endpoints d n and D −n . In fact, 1 Neill (1982) associates to each claims problem (E, d) ∈ C N a coalitional game with set of players N and characteristic function v(S) = max 0, E − d(N \S) , S ∈ 2 N . Thomson (2019) shows that the set of awards vectors for a claims problem coincides with the core of the associated coalitional game, that is, X (E, d) is the set of allocations satisfying the balance requirement that are bounded from below by the minimal rights and are bounded from above by the truncated claims: Then, X (E, d) is a nonempty convex polytope that has, at most, dimension n − 1.
A rule is a function R : C N → R N assigning to each claims problem (E, d) ∈ C N an awards vector R(E, d) ∈ X (E, d). The following rules have been discussed in the literature and will be used throughout the paper.
• Random arrival rule (RA): For each (E, d) ∈ C N and each i ∈ N , Recently The core-center solution was introduced by González-Díaz and Sánchez-Rodríguez (2007) for the class of balanced games as the centroid of the core. Since the set of awards vectors for a claims problem coincides with the core of the associated coalitional game, the average of awards rule corresponds to core-center solution.
We focus now on properties of division rules. We say that a rule R satisfies: A rule satisfies order preservation if it satisfies both order preservation in awards and in losses. Observe that self-duality implies the midpoint property. The weaker version of continuity obtained by considering small changes only in the endowment is called endowment continuity.
With each rule R we can associate a unique dual rule R * , defined by Of the rules listed above, PRO, APRO, T, RA, and AA are self-dual. The CEA and CEL rules are dual. Two properties are dual if, whenever a rule satisfies one of them, its dual satisfies the other. A property is self-dual if it coincides with its dual. The following are pairs of dual properties: order preservations in awards and order preservation in losses; and minimal rights first and claims truncation invariance. The problems (E, d) ∈ C N and (d(N ) − E, d) ∈ C N are dual claims problems. Table 1, adapted from Thomson (2019) andMirás Calvo et al. (2020), summarizes which of the above properties are satisfied by the basic rules. A check mark, , in a cell means that the property in the row is satisfied by the rule indexing the column. A minus sign, −, means the opposite.
One of the most commonly used criteria to rank rules is the Lorenz order. Let x, y ∈ R n ≤ . We say that x Lorenz-dominates y, and write x y, if for each k = 1, . . . , n − 1, The Lorenz order is a partial order in R n ≤ , so it is a binary relation that is reflexive, antisymmetric, and transitive. If x Lorenz-dominates y and x = y, then at least one of the n − 1 inequalities is strict. We have assumed that given a claims problem (E, d) ∈ C N the vector of claims d ∈ R N has its coordinates ordered from small to large, that is, d ∈ R n ≤ . Moreover, the ten rules satisfy order preservation in awards. So if R is any of these rules then R(E, d) ∈ R n ≤ . Therefore, we can use the Lorenz criterion to check whether a rule is more favorable to smaller claimants relative to larger claimants than other. Let R and R be two rules that satisfy order preservation in awards. We say that R Lorenz- Several authors contributed to the ranking of rules. To summarize these results, we borrow from Bosmans and Lauwers (2011) and Thomson (2019) a simple diagram, Fig. 2, that illustrates the ranking of rules using the Lorenz order. An arrow (or a sequence of arrows) from a rule R to a rule R indicates that R Lorenz-dominates R , and the absence of an arrow (or of a sequence of arrows) indicates that there is no relationship. We have added the average of awards rule to the picture, so, in the next section we justify its place in the diagram of Fig. 2.

Ranking the average of awards rule
We have defined the average of awards rule in geometrical terms, as the centroid of the set of awards vectors for a claims problem. Naturally, an alternative and simple way of describing this rule is to assume that all the awards vectors are equally likely and therefore choosing their "average". The average of awards rule assigns to each (E, d) ∈ C N the value AA(E, d) given by the expected value of the (continuous) uniform distribution over the set of awards vectors X (E, d). Besides its intuitive definition, the average of awards rule satisfies a good number of properties, see Table 1. Therefore, if only as a "central" point of reference inside the set of awards vectors, it is worthy to compare it to the basic rules.
Let us see that the ranking of the average of awards rule is, in fact, the one shown in Fig. 2. The absence of arrows connecting the average of awards rule with the Talmud, the random arrival, the adjusted proportional, and the proportional rules is a consequence of the fact that any two self-dual rules are incomparable. Then, we just have to prove that the sequence PIN → AA → MO holds.
Let us introduce three additional properties of rules. Null claims consistency implies that to compute the recommendation made by a rule we can remove the agents whose claims are 0 and apply the rule to the remaining claims problem. Order preservation under endowment variations implies that, given any two agents, if the endowment increases, the smaller claimant should receive a share of the increment that is at most as large as the share received by the larger claimant. Order preservation under claims variations says that if an agent claim increases, given any two claimants whose claim remains the same, the change in the award to the smaller one should be at most as large as the change in the award to the larger one. Formally, we say that a rule R satisfies: The following characterizations of Piniles' and the minimal overlap rules as Lorenzminimal and Lorenz-maximal within some classes of rules were established by Schummer and Thomson (1997) and Bosmans and Lauwers (2011) respectively.
1. Let S 1 be the set of rules that satisfy order preservation in awards, endowment monotonicity, the midpoint property, and order preservation under endowment variations. Piniles' rule is the only rule in S 1 that Lorenz-dominates each rule in S 1 . 2. Let S 2 be the set of rules that satisfy 1 |N | -truncated-claims lower bounds on awards, order preservation, null-claims consistency, and order preservation under claims variations. The minimal overlap rule is the only rule in S 2 that is Lorenz-dominated by each rule in S 2 . Now, according to Table 1, the average of awards rule satisfies the midpoint property, order preservation, endowment monotonicity, and 1 |N | -truncated-claims lower bounds on awards. Let (E, d) ∈ C N and N ⊂ N such that d(N \N ) = 0. It is easy to see that X (E, d) = 0 N \N × X (E, d N ). Then AA j (E, d) = 0 for all j ∈ N \N and AA N (E, d) = AA(E, d N ), so the average of awards rule satisfies null claims consistency. We prove in Appendix A that the average of awards rule also satisfies order preservation under endowment variations and order preservation under claims variations. Therefore, as a direct consequence of the Lorenz-based characterizations of the minimal overlap and Piniles' rules we have that, in fact, the average of awards rule Lorenz-dominates the minimal overlap rule and is Lorenz-dominated by Piniles' rule.

The proportionality deviation index
In 1912, the statistician and sociologist Corrado Gini introduced a coefficient intended to measure the degree of income inequality within a population. To compute the Gini coefficient, first one has to find the Lorenz curve, developed by the economist Max O. Lorenz in 1905, that represents in the horizontal axis the proportion of the population, from lowest to highest income, and in the vertical axis the cumulative percentage of income or wealth owned. A perfectly equal distribution of wealth would have a Lorenz curve equal to the line y = x. The Gini coefficient measures how far the actual Lorenz curve for a population's income is from the line of equality. In our setting, given a claims problem if one plots the cumulative percentage of awards with respect to the proportion of claimants, from lowest to highest claims, the line of equality represents the egalitarian division of the endowment. But, the egalitarian division is not a rule and so this line is not particularly suitable. Therefore, instead of the proportion of population, we represent in the horizontal axis the cumulative percentage of claims, ordered from small to large. As a consequence, since the proportional rule shares the endowment in the same proportion as claims, now the line y = x represents the line of proportionality. In this section, we define a pair of indices aimed at measuring the degree of discrepancy between the division proposed by a rule and the proportional distribution.
Thend = (d 1 , . . . ,d n ) ∈ R n ≤ is the vector of the percentages of the cumulative claims with respect to the total sum of claims d(N ). Naturally, 0 ≤d ≤ 1 andd n = 1. For Let R be a rule that satisfies order preservation in awards. Then, for each claims In what follows assume that (E, d), . . . ,R n (E, d)) ∈ R n ≤ is the vector of the percentages of the cumulative awards assigned by the rule R with respect to the endowment.
Definition 4.1 Given a claims problem (E, d) ∈ C N with d ∈ R n ≤ and a rule R satisfying order preservation in awards, the polygonal path connecting the n + 1 points d i ,R i (E, d) , i = 0, . . . , n, is called the cumulative claims-awards curve.
The continuous piecewise linear function L R E,d : [0, 1] → [0, 1] whose graph is the cumulative claims-awards curve is called the cumulative claims-awards function of R for the problem (E, d): Clearly, L R E,d (0) = 0 and L R E,d (1) = 1 but, contrary to a conventional Lorenz curve, the graph of L R E,d does not necessarily lay below the identity line (see Figs. 3, 4, and 5). Nevertheless, from elementary calculus, we have that L R E,d is a monotonically increasing function so its graph is contained in the unit square, i.e., 0 Basically, the function L R E,d represents the proportion of the initial endowment assigned by the rule R to each cumulative proportion of claims. Since the proportional rule divides the endowment in the same proportions as claims, that is PRO(E, d) =d, we have that its claims-awards curve is always the identity, L PRO E,d (t) = t for all t ∈ [0, 1]. In 10 , 1 2 , 1 , and RA(E, d) = 1 3 , 11 6 , 11 6 . Therefore, RA(E, d) = 1 12 , 13 24 , 1 . Fig. 3 shows the line of proportionality and the claims-awards curve of the random arrival rule for (E, d).
The claims-awards curve allows us to compare the division recommended by the rule R with the division that preserves the proportions of the claims, the proportional rule. The claims-awards curve also captures graphically whether or not two rules are Lorenz-comparable.

Proposition 4.3 Let R and R be two rules satisfying order preservation in awards.
For each (E, d) Figure 3 shows the claims-awards curve of the random arrival rule for the claims problem of Example 4.2. The polygonal curve intersects transversally the line of proportionality indicating that the random arrival rule is not Lorenz-comparable to the proportional rule.
Following the idea underlying the definition of the Gini index, we introduce a pair of coefficients that measure the deviation of the claims-awards curve from the line of proportionality. The signed proportionality deviation index is the ratio of the net signed area that lies between the line of proportionality and the claims-awards curve over the total area under the line of proportionality. The proportionality deviation index is the ratio of the area between the line of proportionality and the claims-awards curve over the area under the line of proportionality.

Definition 4.4 Let (E, d) ∈ C N with d ∈ R N
≤ and let R be a rule satisfying order preservation in awards. The signed proportionality deviation index of R for the problem (E, d) is: The proportionality deviation index of R for the problem (E, d) is: Note that, since the Lorenz curve lies below the identity line, the usual Gini coefficient is a value between 0 and 1. In our context, as it is illustrated in Example 4.7 and Fig. 4, the claims-awards curve is not bounded from above by the identity so, as a consequence, the signed proportionality deviation index can take negative values.

. I(R, E, d) = I + (R, E, d) if and only if R(E, d)is Lorenz-dominated by PRO(E, d). 4. I(R, E, d) = −I + (R, E, d) if and only if R(E, d) Lorenz-dominates PRO(E, d). 5. If R(E, d) Lorenz-dominates R (E, d) then I(R, E, d) ≤ I(R , E, d).
Proposition 4.6 shows that when the recommendation made by a rule for a claims problem and the proportional division are Lorenz-comparable, the corresponding proportionality deviation indices reflect the ordering. If a rule R Lorenz-dominates the proportional rule then its signed proportionality deviation index must be negative, but if it is Lorenz-dominated by the proportional rule it must be positive. But, as we have seen in Example 4.5, even when the awards vectors selected by R and the proportional rule are incomparable, the indices reveal how far from proportionality is the division proposed by the rule. The claims-awards curves of the constrained equal awards and the minimal overlap rules for the problem (9, d) ∈ C N are depicted in Fig. 4. Neither of these two rules satisfies the midpoint property. We know that CEA PRO so L CEA L CEA E,d (t) − t dt, so the signed proportionality deviation index is negative. On the contrary, the signed proportionality deviation index for the MO rule is positive because the shadowed area for the MO rule corresponds to Let (E, d) ∈ C N and R be a rule satisfying order preservation in awards. Since the CEA rule Lorenz-dominates each rule that satisfies order preservation in awards, so, by Proposition 4.6, I(CEA, E, d) ≤ I (R, E, d). Moreover, the CEL rule is Lorenzdominated by each rule that satisfies order preservation in losses, so if R is such a rule then I(R, E, d) ≤ I (CEL, E, d). As a corollary, if R satisfies order preservation then I(CEA, E, d) ≤ I(R, E, d) ≤ I (CEL, E, d).
Let R be a rule and R * its dual. The cumulative claims-awards function L R *  (3,4,5,6) the cumulative "gains" and "losses" curves of a rule are related and so are the signed proportionality deviation indices of a rule and that of its dual.
Proposition 4.8 Let R be a rule satisfying order preservation in awards and R * its dual rule. Then, for all (E, d) ∈ C N , E > 0, we have that: simple algebraic manipulations lead to the following equalities:

Now, from the last equality and Definition 4.4, it is straightforward to obtain that
We compute in Example 4.9 the proportionality deviation indices of the constrained equal awards and the constrained equal losses rules for some particular claims problems. The example also illustrates that the signed proportionality deviation index can be a number as close to −1 or 1 as wanted. But these two coefficients convey more information. When n = 2 we know that CEL and PRO coincide (the indices are zero), but as n increases, the CEL rule selects awards vectors that differ more and more from proportionality (the deviation index tends to 1), and we have a precise measure of that discrepancy.
If n > 2, consider the claims problem (E, d) ∈ C N with E = 1 n−2 and d = (1, . . . , 1, 1 + E Theorem we conclude that there is E * ∈ (4, 5) such that 6 E * , and PRO 2 (E * , d) = 2 9 E * . Therefore, we have an instance of a non proportional division with signed deviation index equal to zero. Nevertheless, we know that the proportional deviation index of the minimal overlap rule for this problem must be strictly positive, I + (MO, E * , d) > 0. Figure 6 shows the signed proportionality deviation index path, its absolute value, and the signed proportionality deviation index path of the minimal overlap rule restricted to the interval [4, 6]. In the subinterval where the signed proportionality deviation index path and its absolute value differ, we know that the minimal overlap and the proportional rules are not Lorenz-comparable, and that they deviate less that 5%.
The cumulative claims-awards curve, the proportionality deviation indices, and the index path can be useful tools to compare rules beyond the information provided by the Lorenz order. For any given claims problem, they are easy to compute from the values of the vector of claims and the rule and convey much information about the rule and its properties in a clear and simple visual way. Nevertheless, as the proportionality deviation indices (and by extension the index path) comprise all the data from the cumulative claims-awards curve in a pair of numbers, some information must be lost in the process. Nevertheless, the combination of both coefficients solves some shortcomings that each of them has when taken alone. Certainly, as Example 4.7 illustrates, the proportionality index does not capture the Lorenz-ranking of awards vectors that is fully reflected by the signed index. On the other hand, Example 4.10 shows that two different divisions can have the same signed proportionality deviation coefficient, but the corresponding proportionality deviation indices must be different. Figure 7 portrays the signed proportionality deviation index paths of the ten rules for the vector of claims d = (3, 4, 5, 6). At first sight, one observes that only the proportional rule and the average of awards rule have smooth paths, because they are the only rules that are endowment differentiable. 1 Moreover, according to Proposi- Fig. 7 Signed index path of the ten rules for the claims vector d = (3, 4, 5, 6) tion 4.6, the ranking of rules is reflected in the graph so, for instance, all the paths lie between those of the CEL and the CEA rules. Whether or not a rule satisfies the midpoint property has a clear implication on its index path. Note that, the index paths of the constrained egalitarian, the Talmud, and Piniles' rules coincide in the interval [0, 1 2 d(N )]. 2 Certainly, for the constrained equal losses rule, both the proportionality deviation index path and the signed proportionality deviation index path coincide. The proportionality deviation index paths of the average of awards, the minimal overlap, and the constrained equal awards rule are compared to the corresponding signed index paths in Fig. 8. For the average of awards and the constrained equal awards rules, the proportionality index path is just the absolute value of the signed proportionality index path. Therefore, according to Proposition 4.6, CEA(E, d) Lorenz-dominates PRO(E, d) for For the minimal overlap rule there is a neighbourhood of E = 5 where the proportionality index path is not the absolute value of the signed proportionality index path, so for these values of the endowment MO(E, d) and PRO(E, d) are not comparable (see Fig. 6).

Generalized deviation indices
The proportionality deviation indices measure the discrepancy of an awards vector with respect to the proportional division. But, depending on the principles of fairness, equity, or justice, that the decision maker wants to apply when facing a particular claims problem, the proportional division may not be the suitable rule of reference. Therefore, we want to generalize the proportionality indices by providing a way to measure the degree of discrepancy between two arbitrary awards vectors. Given a pair of vectors x, y ∈ R n ≤ it is easy to define a cumulative curve L y x representing the vector of cumulative percentages of the coordinates of y against the vector of cumulative percentages of the coordinates of x.
≤ is the vector of cumulative percentages of the coordinates of x with respect to the total sum x(N ). Naturally, 0 ≤x i ≤ 1 for all i ∈ N andx n = 1. Denote  The role of the proportional rule as the benchmark for comparing awards vectors in the analysis of Sect. 4 can be played by any other rule and a deviation index with respect to this new reference rule can be computed. Given two rules R and R satisfying order preservation in awards and a claims problem AA (E,d) are depicted in Fig. 9. Clearly, L Therefore, I PRO(E, d), AA(E, d) = −I(AA, E, d) = 0.45. We conclude that AA(E, d) Lorenz-dominates PRO(E, d) and that the proportional division deviates by 45% from the average of awards rule, the geometrical center of the set of awards vectors. Note that, for this particular claims problem, the adjusted proportional, the constrained egalitarian, Piniles', the random arrival, and the Talmud rules recommend the same division as the average of awards rule.
For a rule R that satisfies order preservation, in addition to the proportionality deviation indices, the coefficients I CEA(E, d), R(E, d) , I CEL(E, d), R(E, d) , and I AA(E, d), R(E, d) are particularly interesting. Since the constrained equal awards and the constrained equal losses rules are Lorenz-maximal and Lorenz-minimal respectively among the rules satisfying order preservation, the signed deviation indices of rule R with respect to the C E A and CEL rules indicate the variation of rule R compared to two extreme rules. The signed deviation index of rule R with respect to the average of awards rule, I AA(E, d), R(E, d) , measures the degree of discrepancy of rule R from a central rule, the geometrical center of the set of awards vectors.  The coefficients that we introduce summarize in a couple of numbers the relative distribution of the endowment recommended by two rules. Therefore, we know not only if they are Lorenz-comparable but also by how much the corresponding awards vectors differ from each other, thus helping the decision maker to select one over the other. Depending on the values of the endowment and the claims, the indices between a giving pair of rules can be very small or very large. If the deviation index is small, both distributions are very similar. However, if the deviation index takes high values then the recommendations made by both rules diverge, and factors like the axiomatics of the rules would play a more significant role. Now, by the induction hypothesis, suppose that the average of awards rule satisfies order preservation under endowment variations for any problem with n − 1 ≥ 2 claimants, and let us show that then it must satisfy the property for problems with n claimants. So, let |N | = n ≥ 3. Since the average of awards satisfies endowment differentiability, given d ∈ R n ≤ and i ∈ N \{n}, it suffices to prove that is an increasing function. Now, using the derivative expressions given above, we obtain: .
It suffices to establish the result for E ∈ d 1 , min 1 2 d(N ), D −n . First, assume that i < n − 1. Then, Since E ≤ D −n , we have that R n (E, d) = E. Then, applying expression (2) and by the induction hypothesis: On the other hand, χ n (E, d) = 1 only if E ≥ d n and then r n (E, d) = E − d n . In that case: Clearly, g n (E, E) ≥ 0 and g n (E, E − d n ) ≥ 0, so from (3) we conclude that indeed Finally, if i = n − 1, we show that ∂(AA n − AA n−1 ) ∂ E (E, d) ≥ 0 by repeating the same arguments as above but applied to the integral representations of AA n and AA n−1 given by (2) in terms of the function g 1 .
A rule R satisfies order preservation under population variation if for each Let us show that the average of awards rule satisfies order preservation under the reduction operation and order preservation under population variation. Indeed, let (E, d) But, r i (E, d) = E − d i and R i (E, d) = E, so by equality (2), We show in Proposition A.1 that AA k (., d −i ) − AA j (., d −i ) is increasing. Then, both properties hold.

Proposition A.2 The average of awards rule satisfies order preservation under claims variations.
Proof Let (E, d) ∈ C N be a claims problem, i ∈ N \{n} and d i < d i ≤ d i+1 . Denote d = (d −i , d i ). It suffices to prove that for each { j, k} ⊂ N \{i} with d j ≤ d k then Observe that if E ≤ d i then X (E, d) = X (E, d ) and the property follows at once. Therefore, assume that The centroid of X (E, d ) is the average of the centroids of each part weighted by its relative measure. Therefore, for each r ∈ N \{i}, we have that, Applying the above equality to the pair { j, k} ⊂ N \{i}, we conclude that AA Since the average of awards rule satisfies order preservation under population variations and order preservation under the reduction operation,

B Areas below the cumulative claims-awards curve
Let x = (x 1 , . . . , x n ) ∈ R N ≤ and y = (y 1 , . . . , y n ) ∈ R N ≤ . Definē For each i ∈ N denote x i =x i −x i−1 = 1 x(N ) x i and ȳ i =ȳ i −ȳ i−1 = 1 y(N ) y i . Consider the continuous piecewise linear function L y x : [0, 1] → [0, 1] connecting the n + 1 points (x i ,ȳ i ), i = 0, . . . , n. that is, The difference between the area inside the unit square below the proportionality line and the area below L y x is: In particular, given a claims problem (E, d) ∈ C N with d ∈ R N ≤ and a rule satisfying order preservation in awards R, taking x = d and y = R(E, d), we have that the signed proportionality deviation index of R for the problem (E, d) is given by: (E, d)). Now, the area between the line of proportionality and the piecewise polygonal curve L y x is given by the integral • If α i ≤ 0 and α i+1 ≤ 0 then 2 x ī • If α i = 1 and α i+1 = −1 then 2 x ī • If α i = −1 and α i+1 = 1 then In particular, given a claims problem (E, d) ∈ C N with d ∈ R N ≤ and a rule satisfying order preservation in awards R, taking x = d and y = R(E, d), we have that the proportionality deviation index of R for the problem (E, d) is given by: