The average-of-awards rule for claims problems

Given a claims problem, the average-of-awards rule (AA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {AA}}$$\end{document}) selects the expected value of the uniform distribution over the set of awards vectors. The AA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {AA}}$$\end{document} rule is the center of gravity of the core of the coalitional game associated with a claims problem, so it corresponds to the core-center. We show that this rule satisfies a good number of properties so as to be included in the inventory of division rules. We also provide several representations of the AA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {AA}}$$\end{document} rule and a procedure to compute it in terms of the parameters that define the problem.


Introduction
A claims problem arises when a scarce resource has to be shared among a set of claimants and the endowment falls short of the sum of individual claims. The question is, how to select a division among the claimants of the amount available. The definition of division rules and the study of different approaches to evaluate and compare them started with O'Neill (1982) and has generated a vast literature. The model has many applications that include bankruptcy problems, taxation systems, rationing problems, or the distribution of the carbon budget.
A division rule must satisfy three natural requirements: no claimant is asked to pay; no claimant receives more than her claim; and the entire endowment is allocated. The set of all the allocations that meet these basic properties is the set of awards vectors for the problem. Therefore, formally, a rule is a function that associates with each problem an awards vector. For an inventory of the principal rules, their properties, and a comprehensive survey on claims problems refer to Thomson (2019).

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An intuitive and simple way of selecting an allocation from the set of awards vectors for a problem is to assume that all the awards vectors are equally likely and therefore choosing their "average". For each problem, the division recommended by the average-of-awards rule ( AA ) is the expected value of the uniform distribution over the set of awards vectors.
Following O'Neill (1982), to each problem one can associate a coalitional game. Curiel et al. (1987) note that the coalitional game associated with a problem is a convex game so its core is nonempty. In fact, the set of awards vectors for a problem coincides with the core of its associated coalitional game. A division rule corresponds to a solution to coalitional games if the choice made by the rule for each problem coincides with the choice made by the solution to coalitional games when applied to the associated coalitional game.
The core-center was defined and characterized for the general class of games with nonempty core by Sánchez-Rodríguez (2007, 2009) and was studied on the domain of airport games by González-Díaz et al. (2015 and Mirás . Since, geometrically, the AA rule is the center of gravity (centroid) of the set of awards vectors, it corresponds to the core-center. There are other rules that select central points of the set of awards vectors. The random arrival rule ( RA ), which corresponds to the Shapley value, is the Steiner point of the set of awards vectors (Pechersky 2015). The adjusted proportional rule ( APRO ), which corresponds to the -value, is a weighted average of the extreme points of the set of awards vectors (González-Díaz et al. 2005). The Talmud rule ( T ), which corresponds to the nucleolus, is the lexicographic center of the set of awards vectors (Schmeidler 1969).
Analyzing the AA rule demands a detailed examination of the structure of the set of awards vectors. This structure is particularly simple: it is a nonempty compact and convex polytope, the intersection of a rectangle with a hyperplane. Many of the geometric features of the set of awards vectors are established in the companion paper Mirás Calvo et al. (2022b). Making use of these characteristics, and providing some decompositions of the set of awards vectors, we prove that the AA rule satisfies a good number of the standard properties for division rules. In this sense, we can say that the rule is well-behaved. Mirás Calvo et al. (2022a) incorporate the average of awards rule to the ranking of the standard rules with the Lorenz order.
There are two particular qualities of this rule that are worth noting. Certainly, the AA rule is a continuous rule: small changes in the parameters of the problem do not lead to large changes in the recommendation made by the rule. In particular, a variation in the endowment, no matter how small, produces a change in the corresponding set of awards vectors. The AA rule is highly sensitive to such changes. In fact, we prove that the recommendation made by the AA rule, for problems with at least three claimants, not only varies continuously with respect to the endowment but also that the rate at which the recommendation changes varies also with continuity, that is, the AA rule is endowment differentiable. Secondly, the AA rule coincides with the concede-and-divide rule ( CD ) for two claimants but, for larger populations, it differs from the standard rules. Therefore, the AA rule is an endowment differentiable extension of the CD rule.

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The average-of-awards rule for claims problems There are simple mechanisms to compute some of the basic rules. But, in general, computing the allocation selected by a rule for a given problem can be computationally hard when the population is large. For example, Aziz (2013) shows that the allocation returned by the RA rule is #P-complete to compute. There are general algorithms to obtain the centroid of higher dimensional polyhedra that can be applied to calculate the AA rule. Here we present a procedure to exactly compute this rule in terms of the parameters of the problem, and provide an analytic expression for threeclaimant problems.
The paper is organized as follows. Section 2 introduces the AA rule. In Sect. 3 we define some properties of division rules and we summarize which of them are satisfied or violated by the AA rule. The specific results are presented in Sect. 4. The computation of the AA rule is addressed in Sect. 5. The purely technical results are left to the Appendix.

The average-of-awards rule
Let N be the set of all finite nonempty subsets of the natural numbers ℕ . Given N ∈ N , z ∈ ℝ N , and S ∈ 2 N let |N| be the number of elements of N and In particular denote z −i = z N�{i} ∈ ℝ N�{i} the vector obtained by neglecting the ith-coordinate of z. For simplicity, we will write z = (z −i , z i ) . Given z, w ∈ ℝ N , the notation z ≤ w means that z i ≤ w i for all i ∈ N.
A claims problem (O'Neill 1982) with set of claimants N ∈ N is a pair (E, d) where E ≥ 0 is the endowment and d ∈ ℝ N is the vector of claims satisfying d ≥ 0 and E ≤ d(N) . We denote the class of claims problems with set of claimants N by C N . Let us assume throughout the paper that N = {1, … , n}.
The minimal right of claimant i ∈ N in (E, d) ∈ C N is the quantity N�{i}) . It is what is left after every one else has been fully compensated, or 0 if that is not possible. The truncated claim of claimant It is the claim truncated by the amount to divide. To simplify, sometimes we write m i = m i (E, d) and Then, the set of awards vectors for (E, d) is a nonempty compact convex polytope that has, at most, dimension n − 1.
A division rule is a function R ∶ C N → ℝ N assigning to each problem (E, d) ∈ C N an awards vector R(E, d) ∈ X(E, d) , that is, a way of associating with each problem a division of the amount available among the claimants. In our definition of a division rule we include three requirements: nonnegativity (no claimant is asked to pay); claims boundedness (no claimant receives more than her claim); and balance (the entire endowment is allocated). Certainly, there are applications in which one (or more) of these requirements could be dropped but they are very natural in most situations.
An intuitive and simple way of selecting an allocation from the set of awards vectors for a problem is to assume that all the awards vectors are equally likely and therefore choosing their "average".

Definition 2.1
The average-of-awards rule is the rule AA ∶ C N → ℝ N that 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).
Let ℝ n ≤ be the set of nonnegative n-dimensional vectors x = (x 1 , … , x n ) such that 0 ≤ x 1 ≤ … ≤ x n . In what follows we assume that given a problem (E, d) ∈ C N , the claims d = (d 1 , … , d n ) ∈ ℝ n are arranged from small to large, i.e., d ∈ ℝ n ≤ . As a consequence we have that So, in order to compute the AA rule we can remove the claimants in N ′ and apply the rule to the problem (E, d N�N � ) ∈ C N�N � whose set of awards vectors is full dimensional.
Let be the (n − 1)-dimensional Lebesgue measure. Fix a vector of claims , the (n − 1)-measure of the set of awards vectors X(E, d). Naturally, X(E, d) is full dimensional if and only if Fig. 2 The set of awards vectors X(3, (1, 2, 2)) and its centroid There are situations where the AA rule is easy to compute. For instance, if all claims are bigger than the endowment, that is E ≤ d 1 , then X(E, d) is the regular simplex spanned by the points , the arithmetic mean of these points, coincides with the egalitarian division, that is, Another interesting case occurs when the largest claim exceeds the aggregate claim of the other claimants and the endowment lies between these two quantities, that is, if E ∈ [d(N�{n}), d n ] . Then the set of awards vectors has a very simple structure: Consequently, the AA rule gives to all the claimants, except for the one with the largest claim, the geometric center of the (n − 1)-rectangle A coalitional game is an ordered pair (N, v) where N ∈ N is a finite set of players and v ∶ 2 N → ℝ , the characteristic function, satisfies v(�) = 0 . Let G N be the set of all coalitional games with player set N. Given a game v ∈ G N , a vector x ∈ ℝ N is said to be an efficient allocation if x(N) = v(N) . A (single-valued) solution is a mapping that associates with each game v in some admissible class of games an efficient allocation. The core of a game v ∈ G N is the set Neill (1982), the coalitional game associated with the claims prob- If for each problem, the recommendation made by a given division rule coincides with the recommendation made by a given solution to coalitional games when applied to the associated coalitional game, we say that the rule corresponds to the solution. The set of awards vectors for a problem coincides with the core of the associated coalitional game [see, for instance, Thomson (2019)], that is, X(E, d)

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The average-of-awards rule for claims problems consists of all allocations satisfying the balance condition that are bounded from below by the minimal rights and bounded from above by the truncated claims: Therefore, the AA rule corresponds to the core-center introduced by González-Díaz and Sánchez-Rodríguez (2007).

Summary of properties
The axiomatic approach has dominated the study of rules. Rules are examined, classified, and characterized according to the properties that they satisfy (or violate). We present some standard properties. We say that a rule R satisfies:

of-claim-and-equal-division lower bounds on awards, if for each
Some of these properties are stronger requirements than others. For instance, minof-claim-and-equal-division lower bounds on awards implies 1 |N| -truncated-claims lower bounds on awards, and other-regarding claim monotonicity implies claim monotonicity. Anonymity implies equal treatment of equals. Order preservation also implies equal treatment of equals. Moreover, composition down (composition up) implies endowment monotonicity.
With each rule R we can associate a unique dual rule R * , the one defined by the right-hand side of the expression in the statement of the self-duality property: Two properties are dual if whenever a rule satisfies one of them then its dual satisfies the other. The following are pairs of dual properties: claims truncation invariance and minimal rights first; 1 |N| -truncated-claims lower bounds on awards and 1 |N| -min-of-claim-anddeficit lower bounds on losses; composition down and composition up; and claim monotonicity and linked claim-endowment monotonicity. Note that order preservation is, in fact, the conjunction of two dual properties: order preservation in awards and order preservation in losses.
We also consider situations in which the population of claimants involved may vary. In this case, a problem is defined by first specifying N ∈ N , then a pair (E, d) ∈ C N . We still denote the class of all problems with claimant set N by C N . So, a rule is a function defined on ⋃ N∈N C N that associates with each N ∈ N and each (E, d) ∈ C N an awards vector for (E, d). We say that a rule R satisfies: Bilateral consistency is the weaker property obtained by considering only subgroups of two remaining agents, that is, when The average-of-awards rule for claims problems ciated partition (N i ) i∈N , then for each i ∈ N and each j ∈ N i we have Table 1 summarizes which of the properties listed above are satisfied or violated by the AA rule. A check mark, m ark , in a cell means that the property is satisfied by the rule and a minus sing, −, means the opposite. The last column indicates the specific result in the next section where the corresponding mark is established. As Table 1 shows, the AA rule satisfies a good number of properties so as to be part of the inventory of rules.
In the next section, we use the following general property of the centroid. Let (E, d) ∈ C N and K be a convex polytope contained in the hyperplane H(E, d). Denote its centroid by c(K). 5 Then c(a + K) = a +c(K) for all a ∈ ℝ N and ∈ ℝ . One of the methods used in obtaining the centroid of a compound shape consists in dividing the shape into a number of parts, that share no common volume, and then finding the overall centroid as the average of the centroid of each part weighted by its relative measure. Formally, if (K) > 0 , K = K 1 ∪ K 2 , (K 1 ∩ K 2 = 0 , and = (K 1 ) (K) , then

Results
We begin by establishing some properties that are a direct consequence of simple geometrical features of the set of awards vectors.
, so the AA rule satisfies null claims consistency. Self-duality follows from the equal- t(E, d)) we conclude that AA satisfies claims truncation invariance. The AA rule also satisfies minimal rights first because it is self-dual and claims truncation invariance and minimal rights first are dual properties. ◻ Self-duality implies that the AA rule satisfies the midpoint property, that is, AA 1 2 d(N), d = 1 2 d. Now, some general properties of the core-center for coalitional games are easy to translate to properties of the AA rule.

Proposition 4.2
The AA rule satisfies equal treatment of equals, homogeneity, and continuity.
Proof The AA rule corresponds to the core-center for coalitional games with nonempty core. González-Díaz and Sánchez-Rodríguez (2007) show that the corecenter treats symmetric players symmetrically and that it is a homogeneous and continuous function of the values of the characteristic function. The AA rule satisfies equal treatment of equals because claimants with the same claims are symmetric players in the associated coalitional game. Since, given > 0 and S ∈ 2 N we have that v (S) = v(S) , where v is the coalitional game associated with the problem ( E, d) ∈ C N , the AA rule satisfies homogeneity. The values, v(S), S ∈ 2 N , of the characteristic function are continuous with respect to the endowment E and the (3) c(K) =c(K 1 ) + (1 − )c(K 2 ).

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The average-of-awards rule for claims problems claims d. Then, the AA rule satisfies continuity because it is a composition of continuous functions. ◻ Next, we consider the properties listed in Table 1 that the AA rule fails.

Proposition 4.3
The AA rule violates composition down, composition up, consistency, min-of-claim-and-equal-division lower bounds on awards, replication invariance, and converse consistency.
Proof Dagan (1996) shows that the constrained equal awards rule ( CEA ) is the only rule satisfying equal treatment of equals, claims truncation invariance and composition up. Then, the AA rule does not satisfy composition up. Since composition down and composition up are dual properties and the AA rule is self-dual, it violates composition down. Aumann and Maschler (1985) prove that the T rule is the only rule to agree with concede-and-divide for two claimants and to be bilateral consistent.

Proposition 4.4 The AA rule satisfies order preservation.
Proof We start by proving that the AA rule satisfies order preservation in awards. Let V (E,d) . By equal treatment of equals, The result also holds if V(E, d) = 0 . Therefore, the AA rule satisfies order preservation in awards. Since order preservation in awards and order preservation in losses are dual properties and the AA rule is self-dual, we conclude that the AA rule satisfies order preservation. ◻ So far, we have relied on simple features or decompositions of the set of awards vectors to establish some properties of the AA rule. Our approach is quite different to deal with endowment monotonicity. We want to prove that if the amount to divide increases, each claimant should receive at least as much as she did initially. Then, we have to understand how the set of awards changes when the endowment increases. Given a rule R and a vector of claims d ∈ ℝ N , the path followed by the awards vector chosen by R as the endowment increases from 0 to d(N), that is, the function R(⋅, d) ∶ [0, d(N)] → ℝ N , is called the path of awards of the rule for the claims vector. The plots of the functions R i (⋅, d) , i ∈ N , are called the schedules of awards of the rule for the claims vector. A rule R satisfies endowment continuity if the path of awards of the rule is continuous for all claims vector. Naturally, continuity implies endowment continuity. Endowment differentiability requires the path of awards R(⋅, d) of the rule to be differentiable on [0, d(N)] for all claims vector d ∈ ℝ N . Some of Young's parametric rules (Young 1987), for instance the PRO rule and Cassel's rule, are endowment differentiable. Fig. 4 The schedules of awards of the AA rule for two-claimant problems 1 3 The average-of-awards rule for claims problems As Fig. 4 illustrates, for problems with two claimants, the CD rule (and therefore the AA rule) violates endowment differentiability. Nevertheless, as we argue next, the AA rule is an endowment differentiable extension of the CD rule to larger populations.
Let |N| ≥ 3 and fix d ∈ ℝ n ≤ with 0 < d 1 . Now, assume that claimant i ∈ N receives an award s ∈ [m i (E, d), t i (E, d)] and leaves. The remaining claimants, the members of N�{i} , face the reduced problem (E − s, d −i ) ∈ C N�{i} . If claimant i ∈ N gets her minimal right m i (E, d) then the remainder d(N)) . Furthermore, in the Appendix (see Theorem A.1), we show that g i (E, ⋅) is a probability density function on the interval and that, for all j ∈ N�{i} , the following integral representations hold: In words, for problems with at least three claimants, the awards chosen by the AA rule for a problem are a weighted average of the choices made by the rule in each of the reduced problems that result when a claimant receives an award, between her minimal right and her truncated claim, and leaves. We prove in the Appendix (see Theorem A.2) the following result: Proposition 4.5 If |N| ≥ 3 then the AA rule is endowment differentiable.
The integral representation of the AA rule given by equality (4) allows us to "lift" some properties satisfied by the CD rule to larger populations. For instance, the CD rule is endowment monotonic (see Fig. 4). We illustrate how the integral representation given by (4) allows us to prove that the AA rule is endowment monotonic. An endowment continuous rule R satisfies endowment monotonicity if and only if for each i ∈ N the function R i (⋅, d) is monotonically increasing (4) on [0, d(N)] for all claims vector d ∈ ℝ N . If a rule is endowment differentiable then endowment monotonicity can be expressed in terms of the derivatives of the rule with respect to the endowment. Indeed, if R(⋅, d) is a differentiable function then R satisfies endowment monotonicity if, for each d ∈ ℝ N , the derivatives of its coordinate functions with respect to E are positive, that is, 0, d(N)] and for all i ∈ N.
Proposition 4.6 The AA rule satisfies endowment monotonicity. Moreover, if |N| ≥ 3 , for each (E, d) ∈ C N with d ∈ ℝ n ≤ , and each i ∈ N: Proof The proof is divided into three cases.
The AA rule, whose awards for each claimant are the continuous piecewise linear functions depicted in Fig. 4, satisfies endowment monotonicity.
, using equality (4) and taking into account the fact that . Now, according to Theorem A.2, in order to see that AA j (⋅, d) is monotonically increasing on [0, d(N)] it suffices to prove that AA j (⋅, d) is monotonically increasing on the interval J = 0, min 1 2 d(N), d(N�{n}) or, equivalently, that AA j E (E, d) ≥ 0 for all E ∈ J . If E < d 1 the result is trivial. But, using the expression for the derivatives of the AA rule given in Theorem A.2 it follows that, for all E ∈ d 1 , min 1 2 d(N), d (N�{n}) , and all j ∈ N�{n} , Therefore, AA j (⋅, d) is monotonically increasing for all j ∈ N.
Let d ∈ ℝ n ≤ with 0 < d 1 , and i ∈ N . We proceed by induction on the number of claimants, so assume that AA j (r and all j ∈ N�{i} . Therefore,

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The average-of-awards rule for claims problems or, equivalently, by (4), From these inequalities and the derivatives obtained in Theorem A.2, we conclude that AA j (⋅, d) is monotonically increasing. ◻ Population monotonicity states that if the population of claimants enlarges but the amount to divide stays the same, each of the claimants initially present should receive at most as much as she did initially.

Proposition 4.7
The AA rule satisfies population monotonicity.
Then, by the inequalities in Proposition 4.6 and endowment monotonicity, . The general case follows applying repeatedly this result. ◻ Recall that, by the balance condition, if a rule satisfies other-regarding claim monotonicity then it also satisfies claim monotonicity.

Proposition 4.8
The AA rule satisfies other-regarding claim monotonicity, claim monotonicity, and linked claim-endowment monotonicity.
Proof We prove other-regarding claim monotonicity. Let (E, d) Moreover, Other-regarding claim monotonicity is a stronger requirement than claim monotonicity. So the AA rule satisfies claim monotonicity. Since claim monotonicity and linked claim-endowment monotonicity are dual properties and the AA rule is selfdual, the AA rule satisfies linked claim-endowment monotonicity. ◻ Finally, let us show that the AA rule guarantees a minimal share to every agent equal to one nth her claim truncated at the amount to be divided.

Proposition 4.9
The AA rule satisfies 1 |N| -truncated-claims lower bounds on awards and 1 |N| -min-of-claim-and-deficit lower bounds on losses.
But if E ≥ d n and j ∈ N�{n} then, by other-regarding claim monotonicity, AA j (E, d) ≥ AA j (E, (d 1 , … , d n−1 , E)) , and we have already shown that Therefore, the AA rule satisfies 1 |N| -truncatedclaims lower bounds on awards. Naturally, the AA rule, being self-dual, also satisfies the corresponding dual property, namely, 1 |N| -min-of-claim-and-deficit lower bounds on losses. ◻

Computation of the AA rule
The definition of the AA rule, though natural, does not provide a mechanism to compute it in terms of the n + 1 parameters that define a problem: the endowment E and the vector of claims d ∈ ℝ N . Here we describe a particular procedure to compute the centroid of the set of awards vectors for a problem. We can assume, by anonymity, that given (E, d) ∈ C N , the claims are sorted in ascending order, that is, d ∈ ℝ n ≤ . By null claims consistency, the agents whose claims are 0 can be removed so we just have to consider problems whose set of awards vectors is full dimensional, that is, 0 < d 1 and 0 < E < d (N) . Moreover, the AA rule is self-dual so we can restrict the algorithm to problems for which the endowment does not exceed the half-sum of claims, E ≤ 1 2 d(N) . In addition, we can also assume that E ≤ d(N�{n}) because if d(N�{n}) ≤ E ≤ d n then AA(E, d) is given by expression (2). Taking all these considerations into account, and exploiting the special structure of the set of awards vectors, Mirás Calvo et al. (2022b) obtain a formula for the AA rule. The following procedure is a direct application of their results.
Procedure to compute the AA rule: Let (E, d) ∈ C N such that d ∈ ℝ n ≤ , 0 < d 1 , and 0 < E ≤ min{ 1 2 d(N), d(N�{n})} . In order to compute AA(E, d) follow these steps: The key step in the above algorithm is the determination of the ratios p i p for each i ∈ I . Therefore, we need a method to compute the volume of the set of awards vectors for a problem. Of course, one can rely on any general algorithm for the volume of a convex polyhedron [such as Lasserre (1983)]. Nevertheless, we use the simple expression for the volume of the set of awards vectors in terms of the endowment and the claims provided by Mirás Calvo et al. (2022b): where F = {S ∈ 2 N ∶ d(S) < E} . Let us illustrate how to combine the algorithm and the volume formula to compute the AA rule. 10 Example 5.1 Let N = {1, 2, 3, 4} and consider the problem (E, d) ∈ C N with E = 7 and d = (2, 4, 6, 8) ∈ ℝ 4 ≤ . Now, I = {i ∈ N ∶ d i < 7} = {1, 2, 3} . Therefore, we consider the problems: We compute the volumes p = V(E, d) and  (5,4,6,8) 3, (2, 3, 6, 8) 1, (2, 4, 1, 8) Since we need to compute the ratios p i p , the factor √ n (n−1)! can be ignored. Nevertheless, for clarity, we carry this factor in the example.

Then,
Though the scope of Lemma 5.2 is restricted to three-claimant problems, it helps to illustrate the schedules of awards of the AA rule as the next example shows.