UNIVERSIDADE DE SANTIAGO DE COMPOSTELA DEPARTAMENTO DE ESTATÍSTICA E INVESTIGACIÓN OPERATIVA

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1 UNIVERSIDADE DE SANTIAGO DE COMPOSTELA DEPARTAMENTO DE ESTATÍSTICA E INVESTIGACIÓN OPERATIVA On the core of an airport game and the properties of its center J. González-Díaz, M. A. Mirás Calvo, C. Quinteiro Sandomingo, E. Sánchez Rodríguez Report 4- Reports in Statistics and Operations Research

2 On the core of an airport game and the properties of its center Julio González-Díaz Departamento de Estatística e Investigación Operativa Universidade de Santiago de Compostela Carmen Quinteiro Sandomingo Departamento de Matemáticas Universidade de Vigo Miguel Ángel Mirás Calvo Departamento de Matemáticas Universidade de Vigo Estela Sánchez Rodríguez Departamento de Estatística e Investigación Operativa Universidade de Vigo February 24, 24 Abstract The airport problems Littlechild and Owen, 973 are a well known class of cost allocation problems. The game-theoretic approach consists of transforming these problems into coalitional games, finding a payoff vector that solves the game and studying the corresponding rule. Since the core of the game is nonempty and it has many allocations at which agents payoffs differ, we study the payoff that is the average expectation of all the stable allocations: the core-center González-Díaz and Sánchez-Rodríguez, 27. The structure of the core is exploited to derive insights on the core-center and its properties. The results in this paper build upon explicit integral formulae for the core-center of the airport game. A thorough analysis of these integrals allows not only to study the monotonicity properties of the core-center, and many other axioms discussed in the survey by Thomson 27, but also to compute in a relatively easy way the core-center of an airport game. Keywords: cooperative TU games, monotonicity, core, core-center, airport games. Introduction The airport problem, introduced by Littlechild and Owen 973, is a classic cost allocation problem that has been widely studied. To get a better idea of the attention it has generated one can refer to the survey by Thomson 27. One standard approach to study this problem consists of associating a cooperative game with it and take advantage of all the machinery developed for cooperative games to gain insights in the original problem. The core, introduced by Gillies 953, stands as one of the most studied solution concepts in the theory of cooperative games. Its properties have been thoroughly analyzed and, when a new class of games is studied, one of the first questions to ask is whether or not the games in that class have an nonempty core. This is because of the desirable stability requirements that underly core allocations. Importantly, the cooperative game associated with an airport problem has a special structure that can be exploited to facilitate the analysis of different solutions. In particular, 2 n parameters are needed to define a general n-player cooperative game, whereas for an airport game one just needs n. This special structure simplifies the geometry of the core of such games, since they turn to be defined by 2n inequality constraints instead of the usual 2 n 2. A related property of airport games is that their Shapley value can be computed in polynomial time, whereas in general the worth of all 2 n coalitions must be used to calculate it. Corresponding author: esanchez@uvigo.es

3 When the core of a game is nonempty, there is a set of alternatives at which agents payoffs differ that are coalitionally stable. Studying the center of gravity of such set may be interesting in some cases. The core-center González-Díaz and Sánchez-Rodríguez, 27 selects the mathematical expectation of the uniform distribution over the core of the game. The intuition provided by its definition is a good reason to be interested in it and to justify the study of its properties. In this paper we try to exploit the aforementioned structure of the core of an airport game to gain insights in the monotonicity properties of its core-center. The formal definition of the core-center is given in terms of integrals over the core of the game. Therefore, in order to study the core and its center of gravity we must extensively employ tools of integral calculus. In fact, our main results build upon particular integral formulae obtained for the core-center of an airport game. Apart from the results on the standard monotonicity properties, our approach also leads to some findings that may be of independent interest and that we enumerate below. First, we show that, to each player i, we can associate a face of the core of an airport game in such a way that the derivative of the volume of the core with respect to the cost associated with player i is proportional to the volume of his associated face. This result, which is crucial to study the core-center, may be of independent interest. Second, we establish that each component of the core-center of an airport game is the ratio of the volumes of the cores of two airport games: the core of the original game and the core of the airport game obtained by replicating agent j. This unexpected result allows to use volume computation algorithms for convex polytopes to develop a method to compute the core-center. Third, the machinery we develop to study the monotonicity properties of the core-center facilitates the study of other axioms. To illustrate, we have studied the behavior of the core-center not only with respect to monotonicity properties, but also with respect to all the axioms listed in the survey by Thomson 27. It turns out that the core-center satisfies, among others, those properties which are, arguably, the most important ones. Finally, as a byproduct of our analysis we get two new natural monotonicity properties that impose conditions on how a change on a cost parameter of a given agent affects the payoffs of the other agents. They are called higher cost decreasing monotonicity and lower cost increasing monotonicity. The first one says that if a cost c i increases while the others are held constant, then the payoff decreases for all the players with costs higher than c i. The second property is a kind of reciprocal, the payoff increases for all the players with costs lower than c i. The paper is structured as follows. In Section 2 we present the basic concepts and notations. Then, in Section 3 we obtain an integral representation of the core-center that exploits the special structure of airport games. In Section 4 we develop our main mathematical results, that build upon thorough exploration of the derivatives of the volumes of the core of an airport game. In sections 5 and 6 we study the properties of the core-center and in Section 7 we present a summary of this analysis, comparing the behavior of the core-center with the behavior of the Shapley value and the nucleolus. We have relegated the most technical results to the Appendix. 2 Preliminaries We assume that there is an infinite set of potential players, indexed by the natural numbers. Then, in each given problem only a finite number of them are present. Let N be the set of all finite subsets of N = {, 2, }. A cost game with transferable utility is a pair N, c, where N N and c: 2 N R is a function assigning, to each coalition S, its cost cs. By convention c =. Given a coalition of players S, S denotes its cardinality. Given N N and S N, a vector x R N is referred to as an allocation and xs = i S x i; also, e S {, } N is defined as e i S = if i S and ei S = otherwise. An allocation is efficient for N, c if xn = cn. A cost game N, c is concave if, for each i N and each S and T such that S T N\{i}, cs {i} cs ct {i} ct. 2

4 For most of the discussion and results, we have a fixed n-player set N = {, 2,, n}. A solution is a correspondence ψ defined on some subdomain of cost games that associates to each game N, c in the subdomain a subset ψn, c of efficient allocations. If a solution is single-valued then it is referred to as an allocation rule. Given a cost game N, c, the imputation set, IN, c, consists of the individually rational and efficient allocations, i.e., IN, c = {x R N : xn = cn and, for each i N, x i c{i}}. The core Gillies, 953, is defined as CN, c = {x IN, c : for each S N, xs cs}. An airport problem Littlechild and Owen, 973 with set of agents N is a positive vector c R N, with c i for each i N. Throughout the paper, given an airport problem c R N, we make the standard assumption that for each pair of agents i and j, if i < j, then c i c j. An allocation for an airport problem is given by a non-negative vector x R N such that xn = c n. An allocation rule selects an allocation for each airport problem in a given subdomain. A complete survey on airport problems is Thomson 27. Given an allocation x, the difference c i x i between agent i s cost parameter and her contribution can be seen as her profit at x. A basic requirement is that at an allocation x no group N N of agents should contribute more that what it would have to pay on its own, max{c i, i N }. Otherwise, the group would unfairly subsidize the other agents. The constraints j i c i are called the no-subsidy constraints. To each airport problem c R N one can associate a cost game N, c defined, for each S N, by setting cs = max i S {c i }; such a game is called an airport game. Airport games are concave and their core coincides with the set of allocations satisfying the no-subsidy constraints. We slightly abuse notation and use CN, c to refer to both the core of the airport problem c R N, hereafter called airport core, and the core of the associated airport game, N, c, CN, c = { x R n : x, xn = c n, and, for each i < n, j i c i }. The core of the airport game is contained in the efficient hyperplane xn = c n and it is defined by, at most, 2n 2 inequality constraints, instead of the maximum number of 2 n 2 inequality constraints of an arbitrary coalitional game. This makes the structure of the core of an airport game more tractable. In particular, whenever c >, the core of an airport game is a n -dimensional convex polytope. Further, because of the no-subsidy constraints, any core payoff for the highest cost agent agent n can be obtained by adding the incremental cost c n c n to any core allocation of the airport game where agents n and n have the same cost c n. Therefore, CN, c = c n c n e {n} + CN, c c n c n e {n}. Now, suppose that the agent with the lowest cost leaves the game paying x, with x c. Since c 2 x c 3 x c n x, we have a new airport problem c,x = c 2 x,, c n x = c N\{} x e N\{} with an associated reduced cost game N\{}, c,x. The problem c,x is known as the downstream-substraction reduced problem of c with respect to N\{} and x Thomson, 27. In general, given i N, and x i c i, the downstream-substraction reduced problem of c with respect to N\{i} and x i, c i,xi, is defined by { c c i,xi k x i k > i k = min{c i x i, c k } k < i. Similarly, the uniform-substraction reduced problem of c with respect to N\{i} and x i is defined by: { c uc i,xi k x i k > i k = max{c k x i, } k < i. The next proposition, whose proof is straightforward, relates the core of the airport game N, c and the core of the N\{}, c,x reduced games. 3

5 Proposition. Let N, c be an airport game. Then, CN, c = { x, x N\{} R n : x c, x N\{} CN\{}, c,x } = x c {x } CN\{}, c,x. Now, if we repeatedly apply the above decomposition, the core of the airport game can be covered with the cores of reduced games of s agents, s n. In particular, we have the following result for s = n. Corollary. Given an airport game N, c, the allocation x,, x n belongs to CN, c if and only if, for each j N\{n}, c j i<j x i, and x n = c n i<n x i. González-Díaz and Sánchez-Rodríguez 28 associate a face game with each face of the core and show that these games have interesting properties for the class of strictly convex games. In terms of cost games, given a coalition T N, the face F T of the core contains the allocations that are worst for T and best for N\T. Geometrically, when < x < c, the core CN\{}, c,x is a cross-section of the airport core, which is also the core of a reduced airport game. The reduced games for the cases x = c and x = are the face games for agent and coalition N\{}, respectively. Remark. In general, the core of an airport game N, c can be described using reduced airport games with respect to other players. Let i N. Then, CN, c = {x i } C N\{i}, c i,x i. x i c i 3 The core-center and its integral representation As we said in the Introduction, one of the main goals of this paper is to gain insights on the monotonicity of the core of an airport problem using its center of gravity as a proxy. Given a cooperative game N, c, its core-center, µn, c, is defined as the center of gravity of the core González-Díaz and Sánchez-Rodríguez, 27. In this section and the next we develop some analytic tools that exploit the structure of the core of an airport game to facilitate the study of the properties of the core-center. Given a convex set A we sometimes use the notation µa to denote its center of gravity. In the case of an airport game N, c with c >, the core is an n -manifold contained in the efficient hyperplane, xn = c n. The latter is, therefore, the tangent space at each point of the manifold. The vector,,, R n is normal to the manifold at each point and it has length n. The transformation g : R n R n, gx,, x n = x,, x n, c n x x n defines a coordinate system for CN, c, so that g CN, c is the projection of the core onto R n that simply drops the n-th coordinate. Let ĈN, c = g CN, c R n. This transformation is illustrated in Figure. Given r n, let m r be the r-dimensional Lebesgue measure. Then, the n -dimensional measure of the core is given by m n CN, c = CN,c dm n = g CN,c n dmn = n m n ĈN, c. Hence, the volume of the core as a subset of R n is n times the volume of its projection onto R n. Analogously, for each i N, the corresponding component of the core-center, µ i N, c, is given by x i dm n = nxi dm n = x i dm n. m n CN, c CN,c m n CN, c ĈN,c m n ĈN, c ĈN,c Example. Consider the airport problem with N = {, 2} and < c c 2. Clearly, the core of the airport game is the segment [, c 2, c, c 2 c ] R 2 x, then µ N, c = dx c = c dx 2 and µ 2N, c = c 2 c 2. 4

6 IN, c c 3 CN, c c 2 ĈN, c c c 2 c Figure : left CN, c and ĈN, c for a 3-player airport game. right ĈN, c for a 4-player airport game. The above integral expression for the core-center can be written in terms of iterated integrals. First, we introduce some notation. Given < c c 2 c n c n and j {,, n }, we define U j n c,, c n = V n c,, c n = c2 x c2 x ˆµ j c,, c n = U j n c,, c n V n c,, c n, n 2 cn x k n 2 cn x k dx n dx 2 dx, dx n dx 2 dx, and with the convention that V =. Clearly, U j n is a homogeneous function of degree n, V n is a homogeneous function of degree n, and ˆµ j is a homogeneous function of degree. Now, applying Corollary, it is easy to derive the following result. Theorem. Let N, c be an airport game such that < c c 2 c n. Then, m n ĈN, c = V n c,, c n. Moreover, for each j {,, n }, µ j N, c = ˆµ j c,, c n and µ n N, c = ˆµ n c,, c n + c n c n. Note that all the coordinates of the core-center, except the last one, are independent of c n. In addition, all the coordinates µ j are homogeneous functions of degree. Remark 2. The decompositions in Remark give rise to alternative integral expressions for the core-center. 4 On the differentiability of the core-center Suppose that < c c 2 c n. From the integral expressions derived in the previous section it is clear that the functions V n and U j n, j N\{n}, can be differentiated with respect to the c i costs, with i N\{n}. As a first consequence, we obtain a result that is fundamental for the analysis in this paper, namely, a representation of the core-center as a ratio of volumes of airport cores. We relegate its proof to the Appendix. Theorem 2. Let N, c be an airport game such that < c c 2 c n and fi N \ {n}. Then: 5

7 . U j n c,, c n = V n c,, c j, c j,, c n 2. ˆµ j c,, c n = V nc,, c j, c j, c n V n c,, c n Remark 3. Apart from being a key tool for the ensuing analysis, Theorem 2 is interesting on its own. There are no known efficient deterministic algorithms for computing the centroid of a convex body. Therefore, the issue of computing the core-center of a general balanced game is very complex. The second statement in Theorem 2 says that the center of gravity of the core of an airport game can be computed just by using the volume of its core and the volume of the core of the airport game obtained by replicating agent j. Then, Theorem 2 opens the door to implementing volume computation algoritms for convex polytopes to compute, in a relatively easy way, the core-center of an airport game. Now, we turn our attention the derivatives of the functions V n and U j n with respect to the costs c i. Proposition 2. For all i N\{n}, V n c i c,, c n = V i c,, c i V n i c i+ c i,, c n c i. Proof. A direct computation using Leibnitz s rule shows that V n c i c,, c n = i 2 ci x k ci+ c i n 2 cn c i x k k=i+ This iterated integral can be split as the product of the following two integrals, dx n dx i+ dx i dx. i 2 ci x k dx i dx and ci+ c i n 2 cn c i x k k=i+ dx n dx i+. The first one coincides with V i c,, c i while the second is equal to V n i c i+ c i,, c n c i. Proposition 3. Let i, j N\{n}. Then, U j n c i c,, c n = V i c,, c i U j i n i c i+ c i,, c n c i U j i c,, c i V n i c i+ c i,, c n c i i < j i > j V i c,, c i V n i c i+,, c n i = j. Moreover, V i c,, c i V n i c i+,, c n can be equivalently written as. i c i V i c,, c i Ui c k,, c i V n i c i+ c i,, c n c i Proof. The computations when either i j are straightforward from the chain rule, Theorem 2 and Proposition 2. The same applies to the first equality in the case j = i. The alternative expression is obtained by directly applying Leibnitz s rule to the integral formulation of U j n. 6

8 Next, we present a series of results that relate the partial derivatives of the functions V n and U j n to the faces of ĈN, c and its centroids. In particular, the derivative of V n with respect to a cost c i is proportional to the volume of the corresponding face of the core. Let N, c be an airport game such that < c c 2 c n. Denote by F i, i N\{n}, the i-th face of ĈN, c, i.e., F i = ĈN, c {x Rn : x + + x i = c i } R n. Let V F i be its n 2-measure and µf i be its centroid. The next decomposition of F i is easily derived. Proposition 4. For all i N\{n}, we have that F i = C{,, i}, c,, c i Ĉ{i +,, n}, c i+ c i,, c n c i. The coordinates of the centroid µf i are { µ j {,, i}, c,, c i if i j µ j F i = ˆµ j i c i+ c i,, c n c i if i < j n. We now show that, whenever the n 2-measure of the face F i is positive, its centroid can be obtained using the partial derivatives computed above. Proposition 5. Let N, c be an airport game with < c c 2 c n. For all i, j N\{n} such that V F i >,. V n c i c,, c n = i V F i. 2. µ j F i = U j n c i c,, c n V n c i c,, c n. Proof. Assume that V F i >. Recall that C{,, i}, c,, c i is an i -dimensional polytope contained in the hyperplane x + + x i = c i, so m i C{,, i}, c,, c i = i m i Ĉ{,, i}, c,, c i = i V i c,, c i. On the other hand, the measure of Ĉ{i+,, n}, c i+ c i,, c n c i as a subset of R n i is V n i c i+ c i,, c n c i. Therefore, the first assertion follows immediately from Proposition 2 and the decomposition of Proposition 4. The proof of the second property is divided in three cases. First, assume that i < j. Then, by Proposition 4, µ j F i = ˆµ j i c i+ c i,, c n c i = U j i n i c i+ c i,, c n c i V n i c i+ c i,, c n c i = where the last equality follows from Propositions 2 and 3. If i > j, then, as above, by Propositions 2, 3 and 4, we have µ j F i = µ j {,, i}, c,, c i = ˆµ j c,, c i = U j i c,, c i V i c,, c i = U j n c i c,, c n V n c i c,, c n. U j n c i c,, c n V n c i c,, c n, 7

9 Finally, when j = i, by Proposition 4 Therefore, i µ i F i = µ i {,, i}, c,, c i = c i µ k {,, i}, c,, c i i i Ui k = c i ˆµ k c,, c i = c i c,, c i V i c,, c i i µ i F i V i c,, c i = c i V i c,, c i Ui c k,, c i. Substituting this expression in Proposition 3 and using Proposition 2, the result follows. Remark 4. The first equality of Proposition 5 admits a generalization for any convex polyhedron Lasserre, Properties of the core-center Following Thomson 27, we recall a list of properties of allocation rules for airport problems. We distinguish between fixed-population axioms and variable-population ones. Fixed population Let ψ be a rule and N, c an airport game. We say that ψ satisfies: Non-negativity if, for each i N, ψ i N, c. Cost boundedness if, for each i N, ψ i N, c c i. Efficiency if i N ψ in, c = c n. No-subsidy if, for each S N, i S ψ in, c max i S c i. Anonymity if, for each permutation π of N and each i N, ψ i πn, c = ψ π N, c. i Equal treatment of equals if, for each i, j N with c i = c j, then ψ i N, c = ψ j N, c. Order preservation for contributions if, for each pair i, j N with c i c j, then ψ i N, c ψ j N, c. Order preservation for benefits if, for each pair i, j N with c i c j, then c i ψ i N, c c j ψ j N, c. We now turn to relational requirements on rules. Let ψ be a rule and N, c, N, c and N, c airport games. We say that ψ satisfies: Homogeneity if, for each α >, ψn, αc = αψn, c. Continuity if, for each sequence {N, c ν } ν N of airport problems such that c ν ψn, c. c, then ψn, c ν Independence of at-least-as-large costs if for each i N such that c i = c i, c j = c j, for each j N\{i} such that c j < c i and 8

10 c j c i, for each j N\{i} such that c j c i, then ψ i N, c = ψ i N, c. Last-agent cost additivity if for each i N such that c i = ma N c j whenever c N\{i} = c N\{i} and c i = c i + γ, then ψ N\{i} N, c = ψ N\{i} N, c and ψ i N, c = ψ i N, c + γ. weak last-agent cost additivity is property weaker than last-agent cost additivity, that demands that the payment required to the last agent should increase by an amount equal to the increase in her cost parameter, nothing being said about the payments required of the others. Conditional cost additivity if ψn, c + c = ψn, c + ψn, c whenever the agents of both airport games are ordered in the same way,. Individual cost monotonicity if for each i N such that c i c i and, for all j N\{i}, c j = c j, then ψ i N, c ψ i N, c. Downstream cost monotonicity if for each i N such that for each j N with c j < c i, c j = c j and for each j N with c j c i, c j c j = c i c i, then for each j N such that c j c i, ψ j N, c ψ j N, c. Marginalism if, under the hypotheses of downstream cost monotonicity, for each j N such that c j < c i, ψ j N, c = ψ j N, c. Strong cost monotonicity if, for each pair N, c and N, c with c c, then ψn, c ψn, c. Weak cost monotonicity if c = c + c, then ψn, c ψn, c. Incremental no subsidy if c = c + c then, for each i N, c j c i ψ j N, c ψ j N, c c i c i. Reciprocity if for each i N such that j i ψ in, c = c i, there is an airport problem c such that c = c + c, and c i j i ψ in, c c n c i c n c i, then there is a pair {j, k} N such that c j c i < c k and ψ j N, c ψ j N, c ψ k N, c ψ k N, c. Others-oriented cost monotonicity if, under the assumptions of individual cost monotonicity, for each j N\{i}, ψ j N, c ψ j N, c. Variable population Let ψ be a rule and N, c an airport game. We say that ψ satisfies: Population monotonicity if, for each N and N with N N, ψ N N, c ψn, c N. First-agent consistency if, for each N, c and j N with j >, then ψ j N, c = ψ j N\{}, c,ψn,c. Downstream-subtraction consistency if, for each N, c, each i N and each j i, then ψ j N, c = ψ j N\{i}, c i,ψin,c. Last-agent consistency if, for each N, c and each j < n, ψ j N, c = ψ j N\{n}, c n,ψnn,c. Uniform-substraction consistency if, for each N, c with c n = c n, each i N with c i = c n and each j i, then ψ j N, c = ψ j N\{i}, uc i,ψin,c. 9

11 Let us begin our analysis of which properties the core-center rule satisfies. A complete recapitulation of our findings will be presented in Section 7. Proposition 6. The core-center satisfies non-negativity, cost-boundedness, efficiency, no-subsidy, anonymity, homogeneity, equal treatment of equals and continuity. Proof. The first six properties follow from the fact that any core allocation satisfies them. A couple of comments on the last two properties are needed. I González-Díaz and Sánchez-Rodríguez 27 prove that the core-center treats symmetric players equally and that it is a continuous function of the values of the characteristic function. In our context, equal treatment of equals holds because agents with the same cost parameter are symmetric players in the associated airport game. Similarly, continuity holds because the values of the characteristic function are continuous with respect to the cost parameters. Then, the core-center satisfies continuity since it is a composition of continuous functions. Proposition 7. The core-center satisfies order preservation for contributions. Proof. Trivially, by Theorem, µ n N, c µ n N, c. Now, take two consecutive agents i and i + where i < n. Then, by Theorem 2, µ i N, c µ i+ N, c if and only if which is immediate since c i c i+. V n c,, c i, c i, c i+,, c n V n c,, c i, c i+, c i+,, c n, Proposition 8. The core-center satisfies order preservations for benefits. Proof. Recall that µ n N, c µ n N, c = c n c n. Let i < n. We have to prove that µ i+ N, c µ i N, c c i+ c i. Applying Theorem 2, the difference µ i+ N, c µ i N, c can be written as follows: µ i+ N, c µ i N, c = = c2 x c2 x i ci i ci+ c i i i ci+ c i i V n c,, c n n cn dx n dx 2 dx i+ i+ V n i ci+,, c n dxi+ dx i dx V n c,, c n Now, by the mean-value theorem for integrals, there exists a point ξ c i i ci+ c i i But, since c s i+ i+ V n i ci+,, c n dxi+ = c i+ c i V n i ci+ i ξ c s c i for all i + s n, we have that V n i ci+ i ξ,, c n i, c i+ i i ξ,, c n i ξ V n i c i+ c i,, c n c i. such that i ξ.

12 Therefore, µ i+ N, c µ i N, c c i+ c i = c i+ c i c i+ c i where the last inequality holds because, i ci i ci i ci V n i c i+ c i,, c n c i dx i dx V n c,, c n V n i c i+ c i,, c n c i dx i dx V n i c i+ i,, c n i x i c i implies that c s c i c s Example 2. The core-center does not satisfy independence of at least-as-large costs. i dx i dx i x i for i + s n. problem with N = {, 2, 3} and < c c 2 c 3. A simple computation shows that µ N, c = c 3 3c 2 2c 2c 2 c. Let c =, 2, 3 and c =, 3, 3. Then, µ N, c = 4 9 < µ N, c = 7 5. Proposition 9. The core-center satisfies last-agent cost additivity. Consider the airport c2 x xdydx = c2 x dydx Proof. Let N, c and N, c be airport games satisfying the hypothesis of last-agent cost additivity. Then, n n x,, x n, c n x i CN, c x,, x n, c n x i CN, c. i= Hence, ĈN, c = ĈN, c and µ N\{n} N, c = µ N\{n} N, c. By Theorem, µ n N, c = µ n N, c + c n c n = µ n N, c + c n + γ c n = µ n N, c + γ. Remark 5. If an allocation rule ψ satisfies efficiency and last-agent cost additivity, then ψ n N, c = 2 n 2 cn ψ i N, c. Example 3. The core-center does not satisfy conditional cost additivity. From Example 2, we know that for an airport problem with N = {, 2, 3} and < c c 2 c 3, µ N, c = c 3c 2 2c 3 2c 2 c. Again, let c =, 2, 3 and c =, 3, 3. Then, c + c = 2, 5, 6 and µ N, c + µ N, c = = 4 45 µ N, c + c = 2. Proposition. The core-center satisfies neither last-agent consistency, nor uniform substraction consistency, nor downstream substraction consistency. Proof. The result follows from the following characterizations and the fact that the core-center satisfies equal treatment of equals, homogeneity and last-agent cost additivity: The slack maximizer rule is the only rule satisfying equal treatment of equals, weak last-agent cost additivity and last-agent consistency Yeh, 23. The constrained equal benefits rule is the only rule satisfying equal treatment of equals, homogeneity, last-agent cost additivity and uniform-subtraction consistency Potters and Sudhölter, 25. i= i=

13 The slack maximizer rule is the only rule satisfying equal treatment of equals, homogeneity, last-agent cost additivity and downstream-subtraction consistency Potters and Sudhölter, 25. Example 4. The core-center does not satisfy first-agent consistency. Let N = {, 2, 3, 4} and c =, 2, 3, 4. Then µ N, c = 27 64, µ 2N, c = and c,µc = c 2 µ c, c 3 µ c, c 4 µ c = 64, 65 64, An easy computation shows that µ 2 N \ {}, c,µc = µ 2N, c. Nevertheless, the core-center satisfies first-agent consistency for 3-player airport games. Indeed, if N, c = {, 2, 3}, c, c 2, c 3, c c 2 c 3, then µ 2 N, c = c 2 µ N, c = µ 2 {2, 3}, c2 µ N, c, c 3 µ N, c 2 µ 3 N, c = c 3 c 2 + µ 2 N, c, c 2, c 2 = c 3 c 2 + µ 2 N, c, c 2, c 3 = c 3 c 2 + c 2 µ N, c 2 = µ 3 {2, 3}, c2 µ N, c, c 3 µ N, c. Example 5. The core-center does not satisfy strong cost monotonicity. Indeed, let N = {, 2, 3} and c =, 2, 4. Then µn, c = 4 9, 7 9, Now, for the airport problem with costs c =, 3, 4, µn, c = 7 5, 9 5, Thus, c c but µ 3 N, c < µ 3 N, c. Example 6. The core-center does not satisfy marginalism. Consider the airport problems with players N = {, 2, 3} and costs c =, 2, 3 and c =, 3, 4 that satisfy the hypothesis of downstream cost monotonicity with i = 2. Their respective core-centers are µn, c = 4 9, 7 9, 6 9 and µn, c = 7 5, 9 5, 34 5 so, in particular, µ N, c µ N, c. 6 On the monotonicity of the core-center In this section we discuss the behavior of the core-center with respect to well known monotonicity properties. Moreover, we also introduce two natural monotonicity properties that have not been studied before in the literature, higher cost decreasing monotonicity and lower cost increasing monotonicity. Throughout this section, when no confusion arises we often use notations such as µ j c,, c p instead of µ j {,, p}, c,, c p. Some of the forthcoming results rely on the following connection between the monotonicity of the core-center components and the centroid of the F i faces. Proposition. Let N, c be an airport game with < c and i, j N\{n}. Then, µ j N, c is increasing with respect to c i if and only if µ j N, c µ j F i. Analogously, it is decreasing if and only if µ j N, c µ j F i. Proof. Recall that µ j N, c = ˆµ j c,, c n and, by Theorem 2, ˆµ j c i c,, c n = U j n c i c,, c n V n c,, c n U j n c,, c n Vn c i c,, c n V n c,, c n 2. Thus, µ j N, c is increasing with respect to c i if and only if the numerator is positive. Now, by Proposition 5 the latter is equivalent to µ j F i = U j n c i c,, c n V n c i c,, c n U j n c,, c n V n c,, c n = µ jn, c. 2

14 We now present two new monotonicity properties for a rule. The first one states that if one single cost c i increases while the others are held constant, then the rule decreases its value for all the players with costs higher than c i. The second property is a kind of reciprocal, the allocation rule increases its value for all the players with costs lower than c i. Definition. Let ψ be a rule. Suppose that we have two airport games N, c and N, c, and i N such that c i c i and for all j N\{i}, c j = c j. Then, ψ satisfies higher cost decreasing monotonicity if ψ j N, c ψ j N, c whenever c j > c i. ψ satisfies lower cost increasing monotonicity if ψ j N, c ψ j N, c whenever c j c i. Proposition 2. Let N, c be an airport game with < c and i, j N\{n}, i < j. The component µ j N, c is decreasing with respect to c i if and only if ˆµ j c,, c n ˆµ j i c i+ c i,, c n c i. Proof. According to Proposition, µ j N, c is decreasing with respect to c i if and only if µ j N, c µ j F i. Now, the result is a direct application of Proposition 4 when i < j. The HCDM property for the core-center is a consequence of Theorem 4, which is a particular case of Theorem 3 below, whose proof is relegated to the Appendix. Theorem 3. For all p, k N such that k p, and all < δ d d p d k, we have that ˆµ p d δ,, d p δ,, d k δ ˆµ p+ δ, d,, d p,, d k. Theorem 4. Given j {2,, n } and costs < c c n, we have that ˆµ j c,, c n ˆµ j c 2 c,, c n c ˆµ c j c j,, c n c j. Proof. This is a particular case of Theorem 3. Given j {2,, n } and r {,, j 2}, the inequality ˆµ j r c r+ c r,, c n c r ˆµ j r c r+2 c r+,, c n c r+ follows by taking k = n r 2, p = j r and δ, d,, d k = c r+ c r, c r+2 c r,, c n c r. Proposition 3. The core-center satisfies higher cost decreasing monotonicity. Proof. As a corollary of Proposition 2 and Theorem 4 we have that if i, j N\{n}, i < j, then µ j N, c is decreasing with respect to c i. We now move to LCIM. First, note that it implies individual monotonicity. The LCIM property for the core-center is a consequence of Theorem 5 below, whose proof is relegated to the Appendix. Theorem 5. Given p, s N and costs < c c p c p+ c p+s, we have that µ p c,, c p µ p c,, c p, c p+ µ p c,, c p, c p+,, c p+s Proposition 4. The core-center satisfies lower cost increasing monotonicity. Proof. Let N, c be an airport game such that < c c 2 c n. Then the core-center satisfies LCIM if and only if µ j N, c is increasing with respect to c i for all j i n. First, assume that j i < n. By Proposition 4, µ j F i = µ j {,, i}, c,, c i. Now, according to Theorem 5, µ j c,, c j µ j c,, c j,, c i µ j c,, c n = µ j N, c, and LCIM is now a direct consequence of Proposition. As for the case j i = n, we already know that µ j N, c, j =,, n, is independent of c n, and that µn c n N, c =. 3

15 Corollary 2. The core-center satisfies individual cost monotonicity. Example 7. The core-center does not satisfy others-oriented cost monotonicity. To see this one can consider, for instance, the two problems in Example 2, where an increase in the cost of player 2 results in a lower core-center payoff for player. The next monotonicity property, downstream cost monotonicity, is a consequence of Theorem 6 below, whose proof is relegated to the Appendix. Theorem 6. Given indices i, j N, j i, a value γ > and costs < c c k, we have that µ j c, c 2,, c i + γ,, c k + γ µ j c, c 2,, c k. Proposition 5. The core-center satisfies downstream cost monotonicity. Proof. Let N, c be an airport game such that < c c 2 c n. Observe that downstream cost monotonicity can be rewritten as follows. If for each pair N, c and N, c and each i N, if for each j N such that c j < c i, c j = c j and each j N such that c j c i, c j = c j + γ γ, then for each j N such that c j c i, ψ j N, c ψ j N, c. Thus, and the result is a direct consequence of Theorem 6. N, c : c c 2 c i c i c n N, c : c c 2 c i c i + γ c n + γ, Proposition 6. Let ψ a rule satisfying downstream cost monotonicity and LCIM. Then, ψ satisfies weak cost monotonicity. Proof. We have to prove that, for each pair N, c and N, c, if there exists N, c such that c = c + c, then ψn, c ψn, c. Consider the following airport problems: Problem Costs c = c c c 2 c i c n c c + c c 2 + c c i + c c n + c c 2 c + c c 2 + c + c 2 c c i + c + c 2 c c n + c + c 2 c.. c i c + c c 2 + c 2 c i + c i c n + c i.. c n = c c + c c 2 + c 2 c i + c i c n + c n Now, noting that c = c n and combining downstream cost monotonicity and LCIM we have ψ N, c n LCIM LCIM ψ N, c i LCIM LCIM ψ N, c DOWN ψ N, c ψ 2 N, c n LCIM LCIM ψ 2 N, c i LCIM DOWN ψ 2 N, c DOWN ψ 2 N, c. ψ i N, c n LCIM LCIM ψ i N, c i DOWN DOWN ψ i N, c DOWN ψ i N, c. ψ n N, c n DOWN DOWN ψ n N, c i DOWN DOWN ψ n N, c DOWN ψ n N, c. Corollary 3. The core-center satisfies weak cost monotonicity. Proposition 7. The core-center satisfies neither reciprocity nor incremental no subsidy. 4

16 Proof. The result follows from the following characterizations in Aadland and Kolpin 998 and the fact that the core-center satisfies no-subsidy, order preservation for contributions and weak cost monotonicity: The constrained equal contribution rule is the only selection from the no-subsidy correspondence satisfying order preservation for contributions, weak cost monotonicity, and reciprocity. The sequential equal contributions rule is the only rule satisfying order preservation for contributions, weak cost monotonicity, and incremental no subsidy. Proposition 8. The core-center satisfies population monotonicity. Proof. We prove the result for the case in which there is k N such that N = N {k}. The general case follows from repeated application of that property. Thus, given N = N\{k}, we prove that µ N N, c µn, c N. We distinguish three cases. Case : c k = c n. So, for each i N, c i c k = c n. By Theorem 5, for each i N, µ i N, c N = µ i c,, c n µ i c,, c n, c n = µ i N, c. Case 2: c k = c. So, for each i N, c i c k = c. Now, for each ε, let c ε = ε, c 2,, c n. Clearly, µn, c N = µ N N, c. By HCDM, for each ε, c ], µ N N, c µ N N, c ε and, by continuity, µ N N, c µ N N, c. Therefore, µ N N, c µ N N, c = µn, c N. Case 3: c < c k < c n. Let i N. We distinguish two subcases. c i > c k : Consider the airport problems N, c ε, with c ε = ε, c,, c k, c k+, c k+2,, c n and ε, c ]. Similarly to Case 2, HCDM and continuity ensure that µ i N, c N = µ i N, c µ i N, c ε. Combining this with a repeated application of HCDM, we have µ i N, c N = µ i N, c µ i c, c, c 2,, c k, c k+, c k+2,, c n µ i c, c 2, c 2,, c k, c k+, c k+2,, c n. µ i c, c 2, c 3,, c k, c k, c k+,, c n = µ i N, c. c i c k : In the case in which c i = c k we assume, without loss of generality, that i < k. Now, applying LCIM repeatedly, and, by Case, µ i N, c N = µ i c,, c k, c k+, c k+2,, c n µ i c,, c k, c k, c k+,, c n, µ i c,, c k, c k, c k+,, c n µ i c,, c k, c k, c k+,, c n, c n = µ i N, c. Combining the inequalities in both equations we get that µ i N, c N µ i N, c. 7 Summary of properties To conclude, we present a table that summarizes the behavior of the core-center with respect to the properties we have studied. 5

17 Acknowledgments Rules Properties Shapley Nucleolus Core-center Fixed population Non negativity Cost boundedness Efficiency No-subsidy Anonymity Equal treatment of equals Order preservation for contributions Order preservation for benefits Homogeneity Continuity Independence at-least-as-large costs - - Last-agent cost additivity Weak last-agent cost additivity Conditional cost additivity - - Individual cost monotonicity Downstream cost monotonicity Marginalism - - Strong cost monotonicity Weak cost monotonicity Incremental no-subsidy - Reciprocity Others-oriented cost monotonicity - - Variable population Population monotonicity First agent consistency - Downstream substraction consistency - - Last-agent consistency - - Uniform substraction consistency We want to thank William Thomson for his encouragement and insight during the writing process of this manuscript. Authors acknowledge the financial of the Spanish Ministry for Science and Innovation through projects MTM C3, ECO C4ECON, and from the Xunta de Galicia through project INCITE PR. References Aadland, D. and Kolpin, V Shared irrigation costs: An empirical and axiomatic analysis. Mathematical Social Sciences, 35: Gillies, D. B Some Theorems on n-person Games. PhD thesis, Princeton. González-Díaz, J. and Sánchez-Rodríguez, E. 27. A natural selection from the core of a tu game: the core-center. International Journal of Game Theory, 36:

18 González-Díaz, J. and Sánchez-Rodríguez, E. 28. Cores of convex and strictly convex games. Games and Economic Behavior, 62: 5. Lasserre, J. B An analytical expression and an algorithm for the volume of a convex polyhedron in R n. Journal of Optimization Theory and Applications, 39: Littlechild, S. C. and Owen, G A simple expression for the Shapley value in a special case. Management Science, 2: Potters, J. and Sudhölter, P. 25. Airport problems and consistent solution rules. Mathematical Social Sciences, 38:83 2. Thomson, W. 27. Cost allocation and airport problems. Rochester Center for Economic Research Working Paper. Yeh, C. 23. Efficienty, consistency, and cost additivity for a class of cost allocation problems. Technical report, Academia Sinica. Appendix In the first part of this Appendix we present the proof of Theorem 2. This result is then crucial to prove also Theorems 3, 5, and 6, which establish the main monotonicity properties satisfied by the core center. Theorem 2. Let N, c be an airport game such that < c c 2 c n and fi N \ {n}. Then:. U j n c,, c n = V n c,, c j, c j,, c n 2. ˆµ j c,, c n = V nc,, c j, c j, c n V n c,, c n Before engaging in the proof of Theorem 2 we need some preliminary results. Recall the convention V =. Lemma. Given < c c k, k N, we have that V c = c, V 2 c, c 2 = c2 2 2 c2 c2 2, and, if k 3, V k c,, c k = ck k k! c k c k k! k Proof. The expressions for V c and V 2 c, c 2 are straightforward. c3 x 2 2 c3 c22 2 i=2 c k c i k i+ V i c,, c i. k i +! V 3 c, c 2, c 3 = c V 2c 2 x, c 3 x dx = c proceed by induction. Let k N, k > 3, and assume that the result holds for k. Then, V k c,, c k = = V k c 2 x,, c k x dx ck x k k! c k c 2 k k! We compute separately the integral of each addend: k 2 i=2 c k c i+ k i k i! The result also holds for k = 3, since dx = c3 3 3! c3 c3 3! c3 c22 2 c. We V i c 2 x,, c i x dx. c k x k k! dx = ck k k! c k c k, k! c k c 2 k k! V i c 2 x,, c i x dx = V i c,, c i dx = c k c 2 k V c, k! and 7

19 By the linearity of the integral, V k c,, c k = ck k k! c k c k k! = ck k k! c k c k k! c k c 2 k k! k 2 i= k 2 V c i=2 c k c i+ k i V i c,, c i, k i! c k c i+ k i V i c,, c i k i! which, after a simple rearrangement of the indices, coincides with the desired expression. Lemma 2. Let < c c k, k N, x c, and denote u k x = V k c x,, c k x. Then, du k dx x = V k c 2 x,, c k x. Proof. The proof is by induction. The property holds for k =, since u x = V c x = c x and du dx x =. It also holds for k = 2, because u 2 x = c2 x2 2 c2 c2 2 and du2 dx x = c 2 x. Now, let k 3 and suppose that the property is true for u i, i k. Then, according to Lemma, u k x = V k c x,, c k x = c k x k Differentiating with respect to x, k! c k c k k! k i=2 c k c i k i+ u i x. k i +! du k x = c k x k dx k! k i=2 c k c i k i+ k i +! du i dx x. By the induction hypothesis, if i 2, dui dx x = V i 2 c 2 x,, c i x. Therefore, du k x = c k x k + c k c 2 k dx k! k! Renumbering the terms, du k ck x k x = c k c 2 k dx k! k! = V k c 2 x,, c k x. where the last equality follows directly from Lemma. Proof of Theorem 2. First observe that V n c,, c j, c j, c n = If u n j = V n j c j j x k,, c n j cj x k k + i=3 k 2 i=2 c k c i k i+ V i 2 c 2 x,, c i x. k i +! V n j cj c k c i+ k i k i! V i c 2 x,, c i x j x k,, c n j x k then, by Lemma 2, du n j d = V n j c j+ 8 j x k,, c n j x k. j x k dxj dx.

20 Integrating by parts, j cj x k [ V n j cj V n j cj j x k,, c n j x k,, c n j ] c j x k j x k dxj = j x k + j cj x k V n j cj+ j x k,, c n The bracketed expression vanishes, since V n j, c j+ c j,, c n c j =. Consequently, j cj x k Finally, V n j c j j x k,, c n V n c,, c j, c j, c n = j x k d = j cj x k j cj x k = U j n c,, c n. V n j c j+ V n j c j+ This last result follows from the previous property and the definition of ˆµ. j x k,, c n j x k,, c n j x k dxj. j x k d. j x k d dx The proofs of the remaining theorems in this Appendix follow the same basic structure. We know, by Proposition, that the monotonicity of the core-center with respect to costs can be established by checking the relative position of the core-center of the game and the centroid of the faces of the core. So, we have to prove inequalities of the type ˆµ p c,, c k ˆµ p+ d,, d k+. But, by Theorem 2, that is equivalent to proving an inequality such as = V k+ c,, c p, c p,, c k V k+ d,, d k+ V k c,, c k V k+2 d,, d p+, d p+,, d k+. Then, we try to decompose each of the four volumes in in terms of volumes of certain manageable types. Finally, we rearrange as a sum of expressions involving these types of volumes and study, by induction, their sign. It turns out that the manageable volumes for Theorem 3 and Theorem 6 are of the same type see Proposition 2 while for Theorem 5 different volumes are needed see Proposition 24. Therefore, we develop the proof of Theorem 3 in full detail and just sketch the one for Theorem 6. We finish with the proof of Theorem 5. An expression like V p+s c,, c p,. s.., c p means that cost c p is repeated s times. When all the costs are the same we write V k c,, c instead of V k c,. k.., c. In order to prove Theorem 3, our first step is to derive some results involving volumes of this type. Lemma 3. For all k N and α, V k α,, α = αk k!. Proof. Clearly, the property holds for k =. Assume that the result is true for k and proceed by induction. Then, α α α x k V k α,, α = V k α x,, α x dx = dx = αk k! k!. Lemma 4. Given k N and < α β c c k, we have that β α V k c x,, c k x dx = V k+ β α, c α,, c k α. 9

21 Proof. The result is true for k = since β α c x dx = c α2 for all i < k. Then, β α V k c x,, c k x dx = β α c k x k k! = c k α k+ k +! β dx α c k β k+ k +! c k c k k! c k c k k! k dx i=2 c k c i k i+ k i +! k β α i=2 2 c β2 β α 2. Assume that the equality holds V i c x,, c i x dx c k c i k i+ V i β α, c α, c i α, k i +! where the first equality holds by Lemma and the second by the induction hypothesis. Again by Lemma, the last expression equals V k+ β α, c α,, c k α. The following result allows to decompose any given volume in terms of volumes with repeated costs. Proposition 9. If < α c c k, k N, then k. V k c,, c k = V i α,, αv k i c i+ α,, c k α, i= 2. V k c α,, c k α = 3. V k c α,, c k α = k V i c α,, c αv k i c i+ c,, c k c, and i= k i=2 Proof.. Let s N such that s k 2 and denote We claim that α Indeed, α I s = s α s α s cs+ I s dx s dx = I s dx s dx = V i c α, c 2 α, i, c 2 αv k i c i+ c 2,, c k c 2. ck k dx k dx s+ = V k s c s+ V s α,, αv k s c s+ α,, c k α + α s α I s+ dx s+ dx + α α s,, c k s α s cs+ α s s. I s+ dx s+ dx 2 I s+ dx s+ dx. Then, in order to prove the claim, we just have to decompose the second addend of the last expression. But, s+ s+ since I s+ = V k s c s+2,, c k, we have, by Lemma 4, s cs+ α s I s+ dx s+ dx = V k s cs+ α, c s+2 α., c k α dx s dx. 2

22 Consequently, α s cs+ α s I s+ dx s+ dx = α s α = V k s cs+ α,, c k α α V k s cs+ α,, c k α dx s dx s α = V k s c s+ α,, c k αv s α,, α. dx s dx Then, Equation 2 holds. Now, we make repeated use of Equation 2 to obtain the first equality of the result. Observe that α V k c,, c k = I dx = I dx + I s dx α = α I dx + α V k c 2 x,, c k x dx = where the last equality holds by Lemma 4. But, now, according to Equation 2, α I dx + V k c α,, c k α, Then, α α α x I dx = I 2 dx 2 dx + V k c 2 α,, c k αv α. V k c,, c k = V k c α,, c k α + V αv k c 2 α,, c k α + Next, decompose equality is reached. α α x α α x I 2 dx 2 dx. I 2 dx 2 dx by applying Equation 2, and repeat the process until the intended 2. To prove the second equality, just take A = c α, so that c i α A = c i c, and apply statement. V k c α,, c k α = k V i c α,, c αv k i c i+ c,, c k c. Now, simply observe that, for i =, V k c c, c 2 c,, c k c =. 3. From Lemma 4 and statement. But, by Lemma 4, V k c α,, c k α = = α k i=2 i= V k c 2 x,, c k x dx V k i c i+ c 2,, c k c 2 V i c 2 x,, c 2 x dx. α α V i c 2 x,, c 2 x dx = V i c α, c 2 α, i, c 2 α. 2

23 Now we need some extra notation. Given k N and < c c k, let Z = and Z α s = V s c k s+ α,, c k α, s =,, k, α < c k s+. When no confusion arises, we write Z s instead of Z α s. Remark 6. Let q, k N, q < k, < c c k, and fix α < c k q. Clearly, Z α = V c k α = c k α. Now, let A = and A r = Z c k q+ r = V r c k r+ c k q+,, c k c k q+, r =,, q. Then, by statement 2 of Proposition 9, Z α q = q V i A q i, with V i = V i c k q+ α,, c k q+ α, i =,, q, i= and by statement 3 of Proposition 9, Z α q+ = q+ i=2 V i A q+ i, with V j = V j c k q α, c k q+ α, j, c k q+ α, j = 2,, q +. Lemma 5. For all q, k N, q < k, and < c c k, fix α < c k q. Then, Z α Z α q Z α q+. Proof. We use the notation and decompositions of Remark 6. Clearly, by Lemma 3, V i = c k q+ α i, i =,, q. 3 i! Besides, applying the definition of V i and Lemma 3, V i = In order to prove that ck q c k q α x i i! dx = ck q+ α i c k q+ c k q i i! = V i X i, where X i = i! c k q+ c k q i, i = 2,, q +. 4 q Z Z q Z q+ = ck αv q i V q i+ Ai, we check that, for all i =,, q, c k αv q i V q i+. Certainly, i= c k αv q i V q i+ = c k αc k q+ α q i q i! c k q+ α q i+ q i +! + c k q+ c k q q i+ q i +!, since c k α c k q+ α and q i +! q i!. Proposition 2. Let t, q, k N be such that t q < k and c c k. Fix α < c k q. Then, Zt α Zq α Zt Z α q+ α. 22

24 Proof. We proceed by induction on t N. The case t = has been proved in Lemma 5. Now, assume that the result holds for any i t, i.e., Z β i Zβ j Zβ i Zβ j+, i j < k, β < c k j, 5 and then, we prove that it also holds for t < k. According to the notation and decompositions of Remark 6, Certainly, A t t q Z t Z q Z t Z q+ = V i A t i i= i= t V i A q i i= q+ V i A t i t q = A s A r Vt s V q r V t s Vq+ r + At s= r= q A q r V r. Then, it suffices to prove that r= First, we claim that i=2 V i A q+ i q A q r V r. t q S = A s A r s,r, where s,r = V t s V q r V t s Vq r+. 6 s= r= V i V j V i Vj+, if i j +. 7 Indeed, applying Equality 4 in Lemma 5, we have that V i V j V i V j+ + V i X j+ and V i X j+. Then, it suffices to prove that V i V j V i V j+ whenever i j +. Let B = c k q+ α and apply Equation 3 in Lemma 5, V i V j V i V j+ = Bi i! B j j! B j+ r= Bi i! j +! = i!j! i!j +! Now Equation 7 is straightforward, since i!j! i!j+! if and only if i j +. q q t + r r = s + b. B i+j. t 2 s Figure 2: The straight line r = s + b, with b = q t +. 23

25 Let b = q t + > and T = {s, r [, t 2] [, q ] : r > s + b} N 2 be the set depicted in Figure 2. According to Equation 7, if s, r [, t ] [, q ] but s, r T then A s A r s,r. Now, take r, s T such that A s A r s,r, then h = r s b > and s + h, r h T, because r h s + h + b. In addition, t s h = q r + and q r + h = t s. Therefore, each negative addend A s A r s,r in Equation 6 can be paired with the corresponding A s+h A r h s+h,r h in the following way A s A r s,r + A s+h A r h s+h,r h = A s A r V t s V q r V t s Vq r+ + A s+h A r h V t s h V q r+h V t s h Vq r+h+ = A s A r V t s V q r V t s Vq r+ X q r+ + A s+h A r h V q r+ V t s V q r Vt s X t s = A s A r Vt s V q r V t s V q r+ + As+h A r h Vq r+ V t s V q r V t s + A s A r V t s X q r+ + A s+h A r h V q r X t s = As+h A r h A s A r Vt s V q r+ V q r V t s + As A r V t s X q r+ + A s+h A r h V q r X t s. Therefore, if we prove that the last expression is positive whenever s, r T, then S. Clearly, the last two terms are positive. Since q r + < t s + then, applying Equation 7, we get that V t s V q r+ V q r V t s. It remains to show that for all s, r T, A s+h A r h A s A r. Now, if s, r T, then. s r h s + h r, since s s + h = r b r, s s + b = r h r and r h < s + h + b s + h. 2. s + h t 2, since s + h = r b q b = t 2. Thus, A r h A s+h A s A r = Ar h A s+h A r h A s+h+ + Ar h A s+h+ A r h 2 A s+h As+ A r A s A r. All the expressions in parentheses are of the form A i A j A i A j+ = Z β i Zβ j Zβ i Zβ j+, with i t 2, i j and β = c k q+. Therefore, we can apply Equation 5, the induction hypotheses, and conclude that all the addends A i A j A i A j+ are positive and then A s+h A r h A s A r as well. The next step consists of providing a way to decompose any given volume in terms of volumes involving only the costs up to a fixed c p. Proposition 2. Let p, k N be such that p < k and < c c k. Then, Proof. First we prove that k p V k c,, c k = V k p i c p++i c p,, c k c p V p+i c,, c p, i+, c p. i= V k c,, c k = V k p c p+ c p,, c k c p V p c,, c p + V k c,, c p, c p, c p+2,, c k. 8 Indeed, we know that V k c,, c k = p cp V k p c p+ p,, c k p dx p dx. 9 24