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An egalitarian approach for sharing the cost of a spanning tree

An egalitarian approach for sharing the cost of a spanning tree

  • José-Manuel Giménez-Gómez, 
  • Josep E. Peris, 
  • Begoña Subiza
PLOS
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Abstract

A minimum cost spanning tree problem analyzes the way to efficiently connect individuals to a source. Hence the question is how to fairly allocate the total cost among these agents. Our approach, reinterpreting the spanning tree cost allocation as a claims problem defines a simple way to allocate the optimal cost with two main criteria: (1) each individual only pays attention to a few connection costs (the total cost of the optimal network and the cost of connecting himself to the source); and (2) an egalitarian criteria is used to share costs. Then, using claims rules, we define an egalitarian solution so that the total cost is allocated as equally as possible. We show that this solutions could propose allocations outside the core, a counter-intuitive fact whenever cooperation is necessary. Then we propose a modification to get a core selection, obtaining in this case an alternative interpretation of the Folk solution.

Introduction

We consider a situation in which some individuals, located at different places, want to be connected to a source in order to obtain a good or a service. Each link connecting any two individuals, or connecting each individual to the source, has a specific fixed cost. This situation is known as the minimum cost spanning tree problem (hereafter, the mcst problem) and it is used to analyze different actual issues, such as telephone, cable TV or water supply networks.

There are several methods for obtaining a way of connecting agents to the source so that the total cost of the selected network is minimum. Once the minimum cost network is built, several solutions have been proposed to allocate the cost among the individuals, such as Bird rule [1], Kar [2], Folk [3, 4], Cycle-complete [5], a family of strict responsive rules [6], or the class of egalitarian Shapley value solutions [7]. Some of these solutions take into account the cost of each link in the network, so all the costs are relevant in order to set the final allocation of the optimal cost, although most of them would never be used. Contrary to this trend, we define a model in which only a few costs in the network are considered.

In doing so, our main assumption is that individuals are not worried about costs of links not being used. They have a local vision of the network and only a few information is relevant to each individual. At this point, it is important to remark that cooperation is needed to build the efficient network. That is, if some individual does not agree with the allocation of the cost, this individual could connect to the source on his own and the cheapest network is not built.

Remark 1 Actual situations reveal that agents do not necessarily agree on how to distribute this cost, and in those cases the social optimum is not implemented. Hence, a more expensive network is built (for an example, see [8]; see also [9] for a discussion about individual and social optimality).

Then, we follow a reductionist approach in which some information is not used, or simplified, when obtaining the cost sharing of a mcst problem. Specifically, we suppose that individuals are only concerned about two particular costs:

  1. Their minimum cost to connect to the source (either directly or throughout other individuals).
  2. The total cost of the optimal network.

If there is no cooperation, each agent will connect to the source by himself and the total cost of the network could be much higher than the optimal one (the minimum cost spanning tree).

Our second key point is the use of an egalitarian criterion. Taking into account that no one should pay more than the cost of connecting to the source by himself (individual rationality), our approach will allocate the cost based on an equal distribution of the benefit of cooperation. This approach leads us to solve the allocation problem by using claims rules. Let us observe two problems that illustrate our idea.

  1. (A). Consider a set of three houses in a row (at the same distance each one from the other). A water supply ω is located at one end of the row. The cost of each link between two adjacent houses is 1 monetary unit. The nearest house to the supply may connect directly with a cost of 10 units; the second house has a direct cost of 11 units; and the cost of directly connecting the farthest house is 12 units. The total (minimum) cost of connecting the three houses to the water supply is 12 units. If each individual connects directly to the source, the total cost is 33 units.
    In this situation, the cooperation provides a benefit (savings) of 21 monetary units. The question is how to share this benefit. Note that individual 1 is in a “better position” and a lower payment of this individual would be reasonable. If the benefit of cooperation is equally shared, each agent obtains a return of 7 unitary units, paying, respectively, 3, 4 and 5 monetary units. But an egalitarian cost sharing will propose an allocation of 4 units for each individual.
  2. (B). Consider a similar situation, but now individual 1 is closer to the source, whereas individuals 2 and 3 remain at the same location and the costs vary accordingly (the new situation is as depicted in the following graph):
    As in the previous case, the total (minimum) cost of connecting the three houses to the water supply is 12 units. But the situation is quite different and now cooperation provides a benefit of 12 monetary units. An equal sharing of the benefits originates a negative allocation to individual 1; that is, he obtains a net benefit from his participation in the network. On the other hand, observe that in the new situation an equal allocation of the cost is not admissible for the first individual, since he may connect to the source with a cost of 1 unit, instead of paying 4 units.

As aforementioned, our main criterion in allocating the cost of the optimal tree is that of egalitarian sharing, which is one of the main criterion supported in the literature. In [10] the egalitarian method appear, jointly with the proportional, as the most important (and simple) ways of sharing a joint cost or benefit.

The rest of the paper is organized as follows. Next Section presents the formal minimum cost spanning tree problem. Then, we relate minimum cost spanning tree problems with claims problems, introduce the egalitarian solution concept and analyze its properties. We show that, in general, our proposal may lie outside the core in some situations. In our last Section we propose a modification fulfilling core stability, and we obtain a new interpretation of the Folk solution. Some final comments close the paper.

Preliminaries: Minimum cost spanning tree problem

A mcst problem involves a finite set of individuals, N = {1, 2, …, n}, who want to be connected to a source ω. Let Nω = N∪{ω}. The agents are connected by edges and for ij, represents the cost of the edge eij connecting agents i, jN. Following the notation in [2], cii represents the cost of the edge connecting agent iN to the source ω. Let C = [cij]n×n the n × n symmetric cost matrix. The mcst problem is represented by the pair (Nω, C). We denote by the set of all mcst problems with n individuals.

A spanning tree over is an undirected graph p with no cycles, which connects all elements of Nω. We can identify a spanning tree with a function p: NNω so that p(i) is the agent (or the source) to whom i connects in his path to the source, and defines the edges In a spanning tree each agent is (directly or indirectly) connected to the source ω. Moreover, given a spanning tree p, there is a single path from any iN to the source ω, given by the edges (i, p(i)), (p(i), p2(i)), …, (pt−1(i), pt(i) = ω), for some integer t < n. Let us denote by the set of all spanning trees in the problem (Nω, C). The cost of building a spanning tree is the sum of the costs of all the edges in this tree; that is (with some abuse of notation, when p(i) = ω, cip(i) = cii)

Given a spanning tree , we denote by p(i, j) the set of edges in the (unique) path in p joining i and j.

Prim [11] provides an algorithm that solves the problem of connecting all the agents to the source at the minimum cost. This method has n steps, as much as the number of individuals in the network. First, it connects to the source the agent i with smallest cost to the source, ciicjj, for all jN. In case that more than one agent fulfills this condition, any of them can be selected. In the second step, an agent in N\{i} with the smallest cost either to the source or to agent i, who is already connected, is selected and this connection is used. The process continues until all agents are connected, at each step connecting an agent still not connected to a connected agent or to the source. We denote by m a tree with minimum cost and by Cm its cost. That is, for all spanning tree p,

Once a network is built, an important issue is how to allocate the associated cost among the agents. A cost sharing rule for mcst problems is a function that proposes for any mcst problem (Nω, C) an allocation such that

Remark 2 In some contexts the non-negativity of the cost αi allocated to each individual is required. This question is related to the assumption of property or non-property rights on the locations that individuals occupy (see, for instance, [6] for a discussion). In the second case, non-property rights approach, the allocations must be necessarily non-negative. In what follows we will consider the non-property rights approach.

Bird [1] proposes a cost allocation so that each individual pays the cost of the edge he directly uses to be connected in the minimum cost spanning tree. In case there are several networks providing the (same) minimum cost, the Bird solution allocates to each individual the average of the cost of the connections he uses in these networks. Since then, several authors have proposed other solution concepts in the mcst literature: for instance, [2, 3, 4, 5, 6, 12], etc. (see [13] for definitions and a comparative analysis of most of these solutions).

Some of these solutions take all the possible connections in the graph into account, although most of these connections are not used in the optimal tree. However, other solutions use a reductionist approach and they are obtained only considering some of the connection costs. Specifically, the Bird solution only considers the cost of the link each individual uses in the optimal network while the costs of other edges are ignored. The Folk and Cycle-complete solutions also take a reductionist approach. As the reductionist approach ignores some of the available information it reduces the parameters of the problem (and the computational complexity).

In order to analyze the cost that any individual would incur without cooperation, we introduce the notion of indirect cost.

Definition 1 Given a mcst problem (Nω, C) the indirect cost to connect individual iN to the source ω is

In this context, demanding that the maximum cost to be allocated to any individual cannot exceed his (indirect) cost to the source (individual rationality) is a compulsory requirement because, in other case, the individual would be better off acting by himself and would not cooperate in building the optimal network.

Definition 2 A cost allocation α = (α1, α2, …, αn) of the minimum cost Cm in a cst problem (Nω, C), is individually rational if for all iN,

Whenever cooperation is necessary, as in mcst situations, the literature on cost sharing singles out core stability as the key property of any allocation rule: no coalition of agents should be charged more than their cost of connecting to the source. Then, given a coalition SN, the stand alone cost for this coalition to be connected to the source (in our non-property rights model) is: where Cm(T) denotes the cost of the optimal tree connecting coalition T to the source. Note that for any iN,

Remark 3 It is important to note that the cost function Cm(S), SN, is not monotonic since the addition of some agents may reduce the cost of the coalition. As we follow the non-property rights approach, any coalition S might use locations of individuals outside S to build their minimum cost spanning tree. So, v(S) represents the minimum cost of connecting all individuals in S to the source ω, possibly using (and paying for) connections of individuals outside S. The stand alone cost function v(S) is monotonic.

Definition 3 A cost allocation α = (α1, α2, …, αn) of the minimum cost Cm in a mcst problem (Nω, C) is a core selection if for all SN, S ≠ ∅, .

Egalitarian cost sharing

We propose an egalitarian treatment of the agents. So, a first attempt to allocate the cost of the optimal network, Cm is to divide it equally among the individuals: , i = 1, 2, …, n. However, as Problem (B) depicts, the equal division may be not individually rational: agent 1 is allocated 4 monetary units, whereas his cost to the source is monetary unit.

Individual rationality provides a way to address the problem of allocating the optimal cost in a mcst problem by transforming it into a surplus sharing problem:

  • First, each agent pays his (indirect) cost to connect the source . So, the individuals jointly contribute with the amount to build a network.
  • Then, the efficient tree may be built, with cooperation, at a cost CmC* and there is a benefit from cooperation given by B = C* − Cm.
  • Any method used to share this benefit B, xi ≥ 0, provides a final allocation of the cost Cm, , which is individually rational.

A possible way is to share equally the benefits obtained from cooperation:

However, as Problem (B) shows, equalizing benefits could end in a negative allocation for some agents, that implies that these agents get a net profit from participating in the network. As mentioned in [6] this possibility only has sense if the individuals have property rights on their location. It is noteworthy that if we want to avoid this possibility (since we are in the non-property rights approach) a constrained equal division should be considered: no one obtains a benefit greater than his initial contribution. Then the benefit of cooperation should be shared as in a claims problem in which each individual claims all his contribution to be returned. A claims problem is a situation involving n individuals who claim some amount di, so that the aggregate demand exceeds the available endowment B, . A claims rule φ divides efficiently the endowment so that agents do not receive a negative amount, nor more than their claim. The following definition transforms mcst problems into claims problems (see [14] for a complete survey on claims problems).

Definition 4 To any mcst problem (Nω, C) with minimum cost Cm, we associate the claims problem (B, d), where B = C* − Cm, and .

That is, each agent claims the amount he paid: his indirect cost to the source. If we use a claims rule φ to solve the problem (B, d), then is an individually rational allocation of the cost Cm. The claim boundedness condition, φidi, implies that the allocations are non-negative, which is coherent with the non-property rights approach. As we are interested in an egalitarian cost sharing, we consider the two main egalitarian claims rules: the Constrained Equal Awards (CEA) and the Constrained Equal Losses (CEL), that equalize, respectively, gains and losses satisfying the restrictions of a claims rule. Formally, given a claims problem

Definition 5 Given a mcst problem (Nω, C) such that the cost of the optimal spanning tree is Cm, let B = C* − Cm and ; then, the constrained equal costs sharing rule assigns to each individual iN the amount

This sharing rule allocates the same amount to all agents, constrained to no one is allocated an amount greater than his indirect cost to the source. Example 1 compares this solution with Bird and Folk proposals in the mcst problems (A) and (B).

Example 1 For problems (A) and (B) the following cost shares are obtained:

Taking the Folk solution as a benchmark, we analyze if its main properties are satisfied or not by our egalitarian proposal αceq. First, we formally define these properties (see [4] and [6] for relationships and interpretations of these properties).

  • A solution α for mcstp satisfies positivity if α(Nω, C)≥0, for any problem (Nω, C).
  • A solution α for mcst problems satisfies continuity if it is a continuous function of the cost matrix C.
  • A solution α for mcstp satisfies symmetry if for any problem (Nω, C) such that there are i, jN with cik = cjk, for all kN\{i, j}, cii = cjj, then αi(Nω, C) = αj(Nω, C).
  • A solution α for mcst problems satisfies strong cost monotonicity if for any pair of problems (Nω, C), (Nω, C′) such that CC′, then αi(Nω, C)≤αi(Nω, C′) for all iN.
  • A solution α for mcst problems satisfies cost monotonicity if for any pair of problems (Nω, C), (Nω, C′) such that C and C′ coincide except that , for some i, kN, then αi(Nω, C)≤αi(Nω, C′).
  • A solution α for mcst problems satisfies population monotonicity if for any problem (Nω, C), any subset SN and any iS, αi(Sω, C|S)≥αi(Nω, C).
  • A solution α for mcst problems satisfies ranking if for any problem (Nω, C) and i, jN such that cikcjk for all kN\{i, j}, ciicjj, then αi(Nω, C)≤αi(Nω, C).
  • A mcst problem (Nω, C) is separable if there are two disjoint subsets ST = N, ST = ∅, such that the mcst in N are union of mcst in each of the sub-problems, m(Nω, C) = m1(Sω, C|S)∪m2(Tω, C|T). A solution α for mcst problems satisfies separability if for any separable problem (Nω, C), N = ST, ST = ∅,
  • A solution α for mcst problems satisfies equal share of extra-costs if for any pair of problems (Nω, C), (Nω, C′) such that:
    1. for all iN, cii = c0, .
    2. for all i, jN, ij,

    then , for all iN.
  • Two mcst problems (Nω, C) and (Nω, C′) are tree equivalent if there is a tree m such that it is a minimum cost spanning tree for both problems, and moreover for all iN. A solution α for mcst problems satisfies independence of irrelevant trees if for any pair of tree equivalent problems (Nω, C) and (Nω, C′), then α(Nω, C) = α(Nω, C′).

Proposition 1 αceq fulfills positivity, continuity, cost monotonicity, strong cost monotonicity, independence of irrelevant trees, ranking, symmetry and equal share of extra-costs. It does not fulfill population monotonicity, separability, nor core stability.

Proof. First, we note that our sharing rule may be written alternatively as:

(1) Positivity is immediately fulfilled, since the CEL rule satisfies claims boundedness and the maximum amount that can be returned is what each individual has paid.

(2) Continuity.

We know that Cm varies continuously with C (see [4]), and the indirect costs, are obviously a continuous function of the cost matrix. On the other hand, CEA is a continuous function on its arguments, which proves that αceq is continuous.

(3) Strong cost monotonicity. Consider a pair of problems (Nω, C), (Nω, C′) such that CC′. Then, and , for all iN. Since the claims rule CEA fulfills endowment monotonicity and claims monotonicity (see [14]), then that implies

Since this property implies cost monotonicity and independence of irrelevant trees (see [4]), all three properties are fulfilled.

(4) Ranking. Given a problem (Nω, C) and i, jN such that cikcjk for all kN, then it is obvious that . Therefore, CEAi(Cm, d)≤CEAj(Cm, d) (order preservation, see [14]) and αceq fulfills ranking. Since ranking implies symmetry this property is also fulfilled.

A core egalitarian proposal

The main objection to αceq> is that, in general, it fails to be a core selection. Then, subsets of agents may have incentives to leave the grand coalition and perform their project by themselves. Next, we show some classes of mcst problems in which αceq> is a core selection. Later, we will propose a modification of our solution (maintaining the egalitarian criteria) in order to achieve core selection for all mcst situations.

Some mcst problems where αceq> is a core-selection

In some families of mcst problems αceq> always provides core allocations. We present two examples of such families.

  1. E1) Let us consider the so-called 2 − mcst problems in which the connection cost between two different individuals (houses, villages, …) can only take one of two possible values: low and high cost (see, for instance, [15]; see also [16] where this class has been generalized to the so-called simple mcst problems).
    Moreover, we assume that cij = k1, ij, cii = k2, and 0 ≤ k1k2. Note that, for this family of mcst problems the property equal share of extra-costs could be applied.
    In this case, if there are n individuals, then Cm = k2 + (n − 1)k1 and . As all claims are identical, then and, for all SN, v(S) = k2+ k1(|S| − 1). Then, since k1k2
    So the allocation provided by αceq> belongs to the core of the cooperative game.
  2. E2) Let us consider linear mcst problems: a group of individuals N = {1, 2, …, n} situated in a row (equally separated) want to connect to a source ω. The cost of connecting one individual with the next one is k monetary units. The cost of connecting individual 1 to the source is M monetary units. If an individual wants to connect to the source, he must do it through all its neighbors on the way towards the source and pay all costs. This is the case of Problem (A).
    Formally, for each i, jN, ij, the connection cost is cij = |ij|k. For each iN, the cost to the source is cii = M + (i − 1)k.
    The minimum cost spanning tree m connects each individual to the next, m(j) = j − 1, j ≥ 2, and the first one with the source, m(1) = ω, with a total cost Cm = M + (n − 1)k. For each iN, and then
    For any coalition SN, v(S) = M + kmax{i − 1, iS}. To obtain the allocation provided by αceq> we distinguish two cases:
    1. If Mk, for all iN. Then, for all SN
    2. If M < k, , , for all k ≥ 2. Then, for all SN
      1. If 1 ∈ S,
      2. If 1 ∉ S, max{i − 1, iS} ≥ |S| and

    So the allocation provided by αceq> belongs to the core of the cooperative game.

A core-egalitarian proposal

In order to obtain an egalitarian solution satisfying core stability, we express the αceq> solution in an alternative way (see [17] to obtain an expression of the CEA claims rule as a minimization problem). Let be the set of non-negative individually rational allocations in a mcst problem (Nω, C) then

Note that, as the distance function is continuous and strictly convex, and set is compact and convex, the minimization problem has always a unique solution. This expression provides us a way of obtaining a core allocation that tries to meet our egalitarian criteria, by defining: where co(Nω, C) denotes the core of the cooperative game defined by the characteristic function v(S), SN. Obviously, this proposal is the most egalitarian core allocation. On the other hand, as , we obtain the following result:

In Example 2, , and coincides with the Folk solution. We show that this coincidence is always true in this kind of mcst problems.

Proposition 2 In the class of 2 − mcst problems βceq coincides with the Folk solution.

Proof. From [16] we know that in this class of problems there is a partition of the set of agents, such that in each subset the Folk solution proposes the same allocation to all individuals in this group (simple components). To simplify the proof we suppose that there are just two components in the 2 − mcst problem (Nω, C) (for more than two components, the reasoning follows an analogous argument):

Let us denote by ni the cardinality of the subset Ni, i = 1, 2, n = n1+ n2 and, as usually, Cm is the cost of the optimal tree. Then, applying separability

Moreover, we know that this allocation is in the core of the monotonic cooperative game. On the other hand, after reordering the agents, we can write such that (x, y) minimizes

Then, by noticing that the function minimizes when all components xi are identical for all iN1, and yj are identical for all jN2. Then, and both solutions coincide.

A piece-wise linear extension

If we denote by the set of all cost matrices involving n individuals, Cn × n, and by the set of elementary cost matrices, i.e., cij ∈ {0, 1} for all i, j = 1, 2, …, n, we know (see [6]) that there is a basis such that any cost matrix can be expressed as

Then, given a mcst solution ψb defined only for elementary cost matrices (a partial solution), its piece-wise linear extension ψ is defined by

As proved in [6], piece-wise linear solutions have the advantage that many normative properties automatically extend from elementary to arbitrary cost matrices. In particular, they show that this is the case with the properties of ranking, cost monotonicity, polynomial complexity, population monotonicity and positivity. Then, we can define the piece-wise linear extension of the βceq solution defined only on elementary problems:

As an immediate consequence of Proposition 2 (elementary problems are a particular case of 2 − mcst problems) we obtain that this extension coincides with the Folk solution.

Corollary 1 For any mcst problem (Nω, C), Υ(Nω, C) = F(Nω, C).

Then, the Folk solution appears as the piece-wise linear extension of βceq, that picks the most egalitarian allocation in the core associated to the mcst problem, showing in this way an alternative interpretation to this solution.

Some further comments

In [18, 19] it is shown that for elementary mcst problems, the Folk solution is the permutation-weighted average of the extreme points of the non-property rights game defined by v(S). Then, for any elementary mcst problem (Nω, C), as a consequence of Proposition 2 we obtain (1) where , k = 1, 2, …, K, denote the extreme points of the core of the cooperative game defined by v(S) (see [18]). Moreover, in this class of mcst problems, βceq also coincides with the nucleolus (see [19]), showing the close relationships between the egalitarian and nucleolus concepts. We could define a non-piecewise linear extension of βceq by using Eq (1).

(5) Equal share of extra-costs. If we consider two problems (Nω, C), (Nω, C′) such that:

  1. for all iN, cii = c0, .
  2. for all i, jN, ij,

then, On the other hand, and di = c0, for all iN. So, as all claims are identical for all the individuals, the CEL rule allocates the same amount B/n, B′/n to each individual and which proves that this solution fulfills equal share of extra-costs.

(7) Example 2 shows that αceq does not fulfill separability nor core selection. Since population monotonicity implies core selection, our proposal does not fulfill population monotonicity.

Example 2 Let us consider the mcst problem defined by the following picture (arcs not depicted have a cost cij = 2):

There are several spanning trees with minimum cost Cm = 6. One of them is given by:

If we denote S = {1}, T = {2, 3, 4}, m = m1m2, where m1 and m2 are the minimum cost spanning trees in problems (S, C|S) and (T, C|T), respectively. On the other hand, , for all iN, and then B = 2 and . We observe that the separability property implies α1 = 2, so αceq> does not fulfill this property. Also note that so this proposal is not a core selection.

Final remarks

We have proposed an egalitarian approach to the problem of sharing the common cost Cm of a mcst. One of the main features in our model is to ignore the “non-relevant” costs, considering only the cost of connecting the individual to the source, and the total cost Cm. Our first attempt to define an egalitarian allocation, the αceq> solution is very simple to understand and to compute. It captures a solidarity approach in which the optimal cost (obtained throughout cooperation) is paid as equally as possible among the agents in the project. Nevertheless, it fails to be coalitionally stable, a crucial property whenever cooperation is necessary. Moreover, the egalitarian criteria will always ignore situations like the one in Example 2, where separability is not fulfilled.

To solve the core stability, we define the βceq solution: the most egalitarian allocation in the core. We have shown that this solution coincides with the Folk solution in elementary problems. This fact allow us to reinterpret the Folk solution as the piece-wise linear extension of βceq applied to elementary problems (the Υ solution).

Finally, an interesting scenario to extend (in future research) both the Folk solution and our egalitarian proposals is the generalized minimum spanning tree problem. In this problem, individuals are grouped into a number of predefined clusters, and the objective is to find a minimum-cost spanning tree of a subset of individuals which includes exactly one individual from each cluster (see, for instance, [20, 21]). As mentioned in those papers, “the generalized minimum spanning tree problem provides an attractive way of modeling various real world applications: in the field of telecommunications, in identifying the position of regional service centers such as stores, warehouses or distribution centers, agricultural settings, energy transportation, physics, etc.” Once the optimal spanning tree is obtained, as in our setting, its cost must be shared by the agents (clusters), and the allocation of each cluster must be shared by the individuals in this cluster.

Acknowledgments

The usual caveat applies. We wish to sincerely thank the referees of this review for their comments and suggestions, which have helped us to improve the paper. This work was carried out in part while Begoña Subiza and Josep E. Peris were visiting the University of New South Wales in Sydney. They appreciate the hospitality and comments received from Haris Aziz and other members of the reading. ADT group at DATA61. They also appreciate the comments from Hervé Moulin and other participants at SECW 2019, Sydney.

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