Obligation Rules for Minimum Cost Spanning Tree Situations and Their Monotonicity Properties

We introduce the class of Obligation Rules for Minimum Cost Spanning Tree Situations. The main result of this paper is that such rules are cost monotonic and induce also population monotonic allocation schemes. Another characteristic of Obligation Rules is that they assign to a minimum cost spanning tree situation a vector of cost contributions which can be obtained as product of a double stochastic matrix with the cost vector of edges in the optimal tree provided by the Kruskal algorithm. It turns out that the Potters value (P-value) is an element of this class.


Introduction
A connection problem arises in the presence of a group of agents, each of which needs to be connected directly or via other agents to a source. If connections among agents are costly, then each agent will evaluate the opportunity of cooperating with other agents in order to reduce costs. In fact, if a group of agents decides to cooperate, a configuration of links which minimizes the total cost of connection is provided by a minimum cost spanning tree.
However, solving the cost-minimization problem is only part of the problem: agents must still support the cost of the minimum cost spanning tree and then a cost allocation problem has to be addressed.
This class of allocation problems has been tackled with the aid of cooperative game theory since the basic paper of Bird (1976). For a detailed discussion of the problem let us refer to the dissertations of Aarts (1994) and Feltkamp (1995), and to the papers of Granot and Huberman (1981).
Many cost allocation methods have been proposed and, as usual, different properties have been considered as well, also in view of the applied economic framework. In many applications the cardinality of the set of agents can vary in time, and also increasing or decreasing of connection costs may occur.
Consider, for instance, a wireless telecommunication network where agents are operators of transmitters for traffic exchange and the source is the central hub station. Agents can decide to communicate directly with the main exchange hub, by means of powerful and very expensive transmitters, or, alternatively, to cooperate and construct a wireless network of less powerful, and consequently, cheaper transmitters. Since transmissions are costly, such a situation can be handled as a minimum cost spanning tree problem. Moreover, new owners of transmitters can be willing to enter the network and the cost of connection can increase (i.e. to improve quality and quantity of services supplied) or decrease (i.e. by improving telecommunication technologies). Of course, in all the connection situations suitable to evolve with time, stability conditions satisfied for the original situation cannot guarantee cooperation among agents also under the new conditions. Therefore many authors have focused their attention in finding allocation methods which can keep, in the most general setting, incentives for cooperation also under modifications in the population of agents and in the structure of connection costs.
In the papers of Kent and Skorin-Kapov (1996), Moretti et al. (2002), and Norde et al. (2004), the question of the existence of population monotonic allocation schemes (pmas) (Sprumont (1990)) is central. A pmas provides a cost allocation vector for every coalition in a monotonic way, i.e. the cost allocated to some player does not increase if the coalition to which he belongs becomes larger.
In the paper of Dutta and Kar (2002), cost monotonic allocation rules have been studied, requiring that the cost allocated to agent i does not increase if the cost of a link involving i goes down, nothing else changing in the network.
In this paper, we introduce a class of allocation rules for minimum cost spanning situations, namely the class of Obligation rules, and show that they have nice monotonicity properties: cost monotonicity and population monotonicity. Actually, our concept is stronger than the concept of cost monotonicity introduced in Dutta and Kar (2002). We simply impose that if some connection costs go down, then no agents will pay more.
It turns out that particular rules in this class are the P -value (Branzei et al. (2003)) and the P τ -values, for each ordering τ of the players ). Moreover it is shown that the P -value is the average of the P τ -values over all the possible orderings τ .
We start with some preliminaries in the next section. In section 3 Obligation rules are introduced starting from the general notion of obligation maps, and some basic properties are studied. In section 4 it is shown that Obligation rules are cost monotonic and induce population monotonic allocation schemes.

Preliminaries and notations
An (undirected) graph is a pair < V, E >, where V is a set of vertices or nodes and E is a set of edges e of the form {i, there exists a path between i and j in E. A connected component of V in < V, E > is a maximal subset of V with the property that any two nodes in this subset are connected in < V, E >. Now, we consider minimum cost spanning tree (mcst) situations. In a mcst situation a set N = {1, . . . , n} of agents is involved willing to be connected as cheap as possible to a source (i.e. a supplier of a service) denoted by 0. In the sequel we use the notation N = N ∪ {0}. An mcst situation can be represented by a tuple < N , E N , w >, where < N , E N > is the complete graph on the set N of nodes or vertices, and w : E N → IR + is a map which assigns to each edge e ∈ E N a nonnegative number w(e) representing the weight or cost of edge e. We call w a weight function. Since in our paper the graph of possible edges is always the complete graph, we simply denote an mcst situation with set of users N , source 0, and weight function w by < N , w >. Often we identify an mcst situation < N , w > with the corresponding weight function w. We denote by W N the set of all mcst situations < N , w > (or w) with node set N . For each S ⊆ N , one can consider the mcst subsituation < S , w |S >, where S = S ∪ {0} and w |S : E S → IR + is the restriction of the weight function w to E S ⊆ E N , i.e. w |S (e) = w(e) for each e ∈ E S . Let < N , w > be an mcst situation. Two nodes i and j are called (w, N )- We define the set Σ E N of linear orders on E N as the set of all bijections σ : {1, . . . , |E N |} → E N , where |E N | is the cardinality of the set E N . For each mcst situation < N , w > there exists at least one linear order σ ∈ Σ E N such that w(σ(1)) ≤ w(σ(2)) ≤ . . . ≤ w(σ(|E N |)). We denote by w σ the column vector w(σ(1)), w(σ(2)), . . . , w(σ(|E N |)) t .
For any σ ∈ Σ E N we define the set Any mcst situation gives rise to two problems: the construction of a network Γ ⊆ E N of minimal cost connecting all users to the source, and a cost sharing problem of distributing this cost in a fair way among users. The cost of a network Γ is w(Γ) = e∈Γ w(e). A network Γ is a spanning network on S ⊆ N if for every e ∈ Γ we have e ∈ E S and for every i ∈ S there is a path in Γ from i to the source. To construct a minimum cost spanning network Γ on N we use in this paper the Kruskal algorithm (Kruskal (1956)), where the edges are considered one by one according to non-decreasing cost, and an edge is either rejected, if it generates a cycle with the edges already constructed, or it is constructed, otherwise.
Let < N , w > be an mcst situation. The minimum cost spanning tree game (N, c w ) (or simply c w ), corresponding to < N , w >, is defined by c w (S) = min{w(Γ)|Γ is a spanning network on S } for every S ∈ 2 N \{∅}, where 2 N stands for the power set of the player set N , with the convention that c w (∅) = 0.
We call a map F : W N → IR N assigning to every mcst situation w a unique cost allocation in Finally a population monotonic allocation scheme or pmas (Sprumont (1990) Note that conditions (i) and (ii) imply that each row of this table is a core element of the corresponding subgame of the game (N, c) (cf. Sprumont (1990)).

Obligation rules
Such an obligation function o on 2 N \ {∅} induces an obligation mapô : , then the resulting empty sum is assumed, by definition, to be the |N |-vector of zeroes:ô(θ) = 0 ∈ IR N .
Then o * is an obligation function and the corresponding obligation map isô * Let w ∈ W N and let σ ∈ Σ E N be such that w ∈ K σ . We can consider a sequence of precisely Remark 2 Note that for each k ∈ {1, . . . , |E N |}, π σ,k is either equal to π σ,k−1 or is obtained from π σ,k−1 by forming the union of two elements of π σ,k−1 .
Definition 1 Letô be an obligation map on Θ(N ∪ {0}). Let σ ∈ Σ E N . The contribution matrix w.r.tô and σ is the matrix D σ,ô ∈ IR N ×E N where the rows correspond to the agents and the columns to the edges, and where for each i ∈ N and each k ∈ {1, . . . , |E N |}.
Some characteristics of the contribution matrix are given in the following proposition.

Definition 2 Letô be an obligation map on
for each mcst situation w in the cone K σ .
Onwards, let e k ∈ IR |E N | be the column vector such that e k i = 1 if i = k and e k i = 0 for each i ∈ {1, . . . , |E N |} \ {k}. From Proposition 1 it follows directly that the matrixD σ,ô ∈ IR N ×|N | defined bȳ for each j ∈ N is a double stochastic matrix (i.e. all entries are nonnegative and each row sum and each column sum are equal to 1), and In order to define Obligation rules properly on the set W N , we need Lemma 1. In the sequel, recall that, for each t ∈ {1, . . . , |E N |}, w σ t is the t-th coordinate of the vector w σ as defined in the Preliminaries.

Proposition 2 Letô be an obligation map on
This proposition makes it possible to define an Obligation rule with respect to an obligation map on Θ(N ∪ {0}) as a map on W N .

Definition 3 Letô be an obligation map on Θ(N ∪ {0}). The Obligation
Remark 3 The P -value (Branzei et al. (2003)) and the P τ -values, with τ ∈ Σ N , introduced in Norde et al. (2004) and studied in Branzei et al.(2003), are Obligation rules. In fact φô * (w) = P (w) and φô τ (w) = P τ (w) for each τ ∈ Σ N , where Σ N is the set of all bijections τ : N → {1, . . . , |N |}. Now we make clear why we chose the name "Obligation rule". Letô an obligation map on Θ(N ∪{0}) and let w ∈ W N . According to the corresponding O-rule φô, each player i ∈ N has to pay fractions of edges summing up to 1, which is the total obligation for player i in the mcst situation w. Stated differently, an O-rule allocates the cost of an edge which forms in some step k, k ∈ {1, . . . , |E N |}, of the Kruskal algorithm to the players in N according to the k-th column of the contribution matrix D σ,ô , with σ ∈ Σ E N such that w ∈ K σ . After step k, by Proposition 1, the quantity of remaining obligations for each player i ∈ N is given by We collect some interesting properties of O-rules in Proposition 3.

Proposition 3 The O-rules are efficient, satisfy the carrier property and form a convex set.
Proof Letô be an obligation map on Θ(N ∪ {0}), let w ∈ W N and let σ ∈ Σ E N be such that w ∈ K σ . i) From (3) and (6) it follows where the second equality follows from Proposition 1 and where Γ is a spanning network on N of minimal cost. So efficiency is proved.
ii) Let i ∈ N be a player who is (w, N )-connected to the source 0. There exists r ∈ {1, . . . , |E N |} such that i is connected to 0 in F σ,r but not in F σ,r−1 and w(σ(r)) = 0. Moreover, by the definition of an obligation map,ô i (π σ,k ) = 0 for k ∈ {r, . . . , |E N |}. It follows by (6) that φô i (w) = 0 and then it is proved that φô satisfies the carrier property.
iii) Letô • ,ô • andô α , with α ∈ [0, 1], be as in Remark 1. Then for every w ∈ W N and σ ∈ Σ E N such that w ∈ K σ , where the third equality follows from Remark 1 and the definition of D σ,ô α . Then it is proved that the set of O-rules is a convex set.
We end this section with a proposition that enlightens the connection between the P -value and the P τ -values, τ ∈ Σ E N , according to Remark 3.
Proposition 4 Let w ∈ W N . Then Proof By Remark 3 and (4) we have only to prove that To prove (8), note that for each i ∈ {1, . . . , |N |}, the edge σ(ρ σ (i)) connects two disconnected subsets of vertices S, T ∈ π σ,ρ σ (i−1) . Then, for each player j ∈ N \ (S ∪ T ), if any, 1 n! τ ∈Σ ND σ,ô τ ji =D σ,ô * ji = 0. On the other hand, for players in S∪T , we have two possibilities regarding the position of the source w.r.t. the sets S and T : i) The source 0 belongs neither to S nor to T implying that for each j ∈ T and for each τ The fraction of orderings τ ∈ Σ N such that arg min{τ (k)|k ∈ S∪T } ∈ S is equal to |S| |S∪T | = |S| |S|+|T | whereas the fraction of such orderings τ ∈ Σ N such that τ (j) = min{τ (k)|k ∈ T } is equal to 1 |T | . Then it follows that for each j ∈ T Similar arguments hold for each j ∈ S too.
ii) The source 0 belongs either to S or to T . Without loss of generality, suppose 0 ∈ S. Then, for each j ∈ S Hence (8) is proved and P (w) = 1 n! τ ∈Σ N P τ (w).
An alternative proof of Proposition 4 is given in Branzei et al. (2004).

Cost Monotonicity and PMAS
In this section we will discuss some nice monotonicity properties of the Orules. First, we provide the definition of cost monotonic solutions for mcst situations.
Definition 4 A solution F : W N → IR N is a cost monotonic solution if for all mcst situations w,w ∈ W N such that w(ē) ≤w(ē) for one edgeē ∈ E N and w(e) =w(e) for each e ∈ E N \ {ē}, it holds that F (w) ≤ F (w).
We prove in Theorem 1 that O-rules are cost monotonic; the main step is the following lemma.
Proof Let K := {e ∈ E N |w(e) = w(ē)} be the set of edges that have the same cost asē. Let σ ∈ Σ E N be such that w ∈ K σ . Without loss of generality we may assume that σ −1 (ē) = max{σ −1 (e)|e ∈ K}, i.e. σ ranks the edges of K withē last. By construction we also havew ∈ K σ and hence where at the inequality we used the fact thatw σ ≥ w σ and the fact that the matrix D σ,ô is nonnegative.

Theorem 1 Obligation rules are cost monotonic.
Proof Letô be an obligation map on Θ(N ∪ {0}) and let φô the O-rule w.r.t o. Let w,w ∈ W N be as in Definition 4.
Applying Lemma 2 for each r ∈ {1, . . . , |H|}, with w r−1 in the role of w and w r in the role ofw, it follows that which finally proves cost monotonicity of O-rules.
By Theorem 1 and Remark 3 the P -value and the P τ -values, for each τ ∈ Σ N , are cost monotonic O-rules. The following example illustrates the cost monotonicity of the P -value.
where λ S = 1 + max{w({i, j})|i, j ∈ S }. Then, in < T ,w > each edge with at least one node not in S is more expensive than in < T , w |T >.
Further, letσ ∈ Σ E T be such thatw ∈ Kσ and let σ S ∈ Σ E S be such that σ S (i) =σ(i) for each i ∈ {1, . . . , |E S |}. Then by (9) it follows that This follows from the fact that in < S , w |S > the edges with at least one node not in S are discarded and in < T ,w > the edges with at least one node not in S are allowed but they are too expensive. The result is that applying the Kruskal procedure on < T ,w > w.r.t.σ the players in S are already connected to 0 before one of the edges with nodes not in S is considered. So, by definition of an obligation map, we have that the contribution matrix with |T | rows and |E T | columns Dσ ,ô T is of the form where the four submatrices D σ S ,ô S , N 1 , N 2 and R are such that: • D σ S ,ô S is the contribution matrix w.r.t. to σ S and toô S with |S| rows and |E S | columns; • N 1 is the null matrix with |S| rows and |E T | − |E S | columns; • N 2 is the null matrix with |T | − |S| rows and |E S | columns; • R is a real valued matrix with |T | − |S| rows and |E T | − |E S | columns obtained according to the definition of the contribution matrix Dσ ,ô T .
Recall that O-rules are cost monotonic. Sincew(e) ≥ w |T (e) for each e ∈ E T , then From (10) and (11) we obtain From (12)

Final remarks
This paper considers the class of Obligation rules and studies their monotonicity properties. They cover old results in Branzei et al. (2003) and Norde et al. (2004). In the former, an axiomatic characterization of the P -value for mcst situations is given. In the latter, existence of a pmas for mcst games is proved. In this paper we introduce a class of solutions for mcst situations, the Obligation rules, which are cost monotonic, induce a pmas and, as already said in Remark 3, include among others the P -value and the P τ -values, for each τ ∈ Σ N . Further, it turns out that the class of O-rules is a subclass of the Construct and Charge rules introduced and studied in Moretti et al. (2004), which are also defined via a matrix product with the unique difference that the columns in the contribution matrix do not necessarily derive from obligation maps.
Of course, other rules which are not of Obligation type can be cost monotonic rules. For instance, the egalitarian rule, which allocates to each player i ∈ N an equal amount of the total cost of the mcst, is cost monotonic but it is not an O-rule, since it does not satisfy the carrier property. Moreover, the cost allocation provided by the egalitarian rule is generically not a core element implying that this rule does not induce any pmas.
In Proposition 4, we illustrate a strong connection between the P -value and the P τ -values, for each τ ∈ Σ N . In Branzei et al. (2004) we characterize the link between these solutions, based on the notion of irreducible core. Roughly speaking, given a mcst situation w ∈ W N , the irreducible core of the mcst game corresponding to w is the core of the concave mcst game corresponding to a mcst situation which is obtained via an adaptation of w introduced in Bird (1976). In Branzei et al. (2004), it is proved that the P τ -values, with τ ∈ Σ N , are extreme points of the irreducible core and that the P -value coincides with the Shapley value of the concave mcst game corresponding to the adaptation of w. This last fact is proven in an alternative way in Bergantinos and Vidal-Puga (2004b).