The Bird Core for Minimum Cost Spanning Tree Problems Revisited: Monotonicity and Additivity Aspects

A new way is presented to define for minimum cost spanning tree (mcst-) games the irreducible core, which is introduced by Bird in 1976. The Bird core correspondence turns out to have interesting monotonicity and additivity properties and each stable cost monotonic allocation rule for mcst-problems is a selection of the Bird core correspondence. Using the additivity property an axiomatic characterization of the Bird core correspondence is obtained.

a minimum cost spanning tree. This inspired independently Bird (1976) and Granot and Claus (1976) to construct and use a cooperative game to tackle this cost allocation problem.
In the seminal paper of Bird (1976) a method is indicated how to find a core element of the minimum cost spanning tree game (mcst game) when a minimum cost spanning tree is given. Further he has introduced, using a fixed mcst, the irreducible core of an mcst game, which is a subset of the core of the game, and which we will call in this paper the Bird core. The Bird core is central in this paper. First, we will give a new "tree free" way to introduce the Bird core by constructing for each mcst-problem a related problem, where the weight function is a non-Archimedean semimetric. The Bird core correspondence turns out to be a crucial correspondence if one is interested in stable cost monotonic allocation rules for mcst-problems. In fact, the Bird core is the "largest" among the correspondences which are cost monotonic and stable. The Bird core has also an interesting additivity property i.e. the Bird core correspondence is additive on each Kruskal cone in the space of mcst-problems with a fixed number of users. The additivity on Kruskal cones can be used to find an axiomatic characterization of the Bird core correspondence.
The outline of the paper is as follows. Section 2 settles notions and notations. In Section 3 the non-Archimedean semimetric is introduced and used to define in a canonical (tree independent) way the reduced game and the Bird core. The relations between stable cost monotonic rules and the Bird core are discussed in Section 4. An axiomatic characterization of the Bird core is given in Section 5. Section 6 concludes.

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, j} with i, j ∈ V , i = j. The complete graph on a set V of vertices is the graph < V, E V >, where E V = {{i, j}|i, j ∈ V and i = j}. A path between i and j in a graph < V, E > is a sequence of nodes (i 0 , i 1 , . . . , i k ), where i = i 0 and j = i k , k ≥ 1, and such that {i s , i s+1 } ∈ E for each s ∈ {0, . . . , k − 1}. A cycle in < V, E > is a path from i to i for some i ∈ V . A path (i 0 , i 1 , . . . , i k ) is without cycles if there do not exist a, b ∈ {0, 1, . . . , k}, a = b, such that i a = i b .
Two nodes i, j ∈ V are connected in < V, E > if i = j or if there exists a path between i and j in < V, 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 >. Given a path P = (i 0 , i 1 , . . . , i k ) between i and j in a graph < V, E >, k ≥ 1, we say that v ∈ V is a node in P if v = i m for some m ∈ {0, . . . , k}; we say that an edge {r, t} ∈ E is on the path P or, equivalently, that i is connected to j via the edge {r, t} in the path P , if there exists m ∈ {0, . . . , k − 1} such that r = i m and t = i m+1 or t = i m and r = i m+1 . Now, we consider minimum cost spanning tree (mcst) situations. In an 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 for 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. If w(e) ∈ {0, 1} for every e ∈ E N , the weight function w is called a simple weight function, and we refer then to < N , E N , w > as a simple mcst situation. Since in our paper the graph of possible edges is always the complete graph, we simply denote an mcst situation with the 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 )-connected if i = j or if there exists a path (i 0 , . . . , i k ) from i to j, with w({i s , i s+1 }) = 0 for every s ∈ {0, . . . , k − 1}. A (w, N )-component of N is a maximal subset of N with the property that any two nodes in this subset are (w, N )-connected. We denote by C i (w) the (w, N )-component to which i belongs and by C(w) the set of all the (w, N )-components of N . Clearly, the collection of (w, N )-components forms a partition of N .
Any mcst situation w ∈ W N 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. Given a spanning network Γ on N we define the set of edges of Γ with nodes in S ⊆ N as the set E Γ S = {{i, j}|{i, j} ∈ Γ and i, j ∈ S }. For any mcst situation w ∈ W N it is possible to determine at least one spanning tree on N , i.e. a spanning network without cycles on N , of minimum cost; each spanning tree of minimum cost is called an mcst for N in w or, shorter, an mcst for w. Two famous algorithms for the determination of minimum cost spanning trees are the algorithm of Prim (Prim (1957)) and the algorithm of Kruskal (Kruskal (1956)). The cost of a minimum cost spanning network Γ on N in a simple mcst situation w equals |C(w)| − 1 (see Lemma 2 in Norde et al. (2004)). Now, let us introduce some basic game theoretical notations. A cooperative cost game is a pair (N, c) where N = {1, . . . , n} is a finite (player -)set and the characteristic function c : 2 N → IR assigns to each subset S ∈ 2 N , called a coalition, a real number c(S), called the cost of coalition S, where 2 N stands for the power set of the player set N , and c(∅) = 0. The core of a game (N, c) is the set of payoff vectors for which no coalition has an incentive to leave the grand coalition N , i.e.
Note that the core of a game can be empty. A game (N, c) is called a concave game if the marginal contribution of any player to any coalition is more than his marginal contribution to a larger coalition, i.e. if it holds that for all i ∈ N and all S ⊆ T ⊆ N \ {i}. An order τ of N is a bijection τ : {1, . . . , |N |} → N . This order is denoted by τ (1), . . . , τ (n), where τ (i) = j means that with respect to τ , player j is in the i-th position. We denote by Σ N the set of possible orders on the set N .
In a coherent way with respect to previous notations, we will indicate the set . . , τ (l − 1)}, which will be denoted shorter as [τ (k)] and (τ (l)) , respectively. 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 \{∅}, with the convention that c w (∅) = 0.
We denote by MCST N the class of all mcst games corresponding to mcst situations in W N . For each σ ∈ Σ E N , we denote by G σ the set {c w | w ∈ K σ } which is a cone. We can express MCST N as the union of all cones G σ , i.e.
The core C(c w ) of an mcst game c w ∈ MCST N is nonempty (Granot and Huberman (1981), Bird (1976)) and, given an mcst Γ (with no cycles) for N in mcst situation w, one can easily find an element in the core looking at the Bird allocation in w corresponding to Γ, i.e. the cost allocation where each player i ∈ N pays the edge in Γ which connects him with his immediate predecessor in < N , Γ >.
We call a map F : W N → IR N assigning to every mcst situation w a unique cost allocation in where Γ is a minimum cost spanning network on N for w.
3 The non-Archimedean semimetric corresponding to an mcst situation Moreover, we call max e∈E(P ) w(e) the top of the path P and denote it by t(P ). We denote by P N ij the set of all paths without cycles from i to j in the graph < N , E N >.
Now we define the key concept of this section, namely the reduced weight function.
Definition 1 Let w ∈ W N . The reduced weight functionw is given bȳ Now, extendingw by puttingw(i, i) = 0 for each i ∈ N , we obtain a nonnegative function on the set of all pairs of elements in N . The obtained reduced weight functionw is a semimetric on N with the sharp triangle inequality, i.e. a non-Archimedean (NA-)semimetric. In formula, for each The proof is left to the reader. If w > 0, thenw is a non-Archimedean metric on the set N .
For the reduced weight functionw we have a special property related to triangles, as the next lemma shows.
This property for NA-semimetrics will be useful in proving that there are many minimum cost spanning trees for (N ,w), as we see in Theorem 1. Unless otherwise clear from the context, in the sequel we simply refer tow as the mcst situation which assigns to each edge {i, j} ∈ E N the reduced weight value as defined in equality (??). Further, we will denote bȳ W N ⊂ W N the set of all NA-semimetric mcst situations which assign to each edge {i, j} ∈ E N the distancew(i, j) provided by a NA-semimetricw on N .
Example 1 Consider the mcst situation < N , w > with N = {0, 1, 2, 3} and w as depicted in Figure 1 The corresponding mcst situationw is depicted in Figure 2.  One main result in this section, Proposition 2, concerns an interesting relation which can be established between the mcst situationw and a minimal mcst situation w Γ as defined by Bird (1976), where Γ is an mcst for N in w.
Recall that given an mcst situation w ∈ W N and an mcst Γ for N in w, the minimal mcst situation w Γ is defined (cf. Bird, 1976) Proposition 2 Let w ∈ W N and i, j ∈ N . Let Γ be an mcst for N in w and P Γ ij be the unique path in Γ from i to j. Then Proof Let P * ∈ arg min P ∈P N ij t(P ) and let e * be an edge on P * such that t(P * ) = w(e * ). Letê = {m, n} be an edge on P Γ ij with w(ê) = t(P Γ ij ). We have to prove that w(ê) = w(e * ). If so, then it follows immediately that min P ∈P N ij t(P ) = w(e * ) = w(ê) = t(P Γ ij ).
Otherwise, first note that by definition of e * w(ê) ≥ w(e * ).
Let S m be the set of all nodes r ∈ N such that n is not on the path from r to m in < N , Γ >; let S n be the set of nodes r ∈ N such that m is not on the path from r to n in < N , Γ >, i.e. S m = {r ∈ N |n / ∈ P Γ mr } and S n = {r ∈ N |m / ∈ P Γ nr }. Note that {S n , S m } is a partition of N and nodes in S n are connected in < N , Γ > to nodes in S m via edge {m, n}. Moreover, by the definition of a path without cycles, i, j must belong to different sets of the partition {S n , S m }. So without loss of generality we suppose that i ∈ S m and j ∈ S n .
Consider the set of edges E + = {{t, v}|t ∈ S m , v ∈ S n }. Then, In order to prove inequality (7), suppose on the contrary that w({m, n}) > w(e) for some e ∈ E + . Then the graph Γ + = (Γ \ {ê}) ∪ {e} would be a spanning network in N cheaper than Γ, which yields a contradiction. By the definition of a path, for each P ∈ P N ij there exists at least one edge e ∈ E + such that e is on the path P . By inequality (7), it follows that t(P ) ≥ w(e) ≥ w(ê). This implies that w(e * ) = min P ∈P N ij t(P ) ≥ w(ê). Together with inequality (6) we have finally w(e * ) = w(ê).
As a direct consequence of Proposition 2 we have that the mcst situation w coincides, for each mcst Γ for w, with the minimal mcst situation w Γ introduced by Bird (1976). So w Γ = w Γ for each pair of mcst Γ, Γ , a fact which is already known (cf. Aarts (1994), Feltkamp (1995), Feltkamp et al.(1994)), but with a complicated proof.
Remark 2 Let w ∈ W N , let Γ be an mcst for w and let τ ∈ Σ N be an order such that Γ and τ fit. The marginal vector m τ (c w ) of the mcst game c w coincides with the Bird allocation in w corresponding to Γ and therefore m τ (c w ) ∈ C(c w ), as is proved in Granot and Huberman (1981).
Remark 3 For each σ ∈ Σ E N there exists a tree Γ which is an mcst for every w ∈ K σ ; further, there exists a τ ∈ Σ N such that Γ and τ fit.
These remarkable considerations together with the next lemma prelude to Theorem 1.
Theorem 1 Let w ∈W N .Then i) for each τ ∈ Σ N there exists an mcst Γ such that Γ and τ fit.
ii) Let c w be the mcst game corresponding to w. Then m τ (c w ) ∈ C(c w ) for all τ ∈ Σ N and c w is a concave game.
Proof i) LetΓ be an mcst for w. Then there is at least oneτ ∈ Σ N such thatΓ andτ fit. Further each τ can be obtained fromτ by a suitable sequence of neighbor switches and so, by applying Lemma 1 repeatedly, we obtain the proof.
ii) Let Γ be an mcst in N for w and let τ ∈ Σ N such that Γ and τ fit. By Remark 2, it follows that m τ (c w ) coincides with the Bird allocation corresponding to Γ. Hence, again by Remark 2, m τ (c w ) ∈ C(c w ). Finally, by the Ichiishi theorem (Ichiishi (1981)) telling that a game is concave iff all marginal vectors are in the core of the game, it follows that c w is a concave game.
Let w ∈ W N . We call the core of the mcst game cw the Bird core of the mcst game c w and denote it by BC(w). By Theorem 1 it directly follows that the Bird core BC(w) of the mcst game c w is the convex hull of all the Bird allocations corresponding to the minimum cost spanning trees forw. Note also that BC(w) ⊆ C(c w ), since cw(S) ≤ c w (S) for each S ∈ 2 N \ {∅} and cw(N ) = c w (N ) (cf. Feltkamp (1995)).
Example 3 Consider the mcst situation w of Figure 1

Monotonicity properties
In Tijs et al.(2004) a class of solutions for mcst situations which are cost monotonic is introduced: the class of obligation rules. Roughly speaking, we define a cost monotonic solution for mcst situations as a solution such that, if the costs of some edges increase, then no agent will pay less. More precisely: Definition 2 A solution F : W N → IR N is a cost monotonic solution if for all mcst situations w, w ∈ W N such that w(e) ≤ w (e) for each e ∈ E N , it holds that F (w) ≤ F (w ).
In this section we introduce a related concept of cost monotonicity for multisolutions on mcst situations. We call a correspondence G : W N IR N assigning to every mcst situation w a set of cost allocations in IR N a multisolution.
Definition 3 A multisolution M : W N IR N is a cost monotonic multisolution if for all mcst situations w, w ∈ W N such that w(e) ≤ w (e) for each e ∈ E N , it holds that Before discussing properties of the Bird core as multisolution for mcst situations, we introduce the following propositions dealing with mcst situations originated by NA-semimetrics.
Proposition 4 Let w, w ∈W N be NA-semimetric mcst situations such that w(e) ≤ w (e) for each e ∈ E N . Then it holds that Proof Let τ ∈ Σ N . By Theorem 1 there exist two mcst's Γ and Γ for w and w , respectively, such that they both fit with τ . First note that , for each j ∈ {2, . . . , |N |}, where the first and the second equality follow by Proposition 3 and the inequality follows from w(e) ≤ w (e) for each e ∈ E N .
Theorem 2 The correspondence BC is a cost monotonic multisolution.
Proof Let w, w ∈ W N be such that w(e) ≤ w (e) for each e ∈ E N . By Theorem 1 and properties of concave games, BC(w) is a convex set whose extreme points are the marginal vectors of the game cw, i.e. each element of BC(w) is a convex combination of marginal vectors of the game cw. Let x ∈ BC(w). There exist numbers α τ , τ ∈ Σ N , with 0 ≤ α τ ≤ 1 for each τ ∈ Σ N , τ ∈Σ N α τ = 1 and where the inequality follows by Proposition 4 and the fact thatw(e) ≤w (e) for each e ∈ E N and the second equality by Theorem 1, which proves BC(w) ⊆ compr − (BC(w )). Using a similar argument the other way around in relations (14), it follows that BC(w ) ⊆ compr + (BC(w)), which concludes the proof.
To connect the cost monotonicity of the Bird core with cost monotonicity of obligation rules, we need Proposition 5.
Proposition 5 Let F : W N → IR N be a cost monotonic and efficient solution. Then i) F (w) = F (w) for every w ∈ W N ; ii) If F is also stable (i.e. F (w ) ∈ C(c w ) for every w ∈ W N ), then F (w) ∈ BC(w) for every w ∈ W N .
Proof Let w ∈ W N . First note that by Definition 1, Let Γ be an mcst for w.
i) By inequality (15) and cost monotonicity of F , F (w) ≤ F (w). On the other hand Γ is an mcst forw too and by efficiency of ii) By inequality (15) Then by stability of F , F (w) ∈ C(cw) = BC(w) ⊆ C(c w ) and by result (i) F (w) ∈ BC(w) too.
Remark 4 Proposition 5 can be extended to multisolutions which are cost monotonic and efficient (Property 1 in next section) multisolutions. From this follows that BC is the "largest" cost monotonic stable multisolution.
Remark 5 As previously said, in Tijs et al.(2004) we introduced the class of obligation rules and proved that they are both cost monotonic and stable solutions for mcst situations. So, by Proposition 5 it follows that for each w ∈ W N , the set F(w) = {φ(w) | φ is an obligation rule} is a subset of the Bird core BC(w) and F(w) = F(w).

An axiomatic characterization of the Bird core
In order to introduce an axiomatic characterization of the Bird core, we need to prove the following fact for NA-semimetric mcst situations.
[The NA-semimetric mcst situationsw,w , αw + α w are obtained via reduction of the weight functions w, w , αw + α w , respectively.] where the second equality follows from the fact that w, w and αw+α w all belong to K σ ; ii) Note that, by Lemma 2, αw, α w , αw + α w ∈ Kσ for someσ ∈ Σ E N .
For each S ∈ 2 N \{∅}, there is, according to Remark 3, a common mcst Γ S for αw, α w and αw + α w . Hence where the third equality follows by (i).
Some interesting properties for multisolutions on mcst situations are the following.
where Γ is a minimum cost spanning network for w on N .

Property 2
The multisolution G has the positive (POS) property if for each w ∈ W N and for each x ∈ G(w) Property 3 The multisolution G has the Upper Bounded Contribution (UBC) property if for each w ∈ W N and every (w, for each x ∈ G(w).
[Here we denote by αG(w) + αG( w) the set {αx + α x|x ∈ G(w), x ∈ G( w}.] Proposition 7 The Bird core BC satisfies the properties EFF, POS, UBC and CPL. Proof Let w ∈ W N and let σ ∈ Σ E N be such that w ∈ K σ . Since BC(w) = C(cw), the following considerations hold: i) For each allocation x ∈ BC(w), i∈N x i = w(Γ) for some mcst Γ by the efficiency property of the core of the game cw. So BC has the EFF property.
ii) For each allocation x ∈ BC(w), x i ≥ 0 for each i ∈ N since the Bird core is the convex hull of all Bird allocations in the mcstw, which are vectors in IR N + . So BC has the POS property.
iii) For each (w, N )-component C = {0} and each x ∈ BC(w) i∈C\{0} by coalitional rationality of the core of the game cw. So BC has the UBC property.
Hence BC has the CPL property.
Inspired by the axiomatic characterization of the P -value ) we provide the following theorem.
Theorem 3 The Bird core BC is the largest multisolution which satisfies EFF, POS, UBC and CPL, i.e. for each multisolution F which satisfies EFF, POS, UBC and CPL, we have F (w) ⊆ BC(w), for each w ∈ W N .
By the UBC property, for each k ∈ {1, . . . , |E N |} and for each (e σ,k , N )- implying that i∈N x e σ,k i = C∈C(e σ,k ) j∈C\{0} x e σ,k j ≤ |C(e σ,k )| − 1 = e σ,k (Γ), where Γ is a minimum spanning network on N for mcst situation e σ,k . By the EFF property, we have i∈N x e σ,k i = e σ,k (Γ), and then inequalities in relation (17) are equalities, that is i∈C\{0} Now, consider the game c e σ,k corresponding to the simple mcst situation e σ,k . Note that for each S ∈ 2 N \ {∅}, c e σ,k (S) = |{C : C is a (e σ,k , N ) − component, C ∩ S = ∅, 0 / ∈ C}|, which is the number of (e σ,k , N )-components not connected to 0 in e σ,k with at least one node in the player set S.

Final remarks
This paper deals mainly with the monotonicity and additivity properties of the Bird core. The attention to monotonicity properties of solutions for cost and reward sharing situations is growing in the literature.
In Sprumont (1990) attention is paid to population monotonic allocations schemes (pmas), in  and Voorneveld et al.(2002) to bimonotonic allocation schemes (bi-mas) and in Branzei et al.(2002) to type monotonic allocation schemes. For mcst-situations, the existence of population monotonic allocation schemes was established in Norde et al.(2004). For special directed mcst-situations also pmas-es exists as is shown in Moretti et al.(2002).
In Tijs et al.(2004) so called obligation rules for mcst-situations turn out to be cost monotonic and induce also a pmas. A special obligation rule is the P -value discussed in Branzei et al.(2004) (see also Feltkamp et al.(1994), Feltkamp (1995)). The P -value can be seen as a special selection of the Bird core: it corresponds to the barycenter of the Bird core (cf. , Bergañtinos and Vidal-Puga (2004)).
For additivity properties of solutions we refer to , Tijs and Branzei (2002).