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Sequential contributions rules for minimum cost spanning tree problems
Mathematical Social Sciences 64 (2012) 136–143
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Sequential contributions rules for minimum cost spanning tree problems Youngsub Chun a,∗ , Joosung Lee a,b a
Department of Economics, Seoul National University, Seoul 151-746, Republic of Korea
b
Department of Economics, Pennsylvania State University, University Park, PA 16802, USA
article
info
Article history: Available online 21 February 2012
abstract We introduce a family of sequential contributions rules for minimum cost spanning tree problems. Each member of the family assigns an agent part of the cost of connecting him to his immediate predecessor, and all of his followers are equally responsible for the remaining part. We characterize the family by imposing the axioms of efficiency, non-negativity, independence of following costs, group independence, and weak first-link consistency. The Bird and the sequential equal contributions rules are two distinguished members of the family. The Bird rule is obtained by requiring an agent to pay the entire cost of connecting him to his immediate predecessor, and the sequential equal contributions rule is obtained by requiring an agent and each of his followers to be equally responsible for this cost. We show how each of these two rules can be singled out from the family. © 2012 Elsevier B.V. All rights reserved.
1. Introduction A minimum cost spanning tree problem is concerned with constructing a minimal cost spanning tree which provides for every agent a connection to the source and allocating the construction cost among the agents. Examples of minimum cost spanning tree problems are abundant: constructing communication networks such as telephone or cable television, or building a drainage system that connects every house in a city with a water purifier. Claus and Kleitman (1973) initiated the study of the problem, and Bird (1976) treated the problem with game-theoretic methods and proposed a cost allocation rule, now called the Bird rule. Ever since, many other rules have been studied: the core and the nucleolus (Granot and Huberman, 1984), the Folk solution (Feltkamp et al., 1994; Branzei et al., 2004; Bergantiños and Vidal-Puga, 2005, 2007; Lorenzo and Lorenzo-Freire, 2009), the Kar rule (Kar, 2002), the Dutta–Kar rule (Dutta and Kar, 2004), and the piecewise linear rules (Bogomolnaia and Moulin, 2010). In this paper, we propose and characterize a new family of rules, which we call the sequential contributions rules: each member of the family assigns each agent part of the cost of connecting him to his immediate predecessor, and all of his followers are equally responsible for the remaining part. Our main axiom is first-link consistency, which requires that upon the departure of a ‘‘first-link agent’’, an agent directly linked to the source, the contribution of his followers should not be affected. Also we formulate a weaker version, weak first-link consistency, which requires that upon the departure of a first-link
∗
Corresponding author. E-mail addresses: [email protected] (Y. Chun), [email protected] (J. Lee).
0165-4896/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2012.02.001
agent, the contribution of his followers should be affected by the same amount. Our main result is a characterization of the sequential contributions rules on the basis of efficiency, non-negativity, independence of following costs, group independence and weak first-link consistency. Efficiency requires that the sum of contributions of all agents should be equal to the cost of constructing an efficient network. Non-negativity requires that the contribution of each agent should not be negative. Independence of following costs requires that the contribution of each agent should not be affected by changes in his followers’ connection costs. Group independence requires that for each group which consists of a first-link agent and all of his followers, the contribution of each member of the group should depend only on the connection costs between themselves. The Bird and the sequential equal contributions rules are two distinguished members of the family of sequential contributions rules.1 The Bird rule is obtained by requiring each agent to pay the entire cost of connecting him to his immediate predecessor, and the sequential equal contributions rule is obtained by making an agent and each of his followers equally responsible for this cost. Next, we show how each of these two rules can be singled out from the family. First, the Bird rule is characterized by efficiency, group independence, and first-link consistency. Also, it is the unique sequential contributions rule satisfying core selection or null responsibility; a rule is a core selection if no coalition of agents can be made better off by building their own network, and null 1 The sequential equal contributions rule is originated from the airport problem, first studied by Littlechild and Owen (1973). For airport problems, the sequential equal contributions rule assigns the cost of each segment equally to all airlines using the segment. The terminology we adopt is borrowed from Thomson (forthcoming).
Y. Chun, J. Lee / Mathematical Social Sciences 64 (2012) 136–143
responsibility requires that if the cost of connecting an agent to his immediate predecessor is zero, then his contribution should be zero. Finally, the Bird rule is characterized by efficiency, null responsibility, and independence of irrelevant costs, which requires that the contribution of each agent should depend only on his path to the source over a minimal cost spanning tree. On the other hand, the sequential equal contributions rule is the unique sequential contributions rule satisfying equal responsibility, which requires that if a connection cost between two agents on the minimum cost spanning tree is zero, then the contribution of the two agents should be the same. The sequential equal contributions rule is also characterized by efficiency, equal responsibility, and independence of irrelevant costs. The paper is organized as follows. Section 2 contains preliminaries and introduces the family of sequential contributions rules. In Section 3, we present an axiomatic characterization of the family. In Section 4, we discuss how the Bird and the sequential equal contributions rules can be characterized by imposing additional axioms. 2. Preliminaries 2.1. The problem Let N = {1, 2, . . .} be a (finite or infinite) universe of ‘‘potential’’ agents. Let N be the family of nonempty and finite subsets of N. A typical element of N is denoted by N = {1, . . . , n}. We are interested in networks whose nodes are elements of a set N0 = N ∪ {0}, where 0 is a special node called the source. Let N0 ≡ {N0 | N ∈ N }. Given N0 ∈ N0 , a cost matrix C = (cij )i,j∈N0 gives the cost of the (direct) link between any two nodes. We assume that cii = 0 for all i ∈ N0 and cij = cji ≥ 0 for all i, j ∈ N0 . We denote the set of all cost matrices over N0 as CN0 and set C ≡ N0 ∈N0 CN0 . Given N0 ∈ N0 , C ∈ CN0 , and S ⊆ N, let C|S0 be the restriction of C to S0 , and c¯ (C ) ∈ R+ be any given number such that for all i, j ∈ N0 , c¯ (C ) > maxi,j∈N0 cij . A graph g over N0 is a subset of {(ij)|i, j ∈ N0 }, whose elements are arcs. Given g and i, j ∈ N0 such that i ̸= j, a path from i to j in g , gij , is a sequence of distinct arcs {(ik−1 ik )}Kk=1 such that (ik−1 ik ) ∈ g for all k ∈ {1, 2, . . . , K }, i = i0 and j = iK . A tree is a graph with a unique path between any twonodes. We denote the set of all trees over N0 as TN0 , and set T ≡ N0 ∈N0 TN0 . Given N0 ∈ N0 , t ∈ TN0 , and i ∈ N, the immediate predecessor of i, p(i; t ), is j ∈ N0 such that (ji) ∈ gi0 and the k-th predecessor of i is pk (i; t ) ≡ pk−1 (p(i; t ); t ). The set of all predecessors of i is P (i; t ) ≡ {pk (i; t ) | k = 1, 2, . . . , K }, where pK (i; t ) = 0. For convenience, let p0 (i; t ) ≡ i. The set of all followers of i is F (i; t ) ≡ {j ∈ N | pk (j; t ) = i for some k = 1, 2, . . .}. When there is no ambiguity, we use the notation p(i), P (i), and F (i) instead of p(i; t ), P (i; t ), and F (i; t ), respectively. We denote the cardinality of F (i) as |F (i)|. For all N0 ∈ N0 and all C ∈ CN0 , t ∈ TN0 is a minimum cost spanning tree for C , or an mcst for C , if for all t ′ ∈ TN0 , (ij)∈t cij ≤ (ij)∈t ′ cij . Let m(C ) = mint ∈TN0 (ij)∈t cij . As established in the literature, an mcst always exists even though it is not necessarily unique. Given N0 ∈ N0 , a minimum cost spanning tree problem, or an mcstp, is a cost matrix C ∈ CN0 . For each N ∈ N , a cost allocation rule, or simply a rule, is a function ϕ : CN0 → RN , which associates to any problem C ∈ CN0 a vector ϕ(C ) ≡ (ϕi (C ))i∈N in RN . For each i ∈ N , ϕi (C ) is the contribution of i.
137
We impose a domain restriction on the set of cost matrices:
CN10
≡ {C ∈ CN0 | C induces a unique mcst. } 1 1 and C 1 ≡ N0 ∈N0 CN0 . For each C ∈ C , we denote the unique mcst for C by g (C ). Given C ∈ C 1 , we denote the immediate connection cost of i, cp(i;g (C ))i , by ci . An agent l is a leaf if F (l; g (C )) = ∅ and a firstlink agent if P (l; g (C )) = {0}. Let Z (C ) be the set of all leaves and A(C ) be the set of all first-link agents. For all N0 ∈ N0 , all C ∈ CN10 , and all l ∈ A(C ), F (l; g (C )) ∪ {l} is a group. Note that N = l∈A(C ) (F (l; g (C )) ∪ {l}) and (F (l; g (C )) ∪ {l}) ∩ (F (m; g (C )) ∪ {m}) = ∅ for all l, m ∈ A(C ) such that l ̸= m. 2.2. Sequential contributions rules We introduce a family of rules, which we call sequential contributions rules. Each member of the family assigns each agent part of his immediate connection cost (the cost of connecting him to his immediate predecessor), and his followers are equally responsible for the remaining part. Formally, for all N0 ∈ N0 , let α N ≡ (αiN )i∈N be a contributions function, where αiN is a function αiN : CN10 → R satisfying the following conditions: (i) for all N0 ∈ N0 and C ∈ CN10 , 0 ≤ αiN (C ) ≤ ci ,
(ii) for all N0 ∈ N0 and C ∈ CN10 , if i ∈ Z (C ), then αiN (C ) = ci ,
(iii) for all N0 , N0′ ∈ N0 such that i ∈ N ∩ N ′ and all C ∈ CN10 and C ′ ∈ CN1 ′ , if F (i; g (C )) = F (i; g (C ′ )) and ci = ci′ , then αiN (C ) = ′
0
αiN (C ′ ).
Let AN be the family of all contributions functions over N. For simplicity of notation, we write α instead of α N when there is no danger of confusion. We are ready to define the sequential contributions rules. Sequential contributions rules: For all N0 ∈ N0 , all C ∈ CN10 , and all α ∈ AN , a sequential contributions rule with a contributions function α , ϕ α ≡ (ϕiα )i∈N , is defined as
ϕiα (C ) = αi (C ) +
ck − αk (C ) . | F (k; g (C ))| k∈P (i;g (C ))
The Bird and the sequential equal contributions rules are two most well-known members of this family. The Bird rule is obtained by requiring each agent to be solely responsible for his immediate connection cost. The sequential equal contributions rule is obtained by requiring an agent and each of his followers to be equally responsible for his immediate connection cost. Bird rule, ϕ B : For all N0 ∈ N0 , all C ∈ CN10 , and all i ∈ N , αi (C ) = ci . Sequential equal contributions rule, ϕ E : For all N0 ∈ N0 , all C ∈ CN10 , i . and all i ∈ N, αi (C ) = |F (i)|+ 1
c
Even when a cost matrix does not induce a unique mcst, we can still define a notion of the sequential contributions rule. For each mcst derived from the cost matrix, we apply a sequential contributions rule and find the corresponding contributions. Each agent’s contribution is calculated by taking an average over all mcsts. 3. The main characterization First, we introduce basic axioms discussed in the literature. Efficiency requires that the sum of the contributions of all agents should be equal to the cost of constructing an efficient network. Non-negativity requires that the contribution of each agent should not be negative.
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Y. Chun, J. Lee / Mathematical Social Sciences 64 (2012) 136–143
Efficiency: For all N0 ∈ N0 and all C ∈ CN0 ,
ϕi (C ) = m(C ). Non-negativity: For all N0 ∈ N0 , all C ∈ CN0 , and all i ∈ N , ϕi (C ) ≥ 0.
i∈N
Next are two independence axioms. Independence of following costs requires that the contribution of each agent should not be affected by his followers’ connection costs. Group independence requires that for each group, the contribution of each member of the group should depend only on the connection costs between themselves.2 In other words, for each group, suppose that there is a change in the connection cost between an agent in some other group and another agent—this agent can be a member of the given group, while the members of the given group remain unchanged. Group independence requires that as long as the connection costs between any pairs in the group remain unchanged, such a change should not affect the contributions of agents in the group. Independence of following costs: For all N0 ∈ N0 , all C , C ′ ∈ CN0 , and all i ∈ N, if F (i; g (C )) = F (i; g (C ′ )) and for all q, w ∈ N0 \ F (i, g (C )), cqw = cq′ w , then ϕi (C ) = ϕi (C ′ ). Group independence: For all N0 , N0′ ∈ N0 , all C ∈ CN0 , and all C ′ ∈ CN ′ , if there exists l ∈ A(C ) ∩ A(C ′ ) such that F (l; g (C )) = 0
First-link consistency: For all N0 ∈ N0 , all C ∈ CN10 , all l ∈ A(C ), and
all i ∈ F (l), ϕi (C ) = ϕi (C −l ). Next is weak first-link consistency, which requires that upon the departure of a first-link agent, all of his followers should be affected by the same amount. It is obvious that first-link consistency implies weak first-link consistency. Weak first-link consistency: For all N0 ∈ N0 , all C ∈ CN10 , all l ∈ A(C ), and all i, j ∈ F (l),
ϕi (C ) − ϕi (C −l ) = ϕj (C ) − ϕj (C −l ). Before presenting our main characterization result, we show that each member of the family of the sequential contributions rules satisfies efficiency, non-negativity, independence of following costs, group independence, and weak first-link consistency. Lemma 1. On C 1 , each sequential contributions rule satisfies efficiency, non-negativity, independence of following costs, group independence, and weak first-link consistency.
F (l; g (C ′ )) and that C|(F (l;g (C ))∪{l})0 = C|(′ F (l;g (C ′ ))∪{l}) , then for all 0 i ∈ F (l, g (C )) ∪ {l}, ϕi (C ) = ϕi (C ′ ).
Proof. Let α ∈ AN . We omit the easy proof that ϕ α satisfies efficiency and non-negativity.
The final axiom for our main characterization is based on a certain formulation of consistency.3 Feltkamp et al. (1999) introduce a consistency axiom for mcstps, namely leaf consistency: it requires that the departure of a leaf should not affect the contribution of other remaining agents. They characterized the Bird rule on the basis of leaf consistency. Recently, Dutta and Kar (2004) propose two other consistency axioms for mcstps. Suppose that an agent leaves, but all the remaining agents are allowed to use the leaving node. These agents can use the leaving node in two different ways:
• Independence of following costs: For all N0 ∈ N0 , all C , C ′ ∈ CN10 , and all i ∈ N, if F (i; g (C )) = F (i; g (C ′ )) = F (i) and for all q, w ∈ N \ F (i), cqw = cq′ w , we obtain: (i) P (i; g (C )) = P (i; g (C ′ )), (ii) for all k ∈ P (i) ∪ {i}, F (k; g (C )) = F (k; g (C ′ )) and ck = ck′ , (iii) since |F (i; g (C ))| = |F (i; g (C ′ ))| and ci = ci′ , αi (C ) = αi (C ′ ).
(i) source consistency, which assumes that the leaving node can be used only to connect to the source, and (ii) tree consistency, which assumes that the leaving node can be used to connect to any node. For each formulation, consistency requires that the contribution of the other agents should be unchanged between the original and the reduced cost matrices. Dutta and Kar characterize the Bird rule on the basis of source consistency and the Dutta–Kar rule on the basis of tree consistency. In this paper, we introduce a fourth consistency axiom, first-link consistency. For all C ∈ C 1 and all l ∈ A(C ), we imagine that a firstlink agent l leaves after constructing the arc (l0), but the remaining agents are allowed to connect to the source through the arc (l0). In this reduced problem, each remaining node has the option to use the node l to connect to the source. Formally, the connection costs on N0 \ {l} are changed as follows:
Therefore,
1 − αk (C )
k∈P (i;g (C ))
|F (k; g (C ))|
ϕiα (C ) = αi (C ) + = αi (C ′ ) +
k∈P (i;g (C ′ ))
ck
1 − αk (C ′ )
|F (k; g (C ′ ))|
ck′
= ϕiα (C ′ ), as desired.
• Group independence: For all N0 , N0′ ∈ N0 such that l ∈ N ∩ N ′ , all C ∈ CN10 , and all C ′ ∈ CN10 , if l ∈ A(C ) ∩ A(C ′ ) and C|F (l;g (C ))∪{l} = C|′F (l;g (C ′ ))∪{l} , we obtain: for all i ∈ F (l, g (C )) ∪ {l}, (i) P (i; g (C )) = P (i; g (C ′ )), (ii) for all k ∈ P (i) ∪ {i}, F (k; g (C )) = F (k; g (C ′ )) and ck = ck′ , (iii) since |F (i; g (C ))| = |F (i; g (C ′ ))| and ci = ci′ , αi (C ) = αi (C ′ ). Therefore, for all i ∈ F (l, g (C )) ∪ {l},
−l (i) for all i ̸= l, ci0 = min{ci0 , cil }, (ii) if {i, j} ∩ {l, 0} = ∅, then cij−l = cij .
ϕiα (C ) = αi (C ) + −l
Let C −l be the reduced cost matrix. Note that for all i ∈ N \{l}, ci = ci and C −l ∈ C 1 . Similarly, let C −P (i) be the reduced cost matrix after the departure of P (i) \ {0}. We abuse the notation by allowing the possibility that P (i) \ {0} = ∅. Of course, if P (i) \ {0} = ∅, then C −P (i) = C . First-link consistency requires that upon the departure of a firstlink agent, none of his followers should be affected. 2 Kar (2002) introduces a weaker version of group independence, which requires that a change in the connection cost within a group should not affect the contribution of agents in some other groups. 3 For a survey on consistency, see Thomson (2000).
1 − αk (C ) ck | F (k; g (C ))| k∈P (i;g (C ))
= αi (C ′ ) +
k∈P (i;g (C ′ ))
1 − αk (C ′ )
|F (k; g (C ′ ))|
ck′
= ϕiα (C ′ ), as desired.
• Weak first-link consistency: For all N0 ∈ N0 , all C ∈ CN10 , all l ∈ A(C ), and all i ∈ F (l), we obtain: (i) P (i; g (C )) = P (i; g (C −l )) ∪ {l}, (ii) since F (i; g (C )) = F (i; g (C −l )) and ci = ci−l , αi (C ) = αi (C −l ).
Y. Chun, J. Lee / Mathematical Social Sciences 64 (2012) 136–143
Therefore, for all i ∈ F (l),
ϕiα (C ) − ϕiα (C −l ) =
αi (C ) +
−
ck − αk (C ) | F (k; g (C ))| k∈P (i;g (C ))
k∈P (i;g (C −l ))
cl − αl (C ) , |F (l; g (C ))|
=
P (i; g (C −l )), F (k; g (C )) = F (k; g (C −l )). Therefore, for all i ∈ (N \ {l}) \ F (l; g (C )),
ϕi (C ) = ϕi (C −l )
We are ready to present our characterization of the sequential contributions rules.
k∈P (i;g (C −l ))
= αi (C ) +
Step 1: For all N0 ∈ N0 , all C ∈ such that
ϕi (C ) = αi (C ) +
CN10 ,
and all i ∈ N, there exists αi (C )
ck − αk (C ) . | F (k; g (C ))| k∈P (i;g (C ))
0
exists αi (C ) such that ′
ϕi (C ) = ϕi (C −l ) + γ
k∈P (i;g (C ′ ))
ck − αk (C ) . |F (k; g (C ′ ))|
ϕi (C ) − cl
ϕi (C ) +
i∈F (l;g (C ))
ϕi (C ) + ϕl (C ) − cl
i∈(N \{l})\F (l;g (C ))
=
ϕi (C ) + |F (l; g (C ))|γ −l
i∈F (l;g (C ))
+
ϕi (C ) + ϕl (C ) − cl −l
i∈(N \{l})\F (l;g (C ))
=
ϕi (C −l ) + |F (l; g (C ))|γ + ϕl (C ) − cl ,
i∈N \{l}
which implies that ϕl (C ) = cl −|F (l; g (C ))|γ . Therefore, by setting αl (C ) ≡ cl − |F (l; g (C ))|γ , we have ϕl (C ) = αl (C ). Since C −l ∈ CN1 \{l} , from the induction hypothesis, let αi (C ) ≡ 0
|F (k; g (C ))|
|F (l; g (C ))|
.
ck − αk (C )
k∈P (i;g (C ))
|F (k; g (C ))|
ϕi (C ) = αi (C ) +
.
Step 2: α defined in Step 1 satisfies condition (iii) in the definition of the sequential contributions rules. Proof. Let N0 , N0′ ∈ N0 be such that i ∈ N ∩ N ′ . Let C ∈ CN10 and C ′ ∈ CN1 ′ be such that F (i; g (C )) = F (i; g (C ′ )) and ci = ci′ . We 0
show that αi (C ) = αi (C ′ ). By the construction of α , for all i ∈ N,
αi (C ) = αi (C −P (i) )
(1)
−P (i)
).
(2)
By group independence, ϕi (C −P (i) ) = ϕi (C|(F (i)∪{i})0 ) and ϕi (C ) = ϕi (C|(′ F (i)∪{i}) ). Since i ∈ A(C −P (i) ), ϕi (C −P (i) ) = αi (C −P (i) ) and 0
−P (i)
ϕi (C|(F (i)∪{i})0 ) = αi (C|(F (i)∪{i})0 ). Similarly, since i ∈ A(C ′ ), −P (i) −P (i) ϕi (C ′ ) = αi (C ′ ) and ϕi (C|(′ F (i)∪{i})0 ) = αi (C|(′ F (i)∪{i})0 ).
Therefore,
αi (C −P (i) ) = αi (C|(F (i)∪{i})0 )
i∈N
=
k∈P (i;g (C ))
cl − αl (C )
′−P (i)
exists γ such that ϕi (C ) = ϕi (C ) + γ . By group independence, for all i ∈ (N \ {l}) \ F (l; g (C )), ϕi (C ) = ϕi (C −l ). By efficiency,
i∈N \{l}
ck − αk (C )
+
Altogether, for all i ∈ N, there exists αi (C ) such that
αi (C ′ ) = αi (C ′
−l
= αi (C ) +
′
Without loss generality, let N ≡ {1, . . . , n}. By weak first-link consistency, for all C ∈ CN10 , all l ∈ A(C ), and all i ∈ F (l; g (C )), there
ϕi (C −l ) =
k∈P (i;g (C −l ))
|F (k; g (C −l ))|
and for all i ∈ N , ′
ck − αk (C −l )
= αi (C −l ) +
′
ϕi (C ′ ) = αi (C ′ ) +
ck − αk (C ) . | F (k; g (C ))| k∈P (i;g (C ))
This proves Step 1.
Proof. First, suppose that |N | = 1. Without loss of generality, let N ≡ {1}. By efficiency, for all C ∈ CN10 , ϕ1 (C ) = c1 , the desired conclusion. We will establish the claim for an arbitrary set of agents by an induction argument. As induction hypothesis, suppose that for all N0′ ∈ N0 such that |N ′ | ≤ n − 1, all C ′ ∈ CN1 ′ , and all i′ ∈ N ′ , there
|F (k; g (C −l ))|
On the other hand, if i ∈ F (l; g (C )), then ϕi (C ) = ϕi (C −l ) + γ . Moreover, P (i; g (C )) = P (i; g (C −l )) ∪ {l}, and for all k ∈ P (i; g (C −l )), F (k; g (C )) = F (k; g (C −l )). Therefore, for all i ∈ F (l; g (C )),
Theorem 1. On C 1 , a rule satisfies efficiency, non-negativity, independence of following costs, group independence, and weak first-link consistency if and only if it is a sequential contributions rule. Proof. By Lemma 1, it suffices to show that if a rule satisfies the five axioms, then it is a sequential contributions rule. Let ϕ be a rule satisfying these axioms. The proof is divided into three steps.
ck − αk (C −l )
= αi (C −l ) +
ck−l − αk (C −l ) − l |F (k; g (C ))|
αi (C −l ) +
as desired.
139
αi (C −l ) for all i ∈ N \ {l}. If i ∈ (N \ {l}) \ F (l; g (C )), then ϕi (C ) = ϕi (C −l ). Moreover, P (i; g (C )) = P (i; g (C −l )), and for all k ∈
(3)
and
αi (C ′
−P (i)
) = αi (C|(′ F (i)∪{i})0 ).
(4)
Since F (i; g (C )) = F (i; g (C ′ )) and ci = ci′ , by independence of following costs, ϕi (C|(F (i)∪{i})0 ) = ϕi (C|(′ F (i)∪{i}) ), which implies that 0
αi (C|(F (i)∪{i})0 ) =
αi (C|(′ F (i)∪{i})0 ).
(5)
Altogether,
αi (C ) = αi (C −P (i) ) = αi (C|(F (i)∪{i})0 ) (1)
(3)
(5)
(4)
= αi (C|(′ F (i)∪{i})0 ) = αi (C ′
This proves Step 2.
−P (i)
(2)
) = αi (C ′ ).
Step 3: α defined in Step 1 satisfies conditions (i) and (ii) in the definition of the sequential contributions rules.
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Y. Chun, J. Lee / Mathematical Social Sciences 64 (2012) 136–143
Proof. For all N0 ∈ N0 , all C ∈ CN10 , and all i ∈ N, let C¯ −P (i) ∈
C(1N \P (i))0 be
c¯q−wP (i)
0
= c¯ (C ) cqw
if q = i and w ∈ F (i), if q ∈ (N \ P (i)) \ (F (i) ∪ {i}) and w ∈ F (i), otherwise.
The remaining proof is divided into two cases: Case 1: i ∈ Z (C ). By efficiency and group independence, αi (C¯ −P (i) ) = ci . Since i ∈ A(C¯ −P (i) ), it follows that ϕi (C¯ −P (i) ) = αi (C¯ −P (i) ). From (1), we conclude that αi (C ) = ci . (Therefore, condition (ii) is satisfied.) Case 2: i ∈ N \ Z (C ). From Step 1, for all j ∈ F (i) such that p(j) = i, ci − αi (C¯ −P (i) ) ϕj (C¯ −P (i) ) = αj (C¯ −P (i) ) + . |F (i; g (C¯ −P (i) ))| Note that αi (C ) = αi (C¯ −P (i) ) and F (i; g (C¯ −P (i) )) = F (i; g (C )). Moreover, by applying efficiency and group independence repeatedly, αj (C¯ −P (i) ) = 0. Therefore,
ϕj (C¯ −P (i) ) =
ci − αi (C )
|F (i; g (C ))|
.
By non-negativity, ϕj (C¯ −P (i) ) ≥ 0 or equivalently αi (C ) ≤ ci . On the other hand, since i ∈ A(C¯ −P (i) ), by the construction of α, ϕi (C¯ −P (i) ) = αi (C ). By non-negativity, ϕi (C¯ −P (i) ) = αi (C ) ≥ 0. Altogether, for all i ∈ N , 0 ≤ αi (C ) ≤ ci , which shows that condition (i) is satisfied. This completes the proof. Remark 1. We show that the axioms of Theorem 1 are independent by presenting additional rules made possible when each axiom is removed from the list. (i) Efficiency: For all C ∈ CN0 and all i ∈ N , ϕiZ (C ) = 0. This zero rule, which assigns all agents zero, satisfies all of the axioms in Theorem 1 except for efficiency. (ii) Non-negativity: For all N0 ∈ N0 and all i ∈ N, let βiN : CN10 → R be a function satisfying the following conditions: (i) for all N0 ∈ N0 and all C ∈ CN10 if i ∈ Z (C ), then βiN (C ) = ci ,
(ii) for all N0 , N0′ ∈ N0 such that i ∈ N ∩ N ′ and all C ∈ CN10 and
C ′ ∈ CN1 ′ , if F (i; g (C )) = F (i; g (C ′ )) and ci = ci′ , then βiN (C ) = 0
′
βiN (C ′ ).
Then, for all C ∈ CN10 and all i ∈ N, βN
ϕi (C ) = βiN (C ) +
ck − βkN (C )
k∈P (i;g (C ))
|F (k; g (C ))|
.
For such a generalized sequential contributions rule, there is no restriction that, for all C ∈ CN10 and all i ∈ N , 0 ≤ βiN (C ) ≤ ci . Each such rule satisfies all the axioms in Theorem 1 except for nonnegativity. (iii) Independence of following costs: For all C ∈ CN10 and all i ∈ N,
ϕiGE (C ) =
1
|F (l; g (C )) ∪ {l}| j∈(F (l;g (C ))∪{l})
cj ,
where l is the unique agent in l ∈ A(C )∩ P (i; g (C )). For each group, this group-wise egalitarian rule assigns each agent in the group an equal portion of the total immediate connection costs of the group. It satisfies all the axioms in Theorem 1 except for independence of following costs. (iv) Group independence: For all C ∈ CN10 and all i ∈ N,
ϕ (C ) = 1 i
1
|Z (C )| j∈Z (C ) ci
cj
if i ∈ Z (C ), otherwise.
This rule assigns each leaf an equal portion of the sum of the immediate connection costs of all leaves, and any other agent his immediate connection cost, as in the Bird rule. It satisfies all the axioms in Theorem 1 except for group independence. (v) Weak first-link consistency: For all C ∈ CN10 and all i ∈ N,
ϕi2 (C ) 1 cj = |(F (l; g (C )) ∪ {l}) ∩ Z (C )| j∈(F (l;g (C ))∪{l}) 0
if i ∈ Z (C ), otherwise.
This rule assigns each leaf an equal portion of the sum of the immediate connection costs of the group, and any other agent zero. It satisfies all the axioms in Theorem 1 except for weak first-link consistency. 4. Further characterizations Now we discuss how the Bird and the sequential equal contributions rules can be singled out from the family. 4.1. The Bird rule Our first characterization of the Bird rule is based on first-link consistency. Theorem 2. On C 1 , the Bird rule is the only rule satisfying efficiency, group independence, and first-link consistency. Proof. It is clear that the Bird rule satisfies efficiency and group independence. Note that for all N0 ∈ N0 , all C ∈ CN10 , all l ∈ A(C ),
and all i ∈ N \ {l}, ci−l = ci . Since ϕiB (C ) = ci = ci−l = ϕiB (C −l ), the Bird rule satisfies first-link consistency. Conversely, let ϕ be a rule satisfying efficiency, group independence, and first-link consistency. Let d(i) be the function such that if i ∈ N \ Z (C ), d(i) ≡ max{h | ph (j) = i for all j ∈ F (i)} and if i ∈ Z (C ), d(i) ≡ 0. We show that for all N0 ∈ N0 , all C ∈ CN10 , and all i ∈ N , ϕi (C ) = ci . First, suppose that d(i) = 0, that is, i ∈ Z (C ). By group independence and first-link consistency, ϕi (C ) = ϕi (C −P (i) ). By efficiency and group independence, ϕi (C −P (i) ) = ci . Therefore, ϕi (C ) = ci . We will establish the claim for an arbitrary set of agents by an induction argument. As induction hypothesis, suppose that for all j ∈ N such that d(j) ≤ H , ϕj (C ) = cj . Now suppose that d(i) = H + 1. It is easy to check that
ϕk (C −P (i) ) =
k∈F (i)∪{i}
ck .
(6)
k∈F (i)∪{i}
Since for all k ∈ F (i), d(k) ≤ H, from the induction hypothesis, ϕk (C ) = ck . Furthermore, by first-link consistency, for all k ∈ F (i), ϕk (C −P (i) ) = ϕk (C ). Therefore, for all k ∈ F (i),
ϕk (C −P (i) ) = ck .
(7) −P (i)
Inserting (7) into (6), we have ϕi (C ) = ci . By first-link consistency, ϕi (C ) = ϕi (C −P (i) ). Therefore, for all i ∈ N such that d(i) = H + 1, ϕi (C ) = ci , the desired conclusion. Remark 2. First-link consistency requires that the departure of a first-link agent should not affect the contributions of his followers. We can formulate a stronger version of the axiom, which requires that the departure should not affect not only his followers but also any other players. With this strengthening of first-link consistency, we can drop group independence from the list appearing in Theorem 2.
Y. Chun, J. Lee / Mathematical Social Sciences 64 (2012) 136–143
Next, we introduce another important property. A rule is a core selection if no coalition of agents can be made better off by building their own network. Core selection: For all N0 ∈ N0 , all C ∈ CN0 , and all S ∈ N , ϕi (C ) ≤ m(C|S0 ).
i∈S
We show that the Bird rule is the only core selection in the family. Theorem 3. On C 1 , the Bird rule is the only sequential contributions rule satisfying core selection. Proof. Bird (1976) shows that the Bird rule is a core selection. Conversely, let ϕ α be a sequential contributions rule satisfying core selection. For all N0 ∈ N0 , all C ∈ CN10 , all i ∈ N \ Z (C ), and all ϵ > 0, let C˜ ϵ−P (i) ∈ C(1N \P (i)) be 0
= cw + ϵ cqw + ϵ
ci − αi (C )
|F (i)|
.
(8)
Since ϕ α is a core selection,
ϕjα (C˜ ϵ−P (i) ) ≤ cj + ϵ. From (8) and (9),
ci −αi (C ) |F (i)|
(9)
(10)
By definition of a contributions function, 0 ≤ αi (C ) ≤ ci .
We will establish the claim for an arbitrary set of agents by an induction argument. As induction hypothesis, suppose that for all N0′ ∈ N0 , all C ′ ∈ CN10 , and all j ∈ N ′ such that DC ′ (j) ≤ H , ϕj (C ′ ) = cj′ . Now suppose that DC (i) = H + 1. Let l = pH (i). By efficiency and group independence,
ϕk (C¯ ) =
k∈(F (l)∪{l})
ck .
(12)
k∈(F (l)∪{l})\F (i)
Since ϕl (C¯ ) = cl from the previous argument, substituting it into (12) gives
ϕk (C¯ ) =
ck .
(13)
k∈F (l)\F (i)
{ϕk (C¯ −l ) + a} =
k∈F (l)
ck .
(14)
k∈F (l)\F (i)
¯ −l By efficiency and group independence, = k∈F (l) ϕk (C ) c , which together with (14) implies that | F ( l )| a = 0. k∈F (l)\F (i) k Since i ∈ F (l), |F (l)| ̸= 0, and therefore, a = 0. Since DC¯ −l (i) = H, from the induction hypothesis,
ϕi (C¯ −l ) = c¯i−l = ci .
ϕi (C ) = ϕi (C¯ ) = ci , the desired conclusion.
(11)
From (10) and (11), for all i ∈ N,
αi (C ) = ci , the desired conclusion.
ϕk (C¯ ) =
By weak first-link consistency, ϕi (C¯ ) = ϕi (C¯ −l ) + a = ci + 0 and by independence of following costs,
≤ ϵ . Since ϵ > 0 is arbitrary,
αi (C ) ≥ ci .
k∈(F (i)∪{i})
By weak first-link consistency, there exists a ∈ R such that for all k ∈ F (l), ϕk (C¯ ) = ϕk (C¯ −l ) + a. Therefore, (13) can be rewritten as
Note that αi (C˜ ϵ−P (i) ) = αi (C ) and j ∈ F (i) implies j ∈ Z (C(1N \P (i)) ). For all j ∈ F (i), 0
ϕjα (C˜ ϵ−P (i) ) = cj +
141
k∈(F (i)∪{i}) ck = ci . Therefore, ϕi (C¯ ) = ci . By independence of following costs, ϕi (C ) = ϕi (C¯ ) = ci .
independence,
k∈F (l)
if [q = i and w ∈ F (i)] or [qw = 0i], if q ∈ N \ (P (i) ∪ {i}) and w ∈ F (i), otherwise.
cw
c˜ϵ−qPw(i)
Our next axiom requires that if an agent’s immediate connection cost is zero, then his contribution be zero. Null responsibility: For all N0 ∈ N0 , all C ∈ CN10 , and all i ∈ N, if ci = 0 then ϕi (C ) = 0. We show that the Bird rule is the only member of the family satisfying null responsibility. Moreover, uniqueness holds even if non-negativity is dropped from the list appearing in Theorem 1. For all N0 ∈ N0 , all C ∈ CN10 , and all i ∈ N, let gi0 (C ) be the (unique) path between i and 0, and DC (i) be the number of arcs in gi0 (C ). Theorem 4. On C 1 , the Bird rule is the only rule satisfying efficiency, independence of following costs, group independence, weak first-link consistency, and null responsibility. Proof. It is obvious that the Bird rule satisfies the five axioms. Conversely, let ϕ be a rule satisfying these axioms. We show that for all i ∈ N , ϕi (C ) = ci . Let C¯ ∈ CN10 be
Our next axiom requires that the contribution of each agent depend only on his path to the source over a minimal cost spanning tree. Independence of irrelevant costs: For all N0 ∈ N0 , all C , C ′ ∈ CN10 such that g (C ) = g (C ′ ), and all i ∈ N, if cqw = cq′ w for all (qw) ∈ gi0 (C ), then ϕi (C ) = ϕi (C ′ ). Our last characterization of the Bird rule is based on efficiency, null responsibility, and independence of irrelevant costs. Note that independence of following costs, group independence, and weak firstlink consistency in Theorem 4 are replaced by independence of irrelevant costs under efficiency and null responsibility. Theorem 5. On C 1 , the Bird rule is the only rule satisfying efficiency, null responsibility, and independence of irrelevant costs. Proof. It can easily be shown that the Bird rule satisfies the three axioms. Conversely, let ϕ be a rule satisfying these axioms. For all N0 ∈ N0 , all C ∈ CN10 , and all i ∈ N, let C i ∈ CN10 be cqw
cqi w
= 0 c¯ (C )
if (qw) ∈ gi0 (C ), if (qw) ∈ g (C ) \ gi0 (C ), otherwise.
if q = i and w ∈ F (i), if q ∈ N \ (F (i) ∪ {i}) and w ∈ F (i), otherwise.
For all i ∈ N, by null responsibility, for all k ∈ N \ (P (i) ∪ {i}), ϕk (C i ) = 0. Furthermore, by efficiency, for all i ∈ N, ck = ϕk (C i ). (15)
First, suppose that DC (i) = 1. If F (i) = ∅, by efficiency and group independence, ϕi (C ) = ci . Now suppose that F (i) ̸= ∅. By null responsibility, for all k ∈ F (i), ϕk (C¯ ) = 0. By efficiency and group
By independence of irrelevant costs, for all i ∈ N , ϕi (C i ) = ϕi (C ). Therefore, it suffices to show that for all i ∈ N , ϕi (C i ) = ci .
0 c¯ (C ) cqw
c¯qw =
k∈P (i)∪{i}
k∈P (i)∪{i}
142
Y. Chun, J. Lee / Mathematical Social Sciences 64 (2012) 136–143
First, suppose that i ∈ A(C ). Since P (i) = ∅, we can rewrite (15) as
ϕi (C i ) = ci , the desired conclusion. Next, suppose that i ∈ N \ A(C ). Since there exists p(i) ∈ N, from (15),
ck =
k∈P (i)
ϕk (C
p(i)
).
k∈P (i)
ck =
k∈P (i)
k∈P (i)∪{i}
(16)
ϕk (C i ) + ϕi (C i )
ϕk (C¯ ) =
ck .
(17)
k∈(F (l)∪{l})\F (i)
l Since ϕl (C¯ ) = |F (l)+ from the previous argument, substituting it 1| into (17) gives
ck + ϕi (C ). i
k∈P (i)
ϕk (C¯ ) =
k∈F (l)
ck + |F (l)|
k∈F (l)\F (i)
cl
|F (l) + 1|
.
(18)
By weak first-link consistency, there exists a ∈ R such that for all k ∈ F (l), ϕk (C¯ ) = ϕk (C¯ −l ) + a. Therefore, (18) can be rewritten as
k∈P (i)
=
. Now suppose that DC (i) = H + 1. Let l = pH (i).
By efficiency and group independence,
ϕk (C i ).
Inserting (16) into (15), ck =
p (j)
|F (ph (j))|+1
h=0
c
k∈P (i)
c′ h
DC ′ (j)−1
k∈(F (l)∪{l})
By independence of irrelevant costs, for all k ∈ P (i), ϕk (C p(i) ) = ϕk (C i ), which yields
We will establish the claim for an arbitrary set of agents by an induction argument. As induction hypothesis, suppose that for all N0′ ∈ N0 , all C ′ ∈ CN10 , and all j ∈ N ′ such that DC ′ (j) ≤ H, ϕj (C ′ ) =
{ϕk (C¯ −l ) + a} =
k∈F (l)
ck + |F (l)|
k∈F (l)\F (i)
cl
|F (l) + 1|
.
(19)
¯ −l By efficiency and group independence, = k∈F (l) ϕk (C ) k∈F (l)\F (i) ck . which together with (19) implies that |F (l)|a =
Therefore,
ϕi (C i ) = ci ,
l |F (l)| |F (lc)+ or 1|
the desired conclusion.
a=
4.2. The sequential equal contributions rule We present two characterizations of the sequential equal contributions rule by additionally requiring that if the immediate connection cost between any two agents is zero, then the contribution of the two agents should be the same.
cl
|F (l) + 1|
.
(20)
Since DC¯ −l (i) = H, from the induction hypothesis,
ϕi (C¯ −l ) =
H −1
cph (i)
h=0
|F ( (i))| + 1 ph
.
(21)
Equal responsibility: For all N0 ∈ N0 and all C ∈ CN10 , if i ∈ N \ A(C ) and ci = 0, then ϕi (C ) = ϕp(i) (C ), and if i ∈ A(C ) and ci = 0, then ϕi (C ) = 0.
Since i ∈ F (l), by weak first-link consistency, ϕi (C¯ ) = ϕi (C¯ −l ) + a, and combined with (20) and (21),
Once again, uniqueness holds even if non-negativity is dropped from the list.
ϕi (C¯ ) =
H −1
cph (i)
h =0
|F ( (i))| + 1
H
Theorem 6. On C 1 , the sequential equal contributions rule is the only rule satisfying efficiency, independence of following costs, group independence, weak first-link consistency, and equal responsibility. Proof. It is obvious that the sequential equal contributions rule satisfies the five axioms. Conversely, let ϕ be a rule satisfying these axioms. We show that for all N0 ∈ N0 , all C ∈ CN10 , and all i ∈ N, DC (i)−1
ϕi (C ) = Let C¯ ∈
cph (i)
h=0
|F (ph (i))| + 1
CN10
.
be
0 c¯ (C ) cqw
c¯qw =
if q = i and w ∈ F (i), if q ∈ N \ (F (i) ∪ {i}) and w ∈ F (i), otherwise.
First, suppose that DC (i) = 1. If F (i) = ∅, by efficiency and group ci independence, ϕi (C ) = ci = |F (i)+ . Now suppose that F (i) ̸= ∅. 1|
By equal responsibility, for all k ∈ F (i), ϕk (C¯ ) = ϕi (C¯ ). By efficiency and group independence, k∈(F (i)∪{i}) ϕk (C¯ ) = k∈(F (i)∪{i}) ck = ci . i Therefore, ϕi (C¯ ) = |F (i)+ . By independence of following costs, 1|
c
ϕi (C ) = ϕi (C¯ ) =
ci
|F (i) + 1|
.
=
+
ph
cph (i)
h=0
|F ( (i))| + 1 ph
cl
|F (l) + 1|
.
By independence of following costs,
ϕi (C ) = ϕi (C¯ ) =
H
cph (i)
h=0
|F ( (i))| + 1
the desired conclusion.
ph
,
Our final characterization of the sequential equal contributions rule is obtained by replacing independence of following costs, group independence, and weak first-link consistency by independence of irrelevant costs. Theorem 7. On C 1 , the sequential equal contributions rule is the only rule satisfying efficiency, equal responsibility, and independence of irrelevant costs. Proof. It can easily be shown that the sequential equal contributions rule satisfies the three axioms. Conversely, let ϕ be a rule satisfying these axioms. We show that for all N0 ∈ N0 , all C ∈ CN10 , and all i ∈ N, DC (i)−1
ϕi (C ) =
cph (i)
h=0
|F ( (i))| + 1 ph
.
Y. Chun, J. Lee / Mathematical Social Sciences 64 (2012) 136–143
Since DC (p(i)) = H, from the induction hypothesis, ϕp(i) (C ) =
Let C i ∈ CN10 be defined by
cqi w
=
c h p (p(i)) h=0 |F (ph (p(i)))|+1 .
H −1
if (qw) ∈ gi0 (C ), if (qw) ∈ g (C ) \ gi0 (C ), otherwise.
cqw 0 c¯ (C )
ϕi (C ) =
i First, suppose that DC (i) = 1. By efficiency, k∈N ϕk (C ) = ci . i Since for all k ∈ F (i), ck = 0, then by equal responsibility, for all k ∈ F (i), ϕk (C i ) = ϕi (C i ) and for all k ∈ N \(F (i)∪{i}), ϕk (C i ) = 0. Therefore,
ci =
k∈F (i)∪{i}
=
ϕk (C i ) +
k∈N \(F (i)∪{i})
= {|F (i)| + 1}ϕi (C i ), or equivalently,
|F (i)| + 1
.
(22)
By independence of irrelevant costs, ϕi (C ) = ϕi (C i ), and from (22),
ϕi (C ) = ϕi (C i ) =
ci
|F (i)| + 1
.
We will establish the claim for an arbitrary set of agents by an induction argument. As induction hypothesis, suppose that for all j ∈ N such that DC (j) ≤ H , ϕj (C ) =
DC (j)−1
suppose that DC (i) = H + 1. By efficiency,
ϕk (C p(i) ) + ci =
h =0
. Now |F (ph (j))|+1
ϕk (C i ).
Since for all k ∈ N \ F (pH (i)), ϕk (C i ) = ϕk (C p(i) ),
ϕk (C p(i) ) + ci =
k∈F (pH (i))
ϕk (C i ).
k∈F (pH (i))
Furthermore, since for all k ∈ {ph (i) | h = 1, 2, . . . , H }, ϕk (C i ) = ϕk (C p(i) ), we have
ϕk (C p(i) ) + ci =
k∈F (i)∪{i}
ϕk (C i ).
(23)
k∈F (i)∪{i}
By equal responsibility, since for all k ∈ F (i), ϕk (C i ) = ϕi (C i ) and ϕk (C p(i) ) = ϕi (C p(i) ), we have
{|F (i)| + 1}ϕi (C p(i) ) + ci = {|F (i)| + 1}ϕi (C i ), or equivalently,
ϕi (C i ) =
ci
|F (i)| + 1
|F (i)| + 1
+
H −1
cph (p(i))
h =0
|F (ph (p(i)))| + 1
H
cph (i)
h=0
|F (ph (i))| + 1
,
We are grateful to William Thomson, a referee, and the participants of the 2009 Public Economic Theory Meeting (PET09 Galway), the sixth biannual conference of the Society for Economic Design (SED 2009 Maastricht), the fifth Spain Italy Netherlands meeting on game theory (SING5), the sixth Asian general equilibrium theory workshop (GETA 2009), the second Taiwan–Dutch and international conference on game theory, and the logic, game theory, and social choice 6 for their comments. Chun’s work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2010-330-B00077). References
c h p (j)
k∈N
k∈N
ci
Acknowledgments
ϕk (C ) i
ci
=
Therefore,
the desired conclusion.
ϕk (C i )
k∈F (i)∪{i}
ϕi (C i ) =
143
+ ϕi (C p(i) ).
By equal responsibility, ϕi (C p(i) ) = ϕp(i) (C p(i) ) and by independence of irrelevant costs, ϕi (C i ) = ϕi (C ) and ϕp(i) (C p(i) ) = ϕp(i) (C ).
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Mathematical Social Sciences journal homepage: www.elsevier.com/locate/econbase
Sequential contributions rules for minimum cost spanning tree problems Youngsub Chun a,∗ , Joosung Lee a,b a
Department of Economics, Seoul National University, Seoul 151-746, Republic of Korea
b
Department of Economics, Pennsylvania State University, University Park, PA 16802, USA
article
info
Article history: Available online 21 February 2012
abstract We introduce a family of sequential contributions rules for minimum cost spanning tree problems. Each member of the family assigns an agent part of the cost of connecting him to his immediate predecessor, and all of his followers are equally responsible for the remaining part. We characterize the family by imposing the axioms of efficiency, non-negativity, independence of following costs, group independence, and weak first-link consistency. The Bird and the sequential equal contributions rules are two distinguished members of the family. The Bird rule is obtained by requiring an agent to pay the entire cost of connecting him to his immediate predecessor, and the sequential equal contributions rule is obtained by requiring an agent and each of his followers to be equally responsible for this cost. We show how each of these two rules can be singled out from the family. © 2012 Elsevier B.V. All rights reserved.
1. Introduction A minimum cost spanning tree problem is concerned with constructing a minimal cost spanning tree which provides for every agent a connection to the source and allocating the construction cost among the agents. Examples of minimum cost spanning tree problems are abundant: constructing communication networks such as telephone or cable television, or building a drainage system that connects every house in a city with a water purifier. Claus and Kleitman (1973) initiated the study of the problem, and Bird (1976) treated the problem with game-theoretic methods and proposed a cost allocation rule, now called the Bird rule. Ever since, many other rules have been studied: the core and the nucleolus (Granot and Huberman, 1984), the Folk solution (Feltkamp et al., 1994; Branzei et al., 2004; Bergantiños and Vidal-Puga, 2005, 2007; Lorenzo and Lorenzo-Freire, 2009), the Kar rule (Kar, 2002), the Dutta–Kar rule (Dutta and Kar, 2004), and the piecewise linear rules (Bogomolnaia and Moulin, 2010). In this paper, we propose and characterize a new family of rules, which we call the sequential contributions rules: each member of the family assigns each agent part of the cost of connecting him to his immediate predecessor, and all of his followers are equally responsible for the remaining part. Our main axiom is first-link consistency, which requires that upon the departure of a ‘‘first-link agent’’, an agent directly linked to the source, the contribution of his followers should not be affected. Also we formulate a weaker version, weak first-link consistency, which requires that upon the departure of a first-link
∗
Corresponding author. E-mail addresses: [email protected] (Y. Chun), [email protected] (J. Lee).
0165-4896/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2012.02.001
agent, the contribution of his followers should be affected by the same amount. Our main result is a characterization of the sequential contributions rules on the basis of efficiency, non-negativity, independence of following costs, group independence and weak first-link consistency. Efficiency requires that the sum of contributions of all agents should be equal to the cost of constructing an efficient network. Non-negativity requires that the contribution of each agent should not be negative. Independence of following costs requires that the contribution of each agent should not be affected by changes in his followers’ connection costs. Group independence requires that for each group which consists of a first-link agent and all of his followers, the contribution of each member of the group should depend only on the connection costs between themselves. The Bird and the sequential equal contributions rules are two distinguished members of the family of sequential contributions rules.1 The Bird rule is obtained by requiring each agent to pay the entire cost of connecting him to his immediate predecessor, and the sequential equal contributions rule is obtained by making an agent and each of his followers equally responsible for this cost. Next, we show how each of these two rules can be singled out from the family. First, the Bird rule is characterized by efficiency, group independence, and first-link consistency. Also, it is the unique sequential contributions rule satisfying core selection or null responsibility; a rule is a core selection if no coalition of agents can be made better off by building their own network, and null 1 The sequential equal contributions rule is originated from the airport problem, first studied by Littlechild and Owen (1973). For airport problems, the sequential equal contributions rule assigns the cost of each segment equally to all airlines using the segment. The terminology we adopt is borrowed from Thomson (forthcoming).
Y. Chun, J. Lee / Mathematical Social Sciences 64 (2012) 136–143
responsibility requires that if the cost of connecting an agent to his immediate predecessor is zero, then his contribution should be zero. Finally, the Bird rule is characterized by efficiency, null responsibility, and independence of irrelevant costs, which requires that the contribution of each agent should depend only on his path to the source over a minimal cost spanning tree. On the other hand, the sequential equal contributions rule is the unique sequential contributions rule satisfying equal responsibility, which requires that if a connection cost between two agents on the minimum cost spanning tree is zero, then the contribution of the two agents should be the same. The sequential equal contributions rule is also characterized by efficiency, equal responsibility, and independence of irrelevant costs. The paper is organized as follows. Section 2 contains preliminaries and introduces the family of sequential contributions rules. In Section 3, we present an axiomatic characterization of the family. In Section 4, we discuss how the Bird and the sequential equal contributions rules can be characterized by imposing additional axioms. 2. Preliminaries 2.1. The problem Let N = {1, 2, . . .} be a (finite or infinite) universe of ‘‘potential’’ agents. Let N be the family of nonempty and finite subsets of N. A typical element of N is denoted by N = {1, . . . , n}. We are interested in networks whose nodes are elements of a set N0 = N ∪ {0}, where 0 is a special node called the source. Let N0 ≡ {N0 | N ∈ N }. Given N0 ∈ N0 , a cost matrix C = (cij )i,j∈N0 gives the cost of the (direct) link between any two nodes. We assume that cii = 0 for all i ∈ N0 and cij = cji ≥ 0 for all i, j ∈ N0 . We denote the set of all cost matrices over N0 as CN0 and set C ≡ N0 ∈N0 CN0 . Given N0 ∈ N0 , C ∈ CN0 , and S ⊆ N, let C|S0 be the restriction of C to S0 , and c¯ (C ) ∈ R+ be any given number such that for all i, j ∈ N0 , c¯ (C ) > maxi,j∈N0 cij . A graph g over N0 is a subset of {(ij)|i, j ∈ N0 }, whose elements are arcs. Given g and i, j ∈ N0 such that i ̸= j, a path from i to j in g , gij , is a sequence of distinct arcs {(ik−1 ik )}Kk=1 such that (ik−1 ik ) ∈ g for all k ∈ {1, 2, . . . , K }, i = i0 and j = iK . A tree is a graph with a unique path between any twonodes. We denote the set of all trees over N0 as TN0 , and set T ≡ N0 ∈N0 TN0 . Given N0 ∈ N0 , t ∈ TN0 , and i ∈ N, the immediate predecessor of i, p(i; t ), is j ∈ N0 such that (ji) ∈ gi0 and the k-th predecessor of i is pk (i; t ) ≡ pk−1 (p(i; t ); t ). The set of all predecessors of i is P (i; t ) ≡ {pk (i; t ) | k = 1, 2, . . . , K }, where pK (i; t ) = 0. For convenience, let p0 (i; t ) ≡ i. The set of all followers of i is F (i; t ) ≡ {j ∈ N | pk (j; t ) = i for some k = 1, 2, . . .}. When there is no ambiguity, we use the notation p(i), P (i), and F (i) instead of p(i; t ), P (i; t ), and F (i; t ), respectively. We denote the cardinality of F (i) as |F (i)|. For all N0 ∈ N0 and all C ∈ CN0 , t ∈ TN0 is a minimum cost spanning tree for C , or an mcst for C , if for all t ′ ∈ TN0 , (ij)∈t cij ≤ (ij)∈t ′ cij . Let m(C ) = mint ∈TN0 (ij)∈t cij . As established in the literature, an mcst always exists even though it is not necessarily unique. Given N0 ∈ N0 , a minimum cost spanning tree problem, or an mcstp, is a cost matrix C ∈ CN0 . For each N ∈ N , a cost allocation rule, or simply a rule, is a function ϕ : CN0 → RN , which associates to any problem C ∈ CN0 a vector ϕ(C ) ≡ (ϕi (C ))i∈N in RN . For each i ∈ N , ϕi (C ) is the contribution of i.
137
We impose a domain restriction on the set of cost matrices:
CN10
≡ {C ∈ CN0 | C induces a unique mcst. } 1 1 and C 1 ≡ N0 ∈N0 CN0 . For each C ∈ C , we denote the unique mcst for C by g (C ). Given C ∈ C 1 , we denote the immediate connection cost of i, cp(i;g (C ))i , by ci . An agent l is a leaf if F (l; g (C )) = ∅ and a firstlink agent if P (l; g (C )) = {0}. Let Z (C ) be the set of all leaves and A(C ) be the set of all first-link agents. For all N0 ∈ N0 , all C ∈ CN10 , and all l ∈ A(C ), F (l; g (C )) ∪ {l} is a group. Note that N = l∈A(C ) (F (l; g (C )) ∪ {l}) and (F (l; g (C )) ∪ {l}) ∩ (F (m; g (C )) ∪ {m}) = ∅ for all l, m ∈ A(C ) such that l ̸= m. 2.2. Sequential contributions rules We introduce a family of rules, which we call sequential contributions rules. Each member of the family assigns each agent part of his immediate connection cost (the cost of connecting him to his immediate predecessor), and his followers are equally responsible for the remaining part. Formally, for all N0 ∈ N0 , let α N ≡ (αiN )i∈N be a contributions function, where αiN is a function αiN : CN10 → R satisfying the following conditions: (i) for all N0 ∈ N0 and C ∈ CN10 , 0 ≤ αiN (C ) ≤ ci ,
(ii) for all N0 ∈ N0 and C ∈ CN10 , if i ∈ Z (C ), then αiN (C ) = ci ,
(iii) for all N0 , N0′ ∈ N0 such that i ∈ N ∩ N ′ and all C ∈ CN10 and C ′ ∈ CN1 ′ , if F (i; g (C )) = F (i; g (C ′ )) and ci = ci′ , then αiN (C ) = ′
0
αiN (C ′ ).
Let AN be the family of all contributions functions over N. For simplicity of notation, we write α instead of α N when there is no danger of confusion. We are ready to define the sequential contributions rules. Sequential contributions rules: For all N0 ∈ N0 , all C ∈ CN10 , and all α ∈ AN , a sequential contributions rule with a contributions function α , ϕ α ≡ (ϕiα )i∈N , is defined as
ϕiα (C ) = αi (C ) +
ck − αk (C ) . | F (k; g (C ))| k∈P (i;g (C ))
The Bird and the sequential equal contributions rules are two most well-known members of this family. The Bird rule is obtained by requiring each agent to be solely responsible for his immediate connection cost. The sequential equal contributions rule is obtained by requiring an agent and each of his followers to be equally responsible for his immediate connection cost. Bird rule, ϕ B : For all N0 ∈ N0 , all C ∈ CN10 , and all i ∈ N , αi (C ) = ci . Sequential equal contributions rule, ϕ E : For all N0 ∈ N0 , all C ∈ CN10 , i . and all i ∈ N, αi (C ) = |F (i)|+ 1
c
Even when a cost matrix does not induce a unique mcst, we can still define a notion of the sequential contributions rule. For each mcst derived from the cost matrix, we apply a sequential contributions rule and find the corresponding contributions. Each agent’s contribution is calculated by taking an average over all mcsts. 3. The main characterization First, we introduce basic axioms discussed in the literature. Efficiency requires that the sum of the contributions of all agents should be equal to the cost of constructing an efficient network. Non-negativity requires that the contribution of each agent should not be negative.
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Efficiency: For all N0 ∈ N0 and all C ∈ CN0 ,
ϕi (C ) = m(C ). Non-negativity: For all N0 ∈ N0 , all C ∈ CN0 , and all i ∈ N , ϕi (C ) ≥ 0.
i∈N
Next are two independence axioms. Independence of following costs requires that the contribution of each agent should not be affected by his followers’ connection costs. Group independence requires that for each group, the contribution of each member of the group should depend only on the connection costs between themselves.2 In other words, for each group, suppose that there is a change in the connection cost between an agent in some other group and another agent—this agent can be a member of the given group, while the members of the given group remain unchanged. Group independence requires that as long as the connection costs between any pairs in the group remain unchanged, such a change should not affect the contributions of agents in the group. Independence of following costs: For all N0 ∈ N0 , all C , C ′ ∈ CN0 , and all i ∈ N, if F (i; g (C )) = F (i; g (C ′ )) and for all q, w ∈ N0 \ F (i, g (C )), cqw = cq′ w , then ϕi (C ) = ϕi (C ′ ). Group independence: For all N0 , N0′ ∈ N0 , all C ∈ CN0 , and all C ′ ∈ CN ′ , if there exists l ∈ A(C ) ∩ A(C ′ ) such that F (l; g (C )) = 0
First-link consistency: For all N0 ∈ N0 , all C ∈ CN10 , all l ∈ A(C ), and
all i ∈ F (l), ϕi (C ) = ϕi (C −l ). Next is weak first-link consistency, which requires that upon the departure of a first-link agent, all of his followers should be affected by the same amount. It is obvious that first-link consistency implies weak first-link consistency. Weak first-link consistency: For all N0 ∈ N0 , all C ∈ CN10 , all l ∈ A(C ), and all i, j ∈ F (l),
ϕi (C ) − ϕi (C −l ) = ϕj (C ) − ϕj (C −l ). Before presenting our main characterization result, we show that each member of the family of the sequential contributions rules satisfies efficiency, non-negativity, independence of following costs, group independence, and weak first-link consistency. Lemma 1. On C 1 , each sequential contributions rule satisfies efficiency, non-negativity, independence of following costs, group independence, and weak first-link consistency.
F (l; g (C ′ )) and that C|(F (l;g (C ))∪{l})0 = C|(′ F (l;g (C ′ ))∪{l}) , then for all 0 i ∈ F (l, g (C )) ∪ {l}, ϕi (C ) = ϕi (C ′ ).
Proof. Let α ∈ AN . We omit the easy proof that ϕ α satisfies efficiency and non-negativity.
The final axiom for our main characterization is based on a certain formulation of consistency.3 Feltkamp et al. (1999) introduce a consistency axiom for mcstps, namely leaf consistency: it requires that the departure of a leaf should not affect the contribution of other remaining agents. They characterized the Bird rule on the basis of leaf consistency. Recently, Dutta and Kar (2004) propose two other consistency axioms for mcstps. Suppose that an agent leaves, but all the remaining agents are allowed to use the leaving node. These agents can use the leaving node in two different ways:
• Independence of following costs: For all N0 ∈ N0 , all C , C ′ ∈ CN10 , and all i ∈ N, if F (i; g (C )) = F (i; g (C ′ )) = F (i) and for all q, w ∈ N \ F (i), cqw = cq′ w , we obtain: (i) P (i; g (C )) = P (i; g (C ′ )), (ii) for all k ∈ P (i) ∪ {i}, F (k; g (C )) = F (k; g (C ′ )) and ck = ck′ , (iii) since |F (i; g (C ))| = |F (i; g (C ′ ))| and ci = ci′ , αi (C ) = αi (C ′ ).
(i) source consistency, which assumes that the leaving node can be used only to connect to the source, and (ii) tree consistency, which assumes that the leaving node can be used to connect to any node. For each formulation, consistency requires that the contribution of the other agents should be unchanged between the original and the reduced cost matrices. Dutta and Kar characterize the Bird rule on the basis of source consistency and the Dutta–Kar rule on the basis of tree consistency. In this paper, we introduce a fourth consistency axiom, first-link consistency. For all C ∈ C 1 and all l ∈ A(C ), we imagine that a firstlink agent l leaves after constructing the arc (l0), but the remaining agents are allowed to connect to the source through the arc (l0). In this reduced problem, each remaining node has the option to use the node l to connect to the source. Formally, the connection costs on N0 \ {l} are changed as follows:
Therefore,
1 − αk (C )
k∈P (i;g (C ))
|F (k; g (C ))|
ϕiα (C ) = αi (C ) + = αi (C ′ ) +
k∈P (i;g (C ′ ))
ck
1 − αk (C ′ )
|F (k; g (C ′ ))|
ck′
= ϕiα (C ′ ), as desired.
• Group independence: For all N0 , N0′ ∈ N0 such that l ∈ N ∩ N ′ , all C ∈ CN10 , and all C ′ ∈ CN10 , if l ∈ A(C ) ∩ A(C ′ ) and C|F (l;g (C ))∪{l} = C|′F (l;g (C ′ ))∪{l} , we obtain: for all i ∈ F (l, g (C )) ∪ {l}, (i) P (i; g (C )) = P (i; g (C ′ )), (ii) for all k ∈ P (i) ∪ {i}, F (k; g (C )) = F (k; g (C ′ )) and ck = ck′ , (iii) since |F (i; g (C ))| = |F (i; g (C ′ ))| and ci = ci′ , αi (C ) = αi (C ′ ). Therefore, for all i ∈ F (l, g (C )) ∪ {l},
−l (i) for all i ̸= l, ci0 = min{ci0 , cil }, (ii) if {i, j} ∩ {l, 0} = ∅, then cij−l = cij .
ϕiα (C ) = αi (C ) + −l
Let C −l be the reduced cost matrix. Note that for all i ∈ N \{l}, ci = ci and C −l ∈ C 1 . Similarly, let C −P (i) be the reduced cost matrix after the departure of P (i) \ {0}. We abuse the notation by allowing the possibility that P (i) \ {0} = ∅. Of course, if P (i) \ {0} = ∅, then C −P (i) = C . First-link consistency requires that upon the departure of a firstlink agent, none of his followers should be affected. 2 Kar (2002) introduces a weaker version of group independence, which requires that a change in the connection cost within a group should not affect the contribution of agents in some other groups. 3 For a survey on consistency, see Thomson (2000).
1 − αk (C ) ck | F (k; g (C ))| k∈P (i;g (C ))
= αi (C ′ ) +
k∈P (i;g (C ′ ))
1 − αk (C ′ )
|F (k; g (C ′ ))|
ck′
= ϕiα (C ′ ), as desired.
• Weak first-link consistency: For all N0 ∈ N0 , all C ∈ CN10 , all l ∈ A(C ), and all i ∈ F (l), we obtain: (i) P (i; g (C )) = P (i; g (C −l )) ∪ {l}, (ii) since F (i; g (C )) = F (i; g (C −l )) and ci = ci−l , αi (C ) = αi (C −l ).
Y. Chun, J. Lee / Mathematical Social Sciences 64 (2012) 136–143
Therefore, for all i ∈ F (l),
ϕiα (C ) − ϕiα (C −l ) =
αi (C ) +
−
ck − αk (C ) | F (k; g (C ))| k∈P (i;g (C ))
k∈P (i;g (C −l ))
cl − αl (C ) , |F (l; g (C ))|
=
P (i; g (C −l )), F (k; g (C )) = F (k; g (C −l )). Therefore, for all i ∈ (N \ {l}) \ F (l; g (C )),
ϕi (C ) = ϕi (C −l )
We are ready to present our characterization of the sequential contributions rules.
k∈P (i;g (C −l ))
= αi (C ) +
Step 1: For all N0 ∈ N0 , all C ∈ such that
ϕi (C ) = αi (C ) +
CN10 ,
and all i ∈ N, there exists αi (C )
ck − αk (C ) . | F (k; g (C ))| k∈P (i;g (C ))
0
exists αi (C ) such that ′
ϕi (C ) = ϕi (C −l ) + γ
k∈P (i;g (C ′ ))
ck − αk (C ) . |F (k; g (C ′ ))|
ϕi (C ) − cl
ϕi (C ) +
i∈F (l;g (C ))
ϕi (C ) + ϕl (C ) − cl
i∈(N \{l})\F (l;g (C ))
=
ϕi (C ) + |F (l; g (C ))|γ −l
i∈F (l;g (C ))
+
ϕi (C ) + ϕl (C ) − cl −l
i∈(N \{l})\F (l;g (C ))
=
ϕi (C −l ) + |F (l; g (C ))|γ + ϕl (C ) − cl ,
i∈N \{l}
which implies that ϕl (C ) = cl −|F (l; g (C ))|γ . Therefore, by setting αl (C ) ≡ cl − |F (l; g (C ))|γ , we have ϕl (C ) = αl (C ). Since C −l ∈ CN1 \{l} , from the induction hypothesis, let αi (C ) ≡ 0
|F (k; g (C ))|
|F (l; g (C ))|
.
ck − αk (C )
k∈P (i;g (C ))
|F (k; g (C ))|
ϕi (C ) = αi (C ) +
.
Step 2: α defined in Step 1 satisfies condition (iii) in the definition of the sequential contributions rules. Proof. Let N0 , N0′ ∈ N0 be such that i ∈ N ∩ N ′ . Let C ∈ CN10 and C ′ ∈ CN1 ′ be such that F (i; g (C )) = F (i; g (C ′ )) and ci = ci′ . We 0
show that αi (C ) = αi (C ′ ). By the construction of α , for all i ∈ N,
αi (C ) = αi (C −P (i) )
(1)
−P (i)
).
(2)
By group independence, ϕi (C −P (i) ) = ϕi (C|(F (i)∪{i})0 ) and ϕi (C ) = ϕi (C|(′ F (i)∪{i}) ). Since i ∈ A(C −P (i) ), ϕi (C −P (i) ) = αi (C −P (i) ) and 0
−P (i)
ϕi (C|(F (i)∪{i})0 ) = αi (C|(F (i)∪{i})0 ). Similarly, since i ∈ A(C ′ ), −P (i) −P (i) ϕi (C ′ ) = αi (C ′ ) and ϕi (C|(′ F (i)∪{i})0 ) = αi (C|(′ F (i)∪{i})0 ).
Therefore,
αi (C −P (i) ) = αi (C|(F (i)∪{i})0 )
i∈N
=
k∈P (i;g (C ))
cl − αl (C )
′−P (i)
exists γ such that ϕi (C ) = ϕi (C ) + γ . By group independence, for all i ∈ (N \ {l}) \ F (l; g (C )), ϕi (C ) = ϕi (C −l ). By efficiency,
i∈N \{l}
ck − αk (C )
+
Altogether, for all i ∈ N, there exists αi (C ) such that
αi (C ′ ) = αi (C ′
−l
= αi (C ) +
′
Without loss generality, let N ≡ {1, . . . , n}. By weak first-link consistency, for all C ∈ CN10 , all l ∈ A(C ), and all i ∈ F (l; g (C )), there
ϕi (C −l ) =
k∈P (i;g (C −l ))
|F (k; g (C −l ))|
and for all i ∈ N , ′
ck − αk (C −l )
= αi (C −l ) +
′
ϕi (C ′ ) = αi (C ′ ) +
ck − αk (C ) . | F (k; g (C ))| k∈P (i;g (C ))
This proves Step 1.
Proof. First, suppose that |N | = 1. Without loss of generality, let N ≡ {1}. By efficiency, for all C ∈ CN10 , ϕ1 (C ) = c1 , the desired conclusion. We will establish the claim for an arbitrary set of agents by an induction argument. As induction hypothesis, suppose that for all N0′ ∈ N0 such that |N ′ | ≤ n − 1, all C ′ ∈ CN1 ′ , and all i′ ∈ N ′ , there
|F (k; g (C −l ))|
On the other hand, if i ∈ F (l; g (C )), then ϕi (C ) = ϕi (C −l ) + γ . Moreover, P (i; g (C )) = P (i; g (C −l )) ∪ {l}, and for all k ∈ P (i; g (C −l )), F (k; g (C )) = F (k; g (C −l )). Therefore, for all i ∈ F (l; g (C )),
Theorem 1. On C 1 , a rule satisfies efficiency, non-negativity, independence of following costs, group independence, and weak first-link consistency if and only if it is a sequential contributions rule. Proof. By Lemma 1, it suffices to show that if a rule satisfies the five axioms, then it is a sequential contributions rule. Let ϕ be a rule satisfying these axioms. The proof is divided into three steps.
ck − αk (C −l )
= αi (C −l ) +
ck−l − αk (C −l ) − l |F (k; g (C ))|
αi (C −l ) +
as desired.
139
αi (C −l ) for all i ∈ N \ {l}. If i ∈ (N \ {l}) \ F (l; g (C )), then ϕi (C ) = ϕi (C −l ). Moreover, P (i; g (C )) = P (i; g (C −l )), and for all k ∈
(3)
and
αi (C ′
−P (i)
) = αi (C|(′ F (i)∪{i})0 ).
(4)
Since F (i; g (C )) = F (i; g (C ′ )) and ci = ci′ , by independence of following costs, ϕi (C|(F (i)∪{i})0 ) = ϕi (C|(′ F (i)∪{i}) ), which implies that 0
αi (C|(F (i)∪{i})0 ) =
αi (C|(′ F (i)∪{i})0 ).
(5)
Altogether,
αi (C ) = αi (C −P (i) ) = αi (C|(F (i)∪{i})0 ) (1)
(3)
(5)
(4)
= αi (C|(′ F (i)∪{i})0 ) = αi (C ′
This proves Step 2.
−P (i)
(2)
) = αi (C ′ ).
Step 3: α defined in Step 1 satisfies conditions (i) and (ii) in the definition of the sequential contributions rules.
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Y. Chun, J. Lee / Mathematical Social Sciences 64 (2012) 136–143
Proof. For all N0 ∈ N0 , all C ∈ CN10 , and all i ∈ N, let C¯ −P (i) ∈
C(1N \P (i))0 be
c¯q−wP (i)
0
= c¯ (C ) cqw
if q = i and w ∈ F (i), if q ∈ (N \ P (i)) \ (F (i) ∪ {i}) and w ∈ F (i), otherwise.
The remaining proof is divided into two cases: Case 1: i ∈ Z (C ). By efficiency and group independence, αi (C¯ −P (i) ) = ci . Since i ∈ A(C¯ −P (i) ), it follows that ϕi (C¯ −P (i) ) = αi (C¯ −P (i) ). From (1), we conclude that αi (C ) = ci . (Therefore, condition (ii) is satisfied.) Case 2: i ∈ N \ Z (C ). From Step 1, for all j ∈ F (i) such that p(j) = i, ci − αi (C¯ −P (i) ) ϕj (C¯ −P (i) ) = αj (C¯ −P (i) ) + . |F (i; g (C¯ −P (i) ))| Note that αi (C ) = αi (C¯ −P (i) ) and F (i; g (C¯ −P (i) )) = F (i; g (C )). Moreover, by applying efficiency and group independence repeatedly, αj (C¯ −P (i) ) = 0. Therefore,
ϕj (C¯ −P (i) ) =
ci − αi (C )
|F (i; g (C ))|
.
By non-negativity, ϕj (C¯ −P (i) ) ≥ 0 or equivalently αi (C ) ≤ ci . On the other hand, since i ∈ A(C¯ −P (i) ), by the construction of α, ϕi (C¯ −P (i) ) = αi (C ). By non-negativity, ϕi (C¯ −P (i) ) = αi (C ) ≥ 0. Altogether, for all i ∈ N , 0 ≤ αi (C ) ≤ ci , which shows that condition (i) is satisfied. This completes the proof. Remark 1. We show that the axioms of Theorem 1 are independent by presenting additional rules made possible when each axiom is removed from the list. (i) Efficiency: For all C ∈ CN0 and all i ∈ N , ϕiZ (C ) = 0. This zero rule, which assigns all agents zero, satisfies all of the axioms in Theorem 1 except for efficiency. (ii) Non-negativity: For all N0 ∈ N0 and all i ∈ N, let βiN : CN10 → R be a function satisfying the following conditions: (i) for all N0 ∈ N0 and all C ∈ CN10 if i ∈ Z (C ), then βiN (C ) = ci ,
(ii) for all N0 , N0′ ∈ N0 such that i ∈ N ∩ N ′ and all C ∈ CN10 and
C ′ ∈ CN1 ′ , if F (i; g (C )) = F (i; g (C ′ )) and ci = ci′ , then βiN (C ) = 0
′
βiN (C ′ ).
Then, for all C ∈ CN10 and all i ∈ N, βN
ϕi (C ) = βiN (C ) +
ck − βkN (C )
k∈P (i;g (C ))
|F (k; g (C ))|
.
For such a generalized sequential contributions rule, there is no restriction that, for all C ∈ CN10 and all i ∈ N , 0 ≤ βiN (C ) ≤ ci . Each such rule satisfies all the axioms in Theorem 1 except for nonnegativity. (iii) Independence of following costs: For all C ∈ CN10 and all i ∈ N,
ϕiGE (C ) =
1
|F (l; g (C )) ∪ {l}| j∈(F (l;g (C ))∪{l})
cj ,
where l is the unique agent in l ∈ A(C )∩ P (i; g (C )). For each group, this group-wise egalitarian rule assigns each agent in the group an equal portion of the total immediate connection costs of the group. It satisfies all the axioms in Theorem 1 except for independence of following costs. (iv) Group independence: For all C ∈ CN10 and all i ∈ N,
ϕ (C ) = 1 i
1
|Z (C )| j∈Z (C ) ci
cj
if i ∈ Z (C ), otherwise.
This rule assigns each leaf an equal portion of the sum of the immediate connection costs of all leaves, and any other agent his immediate connection cost, as in the Bird rule. It satisfies all the axioms in Theorem 1 except for group independence. (v) Weak first-link consistency: For all C ∈ CN10 and all i ∈ N,
ϕi2 (C ) 1 cj = |(F (l; g (C )) ∪ {l}) ∩ Z (C )| j∈(F (l;g (C ))∪{l}) 0
if i ∈ Z (C ), otherwise.
This rule assigns each leaf an equal portion of the sum of the immediate connection costs of the group, and any other agent zero. It satisfies all the axioms in Theorem 1 except for weak first-link consistency. 4. Further characterizations Now we discuss how the Bird and the sequential equal contributions rules can be singled out from the family. 4.1. The Bird rule Our first characterization of the Bird rule is based on first-link consistency. Theorem 2. On C 1 , the Bird rule is the only rule satisfying efficiency, group independence, and first-link consistency. Proof. It is clear that the Bird rule satisfies efficiency and group independence. Note that for all N0 ∈ N0 , all C ∈ CN10 , all l ∈ A(C ),
and all i ∈ N \ {l}, ci−l = ci . Since ϕiB (C ) = ci = ci−l = ϕiB (C −l ), the Bird rule satisfies first-link consistency. Conversely, let ϕ be a rule satisfying efficiency, group independence, and first-link consistency. Let d(i) be the function such that if i ∈ N \ Z (C ), d(i) ≡ max{h | ph (j) = i for all j ∈ F (i)} and if i ∈ Z (C ), d(i) ≡ 0. We show that for all N0 ∈ N0 , all C ∈ CN10 , and all i ∈ N , ϕi (C ) = ci . First, suppose that d(i) = 0, that is, i ∈ Z (C ). By group independence and first-link consistency, ϕi (C ) = ϕi (C −P (i) ). By efficiency and group independence, ϕi (C −P (i) ) = ci . Therefore, ϕi (C ) = ci . We will establish the claim for an arbitrary set of agents by an induction argument. As induction hypothesis, suppose that for all j ∈ N such that d(j) ≤ H , ϕj (C ) = cj . Now suppose that d(i) = H + 1. It is easy to check that
ϕk (C −P (i) ) =
k∈F (i)∪{i}
ck .
(6)
k∈F (i)∪{i}
Since for all k ∈ F (i), d(k) ≤ H, from the induction hypothesis, ϕk (C ) = ck . Furthermore, by first-link consistency, for all k ∈ F (i), ϕk (C −P (i) ) = ϕk (C ). Therefore, for all k ∈ F (i),
ϕk (C −P (i) ) = ck .
(7) −P (i)
Inserting (7) into (6), we have ϕi (C ) = ci . By first-link consistency, ϕi (C ) = ϕi (C −P (i) ). Therefore, for all i ∈ N such that d(i) = H + 1, ϕi (C ) = ci , the desired conclusion. Remark 2. First-link consistency requires that the departure of a first-link agent should not affect the contributions of his followers. We can formulate a stronger version of the axiom, which requires that the departure should not affect not only his followers but also any other players. With this strengthening of first-link consistency, we can drop group independence from the list appearing in Theorem 2.
Y. Chun, J. Lee / Mathematical Social Sciences 64 (2012) 136–143
Next, we introduce another important property. A rule is a core selection if no coalition of agents can be made better off by building their own network. Core selection: For all N0 ∈ N0 , all C ∈ CN0 , and all S ∈ N , ϕi (C ) ≤ m(C|S0 ).
i∈S
We show that the Bird rule is the only core selection in the family. Theorem 3. On C 1 , the Bird rule is the only sequential contributions rule satisfying core selection. Proof. Bird (1976) shows that the Bird rule is a core selection. Conversely, let ϕ α be a sequential contributions rule satisfying core selection. For all N0 ∈ N0 , all C ∈ CN10 , all i ∈ N \ Z (C ), and all ϵ > 0, let C˜ ϵ−P (i) ∈ C(1N \P (i)) be 0
= cw + ϵ cqw + ϵ
ci − αi (C )
|F (i)|
.
(8)
Since ϕ α is a core selection,
ϕjα (C˜ ϵ−P (i) ) ≤ cj + ϵ. From (8) and (9),
ci −αi (C ) |F (i)|
(9)
(10)
By definition of a contributions function, 0 ≤ αi (C ) ≤ ci .
We will establish the claim for an arbitrary set of agents by an induction argument. As induction hypothesis, suppose that for all N0′ ∈ N0 , all C ′ ∈ CN10 , and all j ∈ N ′ such that DC ′ (j) ≤ H , ϕj (C ′ ) = cj′ . Now suppose that DC (i) = H + 1. Let l = pH (i). By efficiency and group independence,
ϕk (C¯ ) =
k∈(F (l)∪{l})
ck .
(12)
k∈(F (l)∪{l})\F (i)
Since ϕl (C¯ ) = cl from the previous argument, substituting it into (12) gives
ϕk (C¯ ) =
ck .
(13)
k∈F (l)\F (i)
{ϕk (C¯ −l ) + a} =
k∈F (l)
ck .
(14)
k∈F (l)\F (i)
¯ −l By efficiency and group independence, = k∈F (l) ϕk (C ) c , which together with (14) implies that | F ( l )| a = 0. k∈F (l)\F (i) k Since i ∈ F (l), |F (l)| ̸= 0, and therefore, a = 0. Since DC¯ −l (i) = H, from the induction hypothesis,
ϕi (C¯ −l ) = c¯i−l = ci .
ϕi (C ) = ϕi (C¯ ) = ci , the desired conclusion.
(11)
From (10) and (11), for all i ∈ N,
αi (C ) = ci , the desired conclusion.
ϕk (C¯ ) =
By weak first-link consistency, ϕi (C¯ ) = ϕi (C¯ −l ) + a = ci + 0 and by independence of following costs,
≤ ϵ . Since ϵ > 0 is arbitrary,
αi (C ) ≥ ci .
k∈(F (i)∪{i})
By weak first-link consistency, there exists a ∈ R such that for all k ∈ F (l), ϕk (C¯ ) = ϕk (C¯ −l ) + a. Therefore, (13) can be rewritten as
Note that αi (C˜ ϵ−P (i) ) = αi (C ) and j ∈ F (i) implies j ∈ Z (C(1N \P (i)) ). For all j ∈ F (i), 0
ϕjα (C˜ ϵ−P (i) ) = cj +
141
k∈(F (i)∪{i}) ck = ci . Therefore, ϕi (C¯ ) = ci . By independence of following costs, ϕi (C ) = ϕi (C¯ ) = ci .
independence,
k∈F (l)
if [q = i and w ∈ F (i)] or [qw = 0i], if q ∈ N \ (P (i) ∪ {i}) and w ∈ F (i), otherwise.
cw
c˜ϵ−qPw(i)
Our next axiom requires that if an agent’s immediate connection cost is zero, then his contribution be zero. Null responsibility: For all N0 ∈ N0 , all C ∈ CN10 , and all i ∈ N, if ci = 0 then ϕi (C ) = 0. We show that the Bird rule is the only member of the family satisfying null responsibility. Moreover, uniqueness holds even if non-negativity is dropped from the list appearing in Theorem 1. For all N0 ∈ N0 , all C ∈ CN10 , and all i ∈ N, let gi0 (C ) be the (unique) path between i and 0, and DC (i) be the number of arcs in gi0 (C ). Theorem 4. On C 1 , the Bird rule is the only rule satisfying efficiency, independence of following costs, group independence, weak first-link consistency, and null responsibility. Proof. It is obvious that the Bird rule satisfies the five axioms. Conversely, let ϕ be a rule satisfying these axioms. We show that for all i ∈ N , ϕi (C ) = ci . Let C¯ ∈ CN10 be
Our next axiom requires that the contribution of each agent depend only on his path to the source over a minimal cost spanning tree. Independence of irrelevant costs: For all N0 ∈ N0 , all C , C ′ ∈ CN10 such that g (C ) = g (C ′ ), and all i ∈ N, if cqw = cq′ w for all (qw) ∈ gi0 (C ), then ϕi (C ) = ϕi (C ′ ). Our last characterization of the Bird rule is based on efficiency, null responsibility, and independence of irrelevant costs. Note that independence of following costs, group independence, and weak firstlink consistency in Theorem 4 are replaced by independence of irrelevant costs under efficiency and null responsibility. Theorem 5. On C 1 , the Bird rule is the only rule satisfying efficiency, null responsibility, and independence of irrelevant costs. Proof. It can easily be shown that the Bird rule satisfies the three axioms. Conversely, let ϕ be a rule satisfying these axioms. For all N0 ∈ N0 , all C ∈ CN10 , and all i ∈ N, let C i ∈ CN10 be cqw
cqi w
= 0 c¯ (C )
if (qw) ∈ gi0 (C ), if (qw) ∈ g (C ) \ gi0 (C ), otherwise.
if q = i and w ∈ F (i), if q ∈ N \ (F (i) ∪ {i}) and w ∈ F (i), otherwise.
For all i ∈ N, by null responsibility, for all k ∈ N \ (P (i) ∪ {i}), ϕk (C i ) = 0. Furthermore, by efficiency, for all i ∈ N, ck = ϕk (C i ). (15)
First, suppose that DC (i) = 1. If F (i) = ∅, by efficiency and group independence, ϕi (C ) = ci . Now suppose that F (i) ̸= ∅. By null responsibility, for all k ∈ F (i), ϕk (C¯ ) = 0. By efficiency and group
By independence of irrelevant costs, for all i ∈ N , ϕi (C i ) = ϕi (C ). Therefore, it suffices to show that for all i ∈ N , ϕi (C i ) = ci .
0 c¯ (C ) cqw
c¯qw =
k∈P (i)∪{i}
k∈P (i)∪{i}
142
Y. Chun, J. Lee / Mathematical Social Sciences 64 (2012) 136–143
First, suppose that i ∈ A(C ). Since P (i) = ∅, we can rewrite (15) as
ϕi (C i ) = ci , the desired conclusion. Next, suppose that i ∈ N \ A(C ). Since there exists p(i) ∈ N, from (15),
ck =
k∈P (i)
ϕk (C
p(i)
).
k∈P (i)
ck =
k∈P (i)
k∈P (i)∪{i}
(16)
ϕk (C i ) + ϕi (C i )
ϕk (C¯ ) =
ck .
(17)
k∈(F (l)∪{l})\F (i)
l Since ϕl (C¯ ) = |F (l)+ from the previous argument, substituting it 1| into (17) gives
ck + ϕi (C ). i
k∈P (i)
ϕk (C¯ ) =
k∈F (l)
ck + |F (l)|
k∈F (l)\F (i)
cl
|F (l) + 1|
.
(18)
By weak first-link consistency, there exists a ∈ R such that for all k ∈ F (l), ϕk (C¯ ) = ϕk (C¯ −l ) + a. Therefore, (18) can be rewritten as
k∈P (i)
=
. Now suppose that DC (i) = H + 1. Let l = pH (i).
By efficiency and group independence,
ϕk (C i ).
Inserting (16) into (15), ck =
p (j)
|F (ph (j))|+1
h=0
c
k∈P (i)
c′ h
DC ′ (j)−1
k∈(F (l)∪{l})
By independence of irrelevant costs, for all k ∈ P (i), ϕk (C p(i) ) = ϕk (C i ), which yields
We will establish the claim for an arbitrary set of agents by an induction argument. As induction hypothesis, suppose that for all N0′ ∈ N0 , all C ′ ∈ CN10 , and all j ∈ N ′ such that DC ′ (j) ≤ H, ϕj (C ′ ) =
{ϕk (C¯ −l ) + a} =
k∈F (l)
ck + |F (l)|
k∈F (l)\F (i)
cl
|F (l) + 1|
.
(19)
¯ −l By efficiency and group independence, = k∈F (l) ϕk (C ) k∈F (l)\F (i) ck . which together with (19) implies that |F (l)|a =
Therefore,
ϕi (C i ) = ci ,
l |F (l)| |F (lc)+ or 1|
the desired conclusion.
a=
4.2. The sequential equal contributions rule We present two characterizations of the sequential equal contributions rule by additionally requiring that if the immediate connection cost between any two agents is zero, then the contribution of the two agents should be the same.
cl
|F (l) + 1|
.
(20)
Since DC¯ −l (i) = H, from the induction hypothesis,
ϕi (C¯ −l ) =
H −1
cph (i)
h=0
|F ( (i))| + 1 ph
.
(21)
Equal responsibility: For all N0 ∈ N0 and all C ∈ CN10 , if i ∈ N \ A(C ) and ci = 0, then ϕi (C ) = ϕp(i) (C ), and if i ∈ A(C ) and ci = 0, then ϕi (C ) = 0.
Since i ∈ F (l), by weak first-link consistency, ϕi (C¯ ) = ϕi (C¯ −l ) + a, and combined with (20) and (21),
Once again, uniqueness holds even if non-negativity is dropped from the list.
ϕi (C¯ ) =
H −1
cph (i)
h =0
|F ( (i))| + 1
H
Theorem 6. On C 1 , the sequential equal contributions rule is the only rule satisfying efficiency, independence of following costs, group independence, weak first-link consistency, and equal responsibility. Proof. It is obvious that the sequential equal contributions rule satisfies the five axioms. Conversely, let ϕ be a rule satisfying these axioms. We show that for all N0 ∈ N0 , all C ∈ CN10 , and all i ∈ N, DC (i)−1
ϕi (C ) = Let C¯ ∈
cph (i)
h=0
|F (ph (i))| + 1
CN10
.
be
0 c¯ (C ) cqw
c¯qw =
if q = i and w ∈ F (i), if q ∈ N \ (F (i) ∪ {i}) and w ∈ F (i), otherwise.
First, suppose that DC (i) = 1. If F (i) = ∅, by efficiency and group ci independence, ϕi (C ) = ci = |F (i)+ . Now suppose that F (i) ̸= ∅. 1|
By equal responsibility, for all k ∈ F (i), ϕk (C¯ ) = ϕi (C¯ ). By efficiency and group independence, k∈(F (i)∪{i}) ϕk (C¯ ) = k∈(F (i)∪{i}) ck = ci . i Therefore, ϕi (C¯ ) = |F (i)+ . By independence of following costs, 1|
c
ϕi (C ) = ϕi (C¯ ) =
ci
|F (i) + 1|
.
=
+
ph
cph (i)
h=0
|F ( (i))| + 1 ph
cl
|F (l) + 1|
.
By independence of following costs,
ϕi (C ) = ϕi (C¯ ) =
H
cph (i)
h=0
|F ( (i))| + 1
the desired conclusion.
ph
,
Our final characterization of the sequential equal contributions rule is obtained by replacing independence of following costs, group independence, and weak first-link consistency by independence of irrelevant costs. Theorem 7. On C 1 , the sequential equal contributions rule is the only rule satisfying efficiency, equal responsibility, and independence of irrelevant costs. Proof. It can easily be shown that the sequential equal contributions rule satisfies the three axioms. Conversely, let ϕ be a rule satisfying these axioms. We show that for all N0 ∈ N0 , all C ∈ CN10 , and all i ∈ N, DC (i)−1
ϕi (C ) =
cph (i)
h=0
|F ( (i))| + 1 ph
.
Y. Chun, J. Lee / Mathematical Social Sciences 64 (2012) 136–143
Since DC (p(i)) = H, from the induction hypothesis, ϕp(i) (C ) =
Let C i ∈ CN10 be defined by
cqi w
=
c h p (p(i)) h=0 |F (ph (p(i)))|+1 .
H −1
if (qw) ∈ gi0 (C ), if (qw) ∈ g (C ) \ gi0 (C ), otherwise.
cqw 0 c¯ (C )
ϕi (C ) =
i First, suppose that DC (i) = 1. By efficiency, k∈N ϕk (C ) = ci . i Since for all k ∈ F (i), ck = 0, then by equal responsibility, for all k ∈ F (i), ϕk (C i ) = ϕi (C i ) and for all k ∈ N \(F (i)∪{i}), ϕk (C i ) = 0. Therefore,
ci =
k∈F (i)∪{i}
=
ϕk (C i ) +
k∈N \(F (i)∪{i})
= {|F (i)| + 1}ϕi (C i ), or equivalently,
|F (i)| + 1
.
(22)
By independence of irrelevant costs, ϕi (C ) = ϕi (C i ), and from (22),
ϕi (C ) = ϕi (C i ) =
ci
|F (i)| + 1
.
We will establish the claim for an arbitrary set of agents by an induction argument. As induction hypothesis, suppose that for all j ∈ N such that DC (j) ≤ H , ϕj (C ) =
DC (j)−1
suppose that DC (i) = H + 1. By efficiency,
ϕk (C p(i) ) + ci =
h =0
. Now |F (ph (j))|+1
ϕk (C i ).
Since for all k ∈ N \ F (pH (i)), ϕk (C i ) = ϕk (C p(i) ),
ϕk (C p(i) ) + ci =
k∈F (pH (i))
ϕk (C i ).
k∈F (pH (i))
Furthermore, since for all k ∈ {ph (i) | h = 1, 2, . . . , H }, ϕk (C i ) = ϕk (C p(i) ), we have
ϕk (C p(i) ) + ci =
k∈F (i)∪{i}
ϕk (C i ).
(23)
k∈F (i)∪{i}
By equal responsibility, since for all k ∈ F (i), ϕk (C i ) = ϕi (C i ) and ϕk (C p(i) ) = ϕi (C p(i) ), we have
{|F (i)| + 1}ϕi (C p(i) ) + ci = {|F (i)| + 1}ϕi (C i ), or equivalently,
ϕi (C i ) =
ci
|F (i)| + 1
|F (i)| + 1
+
H −1
cph (p(i))
h =0
|F (ph (p(i)))| + 1
H
cph (i)
h=0
|F (ph (i))| + 1
,
We are grateful to William Thomson, a referee, and the participants of the 2009 Public Economic Theory Meeting (PET09 Galway), the sixth biannual conference of the Society for Economic Design (SED 2009 Maastricht), the fifth Spain Italy Netherlands meeting on game theory (SING5), the sixth Asian general equilibrium theory workshop (GETA 2009), the second Taiwan–Dutch and international conference on game theory, and the logic, game theory, and social choice 6 for their comments. Chun’s work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2010-330-B00077). References
c h p (j)
k∈N
k∈N
ci
Acknowledgments
ϕk (C ) i
ci
=
Therefore,
the desired conclusion.
ϕk (C i )
k∈F (i)∪{i}
ϕi (C i ) =
143
+ ϕi (C p(i) ).
By equal responsibility, ϕi (C p(i) ) = ϕp(i) (C p(i) ) and by independence of irrelevant costs, ϕi (C i ) = ϕi (C ) and ϕp(i) (C p(i) ) = ϕp(i) (C ).
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