An axiomatic approach in minimum cost spanning tree problems with groups

Abstract

We study minimum cost spanning tree problems with groups, where agents are located in different villages, cities, etc. The groups are formed by agents living in the same village. In Bergantiños and Gómez-Rúa (Economic Theory 43:227–262, 2010) we define the rule F as the Owen value of the irreducible game with groups and we prove that F generalizes the folk rule of minimum cost spanning tree problems. Bergantiños and Vidal-Puga (Journal of Economic Theory 137:326–352, 2007a) give two characterizations of the folk rule. In the first one they characterize it as the unique rule satisfying cost monotonicity, population monotonicity and equal share of extra costs. In the second characterization of the folk rule they replace cost monotonicity by independence of irrelevant trees and population monotonicity by separability. In this paper we extend such characterizations to our setting. Some of the properties are the same (cost monotonicity and independence of irrelevant trees) and the other need to be adapted. In general, we do it by claiming the property twice: once among the groups and the other among the agents inside the same group.

Appendix

We prove Proposition 1, which is a consequence of the following claims.

Claim 5

There exist rules satisfying cost monotonicity (and hence independence of irrelevant trees), population monotonicity over groups (and hence separability among groups), population monotonicity over agents (and hence separability among agents), and failing coalitional share of extra costs: Let define the rule f ¹ as follows. Given $T\subset\mathcal{N}$, let π ^N denote the order induced in N by the index of the agents. Namely, given i,j∈N, π ^N(i)<π ^N(j) if and only if i<j. For each mcstp (N ₀,C) and i∈N we define

$$\psi_{i} ( N_{0},C ) =v_{C^{\ast}} \bigl( \mathit{Pre} \bigl( i, \pi^{N} \bigr) \cup \{ i \} \bigr) -v_{C^{\ast}} \bigl( \mathit{Pre} \bigl( i,\pi^{N} \bigr) \bigr) . $$

Let (N ₀,C,G) be an mcstp with groups and i∈G ^k. Thus,

$$f_{i}^{1} ( N_{0},C,G ) =\psi_{i} \bigl( G_{0}^{k},C^{\varphi } \bigr) . $$

Claim 6

There exist rules satisfying cost monotonicity (and hence independence of irrelevant trees), population monotonicity over groups (and hence separability among groups), coalitional share of extra costs and failing separability among agents (and hence population monotonicity over agents): Let define the rule f ² as follows. For each mcstp (N ₀,C) and i∈N we define the equal division rule as

$$E_{i} ( N_{0},C ) =\frac{m(N_{0},C)}{\vert N\vert }. $$

Let (N ₀,C,G) be a mcstp with groups and i∈G ^k. Thus,

$$f_{i}^{2} ( N_{0},C,G ) =E_{i} \bigl( G_{0}^{k},C^{\varphi } \bigr) . $$

Claim 7

There exist rules satisfying cost monotonicity (and hence independence of irrelevant trees), population monotonicity over agents (and hence separability among agents), coalitional share of extra costs and failing separability among groups (and hence population monotonicity over groups). We define the rule f ³ as follows. Let (N ₀,C,G) be a mcstp with groups and i∈G ^k. Thus,

$$f_{i}^{3} ( N_{0},C,G ) =\varphi_{i} \bigl( G_{0}^{k},C^{E} \bigr) $$

where

Claim 8

There exist rules satisfying population monotonicity over groups (and hence separability among groups), population monotonicity over agents (and hence separability among agents), coalitional share of extra costs and failing cost monotonicity (and hence independence of irrelevant trees): We define the rule f ⁴ as follows. Let (N ₀,C,G) be a mcstp with groups. We say that π∈Π _N is admissible if the following conditions are satisfied: (i) given i∈G ^k, j∈G ^k′, k≠k′, if π(i)<π(j), then $c_{0k}^{G}\leq c_{0k^{\prime}}^{G}$; (ii) given i,j∈G ^k, if π(i)<π(j), then c _0i ≤c _0j and (iii) if π(i)<π(j)<π(l), with i,l∈G ^k, then, j∈G ^k. We denote by Π ^′N the set of all permutations over N that are admissible. In an intuitive way: we first order the groups by decreasing connection cost to the source; and then the agents inside groups following the same idea.

Thus, for all i∈N,

$$f_{i}^{4} ( N_{0},C,G ) =\frac{1}{\vert \varPi_{N}^{\prime }\vert }\sum _{\pi\in\varPi_{N}^{\prime}}v_{C^{\ast}}\bigl(\mathit{Pre}(i,\pi )\cup \{i\} \bigr)-v_{C^{\ast}}(\mathit{Pre}(i,\pi)]. $$

Proof of Claim 5

1. f ¹ satisfies cost monotonicity. Let (N ₀,C,G) and (N ₀,C′,G) be such that C≤C′. Consider the problems among coalitions, (G ₀,C ^G) and (G ₀,C ^′G). Then, C ^G≤C ^′G. Since φ satisfies cost monotonicity (Theorem 0), φ _k (G ₀,C ^G)≤φ _k (G ₀,C ^′G) for all G ^k∈G ₀.

Consider now the problem inside a coalition $G_{0}^{k}$ with k=1,…,r. Then, C ^φ≤C ^′φ. By definition of the irreducible matrix, (C ^φ)^∗≤(C ^′φ)^∗. By Lemma 0(b), for all i∈G ^k,

$$\psi_{i} \bigl( G_{0}^{k},C^{\varphi} \bigr) \leq\psi_{i} \bigl( G_{0}^{k},C^{\prime\varphi} \bigr) . $$

2. f ¹ satisfies population monotonicity over groups. See Bergantiños and Gómez-Rúa (2010).

3. f ¹ satisfies population monotonicity over agents. See Bergantiños and Gómez-Rúa (2010).

4. f ¹ fails coalitional share of extra costs. Consider the mcstp with groups where N={1,2}, G={{i}} _i∈N , and matrices

$$C= \left( \begin{array}{c@{\quad}c@{\quad}c} 0 & 10 & 10 \\ 10 & 0 & 2 \\ 10 & 2 & 0\end{array} \right) \quad \mbox{and} \quad C^{\prime}=\left ( \begin{array}{c@{\quad}c@{\quad}c} 0 & 12 & 12 \\ 12 & 0 & 2 \\ 12 & 2 & 0\end{array} \right) . $$

In this case, f ¹(N ₀,C,G)=(10,2) and f ¹(N ₀,C′,G)=(12,2). □

Proof of Claim 6

1. f ² satisfies cost monotonicity. Let (N ₀,C,G) and (N ₀,C′,G) be such that C≤C′. Then C ^G≤C ^′G. Since φ satisfies cost monotonicity (Theorem 0), φ _k (G ₀,C ^G)≤φ _k (G ₀,C ^′G) for all G ^k∈G ₀.

Consider now the problem inside a coalition $(G_{0}^{k},C^{\varphi})$ with k=1,…,r. Then, C ^φ≤C ^′φ and hence

$$f_{i}^{2} ( N_{0},C,G ) =E_{i} \bigl(G_{0}^{k},C^{\varphi}\bigr)\leq E_{i} \bigl(G_{0}^{k},C^{\prime\varphi}\bigr)=f_{i}^{2} \bigl( N_{0},C^{\prime },G \bigr) $$

for all i∈G ^k.

2. f ² satisfies population monotonicity over groups. Let G ^k∈G. Since φ satisfies population monotonicity (Theorem 0), $\varphi_{l}(G_{0},C^{G})\leq\varphi_{l}((G\backslash G^{k})_{0},C^{G\backslash G^{k}})$ for all $l\not=k$. Furthermore, $((G\backslash G^{k})_{0},C^{G})=((G\backslash G^{k})_{0},C^{G\backslash G^{k}})$.

Let C ^′φ denote the matrix C ^φ associated with the problem ((N∖G ^k)₀,C,G∖G ^k).

Let G ^l∈G∖G ^k. For all i,j∈G ^l, $c_{ij}^{\varphi }=c_{ij}^{\prime\varphi}$. For all i∈G ^l,

$$c_{0i}^{\varphi}=\varphi_{l}\bigl(G_{0},C^{G} \bigr)\leq\varphi_{l}\bigl(\bigl(G\backslash G^{k} \bigr)_{0},C^{G\backslash G^{k}}\bigr)=c_{0i}^{\prime\varphi}. $$

That is C ^φ≤C ^′φ. Then, $E_{i}(G_{0}^{l},C^{\varphi})\leq E_{i}(G_{0}^{l},C^{\prime\varphi})$ and hence,

$$f_{i}^{2}(N_{0},C,G)\leq f_{i}^{2} \bigl( \bigl( N\backslash G^{k} \bigr)_{0},C,G\backslash G^{k}\bigr). $$

3. f ² satisfies coalitional share of extra costs. The proof is similar than the one for F and we omit it.

4. f ² fails separability among agents. Consider the mcstp with groups where N={1,2,3}, G={N}, and matrix

$$C= \left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0 & 1 & 2 & 1 \\ 1 & 0 & 0 & 1 \\ 2 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0\end{array} \right). $$

In this case, it is clear that m(N ₀,C)=m({1,2}₀,C)+m({3}₀,C). However, $f^{2}(N_{0},C,G)= ( \frac{2}{3},\frac{2}{3},\frac {2}{3} )$ and $f^{2}(\{1,2\}_{0},C)= ( \frac{1}{2},\frac{1}{2} )$ and f ²({3}₀,C)=1. □

Proof of Claim 7

1. f ³ satisfies cost monotonicity. Let (N ₀,C,G) and (N ₀,C′,G) such that C≤C′. Then C ^G≤C ^′G. Hence, E _k (G ₀,C ^G)≤E _k (G ₀,C ^′G) for all G ^k∈G ₀.

Consider now the problem inside a coalition $(G_{0}^{k},C^{E})$ with k=1,…,r. Then, C ^E≤C ^′E. Since φ satisfies cost monotonicity (Theorem 0),

$$f_{i}^{3} ( N_{0},C,G ) =\varphi_{i} \bigl(G_{0}^{k},C^{E}\bigr)\leq \varphi_{i}\bigl(G_{0}^{k},C^{\prime E} \bigr)=f_{i}^{3} \bigl( N_{0},C^{\prime},G \bigr) $$

for all i∈G ^k.

2. f ³ satisfies population monotonicity over agents. Let G ^k∈G and i∈G ^k. By convenience, let us denote as C′ the cost matrix C restricted to the problem ((N∖{i})₀,C,G ⁻ⁱ). Notice that C′ coincides with C on the agents of (N∖{i})₀.

We consider several cases:

• (a) Assume that $c_{kl}^{G}=c_{kl}^{\prime G}$ for all l∈{0,1,…,r}. Thus, $E_{l} ( G_{0},C^{G} ) =E_{l} ( G^{-i},C^{\prime G^{-i}} )$ for all l=1,…,r.

  • Let j∈G ^k∖{i}. Since φ satisfies population monotonicity (Theorem 0), $\varphi_{j} ( G_{0}^{k},C^{E} ) \leq\varphi_{j} ( (G^{k}\backslash \{i\})_{0},C^{E} )$. Then,

    

  • Let G ^l∈G such that l≠k. Then, $c_{jj^{\prime }}^{E}=c_{jj^{\prime}}^{\prime E}$ for all j,j′∈G ^l∪{0}. Hence,

    $$f_{j}^{3}(N_{0},C,G)=f_{j}^{3} \bigl(\bigl(N\backslash\{i\}\bigr)_{0},C^{\prime},G^{-i} \bigr). $$

• (b) Assume that $c_{kk^{\prime}}^{G}\not=c_{kk^{\prime }}^{\prime G}$ for some k′∈{0,1,…,r}. Then, $c_{kk^{\prime }}^{G}<c_{kk^{\prime}}^{\prime G}$. Furthermore, $c_{ll^{\prime }}^{G}\leq c_{ll^{\prime}}^{\prime G}$ for all l,l′∈{0,1,…,r}.

  Then, E _k (G ₀,C ^G)≤E _k (G ₀,C ^′G), $c_{jj^{\prime}}^{E}=c_{jj^{\prime}}^{\prime E}$ for all j,j′∈G ^k∖{i}, and $c_{0j}^{E}\leq c_{0j}^{\prime E}$ for all j∈G ^k∖{i}. Since φ satisfies cost monotonicity (Theorem 0), φ _j ((G ^k∖{i})₀,C ^E)≤φ _j ((G ^k∖{i})₀,C ^′E) for all j∈G ^k∖{i}.

  Since φ satisfies population monotonicity (Theorem 0), $\varphi_{j} ( G_{0}^{k},C^{E} ) \leq \varphi_{j} ( (G^{k}\backslash \{i\})_{0},C^{E} )$. Then, for all j∈G ^k∖{i}

  

3. f ³ satisfies coalitional share of extra costs. The proof is similar than the one for F and we omit it.

4. f ³ fails separability among groups, consider the mcstp with groups where N={1,2,3}, G={{i}} _i∈N , and matrix

$$C= \left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 0 & 1 & 2 & 1 \\ 1 & 0 & 0 & 1 \\ 2 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0\end{array} \right). $$

In this case, it is clear that m(N ₀,C)=m({1,2}₀,C)+m({3}₀,C). However, $f^{3}(N_{0},C,G)= ( \frac{2}{3},\frac{2}{3},\frac {2}{3} )$ and $f^{3}(\{1,2\}_{0},C)= ( \frac{1}{2},\frac{1}{2} )$ and f ³({3}₀,C)=1. □

Proof of Claim 8

1. f ⁴ satisfies population monotonicity over groups. We can prove this by performing some computations. Even the proof is not trivial, we do not believe it is especially relevant and we omit it.

2. f ⁴ satisfies population monotonicity over agents. We can prove this by performing some computations. Even the proof is not trivial, we do not believe it is especially relevant and we omit it.

3. f ⁴ satisfies coalitional share of extra costs. Let (N ₀,C,G) and (N ₀,C′,G) be two mcstp with groups under the conditions of coalitional share of extra costs. Then, a permutation π∈Π ^′N is admissible if given i,i′∈G ^k∈G and j∈N with π(i)<π(j)<π(i′), then j∈G ^k. Therefore, $f^{4}(N_{0},C,G)=Ow(N,v_{C^{\ast}},G)$. In Bergantiños and Gómez-Rúa (2010) we prove that $F ( N_{0},C,G ) =Ow ( N,v_{C^{\ast }},G )$. Then, in this case, f ⁴(N ₀,C,G)=F(N ₀,C,G), and we have proved above that F satisfies coalitional share of extra costs.

4. f ⁴ fails cost monotonicity. Consider the mcstp with groups (N ₀,C,G) and (N ₀,C′,G) where N={1,2}, G={{i}} _i∈N , and matrices

$$C= \left( \begin{array}{c@{\quad}c@{\quad}c} 0 & 10 & 20 \\ 10 & 0 & 0 \\ 20 & 0 & 0\end{array} \right) \quad \mbox{and} \quad C^{\prime}= \left( \begin{array}{c@{\quad}c@{\quad}c} 0 & 30 & 20 \\ 30 & 0 & 0 \\ 20 & 0 & 0\end{array} \right) . $$

In this case, f ⁴(N ₀,C,G)=(10,0) and f ⁴(N ₀,C′,G)=(0,20). □