Minimum cost spanning tree problems with groups

Transcription

1 Outline Gustavo Bergantiños 1 and María Gómez-Rúa 2 1 Research Group in Economic Analysis Universidade de Vigo 2 Research Group in Economic Analysis Universidade de Vigo Zaragoza, December 2008 Bergantiños and Gómez-Rúa 1/27

2 Outline Outline 1 Minimum cost spanning tree problems 2 Bergantiños and Gómez-Rúa 2/27

3 Outline Minimum cost spanning tree problems 1 Minimum cost spanning tree problems 2 Bergantiños and Gómez-Rúa 3/27

4 The model Minimum cost spanning tree problems Example 1: A minimum cost spanning tree problem (mcstp) is a pair (N 0, C) where: 3 N 0 = N N is the set of agents. 0 is the source C = (c ij) i,j N0 is the cost matrix. c ij is the cost of direct link between agents i and j. c ij = c ji for all i, j N 0. c ii = 0 for all i N 0. Bergantiños and Gómez-Rúa 4/27

5 Relevant questions Minimum cost spanning tree problems How to compute a minimum cost spanning tree? Kruskal (1956); Proceedings of the American Mathematical Society. Prim (1957); Bell Systems Technology Journal. How to divide the cost among the agents Bird (1976); Networks. Feltkamp, Tijs, and Muto (1994); WP Tilburg. Kar (2002); GEB. Dutta and Kar (2004); GEB. Bergantiños and Vidal-Puga (2007); JET. Bergantiños and Gómez-Rúa 5/27

6 Outline Minimum cost spanning tree problems 1 Minimum cost spanning tree problems 2 Bergantiños and Gómez-Rúa 6/27

7 Motivation Minimum cost spanning tree problems In some cases, as in Dutta and Kar (2004) or Bergantiños and Lorenzo (2004, 2005), agents are located in different villages. This means, in terms of the cost matrix, that the connection cost between two agents of the same village is not larger than the connection cost between an agent of this village and an agent from other village. The classical model of mcstp, as described above, can also model these situations. Nevertheless, it ignores the fact that some group of agents are located in the same village. It could be interesting to include this fact in the model. We do it by considering an extra element in the model. Namely a partition G = G 1,..., G m of the set of agents N. For each k = 1,..., m, G k represents the group of agents located in the same village, city,... Bergantiños and Gómez-Rúa 7/27

8 The model Minimum cost spanning tree problems An mcstp with groups is a triple (N 0, C, G) where: (N 0, C) is an mcstp G = G 1,..., G m is a partition of N and for each k = 1,...m, max {cij} min {cij}. i,j G k i G k,j/ G k Rule A rule in mcstp with groups is a function f such that f (N 0, C, G) R N and P f i (N 0, C, G) = m (N 0, C). i N Bergantiños and Gómez-Rúa 8/27

9 A result in mcstp Minimum cost spanning tree problems The rule ϕ is studied in many papers. For instance, Feltkamp et al (1994), Branzei et al (2004), Bergantiños and Vidal-Puga in several papers. Bergantiños and Vidal-Puga (2008). ϕ is the unique rule in mcstp satisfying Restricted additivity. Population monotonicity. Symmetry. In this paper we try to generalize this result to mcstp with groups. Bergantiños and Gómez-Rúa 9/27

10 Restricted Additivity (RA) Restricted Additivity is an additivity property restricted to a subclass of problems. No rule satisfies additivity over all mcstp. The reason is that in the definition of a rule we are claiming that P f i (N 0, C) = m (N 0, C), which is incompatible with i N additivity over all mcstp. Restricted Additivity (RA) For all mcstp (N 0, C) and (N 0, C ) satisfying that there exists an mt t = {(i 0, i)} i N in (N 0, C), (N 0, C ), and (N 0, C + C ) and an order π = (i 1,..., i N ) such that c i 0 1 i 1 c i 0 2 i 2... c i 0 N i N and c i 0 1 i 1 c i 0 2 i 2... c i 0 N i N, for all i N. f i(n 0, C + C ) = f i(n 0, C) + f i(n 0, C ) Bergantiños and Gómez-Rúa 10/27

11 Population Monotonicity (PM) If new agents join a society, no agent of the initial society can be worse off. Population Monotonicity (PM) For all mcstp (N 0, C), all S N, and all i S, f i (N 0, C) f i (S 0, C). Bergantiños and Gómez-Rúa 11/27

12 Symmetry (SYM) Minimum cost spanning tree problems Two symmetric agents must pay the same. Symmetric Agents Two agents i, j N are symmetric if for all k N 0 \ {i, j}, c ik = c jk. Symmetry (SYM) For all mcstp (N 0, C) and all pair of symmetric agents i, j N, f i (N 0, C) = f j (N 0, C). Bergantiños and Gómez-Rúa 12/27

13 Example Minimum cost spanning tree problems Consider the mcstp with groups (N 0, C, G) where N = {1, 2, 3}, G = G 1, G 2, G 1 = {1, 2}, G 2 = {3}, and the cost matrix is represented in the following figure: m(n 0, C) = units are associated with the cost of connecting cities 1 and 2 with the source and 2 units are associated with the cost of connecting agents 1 and 2 inside city 1. Since we are looking for fair shares it seems reasonable to divide these 2 units equally, between, agents 1 and 2. In order to divide the 12 units among the agents two approaches seems reasonable: m({1, 2} 0, C G ) = 12 Bergantiños and Gómez-Rúa 13/27

14 Example Minimum cost spanning tree problems m(n 0, C) = 14 m({1, 2} 0, C G ) = 12 Approach 1 The cost paid by each city should take into account the number of agents who receive benefits for their connection. Thus, city 1 should pay twice than city 2, i.e., city 1 pays 8 and city 2 pays 4. Since, agents inside city 1 are also symmetric, both pay the same. Then, agent 1 pays = 5, agent 2 pays = 5, agent 3 pays 4. Bergantiños and Gómez-Rúa 14/27

15 Example Minimum cost spanning tree problems m(n 0, C) = 14 m({1, 2} 0, C G ) = 12 Approach 2 The cost paid by each city does not depend on the characteristics of the other city. Assuming this, both cities are symmetric. Thus, each city should pay 6 each. Since agents inside city 1 are also symmetric, both pay the same. Then, agent 1 pays = 4, agent 2 pays = 4, agent 3 pays 6. In this paper we have decided to follow this approach. Thus, some properties that will be introduced further on in the paper will be defined according with this approach. Bergantiños and Gómez-Rúa 15/27

16 Restricted Additivity (RA) The same as in mcstp with groups. The rules must be additive on some subclass of problems. Restricted Additivity (RA) For all (N 0, C, G) y (N 0, C, G) satisfying that there exists an mt t = {(i 0, i)} i N in (N 0, C, G), (N 0, C, G), and (N 0, C + C, G) and an order π = (i 1,..., i N ) such that c i 0 1 i 1 c i 0 2 i 2... c i 0 N i N and c i 0 1 i 1 c i 0 2 i 2... c i 0 N i N, for all i N. f i(n 0, C + C, G) = f i(n 0, C, G) + f i(n 0, C, G) Bergantiños and Gómez-Rúa 16/27

17 Population Monotonicity over Groups (PMG) If a new group joins the society, no agent of the initial society can be worse off. Population Monotonicity over Groups (PMG) For all (N 0, C, G), all G k G, and all i N\G k, f i (N 0, C, G) f i ``N\G k 0, C, G\Gk. Bergantiños and Gómez-Rúa 17/27

18 Population Monotonicity over Agents (PMA) It has two parts: First. If an agent joins a group, no agent of this group can be worse off. Population Monotonicity over Agents (PMA) For all (N 0, C, G), all G k G, and all i G k such that G k \ {i}, f j (N 0, C, G) f j (N\ {i}) 0, C, G\G k G k \ {i} for all j G k \ {i}. Bergantiños and Gómez-Rúa 18/27

19 Population Monotonicity over Agents (PMA) Second. Assume that after the entrance of agent i in group G k the minimum connection cost between group G k and the rest of the groups did not change, then agents of the others groups must pay the same. Population Monotonicity over Agents (PMA) Moreover, if for each G l with l k, min {c j G k,j G l jj } = min {c j G k \{i},j G l jj }, for all j N\G k. f j (N 0, C, G) = f j (N\ {i}) 0, C, G\G k G k \ {i} Bergantiños and Gómez-Rúa 19/27

20 Symmetry among Groups (SYMG) Symmetric groups should pay the same. Symmetric Groups Two groups G k and G k are symmetric if for all G l G 0 \ min {cij} = min {c ij}. i G k,j G l i G k,j G l n G k, G k o, Bergantiños and Gómez-Rúa 20/27

21 Symmetry among Groups (SYMG) The amount paid by group G k is P i G k f i. We can decompose this amount into two parts; the cost of connecting agents inside the group among themselves, m `G k, C, and the cost of connecting the group with the source (possible through other groups), P i G k f i m `G k, C. We are assuming that the amount paid by a group should not depend on the internal characteristics of the other groups. Then, it seems reasonable to say that m `G k, C should be paid by agents in G k. Symmetry among Groups (SYMG) For all (N 0, C, G) and all pair of symmetric groups G k, G k G, X f i (N 0, C, G) m G k, C = X f i (N 0, C, G) m G k, C. i G k i G k Bergantiños and Gómez-Rúa 21/27

22 Symmetry among agents in the same group (SYMA) Two symmetric agents belonging to the same group must pay the same. Symmetry among agents in the same group (SYMA) For all (N 0, C, G) and all pair of symmetric agents i, j G k G, f i (N 0, C, G) = f j (N 0, C, G). Bergantiños and Gómez-Rúa 22/27

23 A rule in mcstp with groups Given (N 0, C, G), we define the mcstp among groups `G 0, C G as follows: G 0 = G 0, G 1,..., G m where G 0 = 0. C G is the cost matrix, and for each G k, G k G 0 the connection cost between G k and G k is denoted by c G kk = min {cij}. i G k,j Gk The rule among groups For each i G k, F i (N 0, C, G) = ϕ i G k 0, C ϕ where c ϕ jj = j cjj si 0 / {j, j } ϕ k `G0, C G si 0 {j, j }. Bergantiños and Gómez-Rúa 23/27

24 Axiomatic Characterization Theorem The rule F is the unique rule for minimum cost spanning tree problems with groups satisfying PMG, PMA, SYMG, SYMA and RA. Remark The properties PMG, PMA, SYMG, SYMA and RA are independent. Bergantiños and Gómez-Rúa 24/27

25 TU game with group structure A TU game with group structure is a triple (N, v, G) where (N, v) is a TU game G = G 1,..., G m is a partition of N. The Owen value A permutation π Π N is admissible with respect to G if given i, i G k G and j N with π(i) < π(j) < π(i ), then j G k. Let Π G denote the set of all permutations over N admissible with respect to G. Given (N, v, G) and i G k G, the Owen value is defined as Ow i (N, v, G) = 1 Π G X π Π G [v (Pre (i, π) {i}) v (Pre (i, π))]. Bergantiños and Gómez-Rúa 25/27

26 The Owen value in mcstp with groups Question The Owen value generalizes the Shapley value. The rule F in mcstp with groups generalizes the rule ϕ in mcstp. The rule ϕ is defined as the Shapley value of the TU game (N, v C ). Is there some relationship between the Owen value of the game (N, v C, G) and the rule F? Theorem For each mcstp with groups (N 0, C, G) and i G k G, F i (N 0, C, G) = Ow i (N, v C, G). Bergantiños and Gómez-Rúa 26/27

27 Gustavo Bergantiños 1 and María Gómez-Rúa 2 1 Research Group in Economic Analysis Universidade de Vigo 2 Research Group in Economic Analysis Universidade de Vigo Zaragoza, December 2008 Bergantiños and Gómez-Rúa 27/27