Annals of Operations Research

, Volume 225, Issue 1, pp 45–63 | Cite as

An axiomatic approach in minimum cost spanning tree problems with groups

Article

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.

Keywords

Minimum cost spanning tree problems Folk rule Cost sharing Axiomatization 

References

  1. Aarts, H., & Driessen, T. (1993). The irreducible core of a minimum cost spanning tree game. Mathematical Methods of Operations Research, 38, 163–174. CrossRefGoogle Scholar
  2. Albizuri, M. J. (2009). Generalized coalitional semivalues. European Journal of Operational Research, 196(2), 578–584. CrossRefGoogle Scholar
  3. Bergantiños, G., & Gómez-Rúa, M. (2010). Minimum cost spanning tree problems with groups. Economic Theory, 43, 227–262. CrossRefGoogle Scholar
  4. Bergantiños, G., & Lorenzo, L. (2004). A non-cooperative approach to the cost spanning tree problem. Mathematical Methods of Operations Research, 59, 393–403. CrossRefGoogle Scholar
  5. Bergantiños, G., & Lorenzo, L. (2005). Optimal equilibria in the non-cooperative game associated with cost spanning tree problems. Annals of Operations Research, 137, 101–115. CrossRefGoogle Scholar
  6. Bergantiños, G., & Lorenzo, L. (2008). Non cooperative cost spanning tree problems with budget restrictions. Naval Research Logistics, 55, 747–757. CrossRefGoogle Scholar
  7. Bergantiños, G., Lorenzo, L., & Lorenzo-Freire, S. (2010). The family of cost monotonic and cost additive rules in minimum cost spanning tree problems. Social Choice and Welfare, 34, 695–710. CrossRefGoogle Scholar
  8. Bergantiños, G., Lorenzo, L., & Lorenzo-Freire, S. (2011). A generalization of obligation rules for minimum cost spanning tree problems. European Journal of Operational Research, 211, 122–129. CrossRefGoogle Scholar
  9. Bergantiños, G., & Lorenzo-Freire, S. (2008). Optimistic weighted Shapley rules in minimum cost spanning tree problems. European Journal of Operational Research, 185, 289–298. CrossRefGoogle Scholar
  10. Bergantiños, G., & Vidal-Puga, J. J. (2007a). A fair rule in minimum cost spanning tree problems. Journal of Economic Theory, 137, 326–352. CrossRefGoogle Scholar
  11. Bergantiños, G., & Vidal-Puga, J. J. (2007b). The optimistic TU game in minimum cost spanning tree problems. International Journal of Game Theory, 36, 223–239. CrossRefGoogle Scholar
  12. Bergantiños, G., & Vidal-Puga, J. J. (2009). Additivity in minimum cost spanning tree problems. Journal of Mathematical Economics, 45, 38–42. CrossRefGoogle Scholar
  13. Bird, C. G. (1976). On cost allocation for a spanning tree: a game theoretic approach. Networks, 6, 335–350. CrossRefGoogle Scholar
  14. Bogomolnaia, A., & Moulin, H. (2010). Sharing the cost of a minimal cost spanning tree: beyond the folk solution. Games and Economic Behavior, 69, 238–248. CrossRefGoogle Scholar
  15. Branzei, R., Moretti, S., Norde, H., & Tijs, S. (2004). The P-value for cost sharing in minimum cost spanning tree situations. Theory and Decision, 56, 47–61. CrossRefGoogle Scholar
  16. Chun, Y., & Lee, J. (2009). Sequential contributions rules for minimum cost spanning tree problems. Mimeo, Seoul University. Google Scholar
  17. Dutta, B., & Kar, A. (2004). Cost monotonicity, consistency and minimum cost spanning tree games. Games and Economic Behavior, 48, 223–248. CrossRefGoogle Scholar
  18. Feltkamp, V., Tijs, S., & Muto, S. (1994). On the irreducible core and the equal remaining obligation rule of minimum cost extension problems. Mimeo, Tilburg University. Google Scholar
  19. Gómez-Rúa, M., & Vidal-Puga, J. (2010). The axiomatic approach to three values in games with coalition structure. European Journal of Operational Research, 207, 795–806. CrossRefGoogle Scholar
  20. Hart, S., & Kurz, M. (1983). Endogenous formation of coalitions. Econometrica, 51(4), 1047–1064. CrossRefGoogle Scholar
  21. Kar, A. (2002). Axiomatization of the Shapley value on minimum cost spanning tree games. Games and Economic Behavior, 38, 265–277. CrossRefGoogle Scholar
  22. Kruskal, J. (1956). On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical Society, 7, 48–50. CrossRefGoogle Scholar
  23. Owen, G. (1977). Values of games with a priori unions. In R. Henn & O. Moeschlin (Eds.) Essays in mathematical economics and game theory (pp. 76–88). New York: Springer. CrossRefGoogle Scholar
  24. Prim, R. C. (1957). Shortest connection networks and some generalizations. Bell Systems Technology Journal, 36, 1389–1401. CrossRefGoogle Scholar
  25. Pulido, M., & Sánchez-Soriano, J. (2009). On the core, the Weber set and convexity in games with a priori unions. European Journal of Operational Research, 193, 468–475. CrossRefGoogle Scholar
  26. Shapley, L. S. (1953). A value for n-person games. In H. W. Kuhn & A. W. Tucker (Eds.), Contributions to the theory of games II (pp. 307–317). Princeton: Princeton University Press Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Facultad de Economía, Campus Lagoas-Marcosende, s/nUniversidade de VigoVigoSpain

Personalised recommendations