Advertisement

International Journal of Game Theory

, Volume 38, Issue 1, pp 107–126 | Cite as

A characterization of Kruskal sharing rules for minimum cost spanning tree problems

  • Leticia Lorenzo
  • Silvia Lorenzo-Freire
Original Paper

Abstract

In Tijs et al. (Eur J Oper Res 175:121–134, 2006) a new family of cost allocation rules is introduced in the context of cost spanning tree problems. In this paper we provide the first characterization of this family by means of population monotonicity and a property of additivity.

Keywords

Minimum cost spanning tree problems Kruskal’s algorithm Sharing rules 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aarts H, Driessen T (1993) The irreducible core of a minimum cost spanning tree game. Math Methods Oper Res 38: 163–174CrossRefGoogle Scholar
  2. Bergantiños G, Lorenzo L (2004) A non-cooperative approach to the cost spanning tree problem. Math Methods Oper Res 59: 393–403Google Scholar
  3. Bergantiños G, Lorenzo-Freire S (2008) “Optimistic” weighted Shapley rules in minimum cost spanning tree problems. Eur J Oper Res 185: 289–298CrossRefGoogle Scholar
  4. Bergantiños G, Vidal-Puga JJ (2005) Several approaches to the same rule in cost spanning tree problems. Working paper. Vigo University. Available at the web page of the authorsGoogle Scholar
  5. Bergantiños G, Vidal-Puga JJ (2007a) The optimistic TU game in minimum cost spanning tree problems. Int J Game Theory 36: 223–239CrossRefGoogle Scholar
  6. Bergantiños G, Vidal-Puga JJ (2007b) A fair rule in cost spanning tree problems. J Econ Theory 137: 326–352CrossRefGoogle Scholar
  7. Bergantiños G, Vidal-Puga JJ (2008) Additivity in cost spanning tree problems. J Math Econ (forthcoming)Google Scholar
  8. Bird CG (1976) On cost allocation for a spanning tree: a game theoretic approach. Networks 6: 335–350CrossRefGoogle Scholar
  9. Dutta B, Kar A (2004) Cost monotonicity, consistency and minimum cost spanning tree games. Games Econ Behav 48: 223–248CrossRefGoogle Scholar
  10. Feltkamp V, Tijs S, Muto S (1994) On the irreducible core and the equal remaining obligations rule of minimum cost spanning extension problems. CentER discussion paper 1994, 106. Tilburg UniversityGoogle Scholar
  11. Granot D, Huberman G (1981) Minimum cost spanning tree games. Math Programming 21: 1–18CrossRefGoogle Scholar
  12. Granot D, Huberman G (1984) On the core and nucleolus of the minimum cost spanning tree games. Math Programming 29: 323–347CrossRefGoogle Scholar
  13. Kalai E, Samet D (1987) On weighted Shapley values. Int J Game Theory 16: 205–222CrossRefGoogle Scholar
  14. Kar A (2002) Axiomatization of the Shapley value on minimum cost spanning tree games. Games Econ Behav 38: 265–277CrossRefGoogle Scholar
  15. Kruskal J (1956) On the shortest spanning subtree of a graph and the traveling salesman problem. Proc Am Math Soc 7: 48–50CrossRefGoogle Scholar
  16. Moretti S, Tijs S, Branzei R, Norde H (2005) Cost monotonic “construct and charge” rules for connection situations. CentER discussion paper 2005, 104. Tilburg UniversityGoogle Scholar
  17. Norde H, Moretti S, Tijs S (2004) Minimum cost spanning tree games and population monotonic allocation schemes. Eur J Oper Res 154: 84–97CrossRefGoogle Scholar
  18. Prim RC (1957) Shortest connection networks and some generalizations. Bell Syst Technol J 36: 1389–1401Google Scholar
  19. Schmeidler D (1969) The nucleolus of a characteristic function game. SIAM J Appl Math 17: 1163–1170CrossRefGoogle Scholar
  20. Shapley LS (1953a) Additive and non-additive set functions. Ph.D. Thesis, Department of Mathematics, Princeton UniversityGoogle Scholar
  21. Shapley LS (1953) A value for n-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the Theory of Games II. Princeton University Press, NJ, pp 307–317Google Scholar
  22. Thomson W (1983) The fair division of a fixed supply among a growing population. Math Oper Res 8: 319–326CrossRefGoogle Scholar
  23. Thomson W (1995) Population-monotonic allocation rules. In: Barnett WA, Moulin H, Salles M, Schofield N (eds) Social choice, welfare and ethics. Cambridge University Press, London, pp 79–124Google Scholar
  24. Tijs S, Branzei R, Moretti S, Norde H (2006) Obligation rules for minimum cost spanning tree situations and their monotonicity properties. Eur J Oper Res 175: 121–134CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Facultade de CC.EE. e Empresariais, Research Group in Economic AnalysisUniversidade de VigoVigoSpain
  2. 2.Facultade de InformáticaUniversidade da CoruñaA CoruñaSpain

Personalised recommendations