Annals of Operations Research

, Volume 243, Issue 1–2, pp 179–198 | Cite as

An Owen-type value for games with two-level communication structure

  • René van den Brink
  • Anna Khmelnitskaya
  • Gerard van der Laan
Article

Abstract

We introduce an Owen-type value for games with two-level communication structure, which is a structure where the players are partitioned into a coalition structure such that there exists restricted communication between as well as within the a priori unions of the coalition structure. Both types of communication restrictions are modeled by an undirected communication graph. We provide an axiomatic characterization of the new value using an efficiency, two types of fairness (one for each level of the communication structure), and a new type of axiom, called fair distribution of the surplus within unions, which compares the effect of replacing a union in the coalition structure by one of its maximal connected components on the payoffs of these components. The relevance of the new value is illustrated by an example. We also show that for particular two-level communication structures the Owen value and the Aumann–Drèze value for games with coalition structure, the Myerson value for communication graph games, and the equal surplus division solution appear as special cases of this new value.

Keywords

TU game Coalition structure Communication graph Owen value  Myerson value 

JEL Classification

C71 

References

  1. Aumann, R. J., & Drèze, J. (1974). Cooperative games with coalitional structures. International Journal of Game Theory, 3, 217–237.CrossRefGoogle Scholar
  2. Béal, S., Rémilla, E., & Solal, P. (2012). Fairness and fairness for neighbors: The difference between the Myerson value and component-wise egalitarian solutions. Economics Letters, 117, 263–267.CrossRefGoogle Scholar
  3. Béal, S., Casajus,A. , & Huettner,F. (2013). Efficient extensions of communication values, Working paper.Google Scholar
  4. van den Brink, R., Khmelnitskaya, A., & van der Laan, G. (2012). An efficient and fair solution for communication graph games. Economics Letters, 117, 786–789.CrossRefGoogle Scholar
  5. Casajus, A. (2007). An efficient value for TU games with a cooperation structure, Working paper, Universität Leipzig, Germany.Google Scholar
  6. Driessen, T. S. H., & Funaki, Y. (1991). Coincidence of and collinearity between game theoretic solutions. OR Spektrum, 13, 15–30.CrossRefGoogle Scholar
  7. Gómez-Rúa, M., & Vidal-Puga, J. (2011). Balanced per capita contributions and level structure of cooperation. Top, 19, 167–176.CrossRefGoogle Scholar
  8. Hamiache, G. (2012). A matrix approach to TU games with coalition and communication structures. Social Choice and Welfare, 38, 857–100.CrossRefGoogle Scholar
  9. Kamijo, Y. (2009). A two-step Shapley value for cooperative games with coalition structures. International Game Theory Review, 11, 207–214.CrossRefGoogle Scholar
  10. Khmelnitskaya, A. (2014). Values for games with two-level communication structures. Discrete Applied Mathematics, 166, 34–50. An earlier version entitled “Values for graph-restricted games with coalition structure” is available as Memorandum 2007–1848 of the Department of Applied Mathematics of University of Twente.CrossRefGoogle Scholar
  11. Kongo, T. (2007). Cooperative games with two-level networks, 21COE-GLOPE Working Paper Series, Waseda University, Tokyo. Japan. A modified version of this paper is published as Kongo, T. (2011), Value of games with two-layered hypergraphs. Mathematical Social Sciences, 62, 114–119.Google Scholar
  12. Myerson, R. B. (1977). Graphs and cooperation in games. Mathematics of Operations Research, 2, 225–229.CrossRefGoogle Scholar
  13. 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). Berlin: Springer.CrossRefGoogle Scholar
  14. Shapley, L. S. (1953). A value for \(n\)-person games. In A. W. Tucker & H. W. Kuhn (Eds.), Contributions to the theory of games II (pp. 307–317). Princeton, NJ: Princeton University Press.Google Scholar
  15. Vázquez-Brage, M., García-Jurado, I., & Carreras, F. (1996). The Owen value applied to games with graph-restricted communication. Games and Economic Behavior, 12, 42–53.CrossRefGoogle Scholar
  16. Winter, E. (1989). A value for games with level structures. International Journal of Game Theory, 18, 227–242.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • René van den Brink
    • 1
  • Anna Khmelnitskaya
    • 2
  • Gerard van der Laan
    • 1
  1. 1.Department of Econometrics and Tinbergen InstituteVU UniversityAmsterdamThe Netherlands
  2. 2.Faculty of Applied MathematicsSaint-Petersburg State UniversityPetergof, Saint PetersburgRussia

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