The Core for Games with Cooperation Structure

  • Inés Gallego
  • Michel Grabisch
  • Andrés Jiménez-Losada
  • Alexandre Skoda
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9760)


A cooperative game consists of a set of players and a characteristic function that determines the maximal profit or minimal cost that each subset of players can get when they decide to cooperate, regardless of the actions of the rest of the players. The relationships among the players can modify their bargaining and therefore their payoffs. The model of cooperation structures in a game introduces a graph on the set of players setting their relations and in which its components indicate the groups of players that are initially formed. In this paper we define the core and the Weber set and the notion of convexity for this family of games.


Cooperative game A priori unions Core Weber set Cooperation structure Convexity 



This research has been partially supported by the Spanish Ministry of Economy and Competitiveness ECO2013-40755-P, and by the FQM237 grant of the Andalusian Government.

The second author thanks the Agence Nationale de la Recherche for financial support under contract ANR-13-BSHS1-0010 (DynaMITE).


  1. 1.
    Adam, L., Kroupa, T.: The intermediate set and limiting superdifferential for coalition games: between the core and the Weber Set. Preprint submitted to Games and Economic Behavior (2015)Google Scholar
  2. 2.
    Aumann, R.J., Drèze, J.H.: Cooperative games with coalition structures. Int. J. Game Theory 3, 217–237 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Casajus A.: Beyond basic structures in game theory. Ph.D. thesis. University of Leipzig, Germany (2007)Google Scholar
  4. 4.
    Faigle, U.: Cores of games with restricted cooperation. Methods Models Oper. Res. 33, 405–422 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fernández, J.R., Gallego, I., Jiménez-Losada, A., Ordóñez, M.: Cooperation among agents with a proximity relation. Eur. J. Oper. Res. 250, 555–565 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gillies, D.B.: Solutions to general non-zero-sum games. In: Tucker, A.W., Luce, R.D. (eds.) Contributions to the Theory of Games IV. Annals of Mathematics, vol. 40, 47th edn, p. 85. Princeton University Press, Princeton (1959)Google Scholar
  7. 7.
    Grabisch, M.: The core of games on ordered structures and graphs. Q. J. Oper. Res. 7, 207–238 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ichiishi, T.: Super-modularity: applications to the convex and to the greedy algorithm for LP. J. Econ. Theory 25, 283–286 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Myerson, R.B.: Graphs and cooperation in games. Math. Oper. Res. 2(3), 225–229 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    van den Nouweland, A., Borm, P.: On the convexity of communication games. Int. J. Game Theory 19, 421–430 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Owen, G.: Values of games with a priori unions. In: Henn, R., Moeschlin, O. (eds.) Mathematical Economics and Game Theory. Lecture Notes in Economics and Mathematical Systems, vol. 141, pp. 76–88. Springer, Heidelberg (1977)CrossRefGoogle Scholar
  12. 12.
    Pulido, M.A., Sánchez-Soriano, J.: Characterization of the core in games with restricted cooperation. Eur. J. Oper. Res. 175, 860–869 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pulido, M.A., Sánchez-Soriano, J.: On the core, the Weber set and convexity in games with a priori unions. Eur. J. Oper. Res. 193, 468–475 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Shapley, L.S.: A value for \(n\)-person games. Ann. Math. Stud. 28, 307–317 (1953)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Shapley, L.S.: Cores and convex games. Int. J. Game Theory 1, 1–26 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Weber, R.J.: Probabilistic values for games. Cowles Foundation Discussion Paper 417R. Yale University, New Haven (1978)Google Scholar
  17. 17.
    Weber, R.J.: Probabilistic values for games. In: Roth, A. (ed.) The Shapley value: Essays in Honor of Lloyd S. Shapley, pp. 101–119. Cambridge University Press, Cambridge (1988)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Inés Gallego
    • 1
  • Michel Grabisch
    • 2
    • 3
  • Andrés Jiménez-Losada
    • 1
  • Alexandre Skoda
    • 3
  1. 1.Universidad de SevillaSevillaSpain
  2. 2.Paris School of EconomicsParisFrance
  3. 3.Université de Paris IParisFrance

Personalised recommendations