The Position Value and the Myerson Value for Hypergraph Communication Situations

  • Erfang ShanEmail author
  • Guang Zhang
Part of the Static & Dynamic Game Theory: Foundations & Applications book series (SDGTFA)


We characterize the position value and the Myerson value for uniform hypergraph communication situations by employing the “incidence graph game” and the “link-hypergraph game” which are induced by the original hypergraph communication situations. The incidence graph game and link-hypergraph game are defined on the “incidence graph” and the “link-hypergraph”, respectively, obtained from the original hypergraph. Using the above tools, we represent the position value by the Shapley value of the incidence graph game and the Myerson value of the link-hypergraph game for uniform hypergraph communication situations, respectively. Also, we represent the Myerson value by the Owen value or the two-step Shapley value of the incidence graph game with a coalition structure for hypergraph communication situations.



This research was supported in part by the National Nature Science Foundation of China (grant number 11571222).


  1. 1.
    Algaba, E., Bilbao, J.M., Borm, P., López, J.J.: The position value for union stable systems. Math. Meth. Oper. Res. 52, 221–236 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Algaba, E., Bilbao, J.M., Borm, P., López, J.J.: The Myerson value for union stable systems. Math. Meth. Oper. Res. 54(3), 359–371 (2001)CrossRefGoogle Scholar
  3. 3.
    Béal, S., Rémila, E., Solal, P.: Fairness and fairness for neighbors: the difference between the Myerson value and component-wise egalitarian solutions. Econ. Lett. 117(1), 263–267 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Borm, P., Owen, G., Tijs, S.: On the position value for communication situations. SIAM J. Discret. Math. 5, 305–320 (1992)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Casajus, A.: The position value is the Myerson value, in a sense. Int. J. Game Theory 36, 47–55 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Herings, P.J.J., van der Laan, G., Talman, A.J.J.: The average tree solution for cycle-free graph games. Games Econ. Behav. 62(1), 77–92 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kamijo, Y.: A two-step Shapley value for cooperative games with coalition structures. Int. Game Theory Rev. 11, 207–214 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kongo, T.: Difference between the position value and the Myerson value is due to the existence of coalition structures. Int. J. Game Theory 39, 669–675 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Meessen, R.: Communication games. Master’s thesis, Department of Mathematics, University of Nijmegen, The Netherlands (1988) (in Dutch)Google Scholar
  10. 10.
    Myerson, R.B.: Graphs and cooperation in games. Math. Oper. Res. 2, 225–229 (1977)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Myerson, R.B.: Conference structures and fair allocation rules. Int. J. Game Theory 9, 169–182 (1980)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Owen, G.: Value of games with a priori unions. In: Henn, R., Moeschlin, O. (eds.) Mathematical Economics and Game Theory, pp. 76–88. Springer, Berlin (1977)CrossRefGoogle Scholar
  13. 13.
    Shan, E., Zhang, G., Dong, Y.: Component-wise proportional solutions for communication graph games. Math. Soc. Sci. 81, 22–28 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Shapley, L.S.: A value for n-person games. In: Kuhn, H., Tucker, A.W. (eds.) Contributions to the Theory of Games II, pp. 307–317. Princeton University Press, Princeton (1953)Google Scholar
  15. 15.
    Slikker, M.: A characterization of the position value. Int. J. Game Theory 33, 505–514 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Slikker, M., van den Nouweland, A.: Social and Economic Networks in Cooperative Game Theory. Kluwer, Norwell (2001)CrossRefGoogle Scholar
  17. 17.
    van den Brink, R., van der Laan, G., Pruzhansky, V.: Harsanyi power solutions for graph-restricted games. Int. J. Game Theory 40, 87–110 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    van den Brink, R., Khmelnitskaya, A., van der Laan, G.: An efficient and fair solution for communication graph games. Econ. Lett. 117(3), 786–789 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    van den Nouweland, A., Borm, P., Tijs, S.: Allocation rules for hypergraph communication situations. Int. J. Game Theory 20, 255–268 (1992)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of ManagementShanghai UniversityShanghaiPeople’s Republic of China

Personalised recommendations