On the Position Value for Special Classes of Networks

Part of the Static & Dynamic Game Theory: Foundations & Applications book series (SDGTFA)


This paper deals with a particular class of TU-games, whose cooperation is restricted by a network structure. We consider a communication situation (or graph game) in which a network is produced by subsequent formation of links among players and at each step of the network formation process, the surplus generated by a link is shared between the players involved, according to some rule. As a consequence, we obtain a family of solution concepts that we investigate on particular network structures. This approach provides a different interpretation of the position value, introduced by Borm et al. (SIAM J Discret Math 5(3):305–320, 1992), since it turns out that a specific symmetric rule leads to this solution concept. Moreover, we investigate the problem of computing the position value on particular classes of networks.


TU-games Networks Communication situations Coalition formation Allocation protocols Position value 



We are grateful to Stefano Moretti and Roberto Lucchetti for helpful comments on a previous version of the paper.


  1. 1.
    Banzhaf, J.F. III: Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev. 19, 317–345 (1964)Google Scholar
  2. 2.
    Baron, R., Bal, S., Rmila, E., Solal, P.: Average tree solutions and the distribution of Harsanyi dividends. Int. J. Game Theory 40 (2), 331–349 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Béal, S., Rmila, E., Solal, P.: Compensations in the Shapley value and the compensation solutions for graph games. Int. J. Game Theory 41 (1), 157–178 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Borm, P., Owen, G., Tijs, S.: On the position value for communication situations. SIAM J. Discret. Math. 5 (3), 305–320 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Demange, G.: On group stability in hierarchies and networks. J. Polit. Econ. 112 (4), 754–778 (2004)CrossRefGoogle Scholar
  6. 6.
    Eisenman, R.L.: A profit-sharing interpretation of shapley value for N-person games. Behav. Sci. 12 (5), 396–398 (1967)CrossRefGoogle Scholar
  7. 7.
    Ferreira, R., Grossi, R., Rizzi, R.: Output-sensitive listing of bounded-size trees in undirected graphs. Algorithms-ESA 2011, pp. 275–286. Springer, Berlin/Heidelberg (2011)Google Scholar
  8. 8.
    Hamiache, G.: A value with incomplete communication. Games Econ. Behav. 26 (1), 59–78 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Herings, P.J.J., van der Laan, G., Talman, D.: The average tree solution for cycle-free graph games. Games Econ. Behav. 62 (1), 77–92 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Herings, P.J.J., van der Laan, G., Talman, A.J.J., Yang, Z.: The average tree solution for cooperative games with communication structure. Games Econ. Behav. 68 (2) 626–633 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jackson, M.O., Wolinsky, A.: A strategic model of social and economic networks. J. Econ. Theory 71 (1), 44–74 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Meessen, R.: Communication games. Master’s thesis (in Dutch), Department of Mathematics, University of Nijmegen, The Netherlands (1988)Google Scholar
  13. 13.
    Myerson, R.B.: Graphs and cooperation in games. Math. Oper. Res. 2 (3), 225–229 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Puente, M.A.: Contributions to the representability of simple games and to the calculus of solutions for this class of games. Ph.D. thesis, Technical University of Catalonia (2000)Google Scholar
  15. 15.
    Ruskey, F.: Listing and counting subtrees of a tree. SIAM J. Comput. 10 (1), 141–150 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Shapley, L.S.: A value for n-person games. Technical report, DTIC Document (1952)zbMATHGoogle Scholar
  17. 17.
    Slikker, M.: Decision making and cooperation restrictions. Technical report, Tilburg University (2000)Google Scholar
  18. 18.
    Slikker, M.: A characterization of the position value*. Int. J. Game Theory 33 (4), 505–514 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Slikker, M.: Link monotonic allocation schemes. Int. Game Theory Rev. 7 (04), 473–489 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Székely, L.A., Wang, H.: On subtrees of trees. Adv. Appl. Math. 34 (1), 138–155 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Van den Nouweland, A., Slikker, M.: An axiomatic characterization of the position value for network situations. Math. Soc. Sci. 64 (3), 266–271 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly
  2. 2.Lamsade, PSL Université Paris-DauphineParisFrance

Personalised recommendations