Skip to main content
SpringerLink
Log in

The Myerson Value and Superfluous Supports in Union Stable Systems

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, the set of feasible coalitions in a cooperative game is given by a union stable system. Well-known examples of such systems are communication situations and permission structures. Two games associated with a game on a union stable system are the restricted game (on the set of players in the game) and the conference game (on the set of supports of the system). We define two types of superfluous support property through these two games and provide new characterizations for the Myerson value. Finally, we analyze inheritance of properties between the restricted game and the conference game.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Although in the beginning of this section, we mentioned that we always take as player set N={1,…,n}, the definition of the conference game is the only occasion where we deviate from this. Note that the player set in a conference game is still derived from a structure on N.

References

  1. Myerson, R.B.: Graphs and cooperation in games. Math. Oper. Res. 2, 225–229 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  2. Owen, G.: Values of graph-restricted games. SIAM J. Algebr. Discrete Methods 7, 210–220 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. Borm, P., Owen, G., Tijs, S.H.: On the position value for communication situations. SIAM J. Discrete Math. 5, 305–320 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. van den Nouweland, A., Borm, P., Tijs, S.H.: Allocation rules for hypergraph communication situations. Int. J. Game Theory 20, 255–268 (1992)

    Article  MATH  Google Scholar 

  5. Potters, J.A.M., Reijnierse, H.: Γ-Component additive games. Int. J. Game Theory 24, 49–56 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Myerson, R.B.: Conference structures and fair allocation rules. Int. J. Game Theory 9, 169–182 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Algaba, E., Bilbao, J.M., Borm, P., López, J.J.: The position value for union stable systems. Math. Methods Oper. Res. 52, 221–236 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Algaba, E., Bilbao, J.M., Borm, P., López, J.J.: The Myerson value for union stable structures. Math. Methods Oper. Res. 54, 359–371 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. van den Brink, R.: An axiomatization of the disjunctive permission value for games with a permission structure. Int. J. Game Theory 26, 27–43 (1997)

    Article  MATH  Google Scholar 

  10. Gilles, R.P., Owen, G., van den Brink, R.: Games with permission structures: the conjunctive approach. Int. J. Game Theory 20, 277–293 (1992)

    Article  MATH  Google Scholar 

  11. Algaba, E., Bilbao, J.M., López, J.J.: The position value in communication structures. Math. Methods Oper. Res. 59, 465–477 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Faigle, U., Grabish, M., Heyne, M.: Monge extensions of cooperation and communication structures. Eur. J. Oper. Res. 206, 104–110 (2010)

    Article  MATH  Google Scholar 

  13. van den Nouweland, A.: Games and graphs in economics situations. Ph.D. Thesis, Tilburg University (1993)

  14. van den Brink, R.: On hierarchies and communication. Soc. Choice Welf. (2011). doi:10.1007/s00355-011-0557-y

  15. Shapley, L.S.: A value for n-person games. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games, vol. II, pp. 307–317. Princeton University Press, Princeton (1953)

    Google Scholar 

  16. Meessen, R.: Communication games. M.S. Thesis, Nijmegen University (1988)

  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)

    Article  MATH  Google Scholar 

  18. Algaba, E., Bilbao, J.M., van den Brink, R.: Harsanyi power solutions for games on union stable systems. Tinbergen Discussion Paper 11/127-1, Tinbergen Institute and VU University, Amsterdam (2011)

  19. van den Nouweland, A., Borm, P.: On the convexity of communication games. Int. J. Game Theory 19, 421–430 (1990)

    Article  Google Scholar 

  20. Bondareva, O.: Some applications of linear programming methods to the theory of cooperative games. Probl. Kibern. 10, 119–139 (1963)

    MathSciNet  Google Scholar 

  21. Shapley, L.S.: On balanced sets and cores. Nav. Res. Logist. Q. 14, 453–460 (1967)

    Article  Google Scholar 

  22. Algaba, E., Bilbao, J.M., López, J.J.: A unified approach to restricted games. Theory Decis. 50, 333–345 (2001)

    Article  MATH  Google Scholar 

  23. Jackson, M.O., Wolinsky, A.: A strategic model of social and economic networks. J. Econ. Theory 71, 44–74 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research was finished while the first author was visiting Tinbergen Institute and VU University Amsterdam, under grant Ref. 24022011 of Seville University. Also, this visit was partially supported by Tinbergen Institute. Moreover, this work was presented in some conferences under financial support of the project ECO201017766.

Author information

Authors and Affiliations

  1. Matemática Aplicada II, Escuela Superior de Ingenieros, Camino de los Descubrimientos, s/n, 41092, Sevilla, Spain

    E. Algaba & J. M. Bilbao

  2. Department of Econometrics, VU University and Tinbergen Institute, De Boelelaan 1105, 1081 HV, Amsterdam, The Netherlands

    R. van den Brink

  3. Matemática Aplicada II, Escuela Universitaria Politécnica, Virgen de África 7, 41011, Sevilla, Spain

    J. J. López

Authors
  1. E. Algaba
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. M. Bilbao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. van den Brink
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. J. López
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Algaba.

Additional information

Communicated by Irinel Chiril Dragan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Algaba, E., Bilbao, J.M., van den Brink, R. et al. The Myerson Value and Superfluous Supports in Union Stable Systems. J Optim Theory Appl 155, 650–668 (2012). https://doi.org/10.1007/s10957-012-0077-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-012-0077-7

Keywords

Search

Navigation