SpringerLink
Log in

A probabilistic position value

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

In this article, we generalize the position value, defined by Meessen (Master’s thesis, 1988) for the class of deterministic communication situations, to the class of generalized probabilistic communication situations (Gómez et al. in European Journal of Operational Research 190:539–556, 2008). We provide two characterizations of this new allocation rule. Following in Slikker’s (International Journal of Game Theory 33:505–514, 2005a) footsteps, we characterize the probabilistic position value using probabilistic versions of component efficiency and balanced link contributions. Then we generalize the notion of link potential, defined by Slikker (International Game Theory Review 7:473–489, 2005b) for the class of deterministic communication situations, to the class of generalized probabilistic communication situations, and use it to characterize our allocation rule.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Observe that \(r_{\gamma}^{v}(\emptyset)=0\) because of the zero normalization of (N,v).

  2. For communication situations where (N, v) is not zero-normalized, the position value of a player is defined as the sum of his individual value and his position value in the communication situation (N, w, γ), where (N, w) is the zero-normalization of (N, v), i.e. w(S)=v(S)−∑ i∈S v({i}) for each S⊂N. Our results hold for this more general setting, but to keep the notation simple, we restrict ourselves to zero-normalized games.

  3. In a natural way, we denote by G p and \(G^{{\gamma}_{p}}\) the linear vector spaces of games with players sets p and γ p respectively.

  4. Recall that N={1, …, n}.

References

  • Borm, P., Owen, G., & Tijs, S. (1992). On the position value for communication situations. SIAM Journal on Discrete Mathematics, 5, 305–320.

    Article  Google Scholar 

  • Calvo, E., Lasaga, J., & van den Nouweland, A. (1999). Values of games with probabilistic graphs. Mathematical Social Sciences, 37, 79–95.

    Article  Google Scholar 

  • Gómez, D., González-Arangüena, E., Manuel, C., & Owen, G. (2008). A value for generalized probabilistic communication situations. European Journal of Operational Research, 190, 539–556.

    Article  Google Scholar 

  • Hamiache, G. (1999). A value with incomplete communication. Games and Economic Behavior, 26, 59–78.

    Article  Google Scholar 

  • Harsanyi, J. (1959). A bargaining model for the n-person game. In A. Tucker & R. Luce (Eds.), Contribution to the theory of games IV (pp. 325–355). Princeton: Princeton University Press.

    Google Scholar 

  • Hart, S., & Mas-Colell, A. (1989). Potential, value, and consistency. Econometrica, 57(3), 589–614.

    Article  Google Scholar 

  • Meessen, R. (1988). Communication games. Master’s thesis. The Netherlands: Department of Mathematics, University of Mijmegen.

  • Myerson, R. B. (1977). Graphs and cooperation in games. Mathematics of Operations Research, 2, 225–229.

    Article  Google Scholar 

  • Myerson, R. B. (1980). Conference structures and fair allocation rules. International Journal of Game Theory, 9(3), 169–182.

    Article  Google Scholar 

  • Shapley, L. S. (1953). A value for n-person games. In H. Kuhn & A. Tucker (Eds.), Contributions to the theory of games II (pp. 307–317). Princeton: Princeton University Press.

    Google Scholar 

  • Slikker, M. (2005a). A characterization of the position value. International Journal of Game Theory, 33, 505–514.

    Article  Google Scholar 

  • Slikker, M. (2005b). Link monotonic allocation scheme. International Game Theory Review, 7, 473–489.

    Article  Google Scholar 

  • Slikker, M., & van den Nouweland, A. (2001). Social and economic networks in cooperative games. Dordrecht: Kluwer Academic.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire EQUIPPE, EA 4018, PRES Université Lille Nord de France, Université Lille 3, Villeneuve d’Ascq, France

    A. Ghintran

  2. Dpto. de Estadística e I.O, III, Escuela Universitaria de Estadística, Universidad Complutense de Madrid, Av. Puerta de Hierro s/n, 28040, Madrid, Spain

    E. González-Arangüena & C. Manuel

Authors
  1. A. Ghintran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. González-Arangüena
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. Manuel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Ghintran.

Additional information

This research has been partially supported by the Plan Nacional de I+D+i of the Spanish Government, under the project MTM2008-06778-C02-02/MTM and by the OTKA (Hungarian Fund for Scientific Research) under the project ‘The strong, the Weak and the Cunning: Power and Strategy in Voting Games”. Authors would like to thank Sylvain Béal, Marc Fleurbaey, Guillaume Haeringer, Agnieszka Rusinowska, Philippe Solal and two anonymous referees for helpful comments and suggestions.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ghintran, A., González-Arangüena, E. & Manuel, C. A probabilistic position value. Ann Oper Res 201, 183–196 (2012). https://doi.org/10.1007/s10479-012-1195-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-012-1195-1

Keywords

Search

Navigation