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Cooperative Games in Graph Structure

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Author Info

  • P. Jean-Jacques Herings

    (University of Maastricht)

  • Gerard van der Laan

    ()
    (Vrije Universiteit Amsterdam)

  • Dolf Talman

    (Tilburg University)

Abstract

In this paper we generalize the concept of coalitional games by allowingfor any organizational structure within coalitions represented by a graphon the set of players ot the coalition. A, possibly empty, set of payoffvectors is assigned to any graph on every subset of players. Such a gamewill be called a graph game. For each graph a power vector is determinedthat depends on the relative positions of the players within the graph.A collection of graphs will be called balanced if to any graph in the collection apositive weight can be assigned such that the weighted power vectorssum up to the vector of ones. We then define the balanced-core as a refinement ofthe core. A payoff vector lies in the balanced-core if it lies in the core andthe payoff vector is an element of payoff sets of all graphs in some balanced collection ofgraphs. We prove that any balanced graph game has a nonempty balanced-core.We conclude by some examples showing the usefulness of the conceptsof graph games and balanced-core. In particular these examples show a closerelationship between solutions to noncooperative games andbalanced-core elements of a well-defined graph game.

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Bibliographic Info

Paper provided by Tinbergen Institute in its series Tinbergen Institute Discussion Papers with number 00-072/1.

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Date of creation: 25 Aug 2000
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Handle: RePEc:dgr:uvatin:20000072

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Web page: http://www.tinbergen.nl

Related research

Keywords: cooperative games; graphs; balancedness; core; Nash program;

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References

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  1. Nash, John, 1953. "Two-Person Cooperative Games," Econometrica, Econometric Society, Econometric Society, vol. 21(1), pages 128-140, April.
  2. Laan, G. van der & Talman, A.J.J. & Yang, Z., 1994. "Intersection theorems on polytopes," Discussion Paper, Tilburg University, Center for Economic Research 1994-20, Tilburg University, Center for Economic Research.
  3. Gerard van der Laan & Zaifu Yang & Dolf Talman, 1998. "Cooperative games in permutational structure," Economic Theory, Springer, Springer, vol. 11(2), pages 427-442.
  4. P. Jean-Jacques Herings, 1997. "An extremely simple proof of the K-K-M-S Theorem," Economic Theory, Springer, Springer, vol. 10(2), pages 361-367.
  5. Ichiishi, Tatsuro & Idzik, Adam, 1991. "Closed Covers of Compact Convex Polyhedra," International Journal of Game Theory, Springer, Springer, vol. 20(2), pages 161-69.
  6. Jean Tirole, 1988. "The Theory of Industrial Organization," MIT Press Books, The MIT Press, The MIT Press, edition 1, volume 1, number 0262200716, December.
  7. Bouyssou, Denis, 1992. "Ranking methods based on valued preference relations: A characterization of the net flow method," European Journal of Operational Research, Elsevier, Elsevier, vol. 60(1), pages 61-67, July.
  8. Nowak Andrzej S. & Radzik Tadeusz, 1994. "The Shapley Value for n-Person Games in Generalized Characteristic Function Form," Games and Economic Behavior, Elsevier, Elsevier, vol. 6(1), pages 150-161, January.
  9. Doup, T.M. & Talman, A.J.J., 1987. "A new simplicial variable dimension algorithm to find equilibria on the product space of unit simplices," Open Access publications from Tilburg University urn:nbn:nl:ui:12-148734, Tilburg University.
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Citations

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Cited by:
  1. Herings, P.J.J. & Laan, G. van der & Talman, A.J.J., 2001. "Measuring the Power of Nodes in Digraphs," Discussion Paper, Tilburg University, Center for Economic Research 2001-72, Tilburg University, Center for Economic Research.
  2. van den Brink, Rene & van der Laan, Gerard, 2005. "A class of consistent share functions for games in coalition structure," Games and Economic Behavior, Elsevier, Elsevier, vol. 51(1), pages 193-212, April.
  3. Michel Grabisch & Agnieszka Rusinowska, 2010. "A model of influence in a social network," Theory and Decision, Springer, Springer, vol. 69(1), pages 69-96, July.
  4. Ruys, P.H.M., 2002. "A Managed Service Economy With an Equilibrium for Marketable Services," Discussion Paper, Tilburg University, Center for Economic Research 2002-1, Tilburg University, Center for Economic Research.

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