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Cooperative Games in Graph Structure

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  • Herings,P. Jean-Jacques
  • Laan,Gerard,van der
  • Talman,Dolf

    (METEOR)

Abstract

By a cooperative game in coalitional structure or shortly coalitional game we mean the standard cooperative non-transferable utility game described by a set of payoffs for each coalition that is a nonempty subset of the grand coalition of all players. It is well-known that balancedness is a sufficient condition for the nonemptiness of the core of such a cooperative non-transferable utility game. For this result any information on the internal organization of the coalition is neglected.In this paper we generalize the concept of coalitional games and allow for organizational structure within coalitions. For a subset of players any arbitrarily given structural relation represented by a graph is allowed for. We then consider non-transferable utility games in which a possibly empty set of payoff vectors is assigned to any graph on every subset of players. Such a game will be called a cooperative game in graph structure or shortly graph game. A payoff vector lies in the core of the game if there is no graph on a group of players which can make all of its members better off.We define the balanced-core of a graph game as a refinement of the core. To do so, for each graph a power vector is determined that 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 a positive weight can be assigned such that the weighted power vectors sum up to the vector of ones. A payoff vector lies in the balanced-core if it lies in the core and the payoff vector is an element of payoff sets of all graphs in some balanced collection of graphs. We prove that any balanced graph game has a nonempty balanced-core and therefore a nonempty core. We conclude by some examples showing the usefulness of the concepts of graph games and balanced-core. In particular these examples show a close relationship between solutions to noncooperative games and balanced-core elements of a well-defined graph game. This places the paper in the Nash research program, looking for a unifying theory in which each approach helps to justify and clarify the other.

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Bibliographic Info

Paper provided by Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR) in its series Research Memorandum with number 011.

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Date of creation: 2002
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Handle: RePEc:unm:umamet:2002011

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Keywords: Economics ;

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References

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  1. Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 1999. "Intersection theorems on polytypes," Open Access publications from Tilburg University urn:nbn:nl:ui:12-78480, Tilburg University.
  2. Jean Tirole, 1988. "The Theory of Industrial Organization," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262200716, December.
  3. Ichiishi, Tatsuro & Idzik, Adam, 1991. "Closed Covers of Compact Convex Polyhedra," International Journal of Game Theory, Springer, vol. 20(2), pages 161-69.
  4. Bouyssou, Denis, 1992. "Ranking methods based on valued preference relations: A characterization of the net flow method," European Journal of Operational Research, Elsevier, vol. 60(1), pages 61-67, July.
  5. 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.
  6. Nowak Andrzej S. & Radzik Tadeusz, 1994. "The Shapley Value for n-Person Games in Generalized Characteristic Function Form," Games and Economic Behavior, Elsevier, vol. 6(1), pages 150-161, January.
  7. Nash, John, 1953. "Two-Person Cooperative Games," Econometrica, Econometric Society, vol. 21(1), pages 128-140, April.
  8. Gerard van der Laan & Zaifu Yang & Dolf Talman, 1998. "Cooperative games in permutational structure," Economic Theory, Springer, vol. 11(2), pages 427-442.
  9. P. Jean-Jacques Herings, 1997. "An extremely simple proof of the K-K-M-S Theorem," Economic Theory, Springer, vol. 10(2), pages 361-367.
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Cited by:
  1. Michel Grabisch & Agnieszka Rusinowska, 2008. "A model of influence in a social network," Documents de travail du Centre d'Economie de la Sorbonne b08066, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
  2. Herings, P.J.J. & Laan, G. van der & Talman, A.J.J., 2001. "Measuring the Power of Nodes in Digraphs," Discussion Paper 2001-72, Tilburg University, Center for Economic Research.
  3. René van den Brink & Gerard van der Laan, 2001. "A Class of Consistent Share Functions for Games in Coalition Structure," Tinbergen Institute Discussion Papers 01-044/1, Tinbergen Institute.
  4. Ruys, P.H.M., 2002. "A Managed Service Economy With an Equilibrium for Marketable Services," Discussion Paper 2002-1, Tilburg University, Center for Economic Research.

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