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The Average Tree Solution for Cooperative Games with Communication Structure

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

    (METEOR)

We study cooperative games with communication structure, represented by an undirectedgraph. Players in the game are able to cooperate only if they can form a network in the graph. A single-valued solution, the average tree solution, is proposed for this class ofgames. Given the graph structure we define a collection of spanning trees, where eachspanning tree specifies a particular way by which players communicate and determines a payoff vector of marginal contributions of all the players. The average tree solution is defined to be the average of all these payoff vectors. It is shown that if a game has acomplete communication structure, then the proposed solution coincides with the Shapleyvalue, and that if the game has a cycle-free communication structure, it is the solutionproposed by Herings, van der Laan and Talman (2008). We introduce the notion of linkconvexity, under which the game is shown to have a non-empty core and the average tree solution lies in the core. In general, link-convexity is weaker than convexity. For games with a cycle-free communication structure, link-convexity is even weaker than super-additivity.

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Paper provided by Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR) in its series Research Memorandum with number 026.

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Date of creation: 2008
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Handle: RePEc:unm:umamet:2008026
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  1. Borm, P.E.M. & Owen, G. & Tijs, S.H., 1992. "On the position value for communication situations," Other publications TiSEM 5a8473e4-1df7-42df-ad53-f, Tilburg University, School of Economics and Management.
  2. Gabrielle Demange, 2004. "On group stability in hierarchies and networks," Post-Print halshs-00581662, HAL.
  3. Le Breton, M. & Owen, G. & Weber, S., 1991. "Strongly Balanced Cooperative Games," Papers 92-3, York (Canada) - Department of Economics.
  4. Talman, A.J.J. & Yamamoto, Y., 2007. "Games With Limited Communication Structure," Discussion Paper 2007-19, Tilburg University, Center for Economic Research.
  5. Demange, Gabrielle, 1994. "Intermediate preferences and stable coalition structures," Journal of Mathematical Economics, Elsevier, vol. 23(1), pages 45-58, January.
  6. René Brink & Gerard Laan & Vitaly Pruzhansky, 2011. "Harsanyi power solutions for graph-restricted games," International Journal of Game Theory, Springer, vol. 40(1), pages 87-110, February.
  7. Ren� van den Brink & Gerard van der Laan & Vitaly Pruzhansky, 2004. "Harsanyi Power Solutions for Graph-restricted Games," Tinbergen Institute Discussion Papers 04-095/1, Tinbergen Institute.
  8. Talman, A.J.J. & Yamamoto, Y., 2008. "Average tree solution and subcore for acyclic graph games," Other publications TiSEM 47c15bd0-3911-429c-8952-7, Tilburg University, School of Economics and Management.
  9. Herings, P.J.J. & van der Laan, G. & Talman, A.J.J., 2008. "The average tree solution for cycle-free graph games," Other publications TiSEM f243609c-2847-415f-ae52-1, Tilburg University, School of Economics and Management.
  10. Baron, Richard & Béal, Sylvain & Remila, Eric & Solal, Philippe, 2008. "Average tree solutions for graph games," MPRA Paper 10189, University Library of Munich, Germany.
  11. Kaneko, Mamoru & Wooders, Myrna Holtz, 1982. "Cores of partitioning games," Mathematical Social Sciences, Elsevier, vol. 3(4), pages 313-327, December.
  12. Marco Slikker, 2005. "A characterization of the position value," International Journal of Game Theory, Springer, vol. 33(4), pages 505-514, November.
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