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Average tree solutions for graph games

Author

Listed:
  • Richard Baron

    (CREUSET - Centre de Recherche Economique de l'Université de Saint-Etienne - UJM - Université Jean Monnet - Saint-Étienne)

  • Sylvain Béal

    (CREUSET - Centre de Recherche Economique de l'Université de Saint-Etienne - UJM - Université Jean Monnet - Saint-Étienne)

  • Éric Rémila

    (LIP - Laboratoire de l'Informatique du Parallélisme - ENS de Lyon - École normale supérieure de Lyon - Université de Lyon - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon - Inria - Institut National de Recherche en Informatique et en Automatique - Université de Lyon - CNRS - Centre National de la Recherche Scientifique)

  • Philippe Solal

    (CREUSET - Centre de Recherche Economique de l'Université de Saint-Etienne - UJM - Université Jean Monnet - Saint-Étienne)

Abstract

In this paper we consider cooperative graph games being TU-games in which players cooperate if they are connected in the communication graph. We focus our attention to the average tree solutions introduced by Herings, van der Laan and Talman [6] and Herings, van der Laan, Talman and Yang [7]. Each average tree solution is defined with re- spect to a set, say T , of admissible rooted spanning trees. Each average tree solution is characterized by efficiency, linearity and an axiom of T - hierarchy on the class of all graph games with a fixed communication graph. We also establish that the set of admissible rooted spanning trees introduced by Herings, van der Laan, Talman and Yang [7] is the largest set of rooted spanning trees such that the corresponding aver- age tree solution is a Harsanyi solution. One the other hand, we show that this set of rooted spanning trees cannot be constructed by a dis- tributed algorithm. Finally, we propose a larger set of spanning trees which coincides with the set of all rooted spanning trees in clique-free graphs and that can be computed by a distributed algorithm.
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Suggested Citation

  • Richard Baron & Sylvain Béal & Éric Rémila & Philippe Solal, 2008. "Average tree solutions for graph games," Post-Print hal-00332570, HAL.
    • Handle: RePEc:hal:journl:hal-00332570
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    References listed on IDEAS

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    1. Lange, Fabien & Grabisch, Michel, 2009. "Values on regular games under Kirchhoff's laws," Mathematical Social Sciences, Elsevier, vol. 58(3), pages 322-340, November.
    2. Roger B. Myerson, 1977. "Graphs and Cooperation in Games," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 225-229, August.
    3. Herings, P.J.J. & van der Laan, G. & Talman, A.J.J. & Yang, Z., 2010. "The average tree solution for cooperative games with communication structure," Games and Economic Behavior, Elsevier, vol. 68(2), pages 626-633, March.
    4. René Brink & Gerard Laan & Valeri Vasil’ev, 2007. "Component efficient solutions in line-graph games with applications," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 33(2), pages 349-364, November.
    5. 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.
    6. Talman, A.J.J. & Yamamoto, Y., 2007. "Games With Limited Communication Structure," Other publications TiSEM e5f3ebff-9aea-4023-9525-b, Tilburg University, School of Economics and Management.
    7. Herings, P.J.J. & van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2010. "The average tree solution for cooperative games with communication structure," Other publications TiSEM 24359ac5-6399-42ee-8f0b-7, Tilburg University, School of Economics and Management.
    8. Gilles, Robert P & Owen, Guillermo & van den Brink, Rene, 1992. "Games with Permission Structures: The Conjunctive Approach," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(3), pages 277-293.
    9. Herings, P. Jean Jacques & van der Laan, Gerard & Talman, Dolf, 2008. "The average tree solution for cycle-free graph games," Games and Economic Behavior, Elsevier, vol. 62(1), pages 77-92, January.
    10. Faigle, U & Kern, W, 1992. "The Shapley Value for Cooperative Games under Precedence Constraints," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(3), pages 249-266.
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    Cited by:

    1. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," Annals of Operations Research, Springer, vol. 204(1), pages 33-64, April.
    2. Herings, P.J.J. & van der Laan, G. & Talman, A.J.J. & Yang, Z., 2010. "The average tree solution for cooperative games with communication structure," Games and Economic Behavior, Elsevier, vol. 68(2), pages 626-633, March.
    3. Herings, P.J.J. & van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2010. "The average tree solution for cooperative games with communication structure," Other publications TiSEM 24359ac5-6399-42ee-8f0b-7, Tilburg University, School of Economics and Management.

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    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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