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Compensations in the Shapley value and the compensation solutions for graph games

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  • Sylvain Béal

    ()

  • Eric Rémila

    ()

  • Philippe Solal

    ()

Abstract

We consider an alternative expression of the Shapley value that reveals a system of compensations: each player receives an equal share of the worth of each coalition he belongs to, and has to compensate an equal share of the worth of any coalition he does not belong to. We give an interpretation in terms of formation of the grand coalition according to an ordering of the players and define the corresponding compensation vector. Then, we generalize this idea to cooperative games with a communication graph. Firstly, we consider cooperative games with a forest (cycle-free graph). We extend the compensation vector by considering all rooted spanning trees of the forest (see Demange 2004) instead of orderings of the players. The associated allocation rule, called the compensation solution, is characterized by component efficiency and relative fairness. The latter axiom takes into account the relative position of a player with respect to his component. Secondly, we consider cooperative games with arbitrary graphs and construct rooted spanning trees by using the classical algorithms DFS and BFS. If the graph is complete, we show that the compensation solutions associated with DFS and BFS coincide with the Shapley value and the equal surplus division respectively.

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File URL: http://hdl.handle.net/10.1007/s00182-011-0277-7
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Bibliographic Info

Article provided by Springer in its journal International Journal of Game Theory.

Volume (Year): 41 (2012)
Issue (Month): 1 (February)
Pages: 157-178

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Handle: RePEc:spr:jogath:v:41:y:2012:i:1:p:157-178

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Related research

Keywords: Shapley value; Compensations; Compensation solutions; DFS ; BFS ; Equal surplus division; Relative fairness;

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  1. Ernst Fehr & Klaus M. Schmidt, 1999. "A Theory Of Fairness, Competition, And Cooperation," The Quarterly Journal of Economics, MIT Press, vol. 114(3), pages 817-868, August.
  2. René Brink & Gerard Laan & Valeri Vasil’ev, 2007. "Component efficient solutions in line-graph games with applications," Economic Theory, Springer, vol. 33(2), pages 349-364, November.
  3. 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.
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  13. van den Brink, Rene, 2007. "Null or nullifying players: The difference between the Shapley value and equal division solutions," Journal of Economic Theory, Elsevier, vol. 136(1), pages 767-775, September.
  14. Evans, Robert A., 1996. "Value, Consistency, and Random Coalition Formation," Games and Economic Behavior, Elsevier, vol. 12(1), pages 68-80, January.
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  16. Gabrielle Demange, 2004. "On group stability in hierarchies and networks," Post-Print halshs-00581662, HAL.
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