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

Author

Listed:
  • Béal, Sylvain
  • Rémila, Eric
  • Solal, Philippe
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.

Suggested Citation

  • Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2010. "Compensations in the Shapley value and the compensation solutions for graph games," MPRA Paper 20955, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:20955
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    References listed on IDEAS

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    Cited by:

    1. Sylvain Béal & André Casajus & Frank Huettner, 2018. "Efficient extensions of communication values," Annals of Operations Research, Springer, vol. 264(1), pages 41-56, May.
    2. Sylvain Béal & Anna Khmelnitskaya & Philippe Solal, 2018. "Two-step values for games with two-level communication structure," Journal of Combinatorial Optimization, Springer, vol. 35(2), pages 563-587, February.
    3. Ramón Flores & Elisenda Molina & Juan Tejada, 2014. "Pyramidal values," Annals of Operations Research, Springer, vol. 217(1), pages 233-252, June.
    4. de Clippel, Geoffroy, 2018. "Membership separability: A new axiomatization of the Shapley value," Games and Economic Behavior, Elsevier, vol. 108(C), pages 125-129.
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    More about this item

    Keywords

    Shapley value ; compensations ; relative fairness ; compensation solution ; DFS ; BFS ; equal surplus division;
    All these keywords.

    JEL classification:

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

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