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

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  • 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.

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Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 20955.

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Date of creation: 24 Feb 2010
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Handle: RePEc:pra:mprapa:20955

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Keywords: Shapley value ; compensations ; relative fairness ; compensation solution ; DFS ; BFS ; equal surplus division;

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  1. René Brink & Gerard Laan & Valeri Vasil’ev, 2007. "Component efficient solutions in line-graph games with applications," Economic Theory, Springer, Springer, vol. 33(2), pages 349-364, November.
  2. Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2010. "Rooted-tree solutions for tree games," European Journal of Operational Research, Elsevier, Elsevier, vol. 203(2), pages 404-408, June.
  3. Fehr, Ernst & Schmidt, Klaus M., 1998. "A Theory of Fairness, Competition and Cooperation," CEPR Discussion Papers, C.E.P.R. Discussion Papers 1812, C.E.P.R. Discussion Papers.
  4. Gabrielle Demange, 2004. "On group stability in hierarchies and networks," Post-Print halshs-00581662, HAL.
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  7. Evans, R.A., 1992. "Value, Consistency and Random Coalition Formation," Papers, Cambridge - Risk, Information & Quantity Signals 169, Cambridge - Risk, Information & Quantity Signals.
  8. Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2009. "Average tree solutions and the distribution of Harsanyi dividends," MPRA Paper 17909, University Library of Munich, Germany.
  9. HERINGS, P. Jean-Jacques & van der LAAN, Gerard & TALMAN, Dolf, . "The average tree solution for cycle-free graph games," CORE Discussion Papers RP, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) -2155, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  10. Aadland, David & Kolpin, Van, 1998. "Shared irrigation costs: An empirical and axiomatic analysis," Mathematical Social Sciences, Elsevier, Elsevier, vol. 35(2), pages 203-218, March.
  11. Teck-Hua Ho & Xuanming Su, 2009. "Peer-Induced Fairness in Games," American Economic Review, American Economic Association, American Economic Association, vol. 99(5), pages 2022-49, December.
  12. van den Brink, Rene, 2007. "Null or nullifying players: The difference between the Shapley value and equal division solutions," Journal of Economic Theory, Elsevier, Elsevier, vol. 136(1), pages 767-775, September.
  13. Aadland, David & Kolpin, Van, 2004. "Environmental determinants of cost sharing," Journal of Economic Behavior & Organization, Elsevier, Elsevier, vol. 53(4), pages 495-511, April.
  14. Anna Khmelnitskaya, 2010. "Values for rooted-tree and sink-tree digraph games and sharing a river," Theory and Decision, Springer, Springer, vol. 69(4), pages 657-669, October.
  15. Aadland, David & Kolpin, Van, 2004. "Erratum to "Environmental determinants of cost sharing"," Journal of Economic Behavior & Organization, Elsevier, Elsevier, vol. 55(1), pages 105-121, September.
  16. Ruiz, Luis M. & Valenciano, Federico & Zarzuelo, Jose M., 1998. "The Family of Least Square Values for Transferable Utility Games," Games and Economic Behavior, Elsevier, Elsevier, vol. 24(1-2), pages 109-130, July.
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