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Players Indifferent to Cooperate and Characterizations of the Shapley Value

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  • Conrado Manuel

    (Universidad Complutense de Madrid)

  • Enrique Gonzalez-Aranguena

    (Universidad Complutense de Madrid)

  • Rene van den Brink|

    (VU University Amsterdam)

Abstract

In this paper we provide new axiomatizations of the Shapley value for TU-games using axioms that are based on relational aspects in the interactions among players. Some of these relational aspects, in particular the economic or social interest of each player in cooperating with each other, can be found embedded in the characteristic function. We define a particular relation among the players that it is based on mutual indifference. The first new axiom expresses that the payoffs of two players who are not indifferent to each other are affected in the same way if they become enemies and do not cooperate with each other anymore. The second new axiom expresses that the payoff of a player is not affected if players to whom it is indifferent leave the game. We show that the Shapley value is characterized by these two axioms together with the well-known efficiency axiom. Further, we show that another axiomatization of the Shapley value is obtained if we replace t he second axiom and efficiency by the axiom which applies the efficiency condition to every class of indifferent players. Finally, we extend the previous results to the case of weighted Shapley values.

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Bibliographic Info

Paper provided by Tinbergen Institute in its series Tinbergen Institute Discussion Papers with number 12-036/1.

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Date of creation: 11 Apr 2012
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Handle: RePEc:dgr:uvatin:20120036

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Web page: http://www.tinbergen.nl

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Keywords: TU-game; Shapley value; axiomatization; indifferent players; weighted Shapley values;

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  1. André Casajus, 2011. "Differential marginality, van den Brink fairness, and the Shapley value," Theory and Decision, Springer, Springer, vol. 71(2), pages 163-174, August.
  2. Winter, Eyal, 2002. "The shapley value," Handbook of Game Theory with Economic Applications, Elsevier, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 53, pages 2025-2054 Elsevier.
  3. NEYMAN, Abraham, 1988. "Uniqueness of the Shapley value," CORE Discussion Papers, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) 1988013, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  4. Kongo, T. & Funaki, Y. & Tijs, S.H., 2007. "New Axiomatizations and an Implementation of the Shapley Value," Discussion Paper, Tilburg University, Center for Economic Research 2007-90, Tilburg University, Center for Economic Research.
  5. Algaba, A. & Bilbao, J.M. & Brink, J.R. van den & Jiménez-Losada, A., 2001. "Axiomatizations of the Shapley Value for Cooperative Games on Antimatroids," Discussion Paper, Tilburg University, Center for Economic Research 2001-99, Tilburg University, Center for Economic Research.
  6. Sergiu Hart, 2006. "Shapley Value," Discussion Paper Series, The Center for the Study of Rationality, Hebrew University, Jerusalem dp421, The Center for the Study of Rationality, Hebrew University, Jerusalem.
  7. Gérard Hamiache, 2001. "Associated consistency and Shapley value," International Journal of Game Theory, Springer, Springer, vol. 30(2), pages 279-289.
  8. Ehud Kalai & Dov Samet, 1983. "On Weighted Shapley Values," Discussion Papers, Northwestern University, Center for Mathematical Studies in Economics and Management Science 602, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  9. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, Econometric Society, vol. 57(3), pages 589-614, May.
  10. M. Albizuri, 2010. "Games with externalities: games in coalition configuration function form," Computational Statistics, Springer, Springer, vol. 72(1), pages 171-186, August.
  11. Anna Khmelnitskaya & Elena Yanovskaya, 2007. "Owen coalitional value without additivity axiom," Computational Statistics, Springer, Springer, vol. 66(2), pages 255-261, October.
  12. Guillermo Owen, 1972. "Multilinear Extensions of Games," Management Science, INFORMS, INFORMS, vol. 18(5-Part-2), pages 64-79, January.
  13. Chun, Youngsub, 1989. "A new axiomatization of the shapley value," Games and Economic Behavior, Elsevier, Elsevier, vol. 1(2), pages 119-130, June.
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