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Null or Zero Players: The Difference between the Shapley Value and the Egalitarian Solution

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  • René van den Brink

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    (Faculty of Economics and Business Administration, Vrije Universiteit Amsterdam)

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    Abstract

    A situation in which a finite set of players can generate certain payoffs by cooperation can be described by a cooperative game with transferable utility. A solution for TU-games assigns to every TU-game a distribution of the payoffs that can be earned over the individual players. Two well-known solutions for TU-games are the Shapley value and the egalitarian solution. The Shapley value is characterized in various ways. Most characterizations use some axiom related to null players, i.e. players who contribute nothing to any coalition. We show that in these characterizations, replacing null players by zero players characterizes the egalitarian solution, where a player is a zero player if every coalition containing this player earns zero worth. We illustrate this difference between these two solutions by applying them to auction games.

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    Bibliographic Info

    Paper provided by Tinbergen Institute in its series Tinbergen Institute Discussion Papers with number 04-127/1.

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    Date of creation: 19 Nov 2004
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    Handle: RePEc:dgr:uvatin:20040127

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    Related research

    Keywords: Null players; zero players; Shapley value; egalitarian solution; strong monotonicity; coalitional monotonicity; auction games;

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    1. Jackson, Matthew O. & Wolinsky, Asher, 1996. "A Strategic Model of Social and Economic Networks," Journal of Economic Theory, Elsevier, vol. 71(1), pages 44-74, October.
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    4. René van den Brink, 2002. "An axiomatization of the Shapley value using a fairness property," International Journal of Game Theory, Springer, vol. 30(3), pages 309-319.
    5. Graham, Daniel A & Marshall, Robert C & Richard, Jean-Francois, 1990. "Differential Payments within a Bidder Coalition and the Shapley Value," American Economic Review, American Economic Association, vol. 80(3), pages 493-510, June.
    6. Sergiu Hart, 2006. "Shapley Value," Discussion Paper Series dp421, The Center for the Study of Rationality, Hebrew University, Jerusalem.
    7. Rothschild, R., 2001. "On the use of a modified Shapley value to determine the optimal size of a cartel," Journal of Economic Behavior & Organization, Elsevier, vol. 45(1), pages 37-47, May.
    8. MANIQUET, François, . "A characterization of the Shapley value in queueing problems," CORE Discussion Papers RP -1662, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    9. Feltkamp, V., 1993. "Alternative Axiomatic Characterizations of the Shapley and Banzhaf Values," Papers 9353, Tilburg - Center for Economic Research.
    10. Chun, Youngsub, 1989. "A new axiomatization of the shapley value," Games and Economic Behavior, Elsevier, vol. 1(2), pages 119-130, June.
    11. Algaba, A. & Bilbao, J.M. & Brink, J.R. van den & Jiménez-Losada, A., 2000. "Cooperative Games on Antimatroids," Discussion Paper 2000-124, Tilburg University, Center for Economic Research.
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    Cited by:
    1. René van den Brink & Yukihiko Funaki, 2004. "Axiomatizations of a Class of Equal Surplus Sharing Solutions for Cooperative Games with Transferable Utility," Tinbergen Institute Discussion Papers 04-136/1, Tinbergen Institute.
    2. Rene van den Brink & Arantza Estevez-Fernandez & Gerard van der Laan & Nigel Moes, 2011. "Independence Axioms for Water Allocation," Tinbergen Institute Discussion Papers 11-128/1, Tinbergen Institute.

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