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Core concepts for share vectors

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Author Info
Brink, R. van den
Laan, G. van der (Tilburg University, Center for Economic Research)

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Abstract

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utilities -or simply a TU-game. A value mapping for TU-games is a mapping that assigns to every game a set of vectors each representing a distribution of the payoffs. A value mapping is effcient if to every game it assigns a set of vectors which components all sum up to the worth that can be obtained by all players cooperating together. An approach to effciently allocating the worth of the `grand coalition' is using share mappings which assign to every game a set of share vectors being vectors which components sum up to one such that every component is the corresponding players' share in the total payoff that is to be distributed among the players. In this paper we discuss a class of share mappings containing the (Shapley) share-core, the Banzhaf share-core and the Large Banzhaf share-core. We provide characterizations of this class of share mappings and show how they are related to the corresponding share functions being functions that assign to every TU-game exactly one share vector.

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Paper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number 64.

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Date of creation: 1999
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Handle: RePEc:dgr:kubcen:199964

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

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References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Tadenuma, K, 1992. "Reduced Games, Consistency, and the Core," International Journal of Game Theory, Springer, vol. 20(4), pages 325-34.
  2. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer, vol. 17(2), pages 89-99.
  3. Peleg, B, 1986. "On the Reduced Game Property and Its Converse," International Journal of Game Theory, Springer, vol. 15(3), pages 187-200.
  4. René van den Brink & Gerard van der Laan, 1998. "The Normalized Banzhaf Value and the Banzhaf Share Function," Tinbergen Institute Discussion Papers 98-042/1, Tinbergen Institute.
  5. Dragan, Irinel, 1996. "New mathematical properties of the Banzhaf value," European Journal of Operational Research, Elsevier, vol. 95(2), pages 451-463, December. [Downloadable!] (restricted)
  6. Sergiu Hart, 2006. "Shapley Value," Discussion Paper Series dp421, Center for Rationality and Interactive Decision Theory, Hebrew University, Jerusalem. [Downloadable!]
  7. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer, vol. 23(3), pages 261-81.
  8. (*), Gerard van der Laan & RenÊ van den Brink, 1998. "Axiomatizations of the normalized Banzhaf value and the Shapley value," Social Choice and Welfare, Springer, vol. 15(4), pages 567-582. [Downloadable!] (restricted)
  9. Brink, R. van den & Laan, G. van der, 1999. "Potentials and reduced games for share functions," Discussion Paper 41, Tilburg University, Center for Economic Research. [Downloadable!]
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  10. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May. [Downloadable!] (restricted)
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  1. René van den Brink & Gerard van der Laan, 2001. "A Class of Consistent Share Functions for Games in Coalition Structure," Tinbergen Institute Discussion Papers 01-044/1, Tinbergen Institute. [Downloadable!]
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