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Potentials and Reduced Games for Share Functions

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Author Info
René van den Brink (Tilburg University)
Gerard van der Laan () (Vrije Universiteit Amsterdam)

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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 function for TU-games is a function that assigns to every game a distribution of the payoffs. A value function is efficient if for every game it exactly distributes the worth that can be obtained by all players cooperating together.

An approach to efficiently allocating the worth of the 'grand coalition'is using share functions which assign to every game a vector 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 give some characterizations of a class of share functions containing the Shapley share function and the Banzhaf share function using generalizations of potentials and of Hart and Mas-Colell's reduced game property.

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Publisher Info
Paper provided by Tinbergen Institute in its series Tinbergen Institute Discussion Papers with number 99-026/1.

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Date of creation: 31 Mar 1999
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Handle: RePEc:dgr:uvatin:19990026

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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. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer, vol. 17(2), pages 89-99.
  2. Sergiu Hart, 2006. "Shapley Value," Discussion Paper Series dp421, Center for Rationality and Interactive Decision Theory, Hebrew University, Jerusalem. [Downloadable!]
  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. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer, vol. 23(3), pages 261-81.
  5. (*), 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)
  6. 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.
  7. 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)
  8. 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. Brink, R. van den & Laan, G. van der, 1999. "Core concepts for share vectors," Discussion Paper 64, Tilburg University, Center for Economic Research. [Downloadable!]
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