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Core concepts for share vectors

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Author Info

  • Gerard van der Laan

    ()
    (Department of Econometrics and Tinbergen Institute, Free University, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands)

  • René van den Brink

    ()
    (Department of Econometrics and Center, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands)

Abstract

A value mapping for cooperative games with transferable utilities is a mapping that assigns to every game a set of vectors each representing a distribution of the payoffs. A value mapping is efficient 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 efficiently allocate 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. Every component of a share vector 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, and provide characterizations of this class of share mappings.

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Bibliographic Info

Article provided by Springer in its journal Social Choice and Welfare.

Volume (Year): 18 (2001)
Issue (Month): 4 ()
Pages: 759-784

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Handle: RePEc:spr:sochwe:v:18:y:2001:i:4:p:759-784

Note: Received: 9 August 1999/Accepted: 25 April 2000
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  1. Dragan, Irinel, 1996. "New mathematical properties of the Banzhaf value," European Journal of Operational Research, Elsevier, vol. 95(2), pages 451-463, December.
  2. Peleg, B, 1986. "On the Reduced Game Property and Its Converse," International Journal of Game Theory, Springer, vol. 15(3), pages 187-200.
  3. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer, vol. 23(3), pages 261-81.
  4. Sergiu Hart, 2006. "Shapley Value," Discussion Paper Series dp421, The Center for the Study of Rationality, Hebrew University, Jerusalem.
  5. Gerard van der Laan & René van den Brink, 1998. "Axiomatization of a class of share functions for n-person games," Theory and Decision, Springer, vol. 44(2), pages 117-148, April.
  6. Brink, J.R. van den & Laan, G. van der, 1999. "Potentials and Reduced Games for Share Functions," Discussion Paper 1999-41, Tilburg University, Center for Economic Research.
  7. Tadenuma, K, 1992. "Reduced Games, Consistency, and the Core," International Journal of Game Theory, Springer, vol. 20(4), pages 325-34.
  8. Ichiishi, Tatsuro, 1981. "Super-modularity: Applications to convex games and to the greedy algorithm for LP," Journal of Economic Theory, Elsevier, vol. 25(2), pages 283-286, October.
  9. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
  10. Brink, J.R. van den & Laan, G. van der, 1998. "The normalized Banzhaf value and the Banzhaf share function," Research Memorandum 764, Tilburg University, Faculty of Economics and Business Administration.
  11. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer, vol. 17(2), pages 89-99.
  12. (*), 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.
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Cited by:
  1. Brink, J.R. van den & Laan, G. van der, 2001. "A Class of Consistent Share Functions For Games in Coalition Structure," Discussion Paper 2001-33, Tilburg University, Center for Economic Research.

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