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Core concepts for share vectors

Author

Listed:
  • 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.

Suggested Citation

  • Gerard van der Laan & René van den Brink, 2001. "Core concepts for share vectors," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(4), pages 759-784.
  • 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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    References listed on IDEAS

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    1. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer;Game Theory Society, vol. 17(2), pages 89-99.
    2. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer;Game Theory Society, vol. 23(3), pages 261-281.
    3. 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.
    4. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
    5. Tadenuma, K, 1992. "Reduced Games, Consistency, and the Core," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(4), pages 325-334.
    6. Peleg, B, 1986. "On the Reduced Game Property and Its Converse," International Journal of Game Theory, Springer;Game Theory Society, vol. 15(3), pages 187-200.
    7. Dragan, Irinel, 1996. "New mathematical properties of the Banzhaf value," European Journal of Operational Research, Elsevier, vol. 95(2), pages 451-463, December.
    8. Sergiu Hart, 2006. "Shapley Value," Discussion Paper Series dp421, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    9. van den Brink, J.R. & van der Laan, G., 1999. "Potentials and Reduced Games for Share Functions," Discussion Paper 1999-41, Tilburg University, Center for Economic Research.
    10. 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.
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    Cited by:

    1. van den Brink, J.R. & van der Laan, G., 1999. "Potentials and Reduced Games for Share Functions," Discussion Paper 1999-41, Tilburg University, Center for Economic Research.
    2. van den Brink, Rene & van der Laan, Gerard, 2005. "A class of consistent share functions for games in coalition structure," Games and Economic Behavior, Elsevier, vol. 51(1), pages 193-212, April.

    More about this item

    JEL classification:

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

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