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Potentials and Reduced Games for Share Functions

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  • Brink, J.R. van den
  • Laan, G. van der

    (Tilburg University, Center for Economic Research)

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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Bibliographic Info

Paper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number 1999-41.

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Date of creation: 1999
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Handle: RePEc:dgr:kubcen:199941

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References

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  1. Haller, Hans, 1994. "Collusion Properties of Values," International Journal of Game Theory, Springer, vol. 23(3), pages 261-81.
  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. Lehrer, E, 1988. "An Axiomatization of the Banzhaf Value," International Journal of Game Theory, Springer, vol. 17(2), pages 89-99.
  4. (*), 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.
  5. Sergiu Hart, 2006. "Shapley Value," Discussion Paper Series dp421, The Center for the Study of Rationality, Hebrew University, Jerusalem.
  6. 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.
  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. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
  9. 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.
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Cited by:
  1. Gerard van der Laan & René van den Brink, 2001. "Core concepts for share vectors," Social Choice and Welfare, Springer, vol. 18(4), pages 759-784.
  2. Carreras, Francesc & Giménez, José Miguel, 2011. "Power and potential maps induced by any semivalue: Some algebraic properties and computation by multilinear extensions," European Journal of Operational Research, Elsevier, vol. 211(1), pages 148-159, May.

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