Abstract
A cooperative game with transferable utility–or simply a TU-game– describes a situation in which players can obtain certain payoffs by cooperation. A value function for these games assigns to every TU-game a distribution of payoffs over the players. Well-known solutions for TU-games are the Shapley and the Banzhaf value. An alternative type of solution is the concept of share function, which assigns to every player in a TU-game its share in the worth of the grand coalition. In this paper we consider TU-games in which the players are organized into a coalition structure being a finite partition of the set of players. The Shapley value has been generalized by Owen to TU-games in coalition structure. We redefine this value function as a share function and show that this solution satisfies the multiplication property that the share of a player in some coalition is equal to the product of the Shapley share of the coalition in a game between the coalitions and the Shapley share of the player in a game between the players within the coalition. Analogously we introduce a Banzhaf coalition structure share function. Application of these share functions to simple majority games show some appealing properties.
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REFERENCES
Aumann, R.J. and Drèze, J.H. (1974), Cooperative games with coalition structure, International Journal of Game Theory, 3: 217–237.
Banzhaf, J.F. (1965), Weighted voting doesn't work: A mathematical analysis, Rutgers Law Review, 19: 317–343.
Brink, R. van den and Laan, G. van der (1998a), Axiomatizations of the normalized Banzhaf value and the Shapley value, Social Choice and Welfare, 15: 567–582.
Brink, R. van den and Laan, G. van der (1998b), The normalized Banzhaf value and the Banzhaf share function, in: L.A. Petrosjan and V.V. Mazalov (eds.), Game Theory and Applications IV, pp. 11–31, Nova Science Publishers, New York.
Brink, R. van den and Laan, G. van der (2001), A class of consistent share functions for cooperative games in coalition structure, Tinbergen Institute, Discussion Paper 2001–44/1.
Carreras, F. and Magaña, A. (1994), The multilinear extension and the modified Banzhaf-Coleman index, Mathematical Social Sciences, 28: 215–222.
Davis, M., and Maschler, M. (1965), The kernel of a cooperative game, Naval Research Logistics Quarterly 12: 223–259.
Dubey, P. and Shapley L.S. (1979), Mathematical properties of the Banzhaf power index, Mathematics of Operations Research 4: 99–131.
Grabisch, M., and Roubens, M. (1999) An axiomatic approach to the concept of interaction among players in cooperative games, International Journal of Game Theory 28: 547–565.
Haller, H. (1994), Collusion properties of values, International Journal of Game Theory, 23: 261–281.
Hammer, P.L. and Holzman, R. (1992) Approximations of pseudo-Boolean functions; applications to game theory, ZOR-Methods and Models of Operations Research 36: 3–21.
Hart, S. and Kurz, M. (1983), Endogenous formation of coalitions, Econometrica 51: 1047–1064.
Laan, G. van der, and Brink, R. van den (1998), Axiomatization of a class of share functions for N-person games, Theory and Decision 44: 117–148.
Lehrer, E. (1988), An axiomatization of the Banzhaf value, International Journal of Game Theory 17: 89–99.
Nowak, A.S (1997), On the axiomatization of the Banzhaf value without the additivity axiom, International Journal of Game Theory 26: 137–141.
Shapley, L.S. (1953), A value for n-person games, in: H.W. Kuhn and A.W. Tucker (eds.), Contributions to the Theory of Games, Vol. II, (Annals of Mathematics Studies 28), pp. 307–317, Princeton University Press, Princeton.
Owen, G. (1972), Multilinear extensions of games, Management Sciences 18: 64–79.
Owen, G. (1975), Multilinear extensions and the Banzhaf value, Naval Research Logistics Quart. 22: 741–750.
Owen, G. (1977), Values of games with a priori unions, in: R. Hein and O. Moeschlin (eds.), Essays in Mathematical Economics and Game Theory, pp. 76–88, Springer, New York.
Owen, G. (1981), Modification of the Banzhaf-Coleman index for games with a priori unions, in: M.J. Holler (ed.), Power, Voting, and Voting Power, pp. 232–238, Physica-Verlag, Würzburg, Germany.
Owen, G. and Winter, E. (1992), The multilinear extension and the coalition structure value, Games and Economic Behavior, 4: 582–587.
Winter, E. (1989), A value for cooperative games with levels structure of cooperation, International Journal of Game Theory 18: 227–240.
Winter, E. (1992), The consistency and potential for values of games with coalition structure, Games and Economic Behavior 4: 132–144.
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van der Laan, G., van den Brink, R. A Banzhaf share function for cooperative games in coalition structure. Theory and Decision 53, 61–86 (2002). https://doi.org/10.1023/A:1020805106965
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DOI: https://doi.org/10.1023/A:1020805106965