Abstract
In games with a permission structure it is assumed that players in a cooperative transferable utility game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. We provide axiomatic characterizations of Banzhaf permission values being solutions that are obtained by applying the Banzhaf value to modified TU-games. In these characterizations we use power- and player split neutrality properties. These properties state that splitting a player’s authority and/or contribution over two players does not change the sum of their payoffs.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Algaba E, Bilbao JM, van den Brink R, Jiménez-Losada A (2004) An axiomatizations of the Banzhaf value for cooperative games on antimatroids. Math Methods Oper Res 59: 147–166
Aumann RJ, Drèze JH (1974) Cooperative Games with Coalition Structure. Int J Game Theory 3: 217–237
Banzhaf JF (1965) Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19: 317–343
Derks JJM, Haller HH (1999) Null players out? linear values for games with variable supports. Int Game Theory Rev 1: 301–314
Dubey P, Shapley LS (1979) Mathematical properties of the Banzhaf power index. Math Oper Res 4: 99–131
Gilles RP, Owen G (1999) Cooperative games and disjunctive permission structures. CentER discussion paper 9920, CentER for Economic Research. Tilburg University, Tilburg
Gilles RP, Owen G, van den Brink R (1992) Games with permission structures: the conjunctive approach. Int J Game Theory 20: 277–293
Haller H (1994) Collusion properties of values. Int J Game Theory 23: 261–281
Harsanyi JC (1959) A bargaining model for cooperative n-person games. In: Tucker AW, Luce RD (eds) Contributions to the theory of games IV. Princeton University Press, Princeton, pp 325–355
Lehrer E (1988) An axiomatization of the Banzhaf value. Int J Game Theory 17: 89–99
Myerson RB (1977) Graphs and cooperation in games. Math Oper Res 2: 225–229
Owen G (1975) Multilinear extensions and the Banzhaf value. Nav Res Logist Q 22: 741–750
Shapley LS (1953) A value for n-person games. In: Kuhn HW, Tucker AW (eds) Annals of Mathematics Studies 28 (Contributions to the Theory of Games Vol. 2). Princeton UP, Princeton, pp 307–317
van den Brink R (1997) An axiomatization of the disjunctive permission value for games with a permission structure. Int J Game Theory 26: 27–43
van den Brink R (1999) An axiomatization of the conjunctive permission value for games with a hierarchical permission structure. In: Swart H (eds) Logic, game theory and social choice. Springer, Berlin, pp 125–139
van den Brink R, Gilles RP (1994) A social power index for hierarchically structured populations of economic agents. In: Gilles RP, Ruys PHM (eds) Imperfections and behaviour in economic organizations. Kluwer, Dordrecht, pp 279–318
van den Brink R, Gilles RP (1996) Axiomatizations of the conjunctive permission value for games with permission structures. Games Econ Behav 12: 113–126
Acknowledgment
I would like to thank two anonymous referees and the editor for useful remarks on a previous draft.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
A preliminary version of this article is published in the ICM Millennium Lectures on Games (eds. L.A. Petrosyan and D.W.K. Yeung), Springer-Verlag, Heidelberg, Germany (2003), without permission from the author.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
van den Brink, R. Axiomatizations of Banzhaf permission values for games with a permission structure. Int J Game Theory 39, 445–466 (2010). https://doi.org/10.1007/s00182-009-0221-2
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00182-009-0221-2