Abstract
The evaluation of the power of the parties of a Parliament is usually carried out following a static approach, counting their numbers of seats and possible ideological relations among them. In this paper we propose a dynamic model in which parties may blackmail each other in order to increase their power at the expense of others. This model may be solved in terms of the Bargaining Set (Aumann and Maschler in Advances in game theory. Princeton University Press, Princeton, 1964), a set valued solution, in contrast to the classical power indices that are point valued; so it may be used to check the robustness of a power division taking into account possible blackmailing activities. The paper is mainly based on the idea of proposing an existing instrument, the Bargaining Set, as power index to analyze this blackmailing situation. The computational complexity limits the possibility of applying it to real-world situations, but the possibility of making the computation easier is shown using some examples.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
This property is called monotonicity.
In the literature there are different equivalent definitions of the objection and counterobjection and also the notation of Bargaining Set may be different. We remark that there exist alternative definitions of the Bargaining Set, the most famous one is due to Mas-Colell (1989).
References
Alonso-Meijide, J. M., & Carreras, F. (2011). The proportional coalitional Shapley value. Expert Systems with Applications, 38, 6967–6979.
Aumann, R. J., & Maschler, M. (1964). The bargaining set for cooperative games. In M. Dresher, L. S. Shapley, & A. W. Tucker (Eds.), Advances in game theory. Princeton: Princeton University Press.
Banzhaf, J. F. (1965). Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Review, 19, 317–343.
Bondareva, O. (1963). Certain applications of the methods of linear programming to the theory of cooperative games. Problemy Kibernetiki, 10, 119–139.
Chessa, M., & Fragnelli, V. (2013). Open problems in Veto theory. International Game Theory Review, 15. doi:10.1142/S0219198913400069.
Deegan, J., & Packel, E. W. (1978). A new index of power for simple n-person games. International Journal of Game Theory, 7, 113–123.
Einy, E., & Wettstein, D. (1996). Equivalence between bargaining sets and the core in simple games. International Journal of Game Theory, 25, 65–71.
Holler, M. J. (1982). Forming coalitions and measuring voting power. Political Studies, 30, 262–271.
Johnston, R. J. (1978). On the measurement of power: some reactions to laver. Environment & Planning A, 10, 907–914.
Maschler, M. (1966). The inequalities that determine the bargaining set \(\mathcal{M}^{1} (i)\). Israel Journal of Mathematics, 4, 127–134.
Maschler, M. (1992). The bargaining set, kernel, and nucleolus. In R.J. Aumann & S. Hart (Eds.), Handbook of game theory (Vol. 1). Amsterdam: Elsevier Science.
Mas-Colell, A. (1989). An equivalent theorem for a bargaining set. Journal of Mathematical Economics, 1, 129–138.
Montero, M. (2005). On the nucleolus as a power index. Homo Oeconomicus, 22, 551–567.
Osborne, M. J., & Rubinstein, A. (1994). A course in game theory. Cambridge: MIT Press.
Shapley, L. S. (1953). A value for n-person games. In H. W. Kuhn & A. W. Tucker (Eds.), Contributions to the theory of games II. Princeton: Princeton University Press.
Shapley, L. S. (1967). On balanced sets and cores. Naval Research Logistics Quarterly, 14, 453–460.
Shapley, L. S., & Shubik, M. (1954). A method for evaluating the distribution of power in a committee system. American Political Science Review, 48, 787–792.
Zhou, L. (1994). A new bargaining set of an n-person game and endogenous coalition formation. Games and Economic Behavior, 6, 512–526.
Acknowledgements
The authors gratefully acknowledge some useful discussions with Guido Ortona on blackmailing issues, with Hannu Nurmi on Bargaining Set and two anonymous reviewers for their remarks and suggestions that allowed to improve the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chessa, M., Fragnelli, V. The bargaining set for sharing the power. Ann Oper Res 215, 49–61 (2014). https://doi.org/10.1007/s10479-013-1411-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-013-1411-7