Abstract
This paper presents some new results about majority games. Isbell (1959) was the first to find a majority game without a minimum normalized integer representation; he needed 12 voters to construct such a game. Since then, it has been an open problem to find the minimum number of voters of a majority game without a minimum normalized integer representation. Our main new results are:
1. All majority games with less than 9 voters have a minimum integer representation.
2. For 9 voters, there are 14 majority games without a minimum integer representation, but all these games admit a minimum normalized integer representation.
3. For 10 voters, there exist majority games with neither a minimum integer representation nor a minimum normalized integer representation.
This research was partially supported by Grant MTM 2006–06064 of “Ministerio de Ciencia y Tecnología y el Fondo Europeo de Desarrollo Regional” and SGRC 2005-00651 of “Generalitat de Catalunya”, and by the Spanish “Ministerio de Ciencia y Tecnología” programme TIN2005-05446 (ALINEX).
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Carreras, F., Freixas, J.: Complete simple games. Mathematical Social Sciences 32, 139–155 (1996)
Dubey, P., Shapley, L.S.: Mathematical properties of the Banzhaf power index. Mathematics of Operations Research 4(2), 99–131 (1979)
Freixas, J.: Structure of simple games. PhD thesis, Technical University of Catalonia, Manresa (Barcelona), Spain, (October 1994) (In Spanish)
Freixas, J.: The dimension for the European Union Council under the Nice rules. European Journal of Operational Research 156(2), 415–419 (2004)
Freixas, J., Molinero, X.: On the existence of a minimum integer representation for weighted voting systems. Annals of Operations Research, (submitted, October 2006)
Freixas, J., Zwicker, W.S.: Weighted voting, abstention, and multiple levels of approval. Social Choice and Welfare 21, 399–431 (2003)
Isbell, J.R.: A class of simple games. Duke. Mathematics Journal 25, 423–439 (1958)
Isbell, J.R.: On the enumeration of majority games. Mathematical Tables and Other Aids. to Computation 13(65), 21–28 (1959)
Maschler, M., Peleg, B.: A characterization, existence proof, and dimension bounds for the kernel of a game. Pacific Journal of Mathematics 18, 289–328 (1966)
Muroga, S.: Threshold logic and its applications. Wiley-Interscience, New York, USA (1971)
Muroga, S., Toda, I., Kondo, M.: Majority decision functions of up to six variables. Mathematics of Computation 16(80), 459–472 (1962)
Muroga, S., Toda, I., Takasu, S.: Theory of majority decision elements. J. Franklin Inst. 271(5), 376–418 (1961)
Muroga, S., Tsuboi, T., Baugh, C.R.: Enumeration of threshold funcitons of eight variables. IEEE Trans. Computers 19(9), 818–825 (1970)
Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton, New Jersey, USA (1944)
Taylor, A.D.: Mathematics and Politics. Springer, New York, USA (1995)
Taylor, A.D., Zwicker, W.S.: Simple games: desirability relations, trading, and pseudoweightings. Princeton University Press, New Jersey, USA (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Freixas, J., Molinero, X., Roura, S. (2007). Minimal Representations for Majority Games. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds) Computation and Logic in the Real World. CiE 2007. Lecture Notes in Computer Science, vol 4497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73001-9_31
Download citation
DOI: https://doi.org/10.1007/978-3-540-73001-9_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73000-2
Online ISBN: 978-3-540-73001-9
eBook Packages: Computer ScienceComputer Science (R0)