On the existence of a minimum integer representation for weighted voting systems
A basic problem in the theory of simple games and other fields is to study whether a simple game (Boolean function) is weighted (linearly separable). A second related problem consists in studying whether a weighted game has a minimum integer realization. In this paper we simultaneously analyze both problems by using linear programming.
For less than 9 voters, we find that there are 154 weighted games without minimum integer realization, but all of them have minimum normalized realization. Isbell in 1958 was the first to find a weighted game without a minimum normalized realization, he needed to consider 12 voters to construct a game with such a property. The main result of this work proves the existence of weighted games with this property with less than 12 voters.
KeywordsSimple games Weighted voting games Minimal realizations Minimum realization Realizations with minimum sum
- Brams, S. J., & Fishburn, P. C. (1976). Approval voting. American Political Science Review, 72, 831–847. Google Scholar
- Boros, E., Hammer, P. L., Ibaraki, T., & Kawakawi, K. (1991). Identifying 2-monotonic positive boolean functions in polynomial time. In W. L. Hsu & R. C. T. Lee (Eds.), LNCS: ISA’91 Algorithms (Vol. 557, pp. 104–115). Berlin: Springer. Google Scholar
- Chavátal, V. (1983). Linear programming. New York: Freeman. Google Scholar
- Fishburn, P. C. (1973). The theory of social choice. Princeton: Princeton University Press. Google Scholar
- Freixas, J., & Molinero, X. (2008, accepted). Simple games and weighted games: a theoretical and computational viewpoint. Discrete Applied Mathematics. Google Scholar
- Glpk (gnu linear programming kit) package. (2005). URL:http://www.gnu.org/software/glpk/.
- Hu, S. T. (1965). Threshold logic. USA: Univ. of California Press. Google Scholar
- Maple. (2005). URL: http://www.maplesoft.com/.
- Matlab. (2005). URL: http://www.mathworks.com/products/matlab/.
- Maschler, M., & Peleg, B. (1966). A characterization, existence proof, and dimension bounds for the kernel of a game. Pacific Journal of Mathematics, 18, 289–328. Google Scholar
- Muroga, S. (1971). Threshold logic and its applications. New York: Wiley. Google Scholar
- Von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior. Princeton: Princeton University Press. Google Scholar
- Ostmann, A. (1985). Decisions by players of comparable strength. Journal of Economics, 45, 267–284. Google Scholar
- Shapiro, J. F. (1979). Mathematical programming: structures and algorithms. New York: Wiley. Google Scholar
- Taylor, A. D. (1995). Mathematics and politics. New York: Springer. Google Scholar
- Taylor, A. D., & Zwicker, W. S. (1999). Simple games: desirability relations, trading, and pseudoweightings. New Jersey: Princeton University Press. Google Scholar