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On the structure of minimal winning coalitions in simple voting Maria Axenovich

Summary: On the structure of minimal winning coalitions in simple voting
Maria Axenovich
and Sonali Roy
April 7, 2009
According to Coleman's index of collective power, a decision rule that generates a larger
number of winning coalitions imparts the collectivity a higher a-priori power to act. By
the virtue of the monotonicity conditions, a decision rule is totally characterized by the set
of minimal winning coalitions. In this paper we investigate the structure of the families
of minimal winning coalitions corresponding to maximal and proper simple voting games
(SVG). We show that if the proper and maximal SVG is swap robust and all the minimal
winning coalitions are of the same size, then the SVG is a specific (up to an isomorphism)
system. We also provide examples of proper SVGs to show that the number of winning
coalitions is not monotone with respect to the intuitively appealing system parameters like
the number of blockers, number of non-dummies or the size of the minimal blocking set.
Keywords: simple voting games, proper games, swap robust, collective power
1 Introduction
In most decision making bodies, or equivalently collectivities, such as the United Nations Security
Council and the Council of Ministers in the European Union, the decision whether to accept or


Source: Axenovich, Maria - Department of Mathematics, Iowa State University


Collections: Mathematics