Abstract
Voting is a commonly applied method for the aggregation of the preferences of multiple agents into a joint decision. If preferences are binary, i.e., “yes” and “no”, every voting system can be described by a (monotone) Boolean function : f0; 1gn ! f0; 1g. However, its naive encoding needs 2n bits. The subclass of threshold functions, which is sufficient for homogeneous agents, allows a more succinct representation using n weights and one threshold. For heterogeneous agents one can represent as an intersection of k threshold functions. Taylor and Zwicker have constructed a sequence of examples requiring k 2 n2 ?1 and provided a construction guaranteeingk ? n bn=2c 2 2n?o(n). The magnitude of the worst case situation was thought to be determined by Elkind et al. in 2008, but the analysis unfortunately turned out to be wrong. Here we uncover a relation to coding theory that allows the determination of the minimum number k for a subclass of voting systems. As an application, we give a construction for k 2n?o(n), i.e., there is no gain from a representation complexity point of view.
Highlights
Simple games [12] are models of voting systems without abstention where voters vote either ”yes” or ”no” when they are presented to a proposal
We show that there are no simple games with dimension n times higher than our games
Elkind et al [3] consider the same game as Taylor and Zwicker but they analyze the dimension from another point of view
Summary
Simple games [12] are models of voting systems without abstention where voters vote either ”yes” or ”no” when they are presented to a proposal. For a simple game we specify the coalitions of voters for which a proposal passes if the voters in the coalition are the ”yes”-voters. Some real world voting systems are constructed using several sets of weights with a weight assigned to every voter for each set. A coalition of voters can make a proposal pass if the sum of the weights of the voters in every set of weights meets or exceeds a quota defined for that set. Any voting system can be implemented in this way. No matter how it is defined [12] and the dimension is defined as the minimum number of weight sets needed to implement a voting system in this way. Unicameral systems have dimension 1, bicameral systems have dimension 2 and there are real world voting systems with dimension 3 or higher [2, 4, 8]
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