On the construction of high dimensional simple games

Titelangaben

Kurz, Sascha ; Molinero, Xavier ; Olsen, Martin:
On the construction of high dimensional simple games.
In: Kaminka, Gal A. ; Fox, Maria ; Bouquet, Paolo ; Hüllermeier, Eyke ; Dignum, Virginia ; Dignum, Frank ; van Harmelen, Frank (Hrsg.): ECAI 2016 : 22nd European Conference on Artificial Intelligence ; proceedings. - New York : IOS Press , 2016 . - S. 880-885 . - (Frontiers in Artificial Intelligence and Applications ; 285 )
ISBN 978-1-61499-671-2

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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. 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 χ as an intersection of k threshold functions. Taylor and Zwicker have constructed a sequence of examples requiring k>=2^(n/2-1). 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 new construction closing the gap from a representation complexity point of view.