On the Construction of High Dimensional Simple Games

Title data

Kurz, Sascha ; Molinero, Xavier ; Olsen, Martin:
On the Construction of High Dimensional Simple Games.
2016
Event: 22. European Conference on Artificial Intelligence (ECAI) 2016 , 29.08-02.09.2016 , The Hague, Holland.
(Conference item: Conference , Speech )

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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.