On the Construction of High-Dimensional Simple Games

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 \chi\colon\{0,1\}^n\rightarrow \{0,1\}. However, its naive encoding needs 2^n 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 \chi as an intersection of k threshold functions. Taylor and Zwicker have constructed a sequence of examples requiring k\ge 2^{\frac{n}{2}-1} and provided a construction guaranteeing k\le {n\choose {\lfloor n/2\rfloor}}\in 2^{n-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 for $k$ for a subclass of voting systems. As an application, we give a construction for k\ge 2^{n-o(n)}, i.e., there is no gain from a representation complexity point of view.