On the Construction of High-Dimensional Simple Games

Martin Olsen, Sascha Kurz, Xavier Molinero

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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 ?: {0, 1} n → {0,1}. 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 ≥ 2n/2-1 and provided a construction guaranteeing k ≤ ([nn/2]) ∈ 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 k for a subclass of voting systems. As an application, we give a construction for k > 2 n-o(n), i.e., there is no gain from a representation complexity point of view.

TitelFrontiers in Artificial Intelligence and Applications
Antal sider6
ForlagIOS Press
ISBN (Trykt)978-1-61499-671-2
ISBN (Elektronisk)978-1-61499-672-9
StatusUdgivet - 2016
NavnFrontiers in Artificial Intelligence and Applications