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# On the Construction of High Dimensional Simple Games

## Title data

Olsen, Martin ; Kurz, Sascha ; Molinero, Xavier:
On the Construction of High Dimensional Simple Games.
Bayreuth , 2016 . - 13 S.

## 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 $\chi\colon\{0,1\}^n\rightarrow \{0,1\}$. 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.

## Further data

Item Type: Preprint, postprint erschienen in: 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 simple games; weighted games; dimension; coding theory; Hamming distance Mathematics Subject Classification Code: 91B12 (91A12 68P30) 000 Computer Science, information, general works > 004 Computer science300 Social sciences > 320 Political science500 Science > 510 Mathematics Faculties > Faculty of Mathematics, Physics und Computer ScienceFaculties > Faculty of Mathematics, Physics und Computer Science > Department of MathematicsFaculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematical EconomicsProfile Fields > Emerging Fields > Governance and ResponsibilityFacultiesProfile FieldsProfile Fields > Emerging Fields English Yes 18 Jul 2016 07:30 28 May 2021 04:48 https://epub.uni-bayreuth.de/id/eprint/2935