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Olsen, Martin ; Kurz, Sascha ; Molinero, Xavier:
On the Construction of High Dimensional Simple Games.
Bayreuth
,
2016
. - 13 S.
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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 $\chi\colon\{0,1\}^n\rightarrow \{0,1\}$. 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 |
|---|---|
| Keywords: | simple games; weighted games; dimension; coding theory; Hamming distance |
| Subject classification: | Mathematics Subject Classification Code: 91B12 (91A12 68P30) |
| DDC Subjects: | 000 Computer Science, information, general works > 004 Computer science 300 Social sciences > 320 Political science 500 Science > 510 Mathematics |
| Institutions of the University: | Faculties > Faculty of Mathematics, Physics und Computer Science Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematical Economics Profile Fields > Emerging Fields > Governance and Responsibility Faculties Profile Fields Profile Fields > Emerging Fields |
| Language: | English |
| Originates at UBT: | Yes |
| URN: | urn:nbn:de:bvb:703-epub-2801-1 |
| Date Deposited: | 19 Apr 2016 06:35 |
| Last Modified: | 19 Apr 2016 06:35 |
| URI: | https://epub.uni-bayreuth.de/id/eprint/2801 |
Available Versions of this Item
-
On the Construction of High Dimensional Simple Games. (deposited 04 Feb 2016 11:19)
- On the Construction of High Dimensional Simple Games. (deposited 19 Apr 2016 06:35) [Currently Displayed]
