Title data
Olsen, Martin ; Kurz, Sascha ; Molinero, Xavier:
On the Construction of High Dimensional Simple Games.
Bayreuth
,
2016
.  13 S.
This is the latest version of this item.
PDF
high_dimension4_arxiv.pdf  Published Version Available under License Creative Commons BY 3.0: Attribution . Download (293kB) 
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 $k\le {n\choose {\lfloor n/2\rfloor}}\in 2^{no(n)}$. 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 $k\ge 2^{no(n)}$, 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 Mathematics in Economy Profile Fields > Emerging Fields > Governance and Responsibility Faculties Profile Fields Profile Fields > Emerging Fields 
Language:  English 
Originates at UBT:  Yes 
Date Deposited:  18 Jul 2016 07:30 
Last Modified:  20 Mar 2019 11:17 
URI:  https://epub.unibayreuth.de/id/eprint/2935 
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)
 On the Construction of High Dimensional Simple Games. (deposited 18 Jul 2016 07:30) [Currently Displayed]

On the Construction of High Dimensional Simple Games. (deposited 19 Apr 2016 06:35)