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On the Construction of High Dimensional Simple Games

13 Pages Posted: 4 Feb 2016 Last revised: 15 Jul 2016

Martin Olsen

University of Aarhus

Sascha Kurz

University of Bayreuth

Xavier Molinero

Polytechnic University of Catalonia (UPC)

Date Written: February 4, 2016

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.

Keywords: simple games, weighted games, dimension, coding theory, Hamming distance

JEL Classification: C71, D72

Suggested Citation

Olsen, Martin and Kurz, Sascha and Molinero, Xavier, On the Construction of High Dimensional Simple Games (February 4, 2016). Available at SSRN: https://ssrn.com/abstract=2727602 or http://dx.doi.org/10.2139/ssrn.2727602

Martin Olsen

University of Aarhus ( email )

Nordre Ringgade 1
DK-8000 Aarhus C, 8000
Denmark

Sascha Kurz (Contact Author)

University of Bayreuth ( email )

Universit├Ątsstr. 30
Lehrstuhl f├╝r Wirtschaftsmathematik
Bayreuth, Bavaria D-95440
Germany
+49 921 55 7353 (Phone)
+49 921 55 7352 (Fax)

HOME PAGE: http://www.wm.uni-bayreuth.de/index.php?id=sascha

Xavier Molinero

Polytechnic University of Catalonia (UPC) ( email )

C. Jordi Girona, 31
Barcelona, 08034
Spain

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