*URN to cite this document:*urn:nbn:de:bvb:703-epub-3882-4

## Title data

Hof, Frits ; Kern, Walter ; Kurz, Sascha ; Pashkovich, Kanstantsin ; Paulusma, Daniël:

**Simple games versus weighted voting games: Bounding the critical threshold value.**

Bayreuth
,
2018
. - 10 S.

PDF
critical_threshold_arxiv.pdf - Published Version Available under License Creative Commons BY 4.0: Attribution . Download (335kB) |

## Abstract

A simple game (N,v) is given by a set N of $n$ players and a partition of the set of subsets of N into a set of losing coalitions L with value v(L)=0 that is closed under taking subsets and a set W of winning coalitions with v(W)=1. Simple games with alpha= min _{p>=0} max_{W' in W, L' in L} p(L')/p(W')<1 are exactly the weighted voting games. We show that alpha<=n/4 for every simple game (N,v), confirming the conjecture of Freixas and Kurz (IJGT, 2014). For complete simple games, Freixas and Kurz conjectured that alpha=O(sqrt(n))$. We prove this conjecture up to a (ln n) factor. We also prove that for graphic simple games, that is, simple games in which every minimal winning coalition has size 2, computing alpha is NP-hard, but polynomial-time solvable if the underlying graph is bipartite. Moreover, we show that for every graphic simple game, deciding if alpha<a is polynomial-time solvable for every fixed a>0.

## Further data

Item Type: | Preprint, postprint |
---|---|

Keywords: | simple game; weighted voting game; graphic simple game; complete simple game |

Subject classification: | Mathematics Subject Classification Code: 91B12 94C10 |

DDC Subjects: | 000 Computer Science, information, general works > 004 Computer 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 |

URN: | urn:nbn:de:bvb:703-epub-3882-4 |

Date Deposited: | 23 Oct 2018 06:16 |

Last Modified: | 14 Mar 2019 14:26 |

URI: | https://epub.uni-bayreuth.de/id/eprint/3882 |