International Journal of Game Theory

, Volume 43, Issue 3, pp 659–692 | Cite as

On \({\alpha }\)-roughly weighted games

  • Josep Freixas
  • Sascha Kurz


Gvozdeva et al. (Int J Game Theory, doi: 10.1007/s00182-011-0308-4, 2013) have introduced three hierarchies for simple games in order to measure the distance of a given simple game to the class of (roughly) weighted voting games. Their third class \({\mathcal {C}}_\alpha \) consists of all simple games permitting a weighted representation such that each winning coalition has a weight of at least \(1\) and each losing coalition a weight of at most \(\alpha \). For a given game the minimal possible value of \(\alpha \) is called its critical threshold value. We continue the work on the critical threshold value, initiated by Gvozdeva et al., and contribute some new results on the possible values for a given number of voters as well as some general bounds for restricted subclasses of games. A strong relation between this concept and the cost of stability, i.e. the minimum amount of external payment to ensure stability in a coalitional game, is uncovered.


Simple game Weighted game Complete simple game   Roughly weighted game Voting theory Hierarchy 

Mathematics Subject Classification

91B12 94C10 



We would like to thank Tatyana Gvozdeva, Stefan Napel and two anonymous referees for their thoughtful comments that helped very much to improve the presentation of the paper. Josep Freixas author acknowledges the support of the Barcelona Graduate School of Economics and of the Government of Catalonia.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Applied Mathematics III and Engineering School of ManresaUniversitat Polytècnica de CatalunyaBarcelonaSpain
  2. 2.Department of Mathematics, Physics, and Computer ScienceUniversity of BayreuthBayreuthGermany

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