Simple Games Versus Weighted Voting Games

  • Frits Hof
  • Walter Kern
  • Sascha Kurz
  • Daniël Paulusma
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11059)


A simple game (Nv) is given by a set N of n players and a partition of \(2^N\) into a set \(\mathcal {L}\) of losing coalitions L with value \(v(L)=0\) that is closed under taking subsets and a set \(\mathcal {W}\) of winning coalitions W with \(v(W)=1\). Simple games with \(\alpha = \min _{p\ge 0}\max _{W\in \mathcal{W},L\in \mathcal{L}} \frac{p(L)}{p(W)}<1\) are exactly the weighted voting games. Freixas and Kurz (IJGT, 2014) conjectured that \(\alpha \le \frac{1}{4}n\) for every simple game (Nv). We confirm this conjecture for two complementary cases, namely when all minimal winning coalitions have size 3 and when no minimal winning coalition has size 3. As a general bound we prove that \(\alpha \le \frac{2}{7}n\) for every simple game (Nv). 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 \(a>0\).



The second and fourth author thank Péter Biró and Hajo Broersma for fruitful discussions on the topic of the paper.


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Frits Hof
    • 1
  • Walter Kern
    • 1
  • Sascha Kurz
    • 2
  • Daniël Paulusma
    • 3
  1. 1.University of TwenteEnschedeThe Netherlands
  2. 2.University of BayreuthBayreuthGermany
  3. 3.Durham UniversityDurhamUK

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