SAGT 2018: Algorithmic Game Theory pp 69-81

# 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)

## Abstract

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 $$a>0$$.

## Keywords

Weighted Voting Game Complete Simple Games Coalition Freixas Matching Game
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Notes

### Acknowledgments

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

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© Springer Nature Switzerland AG 2018

## Authors and Affiliations

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