Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Simple Games versus Weighted Voting Games: Bounding the Critical
  Threshold Value   A simple game $(N,v)$ 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 $\alpha= \min_{p\geq
0}\max_{W\in {\cal W}, L\in {\cal L}} \frac{p(L)}{p(W)}<1$ are exactly the
weighted voting games. We show that $\alpha\leq \frac{1}{4}n$ 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 $a>0$.

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The categories of this paper are: math
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The categories of this paper are: solv-int
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The categories of this paper are: quant-ph
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The categories of this paper are: funct-an
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The categories of this paper are: hep-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
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The categories of this paper are: econ
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The categories of this paper are: math-ph