Weighted voting, abstention, and multiple levels of approval

@article{Freixas2003WeightedVA,
  title={Weighted voting, abstention, and multiple levels of approval},
  author={Josep Freixas and William S. Zwicker},
  journal={Social Choice and Welfare},
  year={2003},
  volume={21},
  pages={399-431}
}
In this paper we introduce the class of simple games with several ordered levels of approval in the input and in the output – the ( j,k) simple games – and propose a definition for weighted games in this context. Abstention is treated as a level of input approval intermediate to votes of yes and no. Our main theorem provides a combinatorial characterization, in terms of what we call grade trade robustness, of weighted ( j,k) games within the class of all ( j,k) simple games. We also introduce… 

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