The final publication is available at Springer via http://dx.doi.org/10.1007/s00355-003-0212-3.

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 other subclasses of ( j,k) simple games and classify several examples. For example, we show the existence of a weighted representation for the UNSC, seen as a voting system in which abstention is permitted. Research partially supported by Grant 1999BEAI400096 of the Commissioner for Universities and Research of the Catalonia Generalitat and by Grant BFM 2000–0968 of the Spanish Ministry of Science and Technology. The authors would like to thank Larry Becker, Clifford Brown, Vin Moscardelli, and Fred Jonas for their assistance. Extensive remarks by Moshé Machover as well as comments by an anonymous referee greatly improved the manuscript.

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