On the complexity of problems on simple games
RAIRO - Operations Research - Recherche Opérationnelle, Volume 45 (2011) no. 4, pp. 295-314.

Simple games cover voting systems in which a single alternative, such as a bill or an amendment, is pitted against the status quo. A simple game or a yes-no voting system is a set of rules that specifies exactly which collections of “yea” votes yield passage of the issue at hand. Each of these collections of “yea” voters forms a winning coalition. We are interested in performing a complexity analysis on problems defined on such families of games. This analysis as usual depends on the game representation used as input. We consider four natural explicit representations: winning, losing, minimal winning, and maximal losing. We first analyze the complexity of testing whether a game is simple and testing whether a game is weighted. We show that, for the four types of representations, both problems can be solved in polynomial time. Finally, we provide results on the complexity of testing whether a simple game or a weighted game is of a special type. We analyze strongness, properness, weightedness, homogeneousness, decisiveness and majorityness, which are desirable properties to be fulfilled for a simple game. Finally, we consider the possibility of representing a game in a more succinct and natural way and show that the corresponding recognition problem is hard.

DOI: 10.1051/ro/2011115
Classification: 68Q, 91A
Keywords: simple, weighted, majority games, NP-completeness
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     title = {On the complexity of problems on simple games},
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Freixas, Josep; Molinero, Xavier; Olsen, Martin; Serna, Maria. On the complexity of problems on simple games. RAIRO - Operations Research - Recherche Opérationnelle, Volume 45 (2011) no. 4, pp. 295-314. doi : 10.1051/ro/2011115. http://archive.numdam.org/articles/10.1051/ro/2011115/

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