Volume 45, Number 4, October-December 2011
|Page(s)||295 - 314|
|Published online||19 January 2012|
On the complexity of problems on simple games
Universitat Politècnica de Catalunya, EPSEM-DMA3,
2 Universitat Politècnica de Catalunya, EPSEM-DMA3, 08240 Manresa, Spain
3 Center for Innovation and Business Development, Institute of Business and Technology, Aarhus University, Birk Centerpark 15, 7400 Herning, Denmark
4 Universitat Politècnica de Catalunya, LSI, 08034 Barcelona, Spain
Accepted: 1 September 2011
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.
Mathematics Subject Classification: 68Q / 91A
Key words: Simple / weighted / majority games / NP-completeness
© EDP Sciences, ROADEF, SMAI, 2012
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