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On the complexity of problems on simple games

Published online by Cambridge University Press:  19 January 2012

Josep Freixas ,
Xavier Molinero ,
Martin Olsen  and
Maria Serna
Josep Freixas
Affiliation:
Universitat Politècnica de Catalunya, EPSEM-DMA3, 08240 Manresa, Spain. josep.freixas@upc.edu
Xavier Molinero
Affiliation:
Universitat Politècnica de Catalunya, EPSEM-DMA3, 08240 Manresa, Spain; xavier.molinero@upc.edu
Martin Olsen
Affiliation:
Center for Innovation and Business Development, Institute of Business and Technology, Aarhus University, Birk Centerpark 15, 7400 Herning, Denmark; martino@hih.au.dk
Maria Serna
Affiliation:
Universitat Politècnica de Catalunya, LSI, 08034 Barcelona, Spain; mjserna@lsi.upc.edu
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Abstract

Simple games cover voting systems in which a single alternative, such as a bill or anamendment, is pitted against the status quo. A simple game or a yes-no voting system is aset of rules that specifies exactly which collections of “yea” votes yield passage of theissue at hand. Each of these collections of “yea” voters forms a winning coalition. We areinterested in performing a complexity analysis on problems defined on such families ofgames. This analysis as usual depends on the game representation used as input. Weconsider four natural explicit representations: winning, losing, minimal winning, andmaximal losing. We first analyze the complexity of testing whether a game is simple andtesting 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 thecomplexity of testing whether a simple game or a weighted game is of a special type. Weanalyze strongness, properness, weightedness, homogeneousness, decisiveness andmajorityness, 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 andshow that the corresponding recognition problem is hard.

Keywords

Simpleweightedmajority gamesNP-completeness
Type
Research Article
Information
RAIRO - Operations Research , Volume 45 , Issue 4 , October 2011 , pp. 295 - 314
Copyright
© EDP Sciences, ROADEF, SMAI, 2012

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