Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-23T19:54:54.876Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

Abstract

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Carreras, F. and Freixas, J., Complete simple games. Math. Soc. Sci. 32 (1996) 139–155. Google Scholar
Deĭneko, V.G. and Woeginger, G.J., On the dimension of simple monotonic games. Eur. J. Oper. Res. 170 (2006) 315–318. Google Scholar
Deng, X. and Papadimitriou, C.H., On the complexity of cooperative solution concepts. Math. Oper. Res. 19 (1994) 257–266. Google Scholar
E. Elkind and D. Pasechnik, Computing the nucleolus of weighted voting games, in SODA ’09 : Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics. Philadelphia, PA, USA (2009) 327–335.
E. Elkind, L.A. Goldberg, P.W. Goldberg and M. Wooldridge, Computational complexity of weighted threshold games, in Proceedings of the Twenty-Second AAAI Conference on Artificial Intelligence. Vancouver, British Columbia, Canada (2007) 718–723.
E. Elkind, L.A. Goldberg, P.W. Goldberg and M. Wooldridge, On the dimensionality of voting games, in Proceedings of the Twenty-Second AAAI Conference on Artificial Intelligence. Hyatt Regency McCormick Place, Chicago (2008) 69–74.
Freixas, J. and Molinero, X., Simple games and weighted games : A theoretical and computational viewpoint. Discrete Appl. Math. 157 (2009) 1496–1508. Google Scholar
Freixas, J. and Zwicker, W.S., Weighted voting, abstention, and multiple levels of approval. Soc. Choice Welfare 21 (2003) 399–431. Google Scholar
M.R. Garey and D.S. Johnson, Computers and Intractability : A Guide to the Theory of NP-Completness, edited by W.H. Freeman. San Francisco, New York, USA (1979).
Harrison, G.W. and McDaniel, T., Voting games and computational complexity. Oxford Econ. Papers 60 2008 546–565. Google Scholar
Hegedüs, T. and Megiddo, N., On the geometric separability of Boolean functions. Discrete Appl. Math. 66 (1996) 205–218. Google Scholar
Karmarkar, N., A new polynomial-time algorithm for linear programming. Combinatorica 4 (1984) 373–395. Google Scholar
Khachiyan, L.G., A polynomial algorithm for linear programming. Dokl. Akad. Nauk. SSSR 244 (1979) 1093–1096; English translation Soviet Math. Dokl. 20 (1979) 191–194. Google Scholar
Matsui, Y. and Matsui, T., NP-completeness for calculating power indices of weighted majority games. Theor. Comput. Sci. 263 (2001) 305–310. Google Scholar
Mehta, D. and Raghavan, V., Decision tree approximations of Boolean functions. Theor. Comput. Sci. 270 (2002) 609–623. Google Scholar
G. Owen, Game Theory, 3th edition. Academic Press, San Diego, USA (1995).
C.H. Papadimitriou, Computational Complexity. Addison Wesley (1994).
Peled, U.N. and Simeone, B., Polynomial-time algorithms for regular set-covering and threshold synthesis. Discrete Appl. Math. 12 (1985) 57–69. Google Scholar
Prasad, K. and Kelly, J.S., NP-completeness of some problems concerning voting games. Int. J. Game Theory 19 (1990) 1–9. Google Scholar
J. Rosenmüller, An algorithm for the construction of homogeneous games, in Ökonomie und Mathematik, edited by O. Opitz and B. Rauhut. Springer-Verlag (1987) 63–74.
Taylor, A.D. and Zwicker, W.S., Simple games and magic squares. J. Comb. Theory, Ser. A 71 (1995) 67–68. Google Scholar
A.D. Taylor and W.S. Zwicker, Simple games : desirability relations, trading, and pseudoweightings. Princeton University Press, New Jersey, USA (1999).
J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behavior. Princeton University Press, Princeton, New Jersey, USA (1944).