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On the use of binary decision diagrams for solving problems on simple games

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  • Berghammer, Rudolf
  • Bolus, Stefan
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    Abstract

    Simple games are yes/no cooperative games which arise in many practical applications. Recently, we have used reduced ordered binary decision diagrams and quasi-reduced ordered binary decision diagrams (abbreviated as Robdds and Qobdds, respectively) for the representation of simple games and for the computation of some power indices. In the present paper, we continue this work. We show how further important computational problems on simple games can be solved using Qobdds, viz. the identification of some key players, the computation of the desirability relation on individuals, the test whether a simple game is proper and strong, respectively, and the computation of Qobdd-representations for the sets of all minimal winning coalitions, all shift-minimal winning coalitions and all blocking coalitions, respectively. Applications of these solutions include the computation of recent power indices based on shift-minimal winning coalitions and the test for linear separability of a directed simple game.

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    Bibliographic Info

    Article provided by Elsevier in its journal European Journal of Operational Research.

    Volume (Year): 222 (2012)
    Issue (Month): 3 ()
    Pages: 529-541

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    Handle: RePEc:eee:ejores:v:222:y:2012:i:3:p:529-541

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    Web page: http://www.elsevier.com/locate/eor

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    Keywords: Binary decision diagrams; Simple games; Key players; Desirability relation; Blocking coalition; Shift-minimal winning coalition;

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    1. Berghammer, Rudolf & Rusinowska, Agnieszka & de Swart, Harrie, 2010. "Applying relation algebra and RelView to measures in a social network," European Journal of Operational Research, Elsevier, vol. 202(1), pages 182-195, April.
    2. Prasad, K & Kelly, J S, 1990. "NP-Completeness of Some Problems Concerning Voting Games," International Journal of Game Theory, Springer, vol. 19(1), pages 1-9.
    3. Berghammer, Rudolf & Rusinowska, Agnieszka & de Swart, Harrie, 2009. "An interdisciplinary approach to coalition formation," European Journal of Operational Research, Elsevier, vol. 195(2), pages 487-496, June.
    4. Berghammer, Rudolf & Rusinowska, Agnieszka & de Swart, Harrie, 2007. "Applying relational algebra and RelView to coalition formation," European Journal of Operational Research, Elsevier, vol. 178(2), pages 530-542, April.
    5. Chakravarty, Nilotpal & Goel, Anand Mohan & Sastry, Trilochan, 2000. "Easy weighted majority games," Mathematical Social Sciences, Elsevier, vol. 40(2), pages 227-235, September.
    6. Freixas, J., 2004. "The dimension for the European Union Council under the Nice rules," European Journal of Operational Research, Elsevier, vol. 156(2), pages 415-419, July.
    7. Bolus, Stefan, 2011. "Power indices of simple games and vector-weighted majority games by means of binary decision diagrams," European Journal of Operational Research, Elsevier, vol. 210(2), pages 258-272, April.
    8. Francesc Carreras, 2009. "Protectionism and blocking power indices," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 17(1), pages 70-84, July.
    9. Moshé Machover & Dan S. Felsenthal, 2001. "The Treaty of Nice and qualified majority voting," Social Choice and Welfare, Springer, vol. 18(3), pages 431-464.
    10. Klinz, Bettina & Woeginger, Gerhard J., 2005. "Faster algorithms for computing power indices in weighted voting games," Mathematical Social Sciences, Elsevier, vol. 49(1), pages 111-116, January.
    11. Carreras, Francesc & Freixas, Josep, 1996. "Complete simple games," Mathematical Social Sciences, Elsevier, vol. 32(2), pages 139-155, October.
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