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Forms of representation for simple games: Sizes, conversions and equivalences

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  • Molinero, Xavier
  • Riquelme, Fabián
  • Serna, Maria

Abstract

Simple games are cooperative games in which the benefit that a coalition may have is always binary, i.e., a coalition may either win or loose. This paper surveys different forms of representation of simple games, and those for some of their subfamilies like regular games and weighted games. We analyze the forms of representations that have been proposed in the literature based on different data structures for sets of sets. We provide bounds on the computational resources needed to transform a game from one form of representation to another one. This includes the study of the problem of enumerating the fundamental families of coalitions of a simple game. In particular we prove that several changes of representation that require exponential time can be solved with polynomial-delay and highlight some open problems.

Suggested Citation

  • Molinero, Xavier & Riquelme, Fabián & Serna, Maria, 2015. "Forms of representation for simple games: Sizes, conversions and equivalences," Mathematical Social Sciences, Elsevier, vol. 76(C), pages 87-102.
    • Handle: RePEc:eee:matsoc:v:76:y:2015:i:c:p:87-102
    DOI: 10.1016/j.mathsocsci.2015.04.008
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    References listed on IDEAS

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    1. 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.
    2. EINY, Ezra & LEHRER, Ehud, 1989. "Regular simple games," LIDAM Reprints CORE 850, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Einy, Ezra & Lehrer, Ehud, 1989. "Regular Simple Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 18(2), pages 195-207.
    4. Sudholter, Peter, 1996. "Star-shapedness of the kernel for homogeneous games," Mathematical Social Sciences, Elsevier, vol. 32(3), pages 179-214, December.
    5. Ehud Kalai & Eitan Zemel, 1982. "Totally Balanced Games and Games of Flow," Mathematics of Operations Research, INFORMS, vol. 7(3), pages 476-478, August.
    6. Michael O. Ball & J. Scott Provan, 1988. "Disjoint Products and Efficient Computation of Reliability," Operations Research, INFORMS, vol. 36(5), pages 703-715, October.
    7. Algaba, E. & Bilbao, J. M. & Fernandez Garcia, J. R. & Lopez, J. J., 2003. "Computing power indices in weighted multiple majority games," Mathematical Social Sciences, Elsevier, vol. 46(1), pages 63-80, August.
    8. Mark Voorneveld & Sofia Grahn, 2002. "Cost allocation in shortest path games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 56(2), pages 323-340, November.
    9. Sascha Kurz & Nikolas Tautenhahn, 2013. "On Dedekind’s problem for complete simple games," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(2), pages 411-437, May.
    10. Berghammer, Rudolf & Bolus, Stefan, 2012. "On the use of binary decision diagrams for solving problems on simple games," European Journal of Operational Research, Elsevier, vol. 222(3), pages 529-541.
    11. Peter C. Fishburn, 1969. "Preferences, Summation, and Social Welfare Functions," Management Science, INFORMS, vol. 16(3), pages 179-186, November.
    12. Markus Behle, 2008. "On threshold BDDs and the optimal variable ordering problem," Journal of Combinatorial Optimization, Springer, vol. 16(2), pages 107-118, August.
    13. Vito Fragnelli & Ignacio García-Jurado & Luciano Méndez-Naya, 2000. "On shortest path games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 52(2), pages 251-264, November.
    14. Carreras, Francesc & Freixas, Josep, 1996. "Complete simple games," Mathematical Social Sciences, Elsevier, vol. 32(2), pages 139-155, October.
    15. Josep Freixas & Dorota Marciniak, 2009. "A minimum dimensional class of simple games," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 17(2), pages 407-414, December.
    16. Deineko, Vladimir G. & Woeginger, Gerhard J., 2006. "On the dimension of simple monotonic games," European Journal of Operational Research, Elsevier, vol. 170(1), pages 315-318, April.
    17. Josep Freixas & Xavier Molinero, 2009. "On the existence of a minimum integer representation for weighted voting systems," Annals of Operations Research, Springer, vol. 166(1), pages 243-260, February.
    18. Molinero, Xavier & Riquelme, Fabián & Serna, Maria, 2015. "Cooperation through social influence," European Journal of Operational Research, Elsevier, vol. 242(3), pages 960-974.
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    2. Sylvain Béal & Eric Rémila & Philippe Solal, 2019. "Coalitional desirability and the equal division value," Theory and Decision, Springer, vol. 86(1), pages 95-106, February.

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