IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v306y2023i2p927-940.html
   My bibliography  Save this article

On the generalized dimension and codimension of simple games

Author

Listed:
  • Molinero, Xavier
  • Riquelme, Fabián
  • Roura, Salvador
  • Serna, Maria

Abstract

Weighted voting games are simple games that can be represented by a collection of integer weights for each player so that a coalition wins if the sum of the player weights matches or exceeds a given quota. It is known that a simple game can be expressed as the intersection or the union of weighted voting games. The dimension (codimension) of a simple game is the minimum number of weighted voting games such that their intersection (union) is the given game. In this work, we analyze some subclasses of weighted voting games and their closure under intersection or union. We introduce generalized notions of dimension and codimension regarding some subclasses of weighted voting games. In particular, we show that not all simple games can be expressed as intersection (union) of pure weighted voting games (those in which dummy players are not allowed) and we provide a characterization of such simple games. Finally, we experimentally study the generalized dimension (codimension) for some subclasses defined by establishing restrictions on the representations of weighted voting games.

Suggested Citation

  • Molinero, Xavier & Riquelme, Fabián & Roura, Salvador & Serna, Maria, 2023. "On the generalized dimension and codimension of simple games," European Journal of Operational Research, Elsevier, vol. 306(2), pages 927-940.
    • Handle: RePEc:eee:ejores:v:306:y:2023:i:2:p:927-940
    DOI: 10.1016/j.ejor.2022.07.045
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0377221722006154
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.ejor.2022.07.045?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. Molinero, Xavier & Riquelme, Fabián & Serna, Maria, 2015. "Cooperation through social influence," European Journal of Operational Research, Elsevier, vol. 242(3), pages 960-974.
    4. Kurz, Sascha, 2021. "A note on the growth of the dimension in complete simple games," Mathematical Social Sciences, Elsevier, vol. 110(C), pages 14-18.
    5. Xavier Molinero & Maria Serna & Marc Taberner-Ortiz, 2021. "On Weights and Quotas for Weighted Majority Voting Games," Games, MDPI, vol. 12(4), pages 1-25, December.
    6. Cheung, Wai-Shun & Ng, Tuen-Wai, 2014. "A three-dimensional voting system in Hong Kong," European Journal of Operational Research, Elsevier, vol. 236(1), pages 292-297.
    7. Freixas, Josep & Puente, Maria Albina, 2008. "Dimension of complete simple games with minimum," European Journal of Operational Research, Elsevier, vol. 188(2), pages 555-568, July.
    8. O’Dwyer, Liam & Slinko, Arkadii, 2017. "Growth of dimension in complete simple games," Mathematical Social Sciences, Elsevier, vol. 90(C), pages 2-8.
    9. 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.
    10. Sascha Kurz, 2016. "The inverse problem for power distributions in committees," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 47(1), pages 65-88, June.
    11. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. 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.
    3. Cheung, Wai-Shun & Ng, Tuen-Wai, 2014. "A three-dimensional voting system in Hong Kong," European Journal of Operational Research, Elsevier, vol. 236(1), pages 292-297.
    4. O’Dwyer, Liam & Slinko, Arkadii, 2017. "Growth of dimension in complete simple games," Mathematical Social Sciences, Elsevier, vol. 90(C), pages 2-8.
    5. Frits Hof & Walter Kern & Sascha Kurz & Kanstantsin Pashkovich & Daniël Paulusma, 2020. "Simple games versus weighted voting games: bounding the critical threshold value," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 54(4), pages 609-621, April.
    6. Gusev, Vasily V., 2023. "Set-weighted games and their application to the cover problem," European Journal of Operational Research, Elsevier, vol. 305(1), pages 438-450.
    7. Josep Freixas & Sascha Kurz, 2014. "Enumeration of weighted games with minimum and an analysis of voting power for bipartite complete games with minimum," Annals of Operations Research, Springer, vol. 222(1), pages 317-339, November.
    8. Mahajan, Aseem & Pongou, Roland & Tondji, Jean-Baptiste, 2023. "Supermajority politics: Equilibrium range, policy diversity, utilitarian welfare, and political compromise," European Journal of Operational Research, Elsevier, vol. 307(2), pages 963-974.
    9. Xavier Molinero & Maria Serna & Marc Taberner-Ortiz, 2021. "On Weights and Quotas for Weighted Majority Voting Games," Games, MDPI, vol. 12(4), pages 1-25, December.
    10. Josep Freixas & Sascha Kurz, 2014. "On $${\alpha }$$ α -roughly weighted games," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(3), pages 659-692, August.
    11. Freixas, Josep & Puente, Maria Albina, 2008. "Dimension of complete simple games with minimum," European Journal of Operational Research, Elsevier, vol. 188(2), pages 555-568, July.
    12. Sascha Kurz & Stefan Napel, 2014. "Heuristic and exact solutions to the inverse power index problem for small voting bodies," Annals of Operations Research, Springer, vol. 215(1), pages 137-163, April.
    13. Guemmegne, Juliette T. & Pongou, Roland, 2014. "A policy-based rationalization of collective rules: Dimensionality, specialized houses, and decentralized authority," Journal of Mathematical Economics, Elsevier, vol. 52(C), pages 182-193.
    14. Molinero, Xavier & Riquelme, Fabián & Serna, Maria, 2015. "Cooperation through social influence," European Journal of Operational Research, Elsevier, vol. 242(3), pages 960-974.
    15. Sascha Kurz, 2020. "Are weighted games sufficiently good for binary voting?," Papers 2006.05330, arXiv.org, revised Jul 2021.
    16. Monisankar Bishnu & Sonali Roy, 2012. "Hierarchy of players in swap robust voting games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 38(1), pages 11-22, January.
    17. Maria Montero & Alex Possajennikov, 2021. "An Adaptive Model of Demand Adjustment in Weighted Majority Games," Games, MDPI, vol. 13(1), pages 1-17, December.
    18. Le Breton, Michel & Montero, Maria & Zaporozhets, Vera, 2012. "Voting power in the EU council of ministers and fair decision making in distributive politics," Mathematical Social Sciences, Elsevier, vol. 63(2), pages 159-173.
    19. repec:gig:joupla:v:3:y:2011:i:3:p:95-126 is not listed on IDEAS
    20. Tatiana Gvozdeva & Lane Hemaspaandra & Arkadii Slinko, 2013. "Three hierarchies of simple games parameterized by “resource” parameters," International Journal of Game Theory, Springer;Game Theory Society, vol. 42(1), pages 1-17, February.
    21. Rusinowska, Agnieszka & Taalaibekova, Akylai, 2019. "Opinion formation and targeting when persuaders have extreme and centrist opinions," Journal of Mathematical Economics, Elsevier, vol. 84(C), pages 9-27.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:ejores:v:306:y:2023:i:2:p:927-940. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/eor .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.