IDEAS home Printed from https://ideas.repec.org/a/eee/matsoc/v76y2015icp53-63.html
   My bibliography  Save this article

The complexity of power indexes with graph restricted coalitions

Author

Listed:
  • Benati, Stefano
  • Rizzi, Romeo
  • Tovey, Craig

Abstract

Coalitions of weighted voting games can be restricted to be connected components of a graph. As a consequence, coalition formation, and therefore a player’s power, depends on the topology of the graph. We analyze the problems of computing the Banzhaf and the Shapley–Shubik power indexes for this class of voting games and prove that calculating them is #P-complete in the strong sense for general graphs. For trees, we provide pseudo-polynomial time algorithms and prove #P-completeness in the weak sense for both indexes.

Suggested Citation

  • Benati, Stefano & Rizzi, Romeo & Tovey, Craig, 2015. "The complexity of power indexes with graph restricted coalitions," Mathematical Social Sciences, Elsevier, vol. 76(C), pages 53-63.
    • Handle: RePEc:eee:matsoc:v:76:y:2015:i:c:p:53-63
    DOI: 10.1016/j.mathsocsci.2015.04.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S016548961500027X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.mathsocsci.2015.04.001?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. Borm, P.E.M. & Owen, G. & Tijs, S.H., 1992. "On the position value for communication situations," Other publications TiSEM 5a8473e4-1df7-42df-ad53-f, Tilburg University, School of Economics and Management.
    2. Dennis Leech, 2003. "Computing Power Indices for Large Voting Games," Management Science, INFORMS, vol. 49(6), pages 831-837, June.
    3. Roger B. Myerson, 1977. "Graphs and Cooperation in Games," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 225-229, August.
    4. 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.
    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. Prasad, K & Kelly, J S, 1990. "NP-Completeness of Some Problems Concerning Voting Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 19(1), pages 1-9.
    7. Shapley, L. S. & Shubik, Martin, 1954. "A Method for Evaluating the Distribution of Power in a Committee System," American Political Science Review, Cambridge University Press, vol. 48(3), pages 787-792, September.
    8. Bilbao, J. M. & Fernandez, J. R. & Jimenez, N. & Lopez, J. J., 2002. "Voting power in the European Union enlargement," European Journal of Operational Research, Elsevier, vol. 143(1), pages 181-196, November.
    9. J. Bilbao & J. Fernández & A. Losada & J. López, 2000. "Generating functions for computing power indices efficiently," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 8(2), pages 191-213, December.
    10. Francesco Passarelli & Jason Barr, 2007. "Preferences, the Agenda Setter, and the Distribution of Power in the EU," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 28(1), pages 41-60, January.
    11. Stefano Benati & Giuseppe Vittucci Marzetti, 2013. "Probabilistic spatial power indexes," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 40(2), pages 391-410, February.
    12. Xiaotie Deng & Christos H. Papadimitriou, 1994. "On the Complexity of Cooperative Solution Concepts," Mathematics of Operations Research, INFORMS, vol. 19(2), pages 257-266, May.
    13. J.R. Fernández & E. Algaba & J.M. Bilbao & A. Jiménez & N. Jiménez & J.J. López, 2002. "Generating Functions for Computing the Myerson Value," Annals of Operations Research, Springer, vol. 109(1), pages 143-158, January.
    14. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Martí Jané Ballarín, 2023. "The complexity of power indices in voting games with incompatible players," UB School of Economics Working Papers 2023/441, University of Barcelona School of Economics.
    2. Stefano Benati & Giuseppe Vittucci Marzetti, 2021. "Voting power on a graph connected political space with an application to decision-making in the Council of the European Union," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 57(4), pages 733-761, November.

    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. Stefano Benati & Giuseppe Vittucci Marzetti, 2021. "Voting power on a graph connected political space with an application to decision-making in the Council of the European Union," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 57(4), pages 733-761, November.
    2. Martí Jané Ballarín, 2023. "The complexity of power indices in voting games with incompatible players," UB School of Economics Working Papers 2023/441, University of Barcelona School of Economics.
    3. Yuto Ushioda & Masato Tanaka & Tomomi Matsui, 2022. "Monte Carlo Methods for the Shapley–Shubik Power Index," Games, MDPI, vol. 13(3), pages 1-14, June.
    4. de Keijzer, B. & Klos, T.B. & Zhang, Y., 2012. "Solving Weighted Voting Game Design Problems Optimally: Representations, Synthesis, and Enumeration," ERIM Report Series Research in Management ERS-2012-006-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    5. Michela Chessa, 2014. "A generating functions approach for computing the Public Good index efficiently," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(2), pages 658-673, July.
    6. Antônio Francisco Neto, 2019. "Generating Functions of Weighted Voting Games, MacMahon’s Partition Analysis, and Clifford Algebras," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 74-101, February.
    7. Antônio Francisco Neto & Carolina Rodrigues Fonseca, 2019. "An approach via generating functions to compute power indices of multiple weighted voting games with incompatible players," Annals of Operations Research, Springer, vol. 279(1), pages 221-249, August.
    8. Gianfranco Gambarelli & Angelo Uristani, 2009. "Multicameral voting cohesion games," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 17(4), pages 433-460, December.
    9. Alonso-Meijide, J.M. & Bilbao, J.M. & Casas-Méndez, B. & Fernández, J.R., 2009. "Weighted multiple majority games with unions: Generating functions and applications to the European Union," European Journal of Operational Research, Elsevier, vol. 198(2), pages 530-544, October.
    10. Tanaka, Masato & Matsui, Tomomi, 2022. "Pseudo polynomial size LP formulation for calculating the least core value of weighted voting games," Mathematical Social Sciences, Elsevier, vol. 115(C), pages 47-51.
    11. Alonso-Meijide, J.M. & Casas-Méndez, B. & Fiestras-Janeiro, M.G., 2015. "Computing Banzhaf–Coleman and Shapley–Shubik power indices with incompatible players," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 377-387.
    12. Konstantin Avrachenkov & Laura Cottatellucci & Lorenzo Maggi, 2014. "Confidence Intervals for the Shapley–Shubik Power Index in Markovian Games," Dynamic Games and Applications, Springer, vol. 4(1), pages 10-31, March.
    13. Alexander Mayer, 2018. "Luxembourg in the Early Days of the EEC: Null Player or Not?," Games, MDPI, vol. 9(2), pages 1-12, May.
    14. Sridhar Mandyam & Usha Sridhar, 2017. "DON and Shapley Value for Allocation among Cooperating Agents in a Network: Conditions for Equivalence," Studies in Microeconomics, , vol. 5(2), pages 143-161, December.
    15. 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.
    16. C. Manuel & D. Martín, 2021. "A value for communication situations with players having different bargaining abilities," Annals of Operations Research, Springer, vol. 301(1), pages 161-182, June.
    17. A. Saavedra-Nieves, 2023. "On stratified sampling for estimating coalitional values," Annals of Operations Research, Springer, vol. 320(1), pages 325-353, January.
    18. M. J. Albizuri & A. Goikoetxea, 2021. "The Owen–Shapley Spatial Power Index in Three-Dimensional Space," Group Decision and Negotiation, Springer, vol. 30(5), pages 1027-1055, October.
    19. M. Musegaas & P. E. M. Borm & M. Quant, 2018. "Three-valued simple games," Theory and Decision, Springer, vol. 85(2), pages 201-224, August.
    20. Tobias Hiller, 2021. "Hierarchy and the size of a firm," International Review of Economics, Springer;Happiness Economics and Interpersonal Relations (HEIRS), vol. 68(3), pages 389-404, September.

    More about this item

    Statistics

    Access and download statistics

    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:matsoc:v:76:y:2015:i:c:p:53-63. 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/inca/505565 .

    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.