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Controlling Weighted Voting Games by Deleting or Adding Players with or Without Changing the Quota

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Combinatorial Algorithms (IWOCA 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13270))

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Abstract

Weighted voting games are a well-studied class of succinct simple games that can be used to model collective decision-making in, e.g., legislative bodies such as parliaments and shareholder voting. Power indices [5, 10, 23, 28] are used to measure the influence of players in weighted voting games. In such games, it has been studied how a distinguished player’s power can be changed, e.g., by merging or splitting players (the latter is a.k.a. false-name manipulation) [2, 24], by changing the quota [31], or via structural control by adding or deleting players [25]. We continue the work on the structural control initiated by Rey and Rothe [25] by solving some of their open problems. In addition, we also modify their model to a more realistic setting in which the quota is indirectly changed during the addition or deletion of players (in a different sense than that of Zuckerman et al. [31] who manipulate the quota directly without changing players’ set), and we study the corresponding problems in terms of their computational complexity.

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Notes

  1. 1.

    Under the assumptions of Theorem 1, their incorrect lower bound of the Shapley-Shubik index [25] is \(-1+\frac{(n-m-1)!}{2(n-2)!}\). .

  2. 2.

    The result comes from the fact that computing the numerator of the Penrose-Banzhaf index is #\(\mathrm {P}\)-parsimonious-complete and computing the numerator of the Shapley-Shubik index is #\(\mathrm {P}\)-many-one-complete.

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Acknowledgments

We thank the reviewers for helpful comments. This work was supported in part by Deutsche Forschungsgemeinschaft under grant RO 1202/21-1.

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Authors and Affiliations

  1. Heinrich-Heine-Universität Düsseldorf, Düsseldorf, Germany

    Joanna Kaczmarek & Jörg Rothe

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  1. Joanna Kaczmarek
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  2. Jörg Rothe
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Correspondence to Joanna Kaczmarek .

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Editors and Affiliations

  1. Université Paris-Dauphine, Paris, France

    Cristina Bazgan

  2. Universität Trier, Trier, Germany

    Henning Fernau

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Kaczmarek, J., Rothe, J. (2022). Controlling Weighted Voting Games by Deleting or Adding Players with or Without Changing the Quota. In: Bazgan, C., Fernau, H. (eds) Combinatorial Algorithms. IWOCA 2022. Lecture Notes in Computer Science, vol 13270. Springer, Cham. https://doi.org/10.1007/978-3-031-06678-8_26

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  • DOI: https://doi.org/10.1007/978-3-031-06678-8_26

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