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Manipulation in Communication Structures of Graph-Restricted Weighted Voting Games

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Algorithmic Decision Theory (ADT 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 13023))

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Weighted voting games are an important class of compactly representable simple games that can be used to model collective decision-making processes. The influence of players in weighted voting games is measured by power indices such as the Shapley-Shubik and the Penrose-Banzhaf power indices. Previous work has studied how such power indices can be manipulated via actions such as merging or splitting players [1, 12], adding or deleting players [13], or tampering with the quota [21]. We study graph-restricted weighted voting games [8, 9, 18], a model in which weighted voting games are embedded into a communication structure (i.e., a graph). We investigate to what extent power indices in such games can be changed by adding or deleting edges in the underlying communication structure and we study the resulting problems in terms of their computational complexity.

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  1. 1.

    Specifically, construct from the given Partition instance \((a_{1},\dots ,a_{n})\) the game \(\mathcal {H}=(1,2a_{1},\dots ,2a_{n},1;\alpha +2)\) with \(n+2\) players, the distinguished player \(p=1\), and the communication structure \(H=(M,C)\), where as before all players but the \((n+2)\)nd player form a complete subgraph and the \((n+2)\)nd player is an isolated vertex. Set the addition limit to 1. Let \(C^{c}\) be the set of edges not in C, which can be added to H. We can show that \((\exists {e \in C^{c}}) [\beta ((\mathcal {H},H_{\cup \{e\}}), 1) - \beta ((\mathcal {H},H), 1) > 0]\) if and only if \(\xi > 0\), which gives the desired \(\mathrm {NP}\)-hardness of our control problem.


  1. Aziz, H., Bachrach, Y., Elkind, E., Paterson, M.: False-name manipulations in weighted voting games. Journal of Artificial Intelligence Research 40, 57–93 (2011)

    Article  MathSciNet  Google Scholar 

  2. Banzhaf, J., III.: Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev. 19, 317–343 (1965)

    Google Scholar 

  3. Chalkiadakis, G., Elkind, E., Wooldridge, M.: Computational aspects of cooperative game theory. Synth. Lect. Artif. Intell. Mach. Learn. 5, 1–168 (2011)

    MATH  Google Scholar 

  4. Chalkiadakis, G., Wooldridge, M.: Weighted voting games. In: Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A. (eds.) Handbook of Computational Social Choice, chap. 16, pp. 377–395. Cambridge University Press (2016)

    Google Scholar 

  5. Dubey, P., Shapley, L.: Mathematical properties of the Banzhaf power index. Mathematics of Operations Research 4(2), 99–131 (1979)

    Article  MathSciNet  Google Scholar 

  6. Elkind, E., Rothe, J.: Cooperative game theory. In: Rothe, J. (ed.) Economics and Computation. An Introduction to Algorithmic Game Theory, Computational Social Choice, and Fair Division, chap. 3, pp. 135-193. Springer, Cham (2015).

  7. Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)

    MATH  Google Scholar 

  8. Myerson, R.: Graphs and cooperation in games. Mathematics of Operations Research 2(3), 225–229 (1977)

    Article  MathSciNet  Google Scholar 

  9. Napel, S., Nohn, A., Alonso-Meijide, J.: Monotonicity of power in weighted voting games with restricted communication. Mathematical Social Sciences 64(3), 247–257 (2012)

    Article  MathSciNet  Google Scholar 

  10. Papadimitriou, C., Yannakakis, M.: The complexity of facets (and some facets of complexity). Journal of Computer and System Sciences 28(2), 244–259 (1984)

    Article  MathSciNet  Google Scholar 

  11. Penrose, L.: The elementary statistics of majority voting. Journal of the Royal Statistical Society 109(1), 53–57 (1946)

    Article  Google Scholar 

  12. Rey, A., Rothe, J.: False-name manipulation in weighted voting games is hard for probabilistic polynomial time. J. Artif. Intell. Res. 50, 573–601 (2014)

    Article  MathSciNet  Google Scholar 

  13. Rey, A., Rothe, J.: Structural control in weighted voting games. B.E. J. Theor. Econ. 18(2), 1–15 (2018)

    Google Scholar 

  14. Riege, T., Rothe, J.: Completeness in the boolean hierarchy: Exact-Four-Colorability, minimal graph uncolorability, and exact domatic number problems - a survey. Journal of Universal Computer Science 12(5), 551–578 (2006)

    MathSciNet  Google Scholar 

  15. Riege, T., Rothe, J.: Complexity of the exact domatic number problem and of the exact conveyor flow shop problem. Theor. Comput. Syst. 39(5), 635–668 (2006).

  16. Rothe, J.: Exact complexity of Exact-Four-Colorability. Information Processing Letters 87(1), 7–12 (2003)

    Article  MathSciNet  Google Scholar 

  17. Shapley, L., Shubik, M.: A method of evaluating the distribution of power in a committee system. The American Political Science Review 48(3), 787–792 (1954)

    Article  Google Scholar 

  18. Skibski, O., Michalak, T., Sakurai, Y., Yokoo, M.: A pseudo-polynomial algorithm for computing power indices in graph-restricted weighted voting games. In: Proceedings of the 24th International Joint Conference on Artificial Intelligence, pp. 631–637. AAAI Press/IJCAI, July 2015

    Google Scholar 

  19. Taylor, A., Zwicker, W.: Simple Games: Desirability Relations, Trading, Pseudoweightings. Princeton University Press, Princeton (1999)

    MATH  Google Scholar 

  20. Wagner, K.: More complicated questions about maxima and minima, and some closures of NP. Theoretical Computer Science 51(1–2), 53–80 (1987)

    Article  MathSciNet  Google Scholar 

  21. Zuckerman, M., Faliszewski, P., Bachrach, Y., Elkind, E.: Manipulating the quota in weighted voting games. Artificial Intelligence 180–181, 1–19 (2012)

    Article  MathSciNet  Google Scholar 

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This work was supported in part by Deutsche Forschungsgemeinschaft under grant RO 1202/21-1.

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Correspondence to Joanna Kaczmarek .

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Kaczmarek, J., Rothe, J. (2021). Manipulation in Communication Structures of Graph-Restricted Weighted Voting Games. In: Fotakis, D., Ríos Insua, D. (eds) Algorithmic Decision Theory. ADT 2021. Lecture Notes in Computer Science(), vol 13023. Springer, Cham.

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