False-Name Manipulation in Weighted Voting Games Is Hard for Probabilistic Polynomial Time
False-name manipulation refers to the question of whether a player in a weighted voting game can increase her power by splitting into several players and distributing her weight among these false identities. Analogously to this splitting problem, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. Aziz et al.  analyze the problem of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley–Shubik and the normalized Banzhaf index, and so do Rey and Rothe  for the probabilistic Banzhaf index. All these results provide merely NP-hardness lower bounds for these problems, leaving the question about their exact complexity open. For the Shapley–Shubik and the probabilistic Banzhaf index, we raise these lower bounds to hardness for PP, “probabilistic polynomial time,” and provide matching upper bounds for beneficial merging and, whenever the new players’ weights are given, also for beneficial splitting, thus resolving previous conjectures in the affirmative. It follows from our results that beneficial merging and splitting for these two power indices cannot be solved in NP, unless the polynomial hierarchy collapses, which is considered highly unlikely.
KeywordsMultiagent System Markov Decision Process Power Index Simple Game Vote Game
Unable to display preview. Download preview PDF.
- 2.Aziz, H., Paterson, M.: False name manipulations in weighted voting games: Splitting, merging and annexation. In: Proceedings of the 8th International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 409–416. IFAAMAS (May 2009)Google Scholar
- 3.Bachrach, Y., Elkind, E.: Divide and conquer: False-name manipulations in weighted voting games. In: Proceedings of the 7th International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 975–982. IFAAMAS (May 2008)Google Scholar
- 5.Banzhaf III, J.: Weighted voting doesn’t work: A mathematical analysis. Rutgers Law Review 19, 317–343 (1965)Google Scholar
- 6.Chalkiadakis, G., Elkind, E., Wooldridge, M.: Computational Aspects of Cooperative Game Theory. Synthesis Lectures on Artificial Intelligence and Machine Learning. Morgan and Claypool Publishers (2011)Google Scholar
- 8.Elkind, E., Chalkiadakis, G., Jennings, N.: Coalition structures in weighted voting games. In: Proceedings of the 18th European Conference on Artificial Intelligence, pp. 393–397. IOS Press (July 2008)Google Scholar
- 14.Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press (1972)Google Scholar
- 17.Papadimitriou, C.: Computational Complexity, 2nd edn. Addison-Wesley (1995)Google Scholar
- 20.Rey, A., Rothe, J.: Complexity of merging and splitting for the probabilistic Banzhaf power index in weighted voting games. In: Proceedings of the 19th European Conference on Artificial Intelligence, pp. 1021–1022. IOS Press (August 2010)Google Scholar
- 21.Shapley, L.: A value for n-person games. In: Kuhn, H., Tucker, A. (eds.) Contributions to the Theory of Games. Annals of Mathematics Studies 40, vol. II. Princeton University Press (1953)Google Scholar
- 26.Zuckerman, M., Faliszewski, P., Bachrach, Y., Elkind, E.: Manipulating the quota in weighted voting games. In: Proceedings of the 23rd AAAI Conference on Artificial Intelligence, pp. 215–220. AAAI Press (July 2008)Google Scholar