Abstract
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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Notes
- 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.
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This work was supported in part by Deutsche Forschungsgemeinschaft under grant RO 1202/21-1.
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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. https://doi.org/10.1007/978-3-030-87756-9_13
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