False-Name Manipulations in Weighted Voting Games
Yoram Bachrach2011, Journal of Artificial Intelligence Research
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Abstract
Weighted voting is a classic model of cooperation among agents in decision-making domains. In such games, each player has a weight, and a coalition of players wins the game if its total weight meets or exceeds a given quota. A player's power in such games is usually not directly proportional to his weight, and is measured by a power index, the most prominent among which are the Shapley-Shubik index and the Banzhaf index.
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False-Name Manipulation in Weighted Voting Games Is Hard for Probabilistic Polynomial Time
Lecture Notes in Computer Science, 2014
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. [ABEP11] 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 [RR10] 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 number of false identities is fixed, 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.
False‐Name Manipulation in Weighted Voting Games: Empirical and Theoretical Analysis
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Weighted voting games are important in multiagent systems because of their usage in automated decision making. However, they are not immune from the vulnerability of false-name manipulation by strategic agents that may be present in the games. False-name manipulation involves an agent splitting its weight among several false identities in anticipation of power increase. Previous works have considered false-name manipulation using the well-known Shapley-Shubik and Banzhaf power indices. Bounds on the extent of power that a manipulator may gain exist when it splits into k D 2 false identities for both the Shapley-Shubik and Banzhaf indices. The bounds when an agent splits into k > 2 false identities, until now, have remained open for the two indices. This article answers this open problem by providing four nontrivial bounds when an agent splits into k > 2 false identities for the two indices. Furthermore, we propose a new bound on the extent of power that a manipulator may gain when it splits into several false identities in a class of games referred to as excess unanimity weighted voting games. Finally, we complement our theoretical results with empirical evaluation. Results from our experiments confirm the existence of beneficial splits into several false identities for the two indices, and also establish that splitting into more than two false identities is qualitatively different than the previously known splitting into exactly two false identities.
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Artificial Intelligence, 2012
Weighted voting games provide a popular model of decision making in multiagent systems. Such games are described by a set of players, a list of players' weights, and a quota; a coalition of the players is said to be winning if the total weight of its members meets or exceeds the quota. The power of a player in such games is traditionally identified with her Shapley-Shubik index or her Banzhaf index, two classical power measures that reflect the player's marginal contributions under different coalition formation scenarios. In this paper, we investigate by how much the central authority can change a player's power, as measured by these indices, by modifying the quota. We provide tight upper and lower bounds on the changes in the individual player's power that can result from a change in quota. We also study how the choice of quota can affect the relative power of the players. From the algorithmic perspective, we provide an efficient algorithm for determining whether there is a value of the quota that makes a given player a dummy, i.e., reduces his power (as measured by both indices) to 0. On the other hand, we show that checking which of the two values of the quota makes this player more powerful is computationally hard, namely, complete for the complexity class PP, which is believed to be significantly more powerful than NP.
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Proceedings of the 23rd National Conference on Artificial Intelligence Volume 1, 2008
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A Search-Based Approach to Annexation and Merging in Weighted Voting Games
Proceedings of the 4th International Conference on Agents and Artificial Intelligence, 2012
Weighted voting games are classic cooperative games which provide a compact representation for coalition formation models in multiagent systems. We consider manipulation in weighted voting games via annexation and merging, which involves an agent or some agents misrepresenting their identities in anticipation of gaining more power at the expense of other agents in a game. We show that annexation and merging in weighted voting games can be more serious than as presented in the previous work. Specifically, using similar assumptions as employed in a previous work, we show that manipulators need to do only a polynomial amount of work to find a much improved power gain, and then present two search-based pseudo-polynomial algorithms that manipulators can use. We empirically evaluate our search-based method for annexation and merging. Our method is shown to achieve significant improvement in benefits for manipulating agents in several numerical experiments. While our search-based method achieves improvement in benefits of over 300% more than those of the previous work in annexation, the improvement in benefits is 28% to 45% more than those of the previous work in merging for all the weighted voting games we considered.
Complexity of Unweighted Coalitional Manipulation under Some Common Voting Rules
International Joint Conference on Artificial Intelligence, 2009
In this paper, we study the computational complexity of the unweighted coalitional manipu- lation (UCM) problem under some common voting rules. We show that the UCM problem under maximin is NP-complete. We also show that the UCM problem under ranked pairs is NP-complete, even if there is only one manipulator. Finally, we present a polynomial-time algorithm for the UCM problem
An approximation method for power indices for voting games
2011
The Shapley value and Banzhaf index are two well known indices for measuring the power a player has in a voting game. However, the problem of computing these indices is computationally hard. To overcome this problem, we analyze approximation methods for computing these indices. Although these methods have polynomial time complexity, finding an approximate Shapley value using them is easier than finding an approximate Banzhaf index. We also find the absolute error for the methods and show that this error for the Shapley value is lower than that for the Banzhaf index.
Coalitional Voting Manipulation: A Game-Theoretic Perspective
2011
Computational social choice literature has successfully studied the complexity of manipulation in various voting systems. However, the existing models of coalitional manipulation view the manipulating coalition as an exogenous input, ignoring the question of the coalition formation process. While such analysis is useful as a first approximation, a richer framework is required to model voting manipulation in the real world more accurately, and, in particular, to explain how a manipulating coalition arises and chooses its action. In this paper, we apply tools from cooperative game theory to develop a model that considers the coalition formation process and determines which coalitions are likely to form and what actions they are likely to take. We explore the computational complexity of several standard coalitional game theory solution concepts in our setting, and study the relationship between our model and the classic coalitional manipulation problem as well as the now-standard bribery model.
Easy weighted majority games
Mathematical Social Sciences, 2000
In a weighted majority game each player has a positive integer weight and there is a positive integer quota. A coalition of players is winning (losing) if the sum of the weights of its members exceeds (does not exceed) the quota. A player is pivotal for a coalition if her omission changes it from a winning to a losing one. Most game theoretic measures of the power of a player involve the computation of the number of coalitions for which that player is pivotal. Prasad and Kelly [Prasad, K., Kelly, J.S., 1990. NP-completeness of some problems concerning voting games. International Journal of Game Theory 19, 1-9] prove that the problem of determining whether or not there exists a coalition for which a given player is pivotal is NP-complete. They also prove that counting the number of coalitions for which a given player is pivotal is [P-complete. In the present paper we exhibit classes of weighted majority games for which these problems are easy.
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