NP-completeness of some problems concerning voting games

Prasad, Kislaya; Kelly, Jerry S.

doi:10.1007/bf01753703

Cited by 67 publications

(53 citation statements)

References 2 publications

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“…The notions of #P-many-one-hardness and #P-many-onecompleteness are defined analogously. It is known that, given a WVG G and a player i, computing the raw Banzhaf index is #P-parsimonious-complete (Prasad & Kelly, 1990), whereas computing the raw Shapley-Shubik index is not (Faliszewski & Hemaspaandra, 2009), although it is #P-manyone-complete. For other recent #P-completeness results, we refer to the work of Aziz, Brandt, and Brill (2013).…”

Section: Basic Notions From Computational Complexity Theorymentioning

confidence: 99%

“…By Equation (3) above, the question of whether the new player's probabilistic Banzhaf index is greater than the original player's probabilistic Banzhaf index is equivalent to the question of whether the annexed player has a positive value in the original game. This property can be decided in nondeterministic polynomial time and is NP-hard by a result due to Prasad and Kelly (1990). K Remark 4.10.…”

Section: Which In Turn Is the Case If And Only If The Original Instanmentioning

confidence: 99%

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False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time

Rey¹,

Rothe²

2014

jair

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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. Relatedly, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. For the problems of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley-Shubik and the normalized Banzhaf index, merely NP-hardness lower bounds are known, 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," a class considered to be by far a larger class than NP. For both power indices, we 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. Relatedly, we consider the beneficial annexation problem, asking whether a single player can increase her power by taking over other players' weights. It is known that annexation is never disadvantageous for the Shapley-Shubik index, and that beneficial annexation is NP-hard for the normalized Banzhaf index. We show that annexation is never disadvantageous for the probabilistic Banzhaf index either, and for both the Shapley-Shubik index and the probabilistic Banzhaf index we show that it is NP-complete to decide whether annexing another player is advantageous. Moreover, we propose a general framework for merging and splitting that can be applied to different classes and representations of games.

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Section: Basic Notions From Computational Complexity Theorymentioning

confidence: 99%

Section: Which In Turn Is the Case If And Only If The Original Instanmentioning

confidence: 99%

False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time

Rey¹,

Rothe²

2014

jair

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“…The problem of computing the Shapley value or the Banzhaf index for voting games is #P-complete [5,12]. In order to overcome this problem, we present new approximation methods to find these indices.…”

Section: Definitionmentioning

confidence: 99%

“…Like Shapley value, the Banzhaf index [2] is another way of measuring a player's power. However, a key drawback of both these power indices is that computing them for voting games 1 is, in general, #P-complete [5,12]. In other words, it is practically infeasible to try to compute the exact Shapley value or Banzhaf index.…”

Section: Introductionmentioning

confidence: 99%

An Approximation Method for Power Indices for Voting Games

Fatima

Wooldridge

Jennings

2010

Innovations in Agent-Based Complex Automated Negotiations

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Abstract. 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.

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“…As demonstrated e.g., in [9,15], a lot of important problems on simple games are known to be intractable in terms of complexity theory. In the recent years we successfully have combined relation algebra (cf.…”

Section: Introductionmentioning

confidence: 99%

Computations on Simple Games Using RelView

Berghammer

Rusinowska

Swart

2011

Computer Algebra in Scientific Computing

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Abstract. Simple games are a powerful tool to analyze decision-making and coalition formation in social and political life. In this paper we present relational models of simple games and develop relational algorithms for solving some game-theoretic basic problems. The algorithms immediately can be transformed into the language of the Computer Algebra system RelView and, therefore, the system can be used to solve the problems and to visualize the results of the computations.

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NP-completeness of some problems concerning voting games

Cited by 67 publications

References 2 publications

False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time

False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time

An Approximation Method for Power Indices for Voting Games

Computations on Simple Games Using RelView

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