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Academic literature on the topic 'Banzhaf index'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 April 2022

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Journal articles on the topic "Banzhaf index"

1

Staudacher, Jochen, Felix Wagner, and Jan Filipp. "Dynamic Programming for Computing Power Indices for Weighted Voting Games with Precoalitions." Games 13, no. 1 (2021): 6. http://dx.doi.org/10.3390/g13010006.

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We study the efficient computation of power indices for weighted voting games with precoalitions amongst subsets of players (reflecting, e.g., ideological proximity) using the paradigm of dynamic programming. Starting from the state-of-the-art algorithms for computing the Banzhaf and Shapley–Shubik indices for weighted voting games, we present a framework for fast algorithms for the three most common power indices with precoalitions, i.e., the Owen index, the Banzhaf–Owen index and the symmetric coalitional Banzhaf index, and point out why our new algorithms are applicable for large numbers of players. We discuss implementations of our algorithms for the three power indices with precoalitions in C++ and review computing times, as well as storage requirements.

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2

Dubey, Pradeep, Ezra Einy, and Ori Haimanko. "Compound voting and the Banzhaf index." Games and Economic Behavior 51, no. 1 (2005): 20–30. http://dx.doi.org/10.1016/j.geb.2004.03.002.

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3

Albizuri, M. Josune, and Jesus Aurrekoetxea. "Coalition Configurations and the Banzhaf Index." Social Choice and Welfare 26, no. 3 (2006): 571–96. http://dx.doi.org/10.1007/s00355-006-0102-6.

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4

Rey, A., and J. Rothe. "False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time." Journal of Artificial Intelligence Research 50 (July 22, 2014): 573–601. http://dx.doi.org/10.1613/jair.4293.

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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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5

Bilbao, J. M., A. Jiménez, and J. J. López. "The Banzhaf power index on convex geometries." Mathematical Social Sciences 36, no. 2 (1998): 157–73. http://dx.doi.org/10.1016/s0165-4896(98)00021-3.

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6

Amer, R., F. Carreras, and A. Maga[Nbar]a. "The banzhaf – coleman index for games withralternatives." Optimization 44, no. 2 (1998): 175–98. http://dx.doi.org/10.1080/02331939808844407.

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7

Liao, Yu-Hsien. "Consonance, Symmetry and Extended Outputs." Symmetry 13, no. 1 (2021): 72. http://dx.doi.org/10.3390/sym13010072.

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In many real-world situations, performers always adopt different energy levels (strategies) to participate. Different from pre-existing results, this paper is devoted to proposing several generalized power outputs of the marginal index, the Banzhaf–Coleman index, and the Banzhaf–Owen index, respectively, by assigning different energy levels to all performers. Since these extended power outputs may not be efficacious, we further define the efficacious extensions of these power outputs, respectively. For each of these efficacious power outputs, we demonstrate that there exists a corresponding reduced game and related consonance property that can be used to characterize it. By focusing on the properties of symmetry and accordance, several axiomatic results are also introduced.

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8

Bernardi, Giulia, and Josep Freixas. "An Axiomatization for Two Power Indices for (3,2)-Simple Games." International Game Theory Review 21, no. 01 (2019): 1940001. http://dx.doi.org/10.1142/s0219198919400012.

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The aim of this work is to give a characterization of the Shapley–Shubik and the Banzhaf power indices for (3,2)-simple games. We generalize to the set of (3,2)-simple games the classical axioms for power indices on simple games: transfer, anonymity, null player property and efficiency. However, these four axioms are not enough to uniquely characterize the Shapley–Shubik index for (3,2)-simple games. Thus, we introduce a new axiom to prove the uniqueness of the extension of the Shapley–Shubik power index in this context. Moreover, we provide an analogous characterization for the Banzhaf index for (3,2)-simple games, generalizing the four axioms for simple games and adding another property.

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9

Bilbao, J. M., J. R. Fernández, N. Jiménez, and J. J. López. "The Banzhaf power index for ternary bicooperative games." Discrete Applied Mathematics 158, no. 9 (2010): 967–80. http://dx.doi.org/10.1016/j.dam.2010.02.007.

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10

Albizuri, M. J. "An axiomatization of the modified Banzhaf Coleman index." International Journal of Game Theory 30, no. 2 (2001): 167–76. http://dx.doi.org/10.1007/s001820100071.

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