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Journal articles on the topic 'Banzhaf index'

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Published: 4 June 2021

Last updated: 25 April 2022

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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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11

Abbas, Jabbar. "The Banzhaf interaction index for bi-cooperative games." International Journal of General Systems 50, no. 5 (2021): 486–500. http://dx.doi.org/10.1080/03081079.2021.1924166.

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12

Aziz, H., Y. Bachrach, E. Elkind, and M. Paterson. "False-Name Manipulations in Weighted Voting Games." Journal of Artificial Intelligence Research 40 (January 20, 2011): 57–93. http://dx.doi.org/10.1613/jair.3166.

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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.In this paper, we investigate by how much a player can change his power, as measured by the Shapley-Shubik index or the Banzhaf index, by means of a false-name manipulation, i.e., splitting his weight among two or more identities. For both indices, we provide upper and lower bounds on the effect of weight-splitting. We then show that checking whether a beneficial split exists is NP-hard, and discuss efficient algorithms for restricted cases of this problem, as well as randomized algorithms for the general case. We also provide an experimental evaluation of these algorithms. Finally, we examine related forms of manipulative behavior, such as annexation, where a player subsumes other players, or merging, where several players unite into one. We characterize the computational complexity of such manipulations and provide limits on their effects. For the Banzhaf index, we describe a new paradox, which we term the Annexation Non-monotonicity Paradox.

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13

Haimanko, Ori. "Composition independence in compound games: a characterization of the Banzhaf power index and the Banzhaf value." International Journal of Game Theory 48, no. 3 (2019): 755–68. http://dx.doi.org/10.1007/s00182-019-00660-w.

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14

BARUA, RANA, SATYA R. CHAKRAVARTY, and SONALI ROY. "A NEW CHARACTERIZATION OF THE BANZHAF INDEX OF POWER." International Game Theory Review 07, no. 04 (2005): 545–53. http://dx.doi.org/10.1142/s0219198905000703.

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This paper develops a new axiomatic characterization of the Banzhaf index of power using four axioms from four different contributions to the area. A nice feature of the characterization is independence of the axioms showing importance of each of them in the exercise.

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15

Huang, Ruey-Rong, and Yu-Hsien Liao. "Axiomatic and Dynamic Results for Power Indexes under Symmetry." Axioms 10, no. 4 (2021): 345. http://dx.doi.org/10.3390/axioms10040345.

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Symmetry exists in a multitude of phenomena in varying forms. The main aim of this article is to analyze the plausibility of the equal allocation non-separable costs, the efficient Banzhaf–Owen index and the efficient Banzhaf–Coleman index from the perspective of symmetry. First, based on the difference between “participation processes” and “allocating results”, different forms of symmetry are proposed. Next, building on these forms of symmetry, axiomatic results are put forth for the three power indexes, whereby the plausibility of the three power indexes is analyzed. Finally, on the basis of these different forms of symmetry and related axiomatic results, this article introduces different dynamic processes to analyze how an initial allocation result approaches the results derived from the three power indexes through dynamically modification.

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16

Yakuba, V. "Evaluation of Banzhaf index with restrictions on coalitions formation." Mathematical and Computer Modelling 48, no. 9-10 (2008): 1602–10. http://dx.doi.org/10.1016/j.mcm.2008.05.017.

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17

Holzman, Ron, Ehud Lehrer, and Nathan Linial. "Some Bounds for the Banzhaf Index and Other Semivalues." Mathematics of Operations Research 13, no. 2 (1988): 358–63. http://dx.doi.org/10.1287/moor.13.2.358.

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18

Carreras, Francesc, and Antonio Magaña. "The multilinear extension and the modified Banzhaf-Coleman index." Mathematical Social Sciences 28, no. 3 (1994): 215–22. http://dx.doi.org/10.1016/0165-4896(94)90004-3.

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19

Leech, Dennis. "An Empirical Comparison of the Performance of Classical Power Indices." Political Studies 50, no. 1 (2002): 1–22. http://dx.doi.org/10.1111/1467-9248.00356.

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Power indices are general measures of the relative a priori voting power of individual members of a voting body. They are useful for both positive and normative analysis of voting bodies particularly those using weighted voting. This paper applies new algorithms for computing the rival Shapley-Shubik and Banzhaf indices for large voting bodies to shareholder voting power in a cross section of British companies. Each company is a separate voting body and there is much variation in ownership between them resulting in different power structures. Because the data are incomplete, both finite and ‘oceanic’ games of shareholder voting are analysed. The indices are appraised, using reasonable criteria, from the literature on corporate control. The results are unfavourable to the Shapley-Shubik index and suggest that the Banzhaf index much better reflects the variations in the power of shareholders between companies as the weights of shareholder blocs vary.

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20

Asif, Waqar, Hassaan Khaliq Qureshi, Muttukrishnan Rajarajan, and Marios Lestas. "Combined Banzhaf & Diversity Index (CBDI) for critical node detection." Journal of Network and Computer Applications 64 (April 2016): 76–88. http://dx.doi.org/10.1016/j.jnca.2015.11.025.

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21

Muto, Shigeo. "The Banzhaf Index in Representative Systems with Multiple Political Parties." Games and Economic Behavior 28, no. 1 (1999): 73–104. http://dx.doi.org/10.1006/game.1999.0692.

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22

Kurz, Sascha. "A note on limit results for the Penrose–Banzhaf index." Theory and Decision 88, no. 2 (2019): 191–203. http://dx.doi.org/10.1007/s11238-019-09726-3.

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23

Garrett, Geoffrey, and Iain Mclean. "On Power Indices and Reading Papers." British Journal of Political Science 26, no. 4 (1996): 600. http://dx.doi.org/10.1017/s000712340000764x.

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Peter Morriss's attack on R. J. Johnston's Note takes some side-swipes at us. We are allegedly ‘not quite up to par’, because (i) we are said to advocate the Banzhaf index; (ii) we are accused of ignoring a simpler route to our conclusion.

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24

Manuel, Conrado M., and Daniel Martín. "A Monotonic Weighted Banzhaf Value for Voting Games." Mathematics 9, no. 12 (2021): 1343. http://dx.doi.org/10.3390/math9121343.

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The aim of this paper is to extend the classical Banzhaf index of power to voting games in which players have weights representing different cooperation or bargaining abilities. The obtained value does not satisfy the classical total power property, which is justified by the imperfect cooperation. Nevertheless, it is monotonous in the weights. We also obtain three different characterizations of the value. Then we relate it to the Owen multilinear extension.

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25

MENG, Fanyong, Qiang ZHANG, and Jiaquan ZHAN. "THE INTERVAL-VALUED INTUITIONISTIC FUZZY GEOMETRIC CHOQUET AGGREGATION OPERATOR BASED ON THE GENERALIZED BANZHAF INDEX AND 2-ADDITIVE MEASURE." Technological and Economic Development of Economy 21, no. 2 (2015): 186–215. http://dx.doi.org/10.3846/20294913.2014.946983.

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Based on the operational laws on interval-valued intuitionistic fuzzy sets, the generalized Banzhaf interval-valued intuitionistic fuzzy geometric Choquet (GBIVIFGC) operator is proposed, which is also an interval-valued intuitionistic fuzzy value. It is worth pointing out that the GBIVIFGC operator can be seen as an extension of some geometric mean operators. Since the fuzzy measure is defined on the power set, it makes the problem exponentially complex. In order to overall reflect the interaction among elements and reduce the complexity of solving a fuzzy measure, we further introduce the GBIVIFGC operator w.r.t. 2-additive measures. Furthermore, if the information about weights of experts and attributes is incompletely known, the models of obtaining the optimal 2-additive measures on criteria set and expert set are given by using the introduced cross entropy measure and the Banzhaf index. Finally, an approach to pattern recognition and multi-criteria group decision making under interval-valued intuitionistic fuzzy environment is developed, respectively.

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26

Fatima, Shaheen, Michael Wooldridge, and Nicholas R. Jennings. "A heuristic approximation method for the Banzhaf index for voting games." Multiagent and Grid Systems 8, no. 3 (2012): 257–74. http://dx.doi.org/10.3233/mgs-2012-0194.

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27

Haimanko, Ori. "The axiom of equivalence to individual power and the Banzhaf index." Games and Economic Behavior 108 (March 2018): 391–400. http://dx.doi.org/10.1016/j.geb.2017.05.003.

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28

Hwang, Yan-An, and Yu-Hsien Liao. "Alternative formulation and dynamic process for the efficient Banzhaf–Owen index." Operations Research Letters 45, no. 1 (2017): 60–62. http://dx.doi.org/10.1016/j.orl.2016.12.004.

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29

Bernardi, Giulia. "A New Axiomatization of the Banzhaf Index for Games with Abstention." Group Decision and Negotiation 27, no. 1 (2017): 165–77. http://dx.doi.org/10.1007/s10726-017-9546-6.

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30

Tarkowski, Mateusz K., Piotr L. Szczepański, Tomasz P. Michalak, Paul Harrenstein, and Michael Wooldridge. "Efficient Computation of Semivalues for Game-Theoretic Network Centrality." Journal of Artificial Intelligence Research 63 (October 10, 2018): 145–89. http://dx.doi.org/10.1613/jair.1.11239.

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   Some game-theoretic solution concepts such as the Shapley value and the Banzhaf index have recently gained popularity as measures of node centrality in networks. While this direction of research is promising, the computational problems that surround it are challenging and have largely been left open. To date there are only a few positive results in the literature, which show that some game-theoretic extensions of degree-, closeness- and betweenness-centrality measures are computable in polynomial time, i.e., without the need to enumerate the exponential number of all possible coalitions. In this article, we show that these results can be extended to a much larger class of centrality measures that are based on a family of solution concepts known as semivalues. The family of semivalues includes, among others, the Shapley value and the Banzhaf index. To this end, we present a generic framework for defining game-theoretic network centralities and prove that all centrality measures that can be expressed in this framework are computable in polynomial time. Using our framework, we present a number of new and polynomial-time computable game-theoretic centrality measures.  

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31

Monroy, Luisa, and Francisco R. Fernández. "Banzhaf index for multiple voting systems. An application to the European Union." Annals of Operations Research 215, no. 1 (2013): 215–30. http://dx.doi.org/10.1007/s10479-013-1374-8.

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32

Levy, Marc. "The Banzhaf index in complete and incomplete shareholding structures: A new algorithm." European Journal of Operational Research 215, no. 2 (2011): 411–21. http://dx.doi.org/10.1016/j.ejor.2011.06.007.

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33

Barua, Rana, Satya R. Chakravarty, Sonali Roy, and Palash Sarkar. "A characterization and some properties of the Banzhaf–Coleman–Dubey–Shapley sensitivity index." Games and Economic Behavior 49, no. 1 (2004): 31–48. http://dx.doi.org/10.1016/j.geb.2003.12.003.

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34

Chen, Xiaoying, Amit Sinha, and Xin Chen. "Two proxies for shareholder influence: A case of payout policy." Corporate Ownership and Control 10, no. 1 (2012): 573–85. http://dx.doi.org/10.22495/cocv10i1c6art2.

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This paper investigates the relative importance of two proxies for shareholder influence over the board: the size of equity ownership and Banzhaf index voting power. Our empirical findings indicate that consideration of voting power in models may result in conclusions different from those that consider the size of equity ownership, on payout policy if firms are either paying dividends or repurchasing shares, or both. Comparing to the size of equity ownership, empirical results by the proxy of voting power are more consistent with theoretical predictions

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35

Fertő, Imre, László Á. Kóczy, Attila Kovács, and Balázs R. Sziklai. "The power ranking of the members of the Agricultural Committee of the European Parliament." European Review of Agricultural Economics 47, no. 5 (2020): 1897–919. http://dx.doi.org/10.1093/erae/jbaa011.

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Abstract We aim to identify the most influential members of the Agricultural Committee of the European Parliament (COMAGRI). Unlike previous studies that were based on case studies or interviews with stakeholders, we analyse the voting power of MEPs using a spatial Banzhaf power index. We identify critical members: members whose votes are necessary to form winning coalitions. We found that rapporteurs, EP group coordinators and MEPs from countries with high relative Committee representations, such as Ireland, Poland or Romania are powerful actors. Italy emerges as the most influential member state, while France seems surprisingly weak.

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36

Kovacic, Matija, and Claudio Zoli. "Ethnic distribution, effective power and conflict." Social Choice and Welfare 57, no. 2 (2021): 257–99. http://dx.doi.org/10.1007/s00355-021-01317-y.

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AbstractThis paper highlights the fact that different distributional aspects of ethnicity matter for conflict. We axiomatically derive a parametric class of indices of conflict potential obtained as the sum of each ethnic group’s relative power weighted by the probability of across group interactions. The power component of an extreme element of this class of indices is given by the Penrose–Banzhaf measure of relative power. This index combines in a non-linear way fractionalization, polarization and dominance. The empirical analysis verifies that it outperforms the existing indices of ethnic diversity in explaining ethnic conflict onset.

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37

Nurmi, Hannu. "The representation of voter groups in the European Parliament: a Penrose-Banzhaf index analysis." Electoral Studies 16, no. 3 (1997): 317–27. http://dx.doi.org/10.1016/s0261-3794(97)00027-9.

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38

Liao, Yu-Hsien. "The excess formulations and related results for the normalized Banzhaf index and the Shapley value." TOP 24, no. 1 (2015): 233–41. http://dx.doi.org/10.1007/s11750-015-0389-5.

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39

KURZ, SASCHA. "IMPORTANCE IN SYSTEMS WITH INTERVAL DECISIONS." Advances in Complex Systems 21, no. 06n07 (2018): 1850024. http://dx.doi.org/10.1142/s0219525918500248.

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Given a system where the real-valued states of the agents are aggregated by a function to a real-valued state of the entire system, we are interested in the influence or importance of different agents for that function. This generalizes the notion of power indices for binary voting systems to decisions over interval policy spaces and has applications in economics, engineering, security analysis, and other disciplines. Here, we study the question of importance in systems with interval decisions. Based on the classical Shapley–Shubik and Penrose–Banzhaf index, from binary voting, we motivate and analyze two importance measures. Additionally, we present some results for parametric classes of aggregation functions.

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40

Heard, Andrew, and Tim Swartz. "The Regional Veto Formula and Its Effects on Canada's Constitutional Amendment Process." Canadian Journal of Political Science 30, no. 2 (1997): 339–56. http://dx.doi.org/10.1017/s0008423900015468.

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AbstractIn early 1996, Canada's federal government enacted a new constitutional amending process to provide provincial and regional vetoes over future amendments. This study compares the new process with the “7 and 50” formula found in the Constitution Act, 1982. Using the Banzhaf Index as well as separate measures for the power to prevent and the power to initiate amendments, the article examines the relative influence of the provinces under the two amending formulae. As well, it examines the relative voting power of each province's citizens in any future constitutional referendum. The results show that profound changes are produced by the regional veto amending formula, and the article discusses some remedies for the most negative effects.

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41

Chen, Yinghong. "Valuation of voting scheme changes the cases of Electrolux AB and SKF AB." Corporate Ownership and Control 1, no. 4 (2004): 131–43. http://dx.doi.org/10.22495/cocv1i4p11.

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This paper studies the effects of the change of voting scheme on the market prices of Electrolux and SKF AB using standard event study methodology and a clinical approach. The economic effect of the voting scheme change is assessed using the market model. We investigate the loss of control due to the change of the voting scheme. The degree of the change of power is calculated using Shapley power index (SPI) and Banzhaf power index. There is a wealth transfer from the high vote shareholders to low vote shareholders in the process since in both cases the high power shareholders required no compensation. We expect that share price to have a positive response to such an announcement due to the reduced power discount and corporate governance improvement. The magnitude of the response on the event day depends also on the information structure of the period leading to the announcement.

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42

Diakonikolas, Ilias, and Chrystalla Pavlou. "On the Complexity of the Inverse Semivalue Problem for Weighted Voting Games." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 1869–76. http://dx.doi.org/10.1609/aaai.v33i01.33011869.

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Weighted voting games are a family of cooperative games, typically used to model voting situations where a number of agents (players) vote against or for a proposal. In such games, a proposal is accepted if an appropriately weighted sum of the votes exceeds a prespecified threshold. As the influence of a player over the voting outcome is not in general proportional to her assigned weight, various power indices have been proposed to measure each player’s influence. The inverse power index problem is the problem of designing a weighted voting game that achieves a set of target influences according to a predefined power index. In this work, we study the computational complexity of the inverse problem when the power index belongs to the class of semivalues. We prove that the inverse problem is computationally intractable for a broad family of semivalues, including all regular semivalues. As a special case of our general result, we establish computational hardness of the inverse problem for the Banzhaf indices and the Shapley values, arguably the most popular power indices.

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43

De Keijzer, B., T. B. Klos, and Y. Zhang. "Finding Optimal Solutions for Voting Game Design Problems." Journal of Artificial Intelligence Research 50 (May 22, 2014): 105–40. http://dx.doi.org/10.1613/jair.4109.

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In many circumstances where multiple agents need to make a joint decision, voting is used to aggregate the agents' preferences. Each agent's vote carries a weight, and if the sum of the weights of the agents in favor of some outcome is larger than or equal to a given quota, then this outcome is decided upon. The distribution of weights leads to a certain distribution of power. Several `power indices' have been proposed to measure such power. In the so-called inverse problem, we are given a target distribution of power, and are asked to come up with a game in the form of a quota, plus an assignment of weights to the players whose power distribution is as close as possible to the target distribution (according to some specied distance measure). Here we study solution approaches for the larger class of voting game design (VGD) problems, one of which is the inverse problem. In the general VGD problem, the goal is to find a voting game (with a given number of players) that optimizes some function over these games. In the inverse problem, for example, we look for a weighted voting game that minimizes the distance between the distribution of power among the players and a given target distribution of power (according to a given distance measure). Our goal is to find algorithms that solve voting game design problems exactly, and we approach this goal by enumerating all games in the class of games of interest. We first present a doubly exponential algorithm for enumerating the set of simple games. We then improve on this algorithm for the class of weighted voting games and obtain a quadratic exponential (i.e., 2^O(n^2)) algorithm for enumerating them. We show that this improved algorithm runs in output-polynomial time, making it the fastest possible enumeration algorithm up to a polynomial factor. Finally, we propose an exact anytime-algorithm that runs in exponential time for the power index weighted voting game design problem (the `inverse problem'). We implement this algorithm to find a weighted voting game with a normalized Banzhaf power distribution closest to a target power index, and perform experiments to obtain some insights about the set of weighted voting games. We remark that our algorithm is applicable to optimizing any exponential-time computable function, the distance of the normalized Banzhaf index to a target power index is merely taken as an example.

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44

Wu, Jian-Zhang, and Gleb Beliakov. "Comprehensive nonmodularity and interaction indices for decision analysis." Journal of Intelligent & Fuzzy Systems 40, no. 6 (2021): 10671–85. http://dx.doi.org/10.3233/jifs-201583.

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Nonmodularity is a prominent property of capacity that deeply links to the internal interaction phenomenon of multiple decision criteria. Following the common architectures of the simultaneous interaction indices as well as of the bipartition interaction indices, in this paper, we construct and study the notion of probabilistic nonmodularity index and also its particular cases, such as Shapely and Banzhaf nonmodularity indices, which can be used to describe the comprehensive interaction situations of decision criteria. The connections and differences among three categories of interaction indices are also investigated and compared theoretically and empirically. It is shown that three types of interaction indices have the same roots in their first and second orders, but meanwhile the nonmodularity indices have involved less amount of subsets and can be adopted to describe the interaction phenomenon in decision analysis.

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Gafni, Yotam, Ron Lavi, and Moshe Tennenholtz. "Worst-case Bounds on Power vs. Proportion in Weighted Voting Games with an Application to False-name Manipulation." Journal of Artificial Intelligence Research 72 (September 23, 2021): 99–135. http://dx.doi.org/10.1613/jair.1.13136.

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Weighted voting games apply to a wide variety of multi-agent settings. They enable the formalization of power indices which quantify the coalitional power of players. We take a novel approach to the study of the power of big vs. small players in these games. We model small (big) players as having single (multiple) votes. The aggregate relative power of big players is measured w.r.t. their votes proportion. For this ratio, we show small constant worst-case bounds for the Shapley-Shubik and the Deegan-Packel indices. In sharp contrast, this ratio is unbounded for the Banzhaf index. As an application, we define a false-name strategic normal form game where each big player may split its votes between false identities, and study its various properties. Together, our results provide foundations for the implications of players’ size, modeled as their ability to split, on their relative power.

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Shi, Ruili, Chunxiang Guo, and Xin Gu. "Power Indices in the Context of Social Learning Behaviour in Social Networks." Discrete Dynamics in Nature and Society 2019 (June 25, 2019): 1–13. http://dx.doi.org/10.1155/2019/4532042.

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This paper puts forward the concept of integrated power, synthetically measures the voters’ ability to influence the results of decision-making by influencing others through social learning, considering the interactions between decision-makers in social networks, and offers a method for measuring integrated power. Based on the theory and model of social learning, we analyze the influence of social learning on the voting process and power indices from the perspective of individuals’ professional level, position within the social network structure, relationship closeness, and learning efficiency. A measurement model of integrated power is constructed, and the variation in integrated power compared with that of the Banzhaf index is analyzed by numerical simulation. The results show that when the individual’s professional level is higher and closeness with neighboring decision-makers is greater, then the integrated power index is higher. An individual’s integrated power index may decrease when he/she changes from an isolated node to a nonisolated node, and then his/her integrated power will increase with the increases of neighbor nodes. Social learning efficiency can promote the integrated power of individuals with lower social impact and relationship closeness, but it is not beneficial for the core and influential members of the social network.

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Wagner, Fabian, and Niklas Höhne. "Influence of national governments for or against the entry into force of the Kyoto Protocol: a Banzhaf index analysis." Climate Policy 1, no. 4 (2001): 517–20. http://dx.doi.org/10.3763/cpol.2001.0151.

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Wagner, F. "Influence of national governments for or against the entry into force of the Kyoto Protocol: a Banzhaf index analysis." Climate Policy 1, no. 4 (2001): 517–20. http://dx.doi.org/10.1016/s1469-3062(01)00036-5.

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Göllner, Ralf Thomas. "The Visegrád Group – A Rising Star Post-Brexit? Changing Distribution of Power in the European Council." Open Political Science 1, no. 1 (2017): 1–6. http://dx.doi.org/10.1515/openps-2017-0001.

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Abstract The portmanteau Brexit describes the withdrawal of the United Kingdom (UK) from the European Union (EU) which will cause a shift of power in the European institutions. The departure of one of the largest EU members will affect the voting power of member countries in the European Council significantly. This Council is the central hub of political decision making in the EU, defining the overall political direction and priorities and setting the policy agenda for the entirety of the EU. Using the Banzhaf power index, we have measured the voting power before and after the Brexit and analyzed the increasing power of the members of the Visegrád Group compared to other European states. We have found that there is growth in the voting power of all Visegrád states, with Poland experiencing the biggest increase. However, the extent by which the Visegrád Group will profit from this statistically growing power depends on the coordination of their voting behavior in the future.

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Rapoport, Amnon, and Esther Golan. "Assessment of Political Power in the Israeli Knesset." American Political Science Review 79, no. 3 (1985): 673–92. http://dx.doi.org/10.2307/1956837.

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Immediately after the election to the tenth Israeli parliament (Knesset), 21 students of political science, 24 Knesset members, and seven parliamentary correspondents were each asked (a) to assess the political power ratios of the 10 parties represented in the Knesset and (b) to judge the ideological similarity between them. As ascertained by Saaty's analytic hierarchy scaling technique, the power ratio judgments proved sufficiently consistent to justify the construction of individual ratio scales of perceived political power. The ideological proximities were adequately represented by two-dimensional ideological spaces. Analyses of the derived power measures showed that the higher the political sophistication of the subject, the higher the combined power attributed to the religious parties and the lower the combined power assigned to the two largest parties Likud and Labor. The derived power measures were then compared to the predictions of six power indices, three of which only consider the ideological space. Of the six models, the generalized Banzhaf power index best accounted for the perceived power of 62% of the subjects, whereas the classical Shapley-Shubik index provided the best fit for 31% of the subjects. The generalized power indices were found only partly satisfactory with a need for further revision.

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