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This study is theoretically anchored in the office-seeking approach and coalition theory of political science. It is based on the most widespread theories of portfolio distribution in executive and legislative bodies (proportionality, power index, voting weights), and it tests these theories using data related to coalition negotiations in the Czech Republic for the new 2021 government coalition, a case of a surplus majority government. The study investigates the distribution of ministerial positions in the government, the office of President (Speaker) of the Chamber of Deputies, and the chairs of permanent parliamentary committees. The analysis also explores the types of electoral coalitions formed; the SPOLU coalition was a superadditive coalition, while the PirSTAN coalition was only an additive coalition. Grounded in the theory of electoral games, the Shapley-Shubik and Banzhaf power indices and the theory of coalition formation are applied in order to analyse the possible minimal winning coalitions that could be formed following the 2021 elections to the Chamber of Deputies. [ABSTRACT FROM AUTHOR]

In this paper, we extend the Banzhaf–Coleman–Dubey–Shapley sensitivity index to the class of dichotomous voting games with several levels of approval in input, also known as (j, 2)-simple games. For previous works, on classical simple games ((2, 2)-simple games), a sensitivity index reflects the volatility or degree of suspense in the voting body. Using a set of independent axioms, we provide an axiomatic characterization of that extension on the class of (j, 2)-simple games. [ABSTRACT FROM AUTHOR]

In this paper, we propose an axiomatization of the totally critical raw Banzhaf index (TCRBI). In the literature, the TCRBI was introduced as a generalization of the Banzhaf power index on the class of yes–no voting games with several ordered levels of approval in input ((j, 2)-simple games). Its principal strength, compared to earlier generalizations of the Banzhaf power index, is the fact that it preserves the desirability relation regardless of the number of input levels of approval. Our main result is that, up to a multiplicative positive real number, the TCRBI is the only power index on the class of (j, 2)-simple games, satisfying the following set of axioms : Equal treatment, Positivity, Null Player, and Linear Transfer. [ABSTRACT FROM AUTHOR]

Copyright of Science Technology & Industry / Keji he Chanye is the property of Chinese Society of Technology Economics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The power of individual EU Member States has been changing over the past decades as a result of revisions to the voting systems and the enlargements of the European Union. The present article analyses the development of the voting power of individual Member States in the Council of the European Union before and after the withdrawal of the United Kingdom of Great Britain and Northern Ireland from the European Union. We use the calculation of the standardized Banzhaf power index to calculate the legislative power of the Member States. The calculations recorded in the table point to changes in the weights of national votes caused by Brexit. The article pays special attention to the Visegrad Group, which we define within the European Union as an informal group consisting of four Central European states - the Czech Republic, Hungary, Poland, and the Slovak Republic. The results indicate a significant growth of the voting power in Poland and more moderate growth in the other three Visegrad Group countries which contributes to the shift in the voting equilibrium within the ordinary legislative procedure of the European Union.

The power of individual EU Member States has been changing over the past decades as a result of revisions to the voting systems and the enlargements of the European Union. The present article analyses the development of the voting power of individual Member States in the Council of the European Union before and after the withdrawal of the United Kingdom of Great Britain and Northern Ireland from the European Union. We use the calculation of the standardized Banzhaf power index to calculate the legislative power of the Member States. The calculations recorded in the table point to changes in the weights of national votes caused by Brexit. The article pays special attention to the Visegrad Group, which we define within the European Union as an informal group consisting of four Central European states - the Czech Republic, Hungary, Poland, and the Slovak Republic. The results indicate a significant growth of the voting power in Poland and more moderate growth in the other three Visegrad Group countries which contributes to the shift in the voting equilibrium within the ordinary legislative procedure of the European Union. [ABSTRACT FROM AUTHOR]

We introduce the axiom of composition independence for power indices and value maps. In the context of compound (two-tier) voting, the axiom requires the power attributed to a voter to be independent of the second-tier voting games played in all constituencies other than that of the voter. We show that the Banzhaf power index is uniquely characterized by the combination of composition independence, four semivalue axioms (transfer, positivity, symmetry, and dummy), and a mild efficiency-related requirement. A similar characterization is obtained as a corollary for the Banzhaf value on the space of all finite games (with transfer replaced by additivity). [ABSTRACT FROM AUTHOR]

We introduce a new generating function based method to compute the Banzhaf, Deegan–Packel, Public Good (a.k.a. the Holler power index) and Shapley–Shubik power indices in the presence of incompatibility among players. More precisely, given a graph G = V , E with V the set of players and E the edge set, our extension involves multiple weighted voting games (MWVG's) and incompatible players, i.e., pairs of players belonging to E are not allowed to cooperate. The route to obtain the aforementioned generating functions comprises the use of a key lemma characterizing the set of minimal winning coalitions of the game with incompatibility due to Alonso-Meijide et al. (Appl Math Comput 252(1):377–387, 2015), a tool from combinatorial analysis, namely, the Omega calculus in partition analysis, and basic tools borrowed from commutative algebra involving the computation of certain quotients of polynomial rings module polynomial ideals. Using partition analysis, we obtain new generating functions to compute the Deegan–Packel and Public Good power indices with incompatibility leading to lower time complexity than previous results of Chessa (TOP 22(2):658–673, 2014) and some results of Alonso-Meijide et al. (Appl Math Comput 219(8):3395–3402, 2012). Using a conjunction of partition analysis and commutative algebra, we extend to MWVG's the generating function approach to compute the Banzhaf and Shapley–Shubik power indices in the presence of incompatibility. Finally, an example taken from the real-world, i.e., the European Union under the Lisbon Treaty, is used to illustrate the usefulness of the Omega package, a symbolic computational package that implements the Omega calculus in Mathematica, due to Andrews et al. (Eur J Comb 22(7):887–904, 2001) in the context of MWVG's by computing the PG power index of the associated voting game. [ABSTRACT FROM AUTHOR]

Software defect prediction (SDP) is a process of predicting whether software has defects through historical data information. The existing feature selection algorithm needs to take more attention on the internal relations of features, which is important for SDP. The grey relational analysis (GRA) is introduced to describe the internal relations from the geometric similarity. The classification ability of the feature combination and the individual feature is discussed. The Banzhaf in cooperative game theory is discussed to make good evaluation on each feature. An improved feature selection approach based on One Rule (OR) algorithm for SDP is presented. The experiments are conducted on 9 datasets which are regularly used in SDP, and the results show that the proposed approach performs better than the other comparable feature selection approaches in terms of classification performance. [ABSTRACT FROM AUTHOR]

MacMahon's Partition Analysis (MPA) is a combinatorial tool used in partition analysis to describe the solutions of a linear diophantine system. We show that MPA is useful in the context of weighted voting games. We introduce a new generalized generating function that gives, as special cases, extensions of the generating functions of the Banzhaf, Shapley-Shubik, Banzhaf-Owen, symmetric coalitional Banzhaf, and Owen power indices. Our extensions involve any coalition formation related to a linear diophantine system and multiple voting games. In addition, we show that a combination of ideas from MPA and Clifford algebras is useful in constructing generating functions for coalition configuration power indices. Finally, a brief account on how to design voting systems via MPA is advanced. More precisely, we obtain new generating functions that give, for fixed coalitions, all the distribution of weights of the players of the voting game such that a given player swings or not. [ABSTRACT FROM AUTHOR]

Random Forests are powerful classification and regression tools that are commonly applied in machine learning and image processing. In the majority of random classification forests algorithms, the Gini index and the information gain ratio are commonly used for node splitting. However, these two kinds of node-split methods may pay less attention to the intrinsic structure of the attribute variables and fail to find attributes with strong discriminate ability as a group yet weak as individuals. In this paper, we propose an innovative method for splitting the tree nodes based on the cooperative game theory, from which some attributes with good discriminate ability as a group can be learned. This new random forests algorithm is called Cooperative Profit Random Forests (CPRF). Experimental comparisons with several other existing random classification forests algorithms are carried out on several real-world data sets, including remote sensing images. The results show that CPRF outperforms other existing Random Forests algorithms in most cases. In particular, CPRF achieves promising results in ocean front recognition.

Abstract Random forests algorithms have been widely used in many classification and regression applications. However, the theory of random forests lags far behind their applications. In this paper, we propose a novel random forests classification algorithm based on cooperative game theory. The Banzhaf power index is employed to evaluate the power of each feature by traversing possible feature coalitions. Hence, we call the proposed algorithm Banzhaf random forests (BRFs). Unlike the previously used information gain ratio, which only measures the power of each feature for classification and pays less attention to the intrinsic structure of the feature variables, the Banzhaf power index can measure the importance of each feature by computing the dependency among the group of features. More importantly, we have proved the consistency of BRFs, which narrows the gap between the theory and applications of random forests. Extensive experiments on several UCI benchmark data sets and three real world applications show that BRFs perform significantly better than existing consistent random forests on classification accuracy, and better than or at least comparable with Breiman’s random forests, support vector machines (SVMs) and k-nearest neighbors (KNNs) classifiers. Highlights • A novel random classification forests algorithm, called Banzhaf random forests (BRFs), is proposed. • The Banzhaf power index is employed to evaluate the power of each feature by traversing possible feature coalitions. • The consistency of BRFs is proved. [ABSTRACT FROM AUTHOR]

We propose a novel ensemble pruning methodology using non-monotone Simple Coalitional Games, termed SCG-Pruning. Our main contribution is two-fold: (1) Evaluate the diversity contribution of a classifier based on Banzhaf power index. (2) Define the pruned ensemble as the minimal winning coalition made of the members that together exhibit moderate diversity. We also provide a new formulation of Banzhaf power index for the proposed game using weighted voting games. To demonstrate the validity and the effectiveness of the proposed methodology, we performed extensive statistical comparisons with several ensemble pruning techniques based on 58 UCI benchmark datasets. The results indicate that SCG-Pruning outperforms both the original ensemble and some major state-of-the-art selection approaches. [ABSTRACT FROM AUTHOR]

It is taken into consideration of the influence on voting power distribution of candidate countries’ European Union membership enlargement is one of the most important issue the for European Union. In this study, the impact of Turkey’s membership on voting power distribution is evaluated for coalition formation determined with hierarchical clustering methods, according to the acts adopted by the Council of the European Union and voting system brought by the Treaty of Lisbon. In the analysis, the effect of Turkey’s membership on voting power distribution is evaluated for four coalition formation determined with hierarchical clustering methods and Banzhaf power index, one of the most frequently used voting power measurement, is used for voting power analysis. Based on the results of the analysis, voting power distribution is affected by Turkey’s European Union membership for four coalition formation. When the Council does not act on a proposal from the Commission or the High Representative of the Union for Foreign Affairs and Security Policy, voting power distribution is not affected by Turkey’s membership for four coalition formation.

No Brasil sempre se discute sobre o número de representantes na Câmara Federal, com indicações de sub ou sobre-representatividade. Esta discussão tem valor político e econômico, pois uma parcela dos recursos públicos é definida nesta instância, sendo influenciada pelo efetivo poder estadual, que pode ser mensurado pelo índice de Banzhaf. Uma proposta, sugerida por Penrose, é analisada para as UF's e regiões brasileiras. A aplicação mostra que, tanto para o atual sistema quanto para o proposto, a região Sudeste é a mais favorecida. Em relação aos Estados, os mais desenvolvidos e populosos possuem maior poder de voto em ambas situações.In Brazil there are always discussions about the number of representatives in the Federal Camera, with indications of sub or super representativeness. This discussion has political and economic value because an amount of the public resources is defined in this body and it can be linked to the effective power of the states, which can be measured by the Banzhaf index. A proposal, suggested by Penrose, is analyzed for Brazilian States and Regions. Application shows that, for the current as for the Penrose system, the most favored region is the Southeast. From the viewpoint of States, it is noticed that are those more developed and populous that possess larger voting power in both situations.

Participants in parliamentary voting-usually political parties-are evaluated with a value that assigns them predicted power with respect to their strength by so called power indices. However, in real world, political parties' representatives act not strictly as predicted in theory. One possibility how the representatives differ from theory is the way they form coalitions; the coalitions can be announced (for example official governmental coalition parties in multiparty parliaments) or hidden. To incorporate the coalition forming influence, Bilal et al. (, ) proposed to consider additional weights to possible coalitions into power indices. This article applies the concept of additional weights to calculate an ex post power distribution using Shapley-Shubik power index together with Banzhaf power index on real voting data, namely the data from the Chamber of Deputies of the Czech Parliament with the emphasis on the State Budget voting issues during 2006-2010 parliamentary period. [ABSTRACT FROM AUTHOR]

The electoral system of Suriname has been analyzed. Suriname has a unicameral parliament, the National Assembly. The 51 seats of the National Assembly are distributed among 10 districts. There are large discrepancies between the numbers of voters represented by a seat in the various districts. Apportionment methods leading to different seat distributions are explored and compared with each other and with the current one. The comparison is done with respect to the number of voters represented by a seat, the mean majority deficit and the probability that a majority deficit will occur, the influence of a voter in a particular district using the Banzhaf power index, and the influence of a political party relative to the percentage of the popular vote that the party obtained. The method of equal proportions turns out to yield the best results in general. [ABSTRACT FROM AUTHOR]

Enlargement is one of the most important issue for European Union. The situation that makes this issue important is the influence of membership of the candidate states on the voting power distribution. In this study, the impact of membership of Turkey on voting power distribution is evaluated according to the acts adopted by Council of the European Union and voting system brought by the Treaty of Lisbon. In the analysis, voting power distribution of the countries' is taken into consideration in the case of voting independent from each other and Banzhaf power index is used for voting power measurement. Based on the results of the voting power analysis, in the case of countries' voting independent from each other populous countries loss power and others gain power with the accession of Turkey. When the Council does not act on a proposal from the Commision or the High Representative of the Union for Foreign Affairs and Security Policy, all member countries loss power. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we analyze ternary bicooperative games, which are a refinement of the concept of a ternary voting game introduced by Felsenthal and Machover. Furthermore, majority voting rules based on the difference of votes are simple bicooperative games. First, we define the concepts of the defender and detractor swings for a player. Next, we introduce the Banzhaf power index and the normalized Banzhaf power index. The main result of the paper is an axiomatization of the Banzhaf power index for the class of ternary bicooperative games. Moreover, we study ternary bicooperative games with two lists of weights and compute the Banzhaf power index using generating functions. [Copyright &y& Elsevier]

Copyright of Brazilian Journal of Applied Economics / Economía Aplicada is the property of FEA-RP, Universidade de Sao Paulo and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Abstract: We present three axioms for a power index defined on the domain of simple (voting) games. Positivity requires that no voter has negative power, and at least one has positive power. Transfer requires that, when winning coalitions are enhanced in a game, the change in voting power depends only on the change in the game, i.e., on the set of new winning coalitions. The most crucial axiom is composition: the value of a player in a compound voting game is the product of his power in the relevant first-tier game and the power of his delegate in the second-tier game. We prove that these three axioms categorically determine the Banzhaf index. [Copyright &y& Elsevier]

This paper reports an analysis of the relationships between shareholding and voting power distributions in a sample of British companies. It applies two standard approaches to the measurement of power in simple games: the Shapley-Shubik and the Banzhaf power indices. The results indicate that power is more concentrated than ownership in every case. A comparison of the two indices reveals that typically the Banzhaf index gives a more concentrated power distribution. For the Shapley-Shubik index the power ratio for the largest shareholder is accurately described in terms of the size of holding and the concentration of the remainder. The corresponding Banzhaf power ratio is less dependent on these variables. There is no association between power concentration and company size. [ABSTRACT FROM AUTHOR]