Banzhaf Power Index Research Articles

Distribution of Seats in the New 2021 Czech Government Coalition in Terms of Coalition Theories and Power Indices of Political Parties

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.

Playing games with QCA: The Banzhaf index as a context-sensitive measure of explanatory power in international management

Qualitative Comparative Analysis (QCA) enables researchers in international management to better understand how the impact of a single explanatory factor depends on the context of other factors. But the analytical toolbox of QCA does not include a parameter for the explanatory power of a single explanatory factor or “condition”. In this paper, we therefore reinterpret the Banzhaf power index, originally developed in cooperative game theory, to establish a goodness-of-fit parameter in QCA. The relative Banzhaf index we suggest measures the explanatory power of one condition averaged across all sufficient combinations of conditions. The paper argues that the index is especially informative in three situations that are all salient in international management and call for a context-sensitive analysis of single conditions, namely substantial limited diversity in the data, the emergence of strong INUS conditions in the analysis, and theorizing with contingency factors. The paper derives the properties of the relative Banzhaf index in QCA, demonstrates how the index can be computed easily from a rudimentary truth table, and explores its insights by revisiting selected papers in international management that apply fuzzy-set QCA. It finally suggests a three-step procedure for utilizing the relative Banzhaf index when the causal structure involves both contingency effects and configurational causation.

A Characterization of the Totally Critical Raw Banzhaf Power Index on Dichotomous Voting Games with Several Levels of Approval in Input

A Characterization of the Totally Critical Raw Banzhaf Power Index on Dichotomous Voting Games with Several Levels of Approval in Input

On computing Banzhaf power index for k-edge connectivity in graphs

On computing Banzhaf power index for k-edge connectivity in graphs

THE POWER OF SMALL EU MEMBER STATES AFTER BREXIT: HOW POWERFUL IS THE VISEGRAD GROUP?

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.

Extending multidimensional poverty identification: from additive weights to minimal bundles

In the popular class of multidimensional poverty measures introduced by Alkire and Foster (2011), a threshold switching function is used to identify who is multidimensionally poor. This paper shows that the weights and cut-off employed in this procedure are generally not unique and that such functions implicitly assume all groups of deprivation indicators of some fixed size are perfect substitutes. To address these limitations, I show how the identification procedure can be extended to incorporate any type of positive switching function, represented by the set of minimal deprivation bundles that define a unit as poor. Furthermore, the Banzhaf power index, uniquely defined from the same set of minimal bundles, constitutes a natural and robust metric of the relative importance of each indicator, from which the adjusted headcount can be estimated. I demonstrate the merit of this approach using data from Mozambique, including a decomposition of the adjusted headcount using a ‘one from each dimension’ non-threshold function.

Voting power on a graph connected political space with an application to decision-making in the Council of the European Union

We analyze the problem of computing the Banzhaf and Shapley power indices for graph restricted voting games, defined in a particular class of graphs, that we called line-clique. A line-clique graph is a model of a uni-dimensional political space in which voters with the same bliss point are the connected vertices of a clique and then other arcs connect nodes of consecutive cliques. The interest to this model comes from its correspondence to the spatial voting game: a model that has been proposed and used by political analysts to understand nations’ behavior and the political outcome of the bargaining process within the EU Council. Broadly speaking, the computation of a power index of a graph restricted game is strongly #P-complete, as it includes the enumeration of all winning coalitions. Nevertheless, we show that in this special class of graph coalitions can be enumerated by dynamic programming, resulting in a pseudo-polynomial algorithm and proving that the problem only weakly #P-complete. After implementing our new algorithms and finding that they are very fast in practice, we analyze the voting behavior in the EU Council, as for this application previous research compiled a large data set concerning nations’ political positions and political outcomes. We will test whether voting power has an effect on the political outcome, more precisely, whether nations that are favored by their weight and position can influence the political outcome to their advantages. Using linear regressions, we will see that unrestricted power indices are not capable of any predictive property, but graph restricted indices are. The statistic evidence shows that the combination of voting weight and network position is a source of power that affects the political outcome to the advantage of a country.

An Appropriate Way to Extend the Banzhaf Index for Multiple Levels of Approval

The Banzhaf power index for games admits several extensions if the players have more than two ordered voting options. In this paper we prove that the most intuitive and recognized extension of the index fails to preserve the desirability relation for games with more than three ordered input levels of approval, a failure that undermines the index to be a good measure of power. This leads us to think of an alternative to the Banzhaf index for several input levels of approval. We propose a candidate for which it is proved that: (1) coincides with the Banzhaf index for simple games, (2) it is proportional to its known extension for three levels of approval, and (3) preserves the desirability relation regardless of the number of input levels of approval. This new index is based on measuring the total capacity the player has to alter the outcome. In addition, it can be expressed through a very appropriate mathematical formulation that greatly facilitates its computation. Defining extensions of well-established notions in a wider context requires a careful analysis. Different extensions can provide complementary nuances and, when this occurs, none of them can be considered to be ‘the’ extension. As shown in this paper, this situation applies when trying to extend the Banzhaf power index from simple games to the broader context of games with several ordered input levels of approval.

A request allocation model for processing data in federated cloud computing

PurposeAs the storage and processing requirement of digital information is increasing on the cloud, it is very difficult for the single cloud provider (CP) to meet the resource requirement. Multiple providers form a federation for the execution of users’ requests. For the federated cloud, this paper aims to address the issue distribution of users’ request for resources and revenue among the providers by offering fair and stable distribution models for the federated cloud.Design/methodology/approachThis paper uses cooperative game (CG)-theoretical models, i.e. Shapley–Shubik power index (SSPI) and Banzhaf power index (BPI) for distribution. Performance is analysed using variance and monotonicity using a case study.FindingsNumerical analysis is done using two scenarios. Monotonicity is evaluated. Results show that SSPI performs better as compared to BPI in terms of fairness accuracy and the framework provide the fair distribution of revenue among providers in the federated cloud.Research limitations/implicationsThe proposed framework works efficiently under the specific defined conditions.Social implicationsPaper provides the fair distribution. It assist the centralised cloud exchange in managing the users’ request in such a way every CPs, in the federated cloud will get an equal chance of serving the users’ request. The framework also provides the stable federation. Proposed work provides less rejection rate of users’ request. Finally, it assists the providers in increasing their profits in the federation.Originality/valueThis paper presents a CG theoretic-based framework for the distribution of resources required and revenue. The framework analysed the performance of distribution models by considering the variance and monotonicity for multiple users’ requests.

The power ranking of the members of the Agricultural Committee of the European Parliament

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.

Average weights and power in weighted voting games

We investigate a class of weighted voting games for which weights are randomly distributed over the standard probability simplex. We provide close-formed formulae for the expectation and density of the distribution of weight of the k-th largest player under the uniform distribution. We analyze the average voting power of the k-th largest player and its dependence on the quota, obtaining analytical and numerical results for small values of n and a general theorem about the functional form of the relation between the average Penrose–Banzhaf power index and the quota for the uniform measure on the simplex. We also analyze the power of a collectivity to act (Coleman efficiency index) of random weighted voting games, obtaining analytical upper bounds therefor.

Digitalizing 360 Degree Employee Competency using Banzhaf Power Index and CDF

Traditionally, 360 degree evaluation of employee competency has been done using ratings given by Subordinates or Direct Reports, Peers, Manager and Self. Based on the rating by all rating group, the competency of the employee is determined. The gap between self-rating and ratings of all other group determines whether an employee has hidden strength or has blind spot in each competency variable. However, whenever larger number of subordinates or peers rates an employee, there is inherent bias and the employee’s overall competency rating can be low if the larger group holds grudge against the employee. To avoid bias and determine true rating, this paper proposes innovative use of Banzhaf Power Index. The complete computation and digitalization of Banzhaf Power Index for Chennai based Wind Energy Company is carried out and presented in this paper. The interactive Wolfram Computable Document Format (CDF) has also been created for wider use by personnel managers of other companies.

The Banzhaf Value for Cooperative and Simple Multichoice Games

This article proposes a value which can be considered an extension of the Banzhaf value for cooperative games. The proposed value is defined on the class of j-cooperative games, i.e., games in which players choose among a finite set of ordered actions and the result depends only on these elections. If the output is binary, only two options are available, then j-cooperative games become j-simple games. The restriction of the value to j-simple games leads to a power index that can be considered an extension of the Banzhaf power index for simple games. The paper provides an axiomatic characterization for the value and the index which is closely related to the first axiomatization of the Banzhaf value and Banzhaf power index in the respective contexts of cooperative and simple games.

An Axiomatization for Two Power Indices for (3,2)-Simple Games

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.

Composition independence in compound games: a characterization of the Banzhaf power index and the Banzhaf value

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).

Banzhaf random forests: Cooperative game theory based random forests with consistency

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.

Manipulation by Merging and Annexation in Weighted Voting Games

The problem of manipulation in voting is fundamental and hasreceived attention in recent research in game theory. In this paper, we considertwo cases of manipulation in weighted voting games done by mergingof coalitions into single players and by annexation of a part or all of thevoting weights of another player viewed from two perspectives: of the effectof swings of players and of the role of the Banzhaf power index. We provetwo theorems for manipulation by merging and annexation, and show severalattractive properties in these two processes.ACM Computing Classification System (1998): J.4, I.2.1.

The Visegrád Group – A Rising Star Post-Brexit? Changing Distribution of Power in the European Council

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.

Choosing Voting Systems Behind the Veil of Ignorance: A Two-Tier Voting Experiment

There are many situations in which different groups make collective decisions by committee voting, with each group represented by a single person. A natural question is what voting system such a committee should use. Concepts based on voting power provide guidelines for this choice. The two most prominent concepts require the Banzhaf power index to be proportional to the square root of group size or the Shapley-Shubik power index to be proportional to group size. Instead of studying the choice of voting systems based on such theoretical concepts, in this paper, I ask which systems individuals actually prefer. To answer this question, I design a laboratory experiment in which participants choose voting systems. I find that people behind the veil of ignorance prefer voting systems following the rule of proportional Shapley-Shubik power; in front of the veil subjects prefer voting systems benefiting their own group. Participants’ choices can only partially be explained by utility maximization or other outcome based concepts.

The axiom of equivalence to individual power and the Banzhaf index

I introduce a new axiom for power indices on the domain of finite simple games that requires the total power of any given pair i,j of players in any given game v to be equivalent to some individual power, i.e., equal to the power of some single player k in some game w. I show that the Banzhaf power index is uniquely characterized by this new “equivalence to individual power” axiom in conjunction with the standard semivalue axioms: transfer (which is the version of additivity adapted for simple games), symmetry or equal treatment, positivity (which is strengthened to avoid zeroing-out of the index on some games), and dummy.