Banzhaf Index Research Articles

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.

Switching-Algebraic Calculation of Banzhaf Voting Indices

This paper employs switching-algebraic techniques for the calculation of a fundamental index of voting powers, namely, the total Banzhaf power. This calculation involves two distinct operations: (a) Boolean differencing or differentiation, and (b) computation of the weight (the number of true vectors or minterms) of a switching function. Both operations can be considerably simplified and facilitated if the pertinent switching function is symmetric or it is expressed in a disjoint sum-of-products form. We provide a tutorial exposition on how to implement these two operations, with a stress on situations in which partial symmetry is observed among certain subsets of a set of arguments. We introduce novel Boolean-based symmetry-aware techniques for computing the Banzhaf index by way of two prominent voting systems. These are scalar systems involving six variables and nine variables, respectively. The paper is a part of our on- going effort for transforming the methodologies and concepts of voting systems to the switching-algebraic domain, and subsequently utilizing switching-algebraic tools in the calculation of pertinent quantities in voting theory.

Logic, Game Theory, and Social Choice: What Do They Have in Common?

The answer to the question above is that in all these domains axiomatic characterizations are given of, respectively, mathematical reasoning, certain notions from game theory, and certain social choice rules. The meaning of the completeness theorem in logic is that mathematical reasoning can be characterized by a handful of certain (logical) axioms and rules. If we apply mathematical reasoning to elementary arithmetic, i.e., the addition and multiplication of natural numbers, it turns out that almost all true arithmetical statements, for instance, ∀x∀y[x+y=y+x], can be logically deduced from the axioms of Peano. However, in 1931 Kurt Gődel showed that the axioms of Peano do not (fully) characterize the addition and multiplication of the natural numbers, more precisely, that there are certain special self-referential arithmetical sentences that, although true, cannot be deduced from Peano’s axioms. There are axiomatic characterizations of several social choice and ranking rules that say that a given rule is the only one satisfying a particular set of axioms. Arrow’s impossibility theorem in social choice theory tells us that a certain set of, at first sight, quite reasonable axioms for a social ranking rule turns out to be inconsistent. Consequently, a social ranking rule that satisfies the axioms in question cannot exist. Finally, many notions from game theory, such as the Shapley–Shubik and the Banzhaf index, may also be characterized by a set of axioms.

The distribution of power within the EU: perspectives on a Ukrainian accession and a Turkish accession

On February 28th, 2022, Ukraine submitted an application concerning accession to the European Union. As Ukraine is already linked to the European Union through an Association Agreement and the Deep and Comprehensive Free Trade Agreement—signed in 2014—there might be a chance for the Ukraine to join the EU within the next decade. With about 41 million inhabitants, Ukraine is a rather large country and the 65% majority requirement under the EU’s Qualified Majority Voting rules would give Ukraine some significant power in a future EU28. Herein, it is shown how relative political power indices—as measured by the Banzhaf index—would change in the case of an EU enlargement to allow accession by Ukraine: The large economies would lose power and, surprisingly, many small countries would gain political power. Such perspectives might lead one to reconsider the voting procedures of the Council of the European Union at the EU level in the course of future negotiations on amendments to the Lisbon Treaty. The case of Turkey’s accession is also considered as is the case of Montenegro joining the European Union.

Cooperative solutions for TU and NTU stochastic game-theoretical approach

The player's payoff in stochastic games is a stochastic variable. In most studies, predicted payout is referred to as a payoff, implying that the players are risk averse. However, there may be risk-averse players that consider "risk" while calculating their stochastic payoffs. We offer a model of stochastic games using mean-variance reward functions, which are the sum of expectation and standard deviation multiplied by a coefficient indicating a player's risk aversion. By adopting a maxmin technique to define the characteristic function, we may create a cooperative version of a stochastic game with mean-variance preferences. This paper contains an analysis of a cooperative solution concept of stochastic games for both TU (transferable utility) and NTU (non-transferable utility) games. We proposed a cooperative game solution method of extension of two-person zero sum strategic game to n-agent stochastic stage game using both Shapley and Banzhaf index functions. Results are proved for both discounted and undiscounted payoff situations. Existing models of discrete-time stochastic games and ways to finding cooperative solutions in these games are extended in this study.

The Banzhaf value for games in formal contexts

Given a group of agents, a formal context makes it possible to describe very general situations of cooperation where the results of collaborations depend on certain attributes. A concept game defines a payment for each coalition of actors that generates a concept in a given context. Thus, in this article, we deal with situations of cooperation restricted by the definition of concepts. The interpretation of payments as the level of importance of the coalition leads us to be able to establish a ranking of the agents in the context. Following the classic Banzhaf index for cooperative games, we define a value for these situations in formal contexts and we give an axiomatization of them. Finally, in order to show the model, we include an application in the field of e-commerce.

Dynamic Programming for Computing Power Indices for Weighted Voting Games with Precoalitions

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.

Evaluation of decision power in multi-dimensional rules

This work deals with the evaluation of decision power in multi-dimensional rules. Courtin and Laruelle (2020) introduced a decision process that specifies the collective acceptance or rejection of a proposal with several dimensions. The decision process is modeled as follows: (i) There are several individuals. (ii) There are several dimensions. (iii) Each of the individuals expresses a binary choice (“Yes” or “No”) on each dimension. (iv) A decision process maps each choice to a final binary decision (“Yes” or “No”). We extend and characterize six well-known power indices within this context: the Shapley–Shubik index (Shapley and Shubik, 1954), the Banzhaf index (Banzhaf, 1965), the Public good index (Holler, 1982), the Null individual free index (Alonso-Meijide et al., 2011), the Shift index (Alonso-Meijide and Freixas, 2010) and the Deegan–Packel index (Deegan and Packel, 1978).

Worst-case Bounds on Power vs. Proportion in Weighted Voting Games with an Application to False-name Manipulation

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.

A Monotonic Weighted Banzhaf Value for Voting Games

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.

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.

Reliability Analysis of a Home-scale Microgrid Based on a Threshold System

The reliability of a microgrid power system is an important aspect to analyze so as to ascertain that the system can provide electricity reliably over a specified period of time. This paper analyzes a home-scale model of a microgrid system by using the threshold system model (inadvertently labeled as the weighted k-out-of-n:G system model), which is a system whose success is treated as a threshold switching function. To analyze the reliability of the system, we first proved that its success is a coherent threshold function, and then identified possible (non-unique) values for its weights and threshold. Two methods are employed for this. The first method is called the unity-gap method and the second is called the fair-power method. In the unity-gap method, we utilize certain dominations and symmetries to reduce the number of pertinent inequalities (turned into equations) to be solved. In the fair-power method, the Banzhaf index is calculated to express the weight of each component as its relative power or importance. Finally, a recursive algorithm for computing system reliability is presented. The threshold success function is verified to be shellable, and the non-uniqueness of the set of weights and thresholds is demonstrated to be of no detrimental consequence, as different correct sets of weights and threshold produce equivalent expressions of system reliability.

Reliability Evaluation of Rooftop Solar Photovoltaics Using Coherent Threshold Systems

The trend for use of rooftop solar photovoltaics (PV) is rising due to their promising economic potential as a source of clean renewable energy. In general, a source of renewable solar energy consists of solar PV, an automatic charge controller, a battery pack, and an inverter. The reliability of a rooftop solar PV system is evaluated herein as that of a coherent threshold system (CTS). First, we utilize the unit-gap method and the fair-power method to verify that a given Boolean function is a threshold one and to identify its threshold and component weights. Both methods utilize specific features of the Karnaugh map (K-map). The unit-gap method uses the map to list all necessary inequalities by inspection, and then reduce them significantly by omitting dominated ones. The fair-power method uses the Karnaugh map to compute Banzhaf indices by appropriate map folding followed by XORing of true cells and false cells. We evaluate the CTS reliability via a recursive algorithm based on the Boole-Shannon’s expansion in the switching domain, which is transformed via the real transform to the total probability law in the probability domain.

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.

The Public Good Spatial Power Index in political games

We propose a new spatial index called the Public Good spatial index, which is the spatial version of the standard Public Good index under independence behaviors (PGI). However, we show that the spatial model used, is not well suited for both, the standard Shapley Shubik index and the standard Public Good index under Homogeneity behaviors (PGH); and consequently they do not have a consistent spatial version with respect to our model. By contrast, the same spatial model is appropriated for the Banzhaf index and the aforementioned PGI index, concomitantly, allowing their spatial versions in political games. We also argue that those two well convenient spatial indices are only understandable under behavioral descriptions with independence assumptions on political issues. Finally, the paper details our findings by means of examples, comparisons, and it also provides relevant ways of computing spatial power indices in real case studies when it comes to lower dimensions.

Impact of Brexit on voting power in Council of the European Union

AbstractThe United Kingdom has left the European Union on 31 January 2020. Discussions that preceded such a move were conducted in three dimensions: they pertained a post-Brexit relationship between the UK and EU, future conduct within the UK and the one within the EU. Whilst public discourse has been dominated by the first two, this paper approaches the third one – on how Brexit has affected relationships between remaining 27 EU Member States. Stemming from the calculation of Banzhaf indices, it assesses the impact of Brexit on the voting power of remaining Member States in the Council of the EU – arguably the most important body within the EU institutional architecture – and identifies which countries are going to record the most significant gains and losses in this respect.

Financial Safety Nets in East Asia and Europe: A Political Economy Assessment

This article aims to measure and compare the voting power of the member states of two financial nets: the ASEAN Plus Three Macroeconomic Research Office (AMRO) - Chiang Mai Initiative Multilateralized (CMIM) and the European Stability Mechanism (ESM). Furthermore, the study observes the changes to the CMIM before and after the increase of its resources in 2012. The literature on the comparison between regional safety nets lacks proper evaluations from a political economy perspective. This work fills the gap in the literature by placing two of the most important and recent regional financial safety nets under scrutiny. The article employs empirical analyses using two typical measurements of voting systems such as the Shapley-Shubik and Banzhaf indices. The article shows that the small ASEAN countries, contrary to assumptions in the literature, have been penalized after changes in 2012. By observing simple voting weights only, these effects are not visible. However, based on the results obtained from Shapley-Shubik and the Banzhaf power measurements, we argue that the voting powers of big countries, such as Japan and China, have increased after the changes in the system in 2012. In contrast to the ASEAN example, results in the case of the ESM show that there are no substantial differences in the voting powers of member states based on the Banzhaf index and Shapley-Shubik Index. Based on the empirical results of the article, the authors suggest that AMRO-CMIM should take into account the ESM experience regarding the voting mechanism.

Interval type-2 fuzzy multi-attribute decision-making approaches for evaluating the service quality of Chinese commercial banks

In today’s world, with increased competition, the service quality of Chinese commercial banks is recognized as a major factor that is responsible for enhancing competitiveness. Therefore, it is necessary to evaluate and analyse the service quality of Chinese commercial banks to realize their stable development. The service quality evaluation could be recognized as a multi-attribute decision-making (MADM) problem with multiple assessment attributes, both being of a qualitative and quantitative nature. Owing to the growing complexity and high uncertainty of the financial environment, the assessments of attributes cannot always possibly express using a real and/or type-1 fuzzy number. Additionally, a heterogeneous relationship often exists among the attributes under many real decision cases. In this study, we create two MADM approaches to handle decision-making problems with interval type-2 fuzzy numbers (IT2FNs) and offer their application to service quality evaluations of commercial banks problems. Specifically, we first define some operations on IT2FNs based on Archimedean T-norms (ATs) and develop a bi-directional projection measure of IT2FNs. Next, by combining the generalized Banzhaf index, the Choquet integral and IT2FNs, we propose the interval type-2 fuzzy Archimedean Choquet (IT2FAC) operator, the Banzhaf IT2FAC (BIT2FAC) operator and the 2-additive BIT2FAC (2ABIT2FAC) operator. Then, we establish two optimal models for deriving the weights of attributes based on a bi-directional projection measure of IT2FNs and Banzhaf function. Finally, we create two novel MADM methods under interval type-2 fuzzy contexts, where an illustrative case concerning the service quality evaluation of Chinese commercial banks is used to explain the created MADM approaches.

A note on limit results for the Penrose–Banzhaf index

It is well known that the Penrose-Banzhaf index of a weighted game can differ starkly from corresponding weights. Limit results are quite the opposite, i.e., under certain conditions the power distribution approaches the weight distribution. Here we provide parametric examples that give necessary conditions for the existence of limit results for the Penrose-Banzhaf index.

On the Complexity of the Inverse Semivalue Problem for Weighted Voting Games

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.