Search

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.

Purpose As 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/approach This 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. Findings Numerical 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/implications The proposed framework works efficiently under the specific defined conditions. Social implications Paper 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/value This 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.

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.

This is a post-peer-review, pre-copyedit version of an article published in Group decision and negotiation. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10726-020-09718-7. 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 intu-itive and recognized extension of the index fails to preserve the desirability rela-tion 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 sim-ple 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 exten-sions of well-established notions in a wider context requires a careful analysis. Dif-ferent 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 situa-tion 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. This research was partially supported by funds from: the Spanish Ministry of Economy and Competitiveness (MINECO) and from the European Union (FEDER funds) under grant MTM2015-66818-P (MINECO/FEDER), and the Spanish Ministry of Science and Innovation grant PID2019-I04987GB-I00.

This is a post-peer-review, pre-copyedit version of an article published in Group Decision and Negotiation. The final authenticated version is available online at: https://doi.org/10.1007/s10726-019-09651-4. 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.

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.

(3, 2)-Simple games are a model for voting situation in which players can vote not only in favour or against a proposal but they can also abstain. Also in this model, power indices are used to evaluate the power of players. In particular, the Banzhaf index and the Shapley–Shubik index have been generalized to define analogous power indices in the context of games with abstention. In this work we provide a new axiomatization of the Banzhaf index for games with abstention, to underline its properties and increase the justification of the use of this index as a solution concept also in the family of games with abstention.

The Banzhaf index is a function determining the power or influence in the decision of a set of agents. The extension of this index to the family of the cooperative games is named Banzhaf value. The relationships of closeness among the agents should modify their power. Games with a priori unions study situations where the closeness relations among the agents are taken into account. In this model the agents are organized in an a priori partition where each element of the partition represents a group of agents with close interests or ideas. The power is determined in two steps, first as a problem among the unions and later, inside each one, the power of each agent is determined. Proximity relations extend this model considering leveled closeness among the agents. In this paper we analyze a version of the Banzhaf value for games with a proximity relation and we show the interest of this value by applying it to the allocation of the power of the political groups in the European Parliament.

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.

The author analyzes the power distribution among political groups in the European Parliament from 1979 till 2014. Agent’s voting power is treated as the capability to influence the result of collective decision-making process. Often the voting power is not equal to votes share the agent possess. We obtain the values of classical and modified power indices for European political groups and compare their performance. The Banzhaf power index is employed to measure European political groups’ a priori power that does not impose any restrictions on coalition formation, and power indices that take into account groups’ preferences in coalition formation in order to measure actual voting power. Classical power indices are calculated using only votes’ distribution in the committee and a quota to pass a decision. Pairwise preferences allow to have respect to different willingness of groups to form a single coalition. In this paper preferences are modeled using the roll-call data from the votings on important issues of European politics. The less popular is the MP’s position, the smaller is the number of coalitions she can enter and the less is the voting power. We demonstrate that factions can exploit their position and acquire higher relative power than both share of seats and the Banzhaf index measures, and the Liberal group of the European Parliament forming coalitions with both partners from the left and the right is the case. Conversely, hard restrictions on formation of coalitions due to ideological differences can lead to less actual voting power than a priori.

Weighted voting games are important in multiagent systems because of their usage in automated decision making. However, they are not immune from the vulnerability of false-name manipulation by strategic agents that may be present in the games. False-name manipulation involves an agent splitting its weight among several false identities in anticipation of power increase. Previous works have considered false-name manipulation using the well-known Shapley–Shubik and Banzhaf power indices. Bounds on the extent of power that a manipulator may gain exist when it splits into k = 2 false identities for both the Shapley–Shubik and Banzhaf indices. The bounds when an agent splits into k > 2 false identities, until now, have remained open for the two indices. This article answers this open problem by providing four nontrivial bounds when an agent splits into k > 2 false identities for the two indices. Furthermore, we propose a new bound on the extent of power that a manipulator may gain when it splits into several false identities in a class of games referred to as excess unanimity weighted voting games. Finally, we complement our theoretical results with empirical evaluation. Results from our experiments confirm the existence of beneficial splits into several false identities for the two indices, and also establish that splitting into more than two false identities is qualitatively different than the previously known splitting into exactly two false identities.

Critical node discovery plays a vital role in assessing the vulnerability of a computer network to malicious attacks and failures and provides a useful tool with which one can greatly improve network security and reliability. In this paper, we propose a new metric to characterize the criticality of a node in an arbitrary computer network which we refer to as the Combined Banzhaf & Diversity Index (CBDI). The metric utilizes a diversity index which is based on the variability of a node׳s attributes relative to its neighbours and the Banzhaf power index which characterizes the degree of participation of a node in forming shortest paths. The Banzhaf power index is inspired from the theory of voting games in game theory. The proposed metric is evaluated using analysis and simulations. The criticality of nodes in a network is assessed based on the degradation in network performance achieved when these nodes are removed. We use several performance metrics to evaluate network performance including the algebraic connectivity which is a spectral metric characterizing the connectivity robustness of the network. Extensive simulations in a number of network topologies indicate that the proposed CBDI index chooses more critical nodes which, when removed, degrade network performance to a greater extent than if critical nodes based on other criticality metrics were removed.

The problem of apportioning seats to member countries of the European Parliament after Brexit and in view of new accessions/exits is delicate, as countries with strong economies (and their consequent large contributions to the European Union) require that they have greater representative weight in the European Parliament. In this paper, we propose a model for seat apportionment in the European Parliament, which assigns seats taking into account both the percentages of the populations and the percentages of the contributions by each member state to the European Union budget by means of a linear combination of these two quantities. The proposed model is a modification of the approach given by Bertini, Gambarelli, and Stach in 2005. Using the new model, we studied the power position of each European Union member state before and after the exit of the United Kingdom using the Banzhaf power index. A short latest-literature review on this topic is given.

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 weighte...

In the context of coalitional games, a power index allows to determine the magnitude of contributions associated to each player, i.e., a power index provides information about how influential or relevant a player is in a cooperative interaction. Nevertheless, if the number of involved players is big, then the computation of a power index might become intractable. In this paper, we show how to construct a fullpotential game whose Nash equilibrium coincides with the Shapley or Banzhaf power index for a family of characteristic functions. Therefore, distributed non-cooperative algorithms can be used for cooperative-game purposes. As a consequence, both the computational time and the information requirements are reduced, allowing the use of power indexes in large-scale systems. As an illustrative example, we present a large-scale social network of smart objects where it is desired to enhance the navigability by means of local decisions.

© Oxford University Press and Foundation for the European Review of Agricultural Economics 2015.We consider how cost heterogeneity and market power affect voting power in producer referenda for mandatory agricultural marketing organisations with generic promotion programmes in the United States. We measure voting power using the Banzhaf Power Index and propose a new version of this index based on the profit-maximising theory of the firm that provides an improved estimate of voting power. Examining several types of demand shifts and voting rules, we find that both Banzhaf Power and our new measure vary considerably depending on the market structure and level of cost heterogeneity.

Stochastic Asymmetric Blotto Games: Some New Results John Duffy ∗ Abstract. Blotto games. Alexander Matros † We develop some new theoretical results for stochastic asymmetric Keywords: Colonel Blotto game, Contests, Resource Allocation, Lotteries. JEL Classification Nos. C72, C73, D72, D74. 1. Introduction The Colonel Blotto game (Borel 1921), is a two-player non-cooperative game in which players decide how to allocate their given resources across  battlefields. In Borel’s original version of this game, the player who allocates the most resources to any given battlefield wins that battlefield with certainty. The players’ objective function is either to maximize the sum of the value of the battlefields won, or to win a majority value of the  battlefields. In this paper we study stochastic asymmetric versions of the Blotto game under both of these objective functions. In an “asymmetric Blotto” game, the values of the  battlefields may differ from one another though these different values are common to all players. In the “stochastic asymmetric” version of the Blotto game, the deterministic rule for determining which player wins each battlefield is replaced by a lottery contest success function where the chances of winning a given battlefield are increasing with the amount of resources devoted to that battlefield. This stochastic lottery specification makes the payoff function continuous; as a result, if a Nash equilibrium exists, it is unique and in pure strategies, as opposed to the multiplicity of (typically mixed strategy) equilibria that arise in deterministic versions of the Blotto game. There are two main theoretical papers about stochastic asymmetric Blotto games. 1 The first one, Friedman (1958), considers two players who seek to maximize their expected total payoff. We show that Friedman’s result can be extended to any number of players. The second paper, Lake (1979), was the first to study the stochastic asymmetric “majority rule” Blotto game. 2 This version of the game is particularly relevant to understanding electoral competitions in two party systems, e.g., the electoral college system for electing the U.S. president. Lake studied only the case of equal budget constraints. We show that if players’ budgets are the same (as in Lake) or if they are sufficiently similar and the number of items (battlefields) is not too large, then resource allocation under the majority rule version of the stochastic, asymmetric Blotto game is proportional to the Banzhaf power index for each item, while more generally, resource allocation for a particular item will not be proportional to each item’s Banzhaf power index. Our findings thus generalize those of Lake (1979). Department of Economics, University of California, Irvine. Email: duffy@uci.edu. Moore School of Business, University of South Carolina and Lancaster University Management School. Email: alexander.matros@gmail.com See Kovenock and Roberson (2012) for a broader survey of the Blotto game literature. We discovered Lake’s (1979) paper only after we had completed our analysis. His proofs are different from ours, but his main result coincides with our prediction for the case of equal budgets. We thank Steve Brams for providing this reference.

According to yes-no voting systems, players (e.g., parties in a parliament) have some influence on making some decisions. In formal voting situations, taking into account that a majority vote is needed for making a decision, the question of political power of parties can be considered. There are some well-known indices of political power e.g., the Shapley- Shubik index, the Banzhaf index, the Johnston index, the Deegan-Packel index. In order to take into account different political nature of the parties, as the main factor for forming a winning coalition i.e., a parliamentary majority, we give a modification of the power indices. For the purpose of comparison of these indices of political power from the empirical point of view, we consider the indices of power in some cases, i.e., in relation to a few parliaments.

This paper describes our work in developing approximation algorithms to calculate the Banzhaf Power Index (BPI) in a bicooperative game (that is, games with two coalitions) with large n for the number of players. Our motivation for this work is applying a cooperative game-theoretic framework to cybersecurity scenarios: our past experience with network defense made us receptive to the principle that differences in the criticality of players or network resources in a coalition setting is not always proportional to their differences in weighting or numbers of votes. Hence, calculating a game-theoretic power measure makes sense as a basis for both assessments and allocation decisions. The challenge is that for most real-world scenarios, the value of n is too high for an exact algorithm to solve in time to be actionable in a network defense scenario. We describe our approximation algorithm, and show empirical results that demonstrate that it produces solid estimates of the BPIs that would result from an exact calculation. Therefore, this approximation approach has utility in scenarios where it is imperative to deliver timely results and network membership can be dynamic.

Recent years have witnessed an increased interest, by competition agencies, in assessing the competitive effects of partial acquisitions. We propose a generalization to a partial horizontal acquisition setting of the two most traditional indicators used to screen unilateral anti-competitive effects: the Helfindahl- Hirschman Index and the Gross Upward Price Pressure Index. The proposed generalized indicators can deal with all types of acquisitions that may lessen competition in the industry: acquisitions by owners that are internal to the industry (rival firms) and engage in cross-ownership, as well as acquisitions by owners that are external to the industry and engage in common-ownership. Furthermore, these indicators can deal with direct and indirect acquisitions, which may or may not correspond to control, and nest full mergers as a special case. We provide an empirical application to several acquisitions in the wet shaving industry. The results seem to suggest that (i) a full merger induces higher unilateral anti-competitive effects than a partial controlling acquisition involving the same firms, (ii) a partial controlling acquisition induces higher unilateral anti-competitive effects than a partial non-controlling acquisition involving the same firms and the same financial stakes, and (iii) an acquisition by owners that are internal to the industry induces higher unilateral anti-competitive effects than an acquisition (involving the same firms and the same stakes) by external owners that participate in more than one competitor firm. JEL Classification: L13, L41, L66 Keywords: Antitrust, Partial Horizontal Acquisitions, Oligopoly, Screening Indicators, HHI, GUPPI

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.

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.

In 2013, Jimenez-Losada et al. introduced several extensions of the Myerson value for games with fuzzy communication structure. In a fuzzy communication structure the membership of the players and the relations among them are leveled. Now we study a Banzhaf value for these situations. The Myerson model is followed to define the fuzzy graph Banzhaf value taking as base point the Choquet integral. We propose an axiomatization for this value introducing leveled amalgamation of players. An algorithm to calculate this value is provided and its complexity is studied. Finally we show an applied example computing by this fuzzy value the power of the groups in the European Parliament.

Voting power theories measure the ability of voters to influence the outcome of an election under a given voting rule. In general, each theory gives a different evaluation of power, raising the question of their appropriateness, and calling for the need to identify classes of rules for which different theories agree. We study the ordinal equivalence of the generalizations of the classical power concepts–the influence relation, the Banzhaf power index, and the Shapley–Shubik power index–to multi-choice organizations and political rules. Under such rules, each voter chooses a level of support for a social goal from a finite list of options, and these individual choices are aggregated to determine the collective level of support for this goal. We show that the power theories analyzed do not always yield the same power relationships among voters. Thanks to necessary and/or sufficient conditions, we identify a large class of rules for which ordinal equivalence obtains. Furthermore, we prove that ordinal equivalence obtains for all linear rules allowing a fixed number of individual approval levels if and only if that number does not exceed three. Our findings generalize all the previous results on the ordinal equivalence of the classical power theories, and show that the condition of linearity found to be necessary and sufficient for ordinal equivalence to obtain when voters have at most three options to choose from is no longer sufficient when they can choose from a list of four or more options.

This paper investigates some connections between individual values, such as the Shapley value and the Banzhaf value, and coalition values, such as blockability value, viability value and profitability value, for games in characteristic function form. It turns out that the Shapley value, the Banzhaf value, blockability value, viability value and profitability value assign the same real number to each coalition where the characteristic function is superadditive and inessential. It is also clarified that there is equalization of the Banzhaf value and profitability value of each player in constant-sum game. A new class of games in characteristic function form, which is called equally average coalition solidarity, is proposed in this paper. A proposition which shows that every player's real number evaluated by the Shapley value corresponds with the real number evaluated by the Banzhaf value of the player in the proposed class of games in characteristic function form is provided.