Banzhaf Power Index Research Articles

Оценка распределения влияния между политическими группами в Европейском парламенте в 1979-2014 гг.

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.

False‐Name Manipulation in Weighted Voting Games: Empirical and Theoretical Analysis

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.

An efficient ensemble pruning approach based on simple coalitional games

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.

Coalition Politics and Inter‐Party Conflict Management: A Theoretical Framework

Inter‐party conflict management is a typical coalitional problem in parliamentary governments. To study how and why conflicts in coalitions emerge and how parties cope with them can enhance our knowledge of coalition governance. Here, I propose a framework for comparative studies on the topic. The framework is based on the conception of coalition politics as politics of exchange. It looks at inter‐party interactions, but also accounts for the impact of intra‐party politics. Moreover, I provide a classification of inter‐party conflicts in coalitions and point out when they are more likely. The process of conflict management is operationalized with two proxies—actors, and arenas—and a taxonomy of conflict terminations is presented. The viability of the framework is tested both by mapping coalition governments according to their modes of managing internal conflicts and, after deriving research hypotheses, through empirical inquiries of conflict management in diverse coalitions.Related ArticlesNwokora,Zim, andRiccardoPelizzo.2015. “.”Politics & Policy43():453‐473.http://onlinelibrary.wiley.com/doi/10.1111/polp.12124/abstractPasquino,Gianfranco.2014. “.”Politics & Policy42():548‐566.http://onlinelibrary.wiley.com/doi/10.1111/polp.12079/abstractForestiere,Carolyn.2009. “.”Politics & Policy37():509‐528.http://onlinelibrary.wiley.com/doi/10.1111/j.1747-1346.2009.00183.x/abstractRelated Media.2013. “In it Together: The Inside Story of the Coalition Government.” YouTube.https://www.youtube.com/watch?v=tVyLfyhH-hgHamblin,James.2011. “Math for Liberal Studies: Banzhaf Power Index.” YouTube.https://www.youtube.com/watch?v=3_C8TK-e274

Combined Banzhaf & Diversity Index (CBDI) for critical node detection

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.

Efficient algorithms for game-theoretic betweenness centrality

Betweenness centrality measures the ability of different nodes to control the flow of information in a network. In this article, we extend the standard definition of betweenness centrality using Semivalues—a family of solution concepts from cooperative game theory that includes, among others, the Shapley value and the Banzhaf power index. Any Semivalue-based betweenness centrality measure (such as, for example, the Shapley value-based betweenness centrality measure) has the advantage of evaluating the importance of individual nodes by considering the roles they each play in different groups of nodes. Our key result is the development of a general polynomial-time algorithm to compute the Semivalue-based betweenness centrality measure, and an even faster algorithm to compute the Shapley value-based betweenness centrality measure, both for weighted and unweighted networks. Interestingly, for the unweighted case, our algorithm for computing the Shapley value-based centrality has the same complexity as the best known algorithm for computing the standard betweenness centrality due to Brandes [15]. We empirically evaluate our measures in a simulated scenario where nodes fail simultaneously. We show that, compared to the standard measure, the ranking obtained by our measures reflects more accurately the influence that different nodes have on the functionality of the network.

Producer heterogeneity and voting power in mandatory US agricultural marketing organisations

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.

Spatial power indices with applications on real voting data from the Chamber of Deputies of the Czech Parliament

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. (http://aei.pitt.edu/2052/1/001591_1.pdf, 2001) 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.

Stochastic asymmetric Blotto games: Some new results

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.

Coalition Formation in the EU Decision-Making Process after the Enlargement

The enlargement of he European Union was expected to increase the effect of national inequalities within the supranational decisional system and significantly difficult the resolution of problems of policy coordination. However, the decision-making process of the EU has shown a remarkable flexibility and capacity of adaptation in the face of the entrance of new members. In this paper, I theorize how coalition formation in the Council of Ministers contributes to this flexibility in the issue area of environmental policy. Using an extension of the Banzhaf Power Index, I explain how coalition formation incorporates the preference variations infused by new member states through the combination of re-weighted voting power and a strategic process of bilateral exchanges between coalitions and selectively targeted member states. The analysis is completed with a procedural model, which takes into account the influence of the EP in the legislative process. Without imposing unanimity, the configuration of a decisive coalition under QMV requires an attention to particularized interests. In contrast with propositions based on axiomatic models of bargaining, I argue that the focus on the power-based coalition formation can better explain outcomes of effective differentiated adaptation to the Eastern enlargement.

Power theories for multi-choice organizations and political rules: Rank-order equivalence

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.

Computing cooperative solution concepts in coalitional skill games

We consider a simple model of cooperation among agents called Coalitional Skill Games (CSGs). This is a restricted form of coalitional games, where each agent has a set of skills that are required to complete various tasks. Each task requires a set of skills in order to be completed, and a coalition can accomplish the task only if the coalitionʼs agents cover the set of required skills for the task. The gain for a coalition depends only on the subset of tasks it can complete.We consider the computational complexity of several problems in CSGs, such as testing if an agent is a dummy or veto agent, computing the core and core-related solution concepts, and computing power indices such as the Shapley value and Banzhaf power index.

Sharing Rewards in Cooperative Connectivity Games

We consider how selfish agents are likely to share revenues derived from maintaining connectivity between important network servers. We model a network where a failure of one node may disrupt communication between other nodes as a cooperative game called the vertex Connectivity Game (CG). In this game, each agent owns a vertex, and controls all the edges going to and from that vertex. A coalition of agents wins if it fully connects a certain subset of vertices in the graph, called the primary vertices. Power indices measure an agent's ability to affect the outcome of the game. We show that in our domain, such indices can be used to both determine the fair share of the revenues an agent is entitled to, and identify significant possible points of failure affecting the reliability of communication in the network. We show that in general graphs, calculating the Shapley and Banzhaf power indices is #P-complete, but suggest a polynomial algorithm for calculating them in trees. We also investigate finding stable payoff divisions of the revenues in CGs, captured by the game theoretic solution of the core, and its relaxations, the epsilon-core and least core. We show a polynomial algorithm for computing the core of a CG, but show that testing whether an imputation is in the epsilon-core is coNP-complete. Finally, we show that for trees, it is possible to test for epsilon-core imputations in polynomial time.

On the inverse power index problem

Weighted voting games are frequently used in decision making. Each voter has a weight and a proposal is accepted if the weight sum of the supporting voters exceeds a quota. One line of research is the efficient computation of so-called power indices measuring the influence of a voter. We treat the inverse problem: Given an influence vector and a power index, determine a weighted voting game such that the distribution of influence among the voters is as close as possible to the given target value. We present exact algorithms and computational results for the Shapley–Shubik and the (normalized) Banzhaf power index.

Rebuilding the Great Pyramids: A Method for Identifying Control Relations in Complex Ownership Structures

Identifying the corporate controller (controlling shareholder, ultimate owner) is an essential prerequisite for any debate on the corporate governance of a specific firm and of entire markets. This paper aims to provide a comprehensive, precise and economically sound method for identifying control relations on the corporate level and especially in complex ownership structures. We apply weighted voting games literature as a theoretical framework for our analysis and use the Shapley-Shubik and the Banzhaf power indices to determine control rights. The core element of the proposed method, distinguishing our study from others, in solving the puzzle of corporate control, is the simultaneous analysis of both the specific ownership map within the corporation and the corporate network in which the firm is embedded. We implemented our algorithm into a Java computer program and tested it on a real-world data set of corporate ownership in the Israeli market. The direct product of the analysis of these data is a comprehensive map of control relations at every time point. We find that the corporate control relations identified by our method are richer and more accurate than those provided by different official sources.

Weighted Banzhaf power and interaction indexes through weighted approximations of games

The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players. It was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo-Boolean functions. Considering certain weighted versions of this approximation problem, we define a class of weighted interaction indexes that generalize the Banzhaf interaction index. We show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties. Finally, we give an interpretation of the Banzhaf and Shapley interaction indexes as centers of mass of this subclass of interaction indexes.

A Comparison Between the Methods of Apportionment Using Power Indices: the Case of the US Presidential Elections

In this paper we compare five well-known methods of apportionment, advanced respectively by Adams, Dean, Hill, Webster and Jefferson. The criterion used for this comparison is the minimization of the distance between a power vector and a population vector. Power is measured with the well-known Banzhaf power index; the populations are those of the constituent states of the USA. We first explain the conditions under which this comparison has plausibility. We then compare apportionment methods in terms of their capacity to move power in states closer to their populations. The election of the US President by an electoral college is studied by examining 22 censuses since 1790. Our analysis is largely based on that used in the book by Balinski and Young [2001]. The empirical findings are linked to theoretical results.*

The Banzhaf power index for ternary bicooperative games

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.

Effort Games and the Price of Myopia

AbstractWe consider Effort Games, a game‐theoretic model of cooperation in open environments, which is a variant of the principal‐agent problem from economic theory. In our multiagent domain, a common project depends on various tasks; carrying out certain subsets of the tasks completes the project successfully, while carrying out other subsets does not. The probability of carrying out a task is higher when the agent in charge of it exerts effort, at a certain cost for that agent. A central authority, called the principal, attempts to incentivize agents to exert effort, but can only reward agents based on the success of the entire project.We model this domain as a normal form game, where the payoffs for each strategy profile are defined based on the different probabilities of carrying out each task and on the boolean function that defines which task subsets complete the project, and which do not. We view this boolean function as a simple coalitional game, and call this game the underlying coalitional game. We suggest the Price of Myopia (PoM) as a measure of the influence the model of rationality has on the minimal payments the principal has to make in order to motivate the agents in such a domain to exert effort.We consider the computational complexity of testing whether exerting effort is a dominant strategy for an agent, and of finding a reward strategy for this domain, using either a dominant strategy equilibrium or using iterated elimination of dominated strategies. We show these problems are generally #P‐hard, and that they are at least as computationally hard as calculating the Banzhaf power index in the underlying coalitional game. We also show that in a certain restricted domain, where the underlying coalitional game is a weighted voting game with certain properties, it is possible to solve all of the above problems in polynomial time. We give bounds on PoM in weighted voting effort games, and provide simulation results regarding PoM in another restricted class of effort games, namely effort games played over Series‐Parallel Graphs (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Power Indices Taking into Account Agents' Preferences

A set of new power indices is introduced extending Banzhaf power index and allowing to take into account agents’ preferences to coalesce. An axiomatic characterization of intensity functions representing a desire of agents to coalesce is given. A set of axioms for new power indices is presented and discussed. An example of use of these indices for Russian parliament is given.