Abstention, multiple issues, and the Banzhaf power index

The Banzhaf Power Index (BPI) is a method of measuring the power of voters in determining the outcome of a voting game. Some voting games exhibit a hierarchical structure, including the US electoral college and ensemble learning methods; we call such games hierarchical voting games. It is generally understood that BPI in hierarchical voting games can be computed via a recursive decomposition of the hierarchy, which can substantially reduce the calculation's complexity. We identify a key (previously undocumented) assumption on which this decomposition is based, namely balance, meaning one group of voters has enough votes to win whenever the complementary group of voters does not, and vice versa. We then introduce a generalization of BPI that we call Extended BPI (EBPI) for all voting games, including those that are not balanced, which simplifies to BPI in balanced games. We show that BPI in unbalanced hierarchical voting games decomposes in terms of EBPI at each level in the hierarchy, which yields computational savings analogous to those achieved in the balanced case. As a sample application, we take advantage of the compositionality of language, and model the impact of individual words on a sentence's sentiment as a voting game. As the complement of a phrase in a sentence does not necessarily have the opposite sentiment, this voting game is unbalanced and requires our decomposition of BPI in terms of EBPI. Our results suggest that EBPI is an effective proxy for BPI (because the meaning of a sentence is not always 100% compositional), and demonstrate a dramatic improvement in run time.

Networked systems are pervasive in the current Internet age and they manifest in several ways such as online and enterprize social networks, data center networks, cloud service networks, and global business networks. Such networked systems generally consist of entities and connections (or edges) among these entities. As services or applications are deployed over these networks, the success of any service or application is crucially dependent on the high availability of the components (namely, entities and connections) in the network. This calls for an important problem of identifying the critical components in the network to improve the quality of the services offered over these networks. We propose a novel and generic game theoretic framework to identify critical components with respect to a given task (or service or application) in the network. In particular, we apply the proposed generic game theoretic framework to determine critical edges in the context of k-edge connectivity between certain pairs of nodes in a given network. We call a pair of nodes in a network k-edge connected if there exists k-edge disjoint (shortest) paths between these nodes. Identifying critical edges in the setting of the k-edge connectivity is an extremely important problem especially in the context of data center networks and cloud service networks. The following are the specific contributions of this paper: (1) We first formally define a game theoretic model for the k-edge connectivity between certain pairs of nodes in the network. We refer to this as k-edge connectivity game. We then define the criticality of any edge to be the value of its Banzhaf power index in the k-edge connectivity game. In this setting, identifying critical edges boils down to the computation of Banzhaf power index in the k-edge connectivity game; (2) We then show that computing the Banzhaf power index for any edge in the k-edge connectivity game is #P-complete. We show this result by reducing the problem of counting the number of perfect matchings in a given graph to an instance of computing the Banzhaf power index for an edge in the k-edge connectivity game. This implies that the computation of Banzhaf power index for an edge in the k-edge connectivity game is computationally hard; (3) To alleviate this computational issue, we propose an approximation algorithm and also we present a simple heuristic based on the notion of Shapley value in cooperative game theory; (4) We then derive a closed form expression for computing the Banzhaf power index of any edge in the k-edge connectivity game, when the network structure is a tree; and (5) We finally evaluate the proposed approach using thorough experimentation on certain real world network data sets.

The Banzhaf power index for ternary bicooperative games

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.

Critical node discovery plays a vital role in assessing the vulnerability of a network to an abrupt change, such as an adversarial attack or human intervention. In this paper, we propose a new metric to characterize the criticality of a node in an arbitrary 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 neighbors 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. We evaluate the performance of the new metric using simulations. Our results indicate that in a number of network topologies, the proposed metric outperforms other proposals which have appeared in the literature. 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.

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

Weighted Banzhaf power and interaction indexes through weighted approximations of games

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.

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

We study the complexity of the following problem: Given two weighted voting games G′ and G′′ that each contain a player p, in which of these games is p’s power index value higher? We study this problem with respect to both the Shapley-Shubik power index [16] and the Banzhaf power index [3,6]. Our main result is that for both of these power indices the problem is complete for probabilistic polynomial time (i.e., is PP-complete). We apply our results to partially resolve some recently proposed problems regarding the complexity of weighted voting games. We also show that, unlike the Banzhaf power index, the Shapley-Shubik power index is not #P-parsimonious-complete. This finding sets a hard limit on the possible strengthenings of a result of Deng and Papadimitriou [5], who showed that the Shapley-Shubik power index is #P-metric-complete.KeywordsWeighted voting gamespower indicescomputational complexity

The complexity of power-index comparison

Many multiagent domains where cooperation among agents is crucial to achieving a common goal can be modeled as coalitional games. However, in many of these domains, agents are unequal in their power to affect the outcome of the game. Prior research on weighted voting games has explored power indices, which reflect how much "real power" a voter has. Although primarily used for voting games, these indices can be applied to any simple coalitional game. Computing these indices is known to be computationally hard in various domains, so one must sometimes resort to approximate methods for calculating them. We suggest and analyze randomized methods to approximate power indices such as the Banzhaf power index and the Shapley-Shubik power index. Our approximation algorithms do not depend on a specific representation of the game, so they can be used in any simple coalitional game. Our methods are based on testing the game's value for several sample coalitions. We also show that no approximation algorithm can do much better for general coalitional games, by providing lower bounds for both deterministic and randomized algorithms for calculating power indices.

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.

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.

In this paper we provide a generalized power index which gives a measurement of voting power in multi-candidate elections with weighted voting using preference ballots. We use the power index to compare the power of various players between an election using plurality and one using the Borda method. The power index is based upon the Banzhaf power index.