Banzhaf Power Index Research Articles

The complexity of power-index comparison

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 and the Banzhaf power index. 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 study the complexity of the raw Shapley–Shubik power index. Deng and Papadimitriou showed that the raw Shapley–Shubik power index is #P -metric-complete. We strengthen this by showing that the raw Shapley–Shubik power index is many–one complete for #P . And our strengthening cannot possibly be further improved to parsimonious completeness, since we observe that, in contrast with the raw Banzhaf power index, the raw Shapley–Shubik power index is not #P -parsimonious-complete.

Evaluation of Banzhaf index with restrictions on coalitions formation

The generating functions representative of the Banzhaf power index of the parties, or political groups of electoral bodies, is applied to the case where particular parties simultaneously (in pairs) cannot be part of any coalition. For such types of restrictions on coalition formation, the two-stage procedure for evaluating the Banzhaf index using generating functions is presented. Throughout the steps of the procedure the adjusted Banzhaf index generating function polynomial is transformed to the standard representation used for the calculation of this index. The presented approach is discussed and the procedure is illustrated by several examples.

Actual voting power of the IMF members based on their political-economic integration

A voting power analysis was made to estimate the power of IMF members within the Executive Board and the Fund in general through the existing constituency system. For this purpose we introduced two absolute power indices extending the classical Penrose (non-normalized Banzhaf) and Coleman (“the power of the body to act”) indices by taking into account the members’ preferences to coalesce, based on their membership in the currently functioning political-economic blocs with varying degrees of economic integration and members’ regional proximity as well. We considered an indirect voting system where countries are contending for power at two levels: within their constituencies and within the Executive Board. Accordingly, each of the proposed power indices determines a member’s power at the appropriate level. The total power of a country is then defined by a compound power index, which is the product of the above-mentioned indices. Experiments displayed the robustness of the indices to small variations in the weight coefficients assigned to the preferences considered. Using our approach, we analysed countries’ voting power within the Executive Board under the three cases of majority voting adopted by the IMF, namely, a simple majority and qualified majorities of 70% and 85% of the total votes. The power indices for all members (except those with voting rights suspended) and their constituencies were calculated and compared with the Penrose and Banzhaf indices. The results show that members’ absolute voting power as measured by our indices produces a power distribution different from that of Penrose and Banzhaf power in both directions. The constituencies’ relative voting power is rather close to their IMF voting weights for the case of the simple majority voting and tends to redistribute between constituencies as the threshold number of votes for a decision to be taken increases; thus, for the 85% supermajority, the maximum power share is that of the Netherlands’s constituency and not that of the US’s, as the Banzhaf power index suggests.

The distribution of power in the European Constitution

The aim of this paper is to analyze the distribution of voting power in the Constitution for the enlarged European Union. By using generating functions, we calculate the Banzhaf power indices for the European countries in the Council of Ministers under the decision rules prescribed by the Treaty of Nice and the new rules proposed by the European Constitution Treaty. Moreover, we analyze the power of the European citizens under the egalitarian model proposed by Felsenthal and Machover [D.S. Felsenthal, M. Machover, The measurement of voting power: Theory and practice, problems and paradoxes, Edward Elgar, Cheltenham, 1998].

A decisiveness index for simple games

The decisiveness index introduced in this paper is designed to provide a normalized measure of the agility of all simple games, primarily viewed as collective decision-making mechanisms. We study the mathematical properties of the index and derive different axiomatic characterizations for it. Moreover, a close relationship is shown to the Banzhaf index of power––for which twice the decisiveness index plays the role of potential function––that gives rise to an effective computational procedure. Some real-world examples illustrate the usefulness of the decisiveness index, together with the Banzhaf power index, in applications to political science.

Faster algorithms for computing power indices in weighted voting games

We consider weighted voting games with n players. We show how to compute the Banzhaf power index for every player within a running time of O( n 2 1.415 n ), and how to compute the Shapley–Shubik power index within a running time of O( n 1.415 n ). Our result improves on the straightforward running times of O( n 2 2 n ) and O( n 2 n ), respectively, that are implicit in the definitions of these power indices.

Voting Power, Control Rents and Corporate Governance: An Integrated Analysis of Owner Control and Corporate Governance

This paper integrates existing knowledge in owner control and corporate governance. First it provides different measurements of voting power, and then links the voting power with private benefits of control in an analysis of control rent. The implication of dual classes of shares in a takeover contest is explored. As an application, the power structures of a group of Swedish listed firms are examined using the Shapely-Shubik power indices and the Banzhaf power indices. This paper provides a tool in conducting corporate governance studies, such as degree of control and corporate performance, takeover probability and private benefit of control, etc. Degree of control as an endogenous choice is partly determined by laws and cultural heritage of a country. This needs to be given special care when cross-country comparison is to be conducted. The consistency among corporate governance measures and the harmonization of corporate governance rules in a country setting are most important for the corporate governance system to work efficiently. As a final note, corporate governance rules should be adapted to different types of firms.

Valuation of voting scheme changes the cases of Electrolux AB and SKF AB

This paper studies the effects of the change of voting scheme on the market prices of Electrolux and SKF AB using standard event study methodology and a clinical approach. The economic effect of the voting scheme change is assessed using the market model. We investigate the loss of control due to the change of the voting scheme. The degree of the change of power is calculated using Shapley power index (SPI) and Banzhaf power index. There is a wealth transfer from the high vote shareholders to low vote shareholders in the process since in both cases the high power shareholders required no compensation. We expect that share price to have a positive response to such an announcement due to the reduced power discount and corporate governance improvement. The magnitude of the response on the event day depends also on the information structure of the period leading to the announcement.

Power Relations in an Embedded Institution: The European Bank for Reconstruction and Development

The European Bank for Reconstruction and Development (EBRD) is unique among the set of regional development banks in that non-regional members have substantial influence in decision making. Moreover, two other European institutions have direct influence on decision making in the EBRD's Board of Directors. This paper analyses the relative influence of members of the Board of Directors by applying three measures of a priori voting power - the Banzhaf, Johnston, and Shapley-Shubik power indices. Weighted voting schemes, such as the one used by the EBRD, have very distinct properties that allow for the calculation of indicators of voting power using simple, cooperative games in order to determine how voting weight translates into the ability to affect outcomes. Voting power analysis reveals that the United States and Japan have significant influence over electoral outcomes. Influence is concentrated in a subset of members in large part because of the presence of 'dictators' in the EBRD's voting groups. The results demonstrate that relative voting power often diverges from intended influence due to the way decision rules are structured and because of the use of weighted voting.

Computing power indices in weighted multiple majority games

The Shapley–Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. The Banzhaf power index depends on the number of ways in which each voter can effect a swing. If the input size of the problem is n, then the function which measures the worst case running time for computing these indices is in O n2 n . We present a method based on generating functions to compute these power indices efficiently for weighted multiple majority games and we study the temporal complexity of the algorithms. Finally, we apply the algorithms obtained with this method to compute the Banzhaf and the Shapley–Shubik indices under the two decision rules adopted in the Nice European Union summit.

Voting power in the European Union enlargement

The Shapley–Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. The Banzhaf power index depends on the number of ways in which each voter can effect a swing. If there are n players in a voting situation, then the function which measures the worst case running time for computing these indices is in O( n2 n ). We present a combinatorial method based in generating functions to compute these power indices efficiently in weighted double or triple majority games and we study the time complexity of the algorithms. Moreover, we calculate these power indices for the countries in the Council of Ministers of the European Union under the new decision rules prescribed by the Treaty of Nice.

NP-completeness for calculating power indices of weighted majority games

In this paper, we prove that both problems for calculating the Banzhaf power index and the Shapley-Shubik power index for weighted majority games are NP-complete.

NP-completeness for calculating power indices of weighted majority games

In this paper, we prove that both problems for calculating the Banzhaf power index and the Shapley–Shubik power index for weighted majority games are NP-complete.

Shapley-Shubik and Banzhaf Indices Revisited

We provide a new axiomatization of the Shapley-Shubik and the Banzhaf power indices in the domain of simple superadditive games by means of transparent axioms. Only anonymity is shared with the former characterizations in the literature. The rest of the axioms are substituted by more transparent ones in terms of power in collective decision-making procedures. In particular, a clear restatement and a weaker alternative for the transfer axiom are proposed. Only one axiom differentiates the characterization of either index, and these differentiating axioms provide a new point of comparison. In a first step both indices are characterized up to a zero and a unit of scale. Then both indices are singled out by simple normalizing axioms.

The influence of variables on pseudo-Boolean functions with applications to game theory and multicriteria decision making

The power of players in a collective decision process is a central issue in game theory. For this reason, the concept of influence of players on a simple game has been introduced. More generally, the influence of variables on Boolean functions has been defined and studied. We extend this concept to pseudo-Boolean functions, thus making it possible to appraise the degree of influence of any coalition of players in cooperative games. In the case of monotone games, we also point out the links with the concept of interaction among players. Although they do not have the same meaning at all, both influence and interaction functions coincide on singletons with the so-called Banzhaf power index. We also define the influence of variables on continuous extensions of pseudo-Boolean functions. In particular, the Lovász extension, also called discrete Choquet integral, is used in multicriteria decision making problems as an aggregation operator. In such problems, the degree of influence of decision criteria on the aggregation process can then be quite relevant information. We give the explicit form of this influence for the Choquet integral and its classical particular cases.

The Banzhaf power index on convex geometries

In this paper, we introduce the Banzhaf power indices for simple games on convex geometries. We define the concept of swing for these structures, obtaining convex swings. The number of convex swings and the number of coalitions such that a player is an extreme point are the basic tools to define the convex Banzhaf indices, one normalized and other probabilistic. We obtain a family of axioms that give rise to the Banzhaf indices. In the last section, we present a method to calculate the convex Banzhaf indices with the computer program Mathematica, and we apply this to compute power indices in the Spanish and Catalan parliaments and in the Council of Ministers of the European Union.

Voting strength in the European Parliament: The influence of national and of partisan actors

Abstract. Most decisions by the European Parliament are taken by an absolute majority of its members. Some decisions however – such as the approval of the budget of the European Union – require a two‐thirds majority. The paper analyzes the a priori voting strength of the member states when their representatives vote coherently. It is shown that the increase in votes for Germany in the 1994 reallocation enhanced its position. A less favourable effect, however, can be seen for the other large members (France, Italy, the United Kingdom, and Spain). However, since votes in the EP tend to be cast according to partisan rather than national affiliation, the relative voting power of the political groups with respect to the two quotas is also analyzed. The tool to measure this is the (normalized) Banzhaf power index, partially extended to account for connected coalitions. The paper demonstrates that the distribution of votes between the present EU member states as allocated in proportion to their population size indeed roughly corresponds to their a priori voting power. However, the relative influence of the largest political groups, the European Socialists and the European People's Party, tends to be overestimated by their share of seats in the framework of the simple majority rule, but it is considerable if the quota is two‐thirds. Finally, under the two‐thirds majority rule, the European Liberal, Democratic and Reformist Party as well as the small groups appear to be almost powerless. The more the EP gains political leverage – a further increase in its institutional powers is to be expected in the framework of the ongoing Intergovernmental Conference – the more the distribution of voting power between the member states and between the political groups will be a crucial factor in the shaping of EU policies.

Voting power when using preference ballots

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.

The Balance between Small and Large: Effects of a Double-Majority System on Voting Power in the European Union

In the beginning of 1995, the European Union (EU) has been enlarged by three countries: Austria, Finland, and Sweden. In view of the implications on the EU's institutional balance and the possible accession of more states in the future, different proposals are currently made to change the system of decision making to one that is more transparent and that more adequately represents member-states' population sizes. The issue will certainly be intensely debated in the framework of the 1996 Review Conference of the Maastricht Treaty. One of the current suggestions aims at establishing a double-majority voting system in the Council of the EU—formerly the Council of Ministers—requiring a majority of both member-states and their populations within that institution. Another one aims at introducing a bicameral system. This article analyzes what effects the resort to a double-majority voting system and the introduction of a bicameral structure would have in terms of the member-states' quantitative influence in the EU's decision-making process. Voting under the double-majority requirement is modeled by a simple game and relative voting power calculated by a (composite) Banzhaf power index.

Voting power in the EC decision making and the consequences of two different enlargements

In this paper, we study the voting power in the Council of Ministers of the EC. The purpose of this paper is twofold: first, to evaluate the voting power for the member states of today's Community and, second, to analyse the balance of power after two different enlargements of the Community. The analysis is based on the Shapley-Shubik and the Banzhaf power indices of simple cooperative games. The modified version of the Shapley-Shubik index is used to analyse how the sub-systems of the EC can change the distribution of power in the decision making process.