Banzhaf Index Research Articles

Compound voting and the Banzhaf index

We present three axioms for a power index defined on the domain of simple (voting) games. Positivity requires that no voter has negative power, and at least one has positive power. Transfer requires that, when winning coalitions are enhanced in a game, the change in voting power depends only on the change in the game, i.e., on the set of new winning coalitions. The most crucial axiom is composition: the value of a player in a compound voting game is the product of his power in the relevant first-tier game and the power of his delegate in the second-tier game. We prove that these three axioms categorically determine the Banzhaf index.

Indikátory hlasovací síly v Evropské unii

The article is concerned with voting indicators in the European Union. The first chapter constructs a model of voting power and defines four indicators: A. Simple relative power, B. Shapley-Shubik index, C. Banzhaf index, D. Coleman index. The second chapter defines data: the voting structure according the Nice Treaty and nine algorithms - variants of the EU enlargement. The third chapter summarizes empirical results. The fourth chapter discusses statistical relations between voting indicators. The fifth chapter brings conclusions.

A Priori versus Empirical Voting Power in the EU Council of Ministers

Numerous scholarly articles have investigated member states’ a priori voting power in past, present and possible future Councils of the European Union (EU). This article introduces three empirically oriented variants of the standardized Banzhaf index and compares them with the standardized Banzhaf index itself. The comparison suggests that the averages of the empirically oriented indices converge towards the Banzhaf index, and hence that there is no dominant ordering of the member states or the issues and, consequently, no stable minimal winning coalitions over time. However, statistical analysis reveals that the country-bycountry average difference from the Banzhaf index, considering all issues, remains significant in all three variants of empirical power.

Voting Power and Voting Blocs

We investigate the method of power indices to study voting power of members of a legislature that has voting blocs. Our analysis is theoretical, intended to contribute to a theory of positive political science in which social actors are motivated by the pursuit of power as measured by objective power indices. Our starting points are the papers by Riker (Behavioural Science, 1959, A test of the adequacy of the power and Coleman (American Sociological Review, 1973, Loss of Power). We argue against the Shapley-Shubik index and show that anyway the Shapley-Shubik index per head is inappropriate for voting blocs. We apply the Penrose index (the absolute Banzhaf index) to a hypothetical voting body with 100 members. We show how the power indices of individual bloc members can be used to study the implications of the formation of blocs and how voting power varies as bloc size varies. We briefly consider incentives to migrate between blocs. This technique of analysis has many real world applications to legislatures and international bodies. It can be generalised in many ways: our analysis is a priori (assuming formal voting and ignoring actual voting behaviour) but can be made empirical with voting data reflecting behaviour; it examines the consequences of two blocs but can easily be extended to more.

L.S. Penrose's limit theorem: proof of some special cases

L.S. Penrose was the first to propose a measure of voting power (which later came to be known as ‘the [absolute] Banzhaf (Bz) index’). His limit theorem—which is implicit in his booklet (1952) and for which he gave no rigorous proof—says that in simple weighted voting games (WVGs), if the number of voters increases indefinitely while the quota is pegged at half the total weight, then—under certain conditions—the ratio between the voting powers (as measured by him) of any two voters converges to the ratio between their weights. We conjecture that the theorem holds, under rather general conditions, for large classes of variously defined WVGs, other values of the quota, and other measures of voting power. We provide proofs for some special cases.

The Treaty of Nice and the Distribution of Votes in the Council - Voting Power Consequences for the EU after the Oncoming Enlargement

The IGC 2000 had the objective to prepare the European Union for the enlargement. The aim of the present paper is to evaluate the Nice solution of the Council's decision-making rules from the perspective of the decision on the number of candidate countries which should join the EU in 2004, as proposed by the Brussels European Council in October 2002 and approved by the Copenhagen European Council last December. The evaluation is based on the comparison of the Nice solution with all the relevant proposals, which were presented to the IGC 2000. The results bring an assessment of the voting power effect of both the Nice rules and the proposals. The reason for this comparison consists in the fact that these proposals represented the spectrum of approaches considered as plausible, during the last IGC and some of them much earlier. Since ten designated candidate countries should be regular participants of the next IGC and since these proposals might be picked up again if the question of decision-making rules in the Council is reopened by the next IGC, the question of respective voting power consequences of the proposals for the EU25 should be answered. The method used for evaluation of voting power consequences of individual solutions is based on the Banzhaf index. The comparison leads to the conclusions that the distribution of voting power in the Council after enlargement to 25 members will be more even in terms of equitable representation of population of the member countries than most of other solutions proposed to the IGC, but is far from optimal as it disadvantages a group of smaller members to a greater degree than it disadvantages the most populous countries. Another conclusion is that the population criterion will not influence the voting power of member states after oncoming enlargement and would not be effective in any other dual weighted majority solution.

Power Analysis of the Nice Treaty on the Future of European Integration

I carry out a power analysis of changes in voting weights and rules in the Nice Treaty of the EU on the widening and deepening of European integration, by applying methods that use Shapley-Shubik and Banzhaf indices. Significant decrease in voting power of small countries makes widening of integration more acceptable to incumbent members due to small size of the applicants. Relative increase in the conciliatory power of smaller members, and relative increase in the independent power of bigger members make smaller members compromise more in the coalitions they form, and improve the position of large members for further deepening of the integration. Lastly, the fairness analysis reveals a more federalist face for the EU in the way votes are distributed in Nice.

Power indices and the veil of ignorance

We provide an axiomatic foundation of the expected utility preferences over lotteries on roles in simple superadditive games represented by the two main power indices, the Shapley-Shubik index and the Banzhaf index, when they are interpreted as von Neumann-Morgenstern utility functions. Our axioms admit meaningful interpretations in the setting proposed by Roth in terms of different attitudes toward risk involving roles in collective decision procedures under the veil of ignorance. In particular, an illuminating interpretation of `efficiency', up to now missing in this set up, as well as of the corresponding axiom for the Banzhaf index, is provided.

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.

Paradoxes of voting power in Dutch politics

In this paper we first evaluate thirteen seat distributions inthe Second Chamber of the Dutch parliament by means of severalindices of voting power. Subsequently, we search for theoccurrence of the paradox of redistribution, the paradox ofnew members, and the paradox of large size for each powerindex. The indices used are the Shapley-Shubik index, thenormalized Banzhaf index, the Penrose-Banzhaf index, theHoller index, and the Deegan-Packel index.

Characterizations of two power indices for voting games with r alternatives

In the first three sections of this paper we present a set of axioms which provide a characterization of an extension of the Banzhaf index to voting games with r alternatives, such as the United Nations Security Council where a nation can vote “yes”, “no”, or “abstain”. The fourth section presents a set of axioms which characterizes a power index based on winning sets instead of pivot sets.

A method for evaluating the behavior of power indices in weighted plurality games

In this paper, a systematic method to facilitate the comparison of a priori measures of power in an n-player r-candidate (n, r) weighted plurality game is proposed. This method, which exploits the notion of a structure of embedded winning coalitions (SEWC), enables the listing of all power profiles relevant to an (n, r) game under a given index and permits the computation of the probability of occurrence of each of these profiles. The vulnerability of an index to different paradoxes of power can also be systematically studied. For the purpose of illustration, we apply this method to the analysis of four well-known 2-candidate power indices namely the Shapley-Shubik index, the Banzhaf index, the Johnston index and the Deegan-Packel index. In each case, the set of power profiles and the likelihood of occurrence of each of these profiles are enumerated. The superadditivity property of these indices is also studied.

Inequality among EU citizens in the EU's Council decision procedure

This paper provides a normative assessment of the inequality among EU citizens imputable to the decision-making rule used by the European Council of Ministers. To this end, the EU decision-making process is modeled by a compound two-stage game among the EU citizens, whose respective voting power is evaluated by their Banzhaf index in the years 1958–1999. Two inequality indices measure the resulting inequality in the distribution of power among citizens. This permits us to evaluate the actual qualified majority rule and to compare this rule with an alternative rule, the so-called double majority rule.

European Union enlargement: Power distribution implications of the new institutional arrangements

Abstract. It is argued that enlargement challenges institutional balances and in particular relative powers of national actors within the European Union (EU). This article concentrates on the impact of future enlargement (with the current negotiating 12 candidates) on power distribution in the Council of Ministers of the European Union and the European Parliament based on the decisions taken at the Nice Summit in December 1800. It uses the Shapley‐Shubik and Banzhaf indices to evaluate past and emerging power distributions in both the Council and in the Parliament. A brief section on Turkey (the thirteenth, non‐negotiating, official candidate) is included to evaluate its possible impact in the case of admission to the Union.

An Empirical Comparison of the Performance of Classical Power Indices

Power indices are general measures of the relative a priori voting power of individual members of a voting body. They are useful for both positive and normative analysis of voting bodies particularly those using weighted voting. This paper applies new algorithms for computing the rival Shapley-Shubik and Banzhaf indices for large voting bodies to shareholder voting power in a cross section of British companies. Each company is a separate voting body and there is much variation in ownership between them resulting in different power structures. Because the data are incomplete, both finite and ‘oceanic’ games of shareholder voting are analysed. The indices are appraised, using reasonable criteria, from the literature on corporate control. The results are unfavourable to the Shapley-Shubik index and suggest that the Banzhaf index much better reflects the variations in the power of shareholders between companies as the weights of shareholder blocs vary.

Inferior players in simple games

Power indices like those of Shapley and Shubik (1954) or Banzhaf (1965) measure the distribution of power in simple games. This paper points at a deficiency shared by all established indices: players who are inferior in the sense of having to accept (almost) no share of the spoils in return for being part of a winning coalition are assigned substantial amounts of power. A strengthened version of the dummy axiom based on a formalized notion of inferior players is a possible remedy. The axiom is illustrated first in a deterministic and then a probabilistic setting. With three axioms from the Banzhaf index, it uniquely characterizes the Strict Power Index (SPI). The SPI is shown to be a special instance of a more general family of power indices based on the inferior player axiom.

Influence of national governments for or against the entry into force of the Kyoto Protocol: a Banzhaf index analysis

The recent announcement by the USA to withdraw their support for the Kyoto Protocol has drawn attention to the conditions under which the Protocol will enter into force: at least 55 member states to the United Nations Framework Convention on Climate Change (UNFCCC) have to ratify the Protocol, and those industrialized countries (Annex-I parties) that ratify it have to make up for at least 55% of the carbon dioxide emissions of this group in 1990. In this article, we analyze the a priori decision making power of individual parties, or groups of these, for the entry into force of the Kyoto Protocol, utilizing a mathematical approach known as the Banzhaf index. The results of our analysis suggest that the conditions for the entry into force of the Kyoto Protocol give more influence to the USA than might have been anticipated: while the USA has more voting power than the relative share of emissions suggests, all other Annex-I parties have less. Based on a mathematical model, this analysis shows that the Banzhaf index is a more realistic indicator of voting power than the bare numbers fixed in the Kyoto Protocol. This method could be used more extensively in the future for the analysis of voting procedures, in particular within the UNFCCC.

The Most Dangerous Justice Rides Again: Revisiting the Power Pageant of the Justices

Who is the most powerful Supreme Court Justice? In 1996 we measured voting power on the Court according to each Justice's ability to form five-member coalitions. From the set of all coalitions formed by the Court during its 1994 and 1995 Terms, we developed a generalized Banzhaf index of the Justices' relative strength. Generally speaking, participating in a greater number of unique coalitions translates into greater judicial voting power. To supplement the small number of decisions then available, we derived hypothetical five-Justice coalitions from the intersections of actually observed coalitions involving more than five members. Professor Lynn Baker contested our model, favoring instead an additive measure based on the number of times each Justice participated in any winning coalition. This Article stages a new Power Pageant of the Justices in light of the cases decided by the Court since 1996. For more than seven Terms, the Court has retained the same personnel. This stability provides a unique opportunity to test competing measures of judicial voting power. We hypothesized in 1996 that a larger set of cases might obviate the need to resort to inferred coalitions. Analysis of this larger data set in fact undermines the validity of both our measure and that of Professor Baker over the long run. We conclude that there are three different measures of voting power each reflecting a different aspect of judicial power.

Influence of national governments for or against the entry into force of the Kyoto Protocol: a Banzhaf index analysis

The recent announcement by the USA to withdraw their support for the Kyoto Protocol has drawn attention to the conditions under which the Protocol will enter into force: at least 55 member states to the United Nations Framework Convention on Climate Change (UNFCCC) have to ratify the Protocol, and those industrialized countries (Annex-I parties) that ratify it have to make up for at least 55% of the carbon dioxide emissions of this group in 1990. In this article, we analyze the a priori decision making power of individual parties, or groups of these, for the entry into force of the Kyoto Protocol, utilizing a mathematical approach known as the Banzhaf index. The results of our analysis suggest that the conditions for the entry into force of the Kyoto Protocol give more influence to the USA than might have been anticipated: while the USA has more voting power than the relative share of emissions suggests, all other Annex-I parties have less. Based on a mathematical model, this analysis shows that the Banzhaf index is a more realistic indicator of voting power than the bare numbers fixed in the Kyoto Protocol. This method could be used more extensively in the future for the analysis of voting procedures, in particular within the UNFCCC.

Constitutional analysis: The power index approach

Abstract. A power index approach to the EU institutions gives a new perspective upon the EU institutions and their future reform. Using a standard power measure, the Banzhaf index, we show that in a group of states as the EU council, the voting power of a member state is equal to twice the individual power to block multiplied by the group's collective power to change. Political power of the member states is calculated under alternative constitutional rules for the EU, where cooperative game theory allows the derivation of power equivalences. We suggest that simple majority should be used more often in the EU council.