Voting Power Derives from the Poll Distribution. Shedding Light on Contentious Issues of Weighted Votes and the Constitutional Treaty

Analysis of the Constitutional Treaty of the European Union shows that there is a serious discrepancy between the voting power gradient of Member States computed by the Shapley-Shubik and Banzhaf indices. Given the lack of compelling arguments to choose between these indices on purely axiomatic grounds, we turn to a probabilistic approach as pioneered by Straffin (1977) focusing on the probability distribution of voting poll outcomes. We present a unifying model of power indices as expected decisiveness, which shows that the defining feature of each approach is a particular distribution of the voting poll. Empirical evidence drawn from voting situations, in addition to a consideration of first principles, leads us to reject one of these approaches. The unified formulation allows us to develop useful related concepts of efficiency and blocking leverage, previously used solely by a 'Banzhaf' approach, for the case of Shapley-Shubik, and a comparison of results is shown.Voting power indices, Power gradient, Coefficient of representation, Expected decisiveness, Efficiency, Blocking leverage, Constitution of the European Union

THE USE OF COLEMAN'S POWER INDICES TO INFORM THE CHOICE OF VOTING RULE WITH REFERENCE TO THE IMF GOVERNING BODY AND THE EU COUNCIL OF MINISTERS

In his well known 1971 paper the mathematical sociologist James S. Coleman, proposed three measures of voting power : (1) "the power of a collectivity to act", (2) "the power to prevent action" and (3) "the power to initiate action". (1) is a measure of the overall decisiveness of a voting body taking into account its size, decision rule and the weights of its members, while (2) and (3) are separate indices of the power of individual members, in being able to block or achieve decisions. These measures seem to have been little used for a variety of reasons, although the paper itself is widely cited. First, much of the power indices literature has focused on normalised indices which gives no role to (1) and means that (2) and (3) are identical. Second, Coleman's coalition model is different from that of Shapley and Shubik which has sometimes tended to dominate in discussions of voting power. Third, (2) and (3) are indistinguishable when the decision quota is a simple majority, the distinction becoming important in other voting situations. In this paper I propose that these indices, which are based on a fundamentally different notion of power than that assumed by game-theoretic approaches, have a useful role in aiding a better understanding of collective institutions in which decisions are taken by voting. I use them to illustrate different aspects of the design of a weighted voting system such as the governing body of the IMF or World Bank, or the system of QMV in the European Council.

Measuring influence in command games

In the paper, we study a relation between command games proposed by Hu and Shapley and an influence model. We show that our framework of influence is more general than the framework of the command games. We define several influence functions which capture the command structure. These functions are compatible with the command games, in the sense that each commandable player for a coalition in the command game is a follower of the coalition under the command influence function. Some of the presented influence functions are equivalent to the command games, that is, they are compatible with the command games, and additionally each follower of a coalition under the command influence function is also a commandable player for that coalition in the command games. For some influence functions we define the equivalent command games. We show that not for all influence functions the compatible command games exist. Moreover, we propose a more general definition of the influence index and show that under some assumptions, some power indices, which can be used in the command games, coincide with some expressions of the weighted influence indices. Both the Shapley-Shubik index and the Banzhaf index are equal to a difference between the weighted influence indices under some influence functions, and the only difference between these two power indices lies in the weights for the influence indices. An example of the Confucian model of society is broadly examined.influence function; follower; influence index; command game; commandable player; Shapley-Shubik index; Banzhaf index

Efficiency of Fairness in Voting Systems

Fair representation of voters in a committee representing different voters’ groups is being broadly discussed during last few years. Assuming we know what the fair representation is, there exists a problem of optimal quota: given a “fair” distribution of voting weights, how to set up voting rule (quota) in such a way that distribution of relative a priori voting power is as close as possible to distribution of relative voting weights. Together with optimal quota problem a problem of trade-off between fairness and efficiency (ability of a voting body to change status quo) is formalized by a fairness-efficiency matrix.Committee system, efficiency, fairness, fairness-efficiency matrix, indirect voting power, optimal quota, power indices, voting system

Power measures derived from the sequential query process

We study a basic sequential model for the discovery of winning coalitions in a simple game, well known from its use in defining the Shapley-Shubik power index. We derive in a uniform way a family of measures of collective and individual power in simple games, and show that, as for the Shapley-Shubik index, they extend naturally to measures for TU-games. In particular, the individual measures include all weighted semivalues. We single out the simplest measure in our family for more investigation, as it is new to the literature as far as we know. Although it is very different from the Shapley value, it is closely related in several ways, and is the natural analogue of the Shapley value under a nonstandard, but natural, definition of simple game. We illustrate this new measure by calculating its values on some standard examples.Comment: 13 pages, to appear in Mathematical Social Science

"Decision Making in Europe: Were Spain and Poland Right to Stop the Constitution in December 2003?"

This paper tries to explain why Spain and Poland stopped the Draft Constitution for the European Union in December 2003 and discusses whether this action was compatible with these countries long term interests. The author finds that the decline in power – measured by a power index – arising for Spain and Poland when going from the Nice Treaty to the Draft Constitution cannot explain their veto. While the two countries lose power when shifting from Nice to the Draft Constitution other countries’ power shrinks even more. Other measures - passage probability, blocking leverage and fairness - cannot explain the two countries’ opposition either. This paper contends that the Spanish and Polish rejection can be explained by the weakness of government in the Polish and the need for a reelection topic in the Spanish case. Furthermore this paper asserts that the Spanish and Polish government’s veto was against the medium and long term interest of their own countries. Poland and Spain must have been able to anticipate that the Nice Treaty would not last due to most EU countries’ dislike of it. An analysis of reasonable alternative voting schemes in the EU finds that Spain and Poland would not have been better off in any of these cases and worse off in most; under the voting rules agreed upon under the Irish presidency in June 2004 the two countries are weaker than under the Draft Constitution

The use of Coleman's power indices to inform the choice of voting rule with reference to the IMF governing body and the EU Council of Ministers

In his well known 1971 paper the mathematical sociologist James S. Coleman, proposed three measures of voting power: (1) "the power of a collectivity to act", (2) "the power to prevent action" and (3) "the power to initiate action". (1) is a measure of the overall decisiveness of a voting body taking into account its size, decision rule and the weights of its members, while (2) and (3) are separate indices of the power of individual members, in being able to block or achieve decisions. These measures seem to have been little used for a variety of reasons, although the paper itself is widely cited. First, much of the power indices literature has focused on normalised indices which gives no role to (1) and means that (2) and (3) are identical. Second, Coleman's coalition model is different from that of Shapley and Shubik which has sometimes tended to dominate in discussions of voting power. Third, (2) and (3) are indistinguishable when the decision quota is a simple majority, the distinction becoming important in other voting situations. In this paper I propose that these indices, which are based on a fundamentally different notion of power than that assumed by game-theoretic approaches, have a useful role in aiding a better understanding of collective institutions in which decisions are taken by voting. I use them to illustrate different aspects of the design of a weighted voting system such as the governing body of the IMF or World Bank, or the system of QMV in the European Council

Average Weights and Power in Weighted Voting Games

We investigate a class of weighted voting games for which weights are randomly distributed over the standard probability simplex. We provide close-formed formulae for the expectation and density of the distribution of weight of the $k$-th largest player under the uniform distribution. We analyze the average voting power of the $k$-th largest player and its dependence on the quota, obtaining analytical and numerical results for small values of $n$ and a general theorem about the functional form of the relation between the average Penrose--Banzhaf power index and the quota for the uniform measure on the simplex. We also analyze the power of a collectivity to act (Coleman efficiency index) of random weighted voting games, obtaining analytical upper bounds therefor.Comment: 12 pages, 7 figure

Measuring influence in command games

URL des Documents de travail : http://ces.univ-paris1.fr/cesdp/CESFramDP2008.htmClassification JEL : C7, D7.Documents de travail du Centre d'Economie de la Sorbonne 2008.78 - ISSN : 1955-611XIn the paper, we study a relation between command games proposed by Hu and Shapley and an influence model. We show that our framework of influence is more general than the framework of the command games. We define several influence functions which capture the command structure. These functions are compatible with the command games, in the sense that each commandable player for a coalition in the command game is a follower of the coalition under the command influence function. Some of the presented influence functions are equivalent to the command games, that is, they are compatible with the command games, and additionally each follower of a coalition under the command influence function is also a commandable player for that coalition in the command games. For some influence functions, we define the equivalent command games. We show that not for all influence functions the compatible command games exist. Moreover, we propose a more general definition of the influence index and show that under some assumptions, some power indices, which can be used in the command games, coincide with some expressions of the weighted influence indices. Both the Shapley-Shubik index and the Banzhaf index are equal to a difference between the weighted influence indices under some influence functions, and the only difference between thes two power indices lies in the weights for the influence indices. An example of the Confucian model od society is broadly examined.Nous étudions la relation entre un jeu de commande de Hu et Shapley et un modèle d'influence. Nous montrons que notre modèle d'influence est plus général. Nous définissons plusieurs fonctions d'influence qui modélisent un jeu de commande. Pour certaines fonctions d'influence, nous définissons les jeux de commande équivalents. Nous proposons également une forme générale de l'indice d'influence et montrons qu'ils permettent de retrouver des indices de pouvoir liés aux jeux de commande

The Double Majority Voting Rule of the EU Reform Treaty as a Democratic Ideal for an Enlarging Union : an Appraisal Using Voting Power Analysis

The Double Majority rule in the Treaty is claimed to be simpler, more transparent and more democratic than the existing rule. We examine these questions against the democratic ideal that the votes of all citizens in whatever member country should be of equal value using voting power analysis considering possible future enlargements involving candidate countries and then to a number of hypothetical future enlargements. We find the Double Majority rule to fails to measure up to the democratic ideal in all cases. We find the Jagiellonian compromise to be very close to this ideal.European Union ; Reform Treaty ; Nice Treaty ; Qualified Majority Voting ; Power Indices