Combinatorial and computational aspects of multiple weighted voting games

Weighted voting games are ubiquitous mathematical models which are used in economics, political science, neuroscience, threshold logic, reliability theory and distributed systems. They model situations where agents with variable voting weight vote in favour of or against a decision. A coalition of agents is winning if and only if the sum of weights of the coalition exceeds or equals a specified quota. We provide a mathematical and computational characterization of multiple weighted voting games which are an extension of weighted voting games1. We analyse the structure of multiple weighted voting games and some of their combinatorial properties especially with respect to dictatorship, veto power, dummy players and Banzhaf indices. Among other results we extend the concept of amplitude to multiple weighted voting games. An illustrative Mathematica program to compute voting power properties of multiple weighted voting games is also provided

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

El reparto del poder en la Constitución europea

Se analiza el poder decisional de los países de la Unión Europea, con respecto a la regla de votación incluida en el Tratado de la Constitución Europea, aprobado en Bruselas. En la cumbre de Bruselas, celebrada el 18 de junio de 2004, los Jefes de Estado y de Gobierno de la Unión Europea aprobaron la siguiente modificación del texto de la Constitución Europea: “Una mayoría cualificada se definirá como al menos el 55% de los miembros del Consejo, incluyendo al menos a quince de ellos y representando a Estados miembros que tengan al menos el 65% de la población de la Unión. Una minoría de bloqueo debe incluir al menos cuatro miembros del Consejo”. En este análisis se exponen los porcentajes de poder de Banzhaf de los países de la Unión (25 en la actualidad y 27 cuando se incorporen Bulgaria y Rumania) con las reglas de votación de Niza, de la Convención Europea y de la cumbre de Bruselas. Además, se estudia el poder decisional de cada ciudadano europeo con el modelo de Penrose

A Note on the Shapley Value for Characteristic Functions on Bipartitions

We consider a cooperative game with a bipartition that indicates which players are participating. This paper provides an analytical solution for the Shapley value when the worth of a coalition only depends on the number of participating coalition players. The computational complexity grows linearly in the number of players, which contrasts with the usual exponential increase. Our result remains true when we introduce (i) randomization of the bipartition, and (ii) randomly draw a characteristic function

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

El poder de las naciones en la Unión Europea

En este trabajo se definen algoritmos, basados en funciones generatrices, para calcular el índice de poder de Banzhaf en juegos simples de votación ponderada y en juegos de doble y triple mayoría. La utilización de funciones generatrices permite un cálculo exacto del índice de Banzhaf con una reducción sensible de la complejidad temporal. Además se calculan los índices de Banzhaf para las reglas de decisión, aprobadas en la cumbre de Niza, que se utilizan en la Unión Europea ampliada a 25 países. Finalmente, se demuestra que los sistemas de triple mayoría adoptados son equivalentes en la práctica a juegos de mayoría simple o doble, porque la cuota de población exigida para aprobar una decisión no cambia el índice de Banzhaf de los países de la Unión Europea ampliada.Juegos de votación; índice de Banzhaf; Unión Europea

The Inter-Institutional Distribution of Power in EU Codecision

This paper analyzes the a priori influence of the European Parliament (EP) and the Council of Ministers (CM) on legislation of the European Union adopted under its codecision procedure. In contrast to studies which use conventional power indices, both institutions are assumed to act strategically. Predicted bargaining outcomes of the crucial Conciliation stage of codecision are shown to be strongly biased towards the legislative status quo. Making symmetric preference assumptions for members of CM and EP, CM is on average much more conservative because of its internal qualified majority rule. This makes CM by an order of magnitude more influential than EP, in contrast to a seeming formal parity between the two ‘co-legislators’.power measurement, European Union codecision procedure, bargaining, spatial voting, decision procedures