Penrose voting system and optimal quota

Systems of indirect voting based on the principle of qualified majority can be analysed using the methods of game theory. In particular, this applies to the voting system in the Council of the European Union, which was recently a subject of a vivid political discussion. The a priori voting power of a voter measures his potential influence over the decisions of the voting body under a given decision rule. We investigate a system based on the law of Penrose, in which each representative in the voting body receives the number of votes (the voting weight) proportional to the square root of the population he or she represents. Here we demonstrate that for a generic distribution of the population there exists an optimal quota for which the voting power of any state is proportional to its weight. The optimal quota is shown to decrease with the number of voting countries.Comment: 12 pages, 2 figure

Square root voting system, optimal threshold and \pi

The problem of designing an optimal weighted voting system for the two-tier voting, applicable in the case of the Council of Ministers of the European Union (EU), is investigated. Various arguments in favour of the square root voting system, where the voting weights of member states are proportional to the square root of their population are discussed and a link between this solution and the random walk in the one-dimensional lattice is established. It is known that the voting power of every member state is approximately equal to its voting weight, if the threshold q for the qualified majority in the voting body is optimally chosen. We analyze the square root voting system for a generic 'union' of M states and derive in this case an explicit approximate formula for the level of the optimal threshold: q \simeq 1/2+1/\sqrt{{\pi} M}. The prefactor 1/\sqrt{{\pi}} appears here as a result of averaging over the ensemble of unions with random populations.Comment: revised version, 21 pages in late

Entropy computing via integration over fractal measures

We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and multifractal spectra. It is shown that with certain dynamical systems one can associate the corresponding IFSs in such a way that their generalized entropies are equal. This provides a new method of computing entropy for some classical and quantum dynamical systems. Numerical techniques are based on integration over the fractal measures.Comment: 14 pages in Latex, Revtex + 4 figures in .ps attached (revised version, new title, several changes, to appear in CHAOS

Mathematical aspects of degressive proportionality

We analyze properties of apportionment functions in context of the problem of allocating seats in the European Parliament. Necessary and sufficient conditions for apportionment functions are investigated. Some exemplary families of apportionment functions are specified and the corresponding partitions of the seats in the European Parliament among the Member States of the European Union are presented. Although the choice of the allocation functions is theoretically unlimited, we show that the constraints are so strong that the acceptable functions lead to rather similar solutions.Comment: several minor corrections, revised version 10 pages in two column style, one figure and two tables include

The allocation between EU member states of seats in the European Parliament

This note contains the recommendation for a mathematical basis for the apportionment of the seats in the European Parliament between the Member States of the European Union. This is the unanimous recommendation of the Participants in the Cambridge Apportionment Meeting, held at the instigation of the Committee on Constitutional Affairs at the Centre for Mathematical Sciences, University of Cambridge, on 28–29 January 2011

The allocation between the EU member states of the seats in the European Parliament Cambridge Compromise

This Note contains the recommendation for a mathematical basis for the apportionment of the seats in the European Parliament between the Member States of the European Union. This is the unanimous recommendation of the Participants in the Cambridge Apportionment Meeting, held at the instigation of the Committee on Constitutional Affairs at the Centre for Mathematical Sciences, University of Cambridge, on 28-29 January 2011.Proportional Representation, degressive proportionality, apportionment, European Parliament. Classification

The allocation between the EU member states of the seats in the European Parliament Cambridge Compromise

This Note contains the recommendation for a mathematical basis for the apportionment of the seats in the European Parliament between the Member States of the European Union. This is the unanimous recommendation of the Participants in the Cambridge Apportionment Meeting, held at the instigation of the Committee on Constitutional Affairs at the Centre for Mathematical Sciences, University of Cambridge, on 28-29 January 2011

Coherent States Measurement Entropy

Coherent states (CS) quantum entropy can be split into two components. The dynamical entropy is linked with the dynamical properties of a quantum system. The measurement entropy, which tends to zero in the semiclassical limit, describes the unpredictability induced by the process of a quantum approximate measurement. We study the CS--measurement entropy for spin coherent states defined on the sphere discussing different methods dealing with the time limit $n \to \infty$. In particular we propose an effective technique of computing the entropy by iterated function systems. The dependence of CS--measurement entropy on the character of the partition of the phase space is analysed.Comment: revtex, 22 pages, 14 figures available upon request (e-mail: [email protected]). Submitted to J.Phys.