16 research outputs found
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
. 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.