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Wojciech Slomczynski

Publications and source records attributed to Wojciech Slomczynski.

13 recordsLinked to original sources
arXiv2012

Square root voting system, optimal threshold and π

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{π M}. The prefactor 1/\sqrt{π} appears here as a result of averaging over the ensemble of unions with random populations.

physics.soc-ph↗
arXiv2012

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.

physics.soc-ph↗
arXiv2007

From a toy model to the double square root voting system

We investigate systems of indirect voting based on the law of Penrose, in which each representative in the voting body receives the number of votes (voting weight) proportional to the square root of the population he or she represents. For a generic population distribution the quota required for the qualified majority can be set in such a way that the voting power of any state is proportional to its weight. For a specific distribution of population the optimal quota has to be computed numerically. We analyse a toy voting model for which the optimal quota can be estimated analytically as a function of the number of members of the voting body. This result, combined with the normal approximation technique, allows us to design a simple, efficient, and flexible voting system which can be easily adopted for varying weights and number of players.

physics.soc-ph↗
arXiv2006

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.

physics.soc-ph↗
arXiv2004

Utility maximizing entropy and the second law of thermodynamics

Expected utility maximization problems in mathematical finance lead to a generalization of the classical definition of entropy. It is demonstrated that a necessary and sufficient condition for the second law of thermodynamics to operate is that any one of the generalized entropies should tend to its minimum value of zero.

math.PR↗
arXiv2004

Voting in the European Union: The square root system of Penrose and a critical point

The notion of the voting power is illustrated by examples of the systems of voting in the European Council according to the Treaty of Nice and the more recent proposition of the European Convent. We show that both systems are not representative, in a sense that citizens of different countries have not the same influence for the decision taken by the Council. We present a compromise solution based on the law of Penrose, which states that the weights for each country should be proportional to the square root of its population. Analysing the behaviour of the voting power as a function of the quota we discover a critical point, which allows us to propose the value of the quota to be 62%. The system proposed is simple (only one criterion), representative, transparent, effective and objective: it is based on a statistical approach and does not favour nor handicap any European country.

cond-mat.other↗
arXiv2003

Quantum Iterated Function Systems

Iterated functions system (IFS) is defined by specifying a set of functions in a classical phase space, which act randomly on an initial point. In an analogous way, we define a quantum iterated functions system (QIFS), where functions act randomly with prescribed probabilities in the Hilbert space. In a more general setting a QIFS consists of completely positive maps acting in the space of density operators. We present exemplary classical IFSs, the invariant measure of which exhibits fractal structure, and study properties of the corresponding QIFSs and their invariant states.

quant-ph↗
arXiv2001

Monge Metric on the Sphere and Geometry of Quantum States

Topological and geometrical properties of the set of mixed quantum states in the N-dimensional Hilbert space are analysed. Assuming that the corresponding classical dynamics takes place on the sphere we use the vector SU(2) coherent states and the generalised Husimi distributions to define the Monge distance between arbitrary two density matrices. The Monge metric has a simple semiclassical interpretation and induces a non-trivial geometry. Among all pure states the distance from the maximally mixed state ρ_*, proportional to the identity matrix, admits the largest value for the coherent states, while the delocalized 'chaotic' states are close to ρ_*. This contrasts the geometries induced by the standard (trace, Hilbert-Schmidt or Bures) metrics, where the distance from ρ_* is the same for all pure states. We discuss possible physical consequences including unitary time evolution and the process of decoherence. We introduce also a simplified Monge metric, defined in the space of pure quantum states, and more suitable for numerical computation.

quant-ph↗
arXiv2000

Dynamical entropy for systems with stochastic perturbation

Dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively. Since for such systems the KS-entropy diverges we analyse the difference between the total entropy of a noisy system and the entropy of the noise itself. We show that this quantity is non negative and in the weak noise limit is conjectured to tend to the KS-entropy of the deterministic system. In particular, we consider one-dimensional systems with noise described by a finite-dimensional kernel, for which the Frobenius-Perron operator can be represented by a finite matrix.

chao-dyn↗
arXiv1999

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.

chao-dyn↗
arXiv1997

Monge Distance between Quantum States

We define a metric in the space of quantum states taking the Monge distance between corresponding Husimi distributions (Q--functions). This quantity fulfills the axioms of a metric and satisfies the following semiclassical property: the distance between two coherent states is equal to the Euclidean distance between corresponding points in the classical phase space. We compute analytically distances between certain states (coherent, squeezed, Fock and thermal) and discuss a scheme for numerical computation of Monge distance for two arbitrary quantum states.

quant-ph↗
arXiv1996

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.

chao-dyn↗