Jean-Louis Rouet

On the voting power of an alliance and the subsequent power of its members

Even, and in fact chiefly, if two or more players in a voting game have on a binary issue independent opinions, they may have interest to form a single voting alliance giving an average gain of influence for all of them. Here, assuming the usual independence of votes, we first study the alliance voting power and obtain new results in the so-called asymptotic limit for which the number of players is large enough and the alliance weight remains a small fraction of the total of the weights. Then, we propose to replace the voting game inside the alliance by a random game which allows new possibilities. The validity of the asymptotic limit and the possibility of new alliances are examined by considering the decision process in the Council of Ministers of the European Union.

Influence of expansion on hierarchical structure

We study a one-dimensional model of gravitational instability in an Einstein-de Sitter universe. Scaling in both space and time results in an autonomous set of coupled Poisson-Vlasov equations for both the field and phase space density, and the N-body problem. Using dynamical simulation, we find direct evidence of hierarchical clustering. A multifractal analysis reveals a bifractal geometry similar to that observed in the distribution of galaxies. To demonstrate the role of scaling, we compare the system to other one-dimensional models recently employed to study structure formation. Finally we show that the model yields an estimate of the time of galaxy formation of the correct order.

Cosmology in one dimension: fractal geometry, power spectra and correlation

Concentrations of matter, such as galaxies and galactic clusters, originated as very small density fluctuations in the early universe. The existence of galaxy clusters and super-clusters suggests that a natural scale for the matter distribution may not exist. A point of controversy is whether the distribution is fractal and, if so, over what range of scales. One-dimensional models demonstrate that the important dynamics for cluster formation occur in the position?velocity plane. Here the development of scaling behavior and multifractal geometry is investigated for a family of one-dimensional models for three different, scale-free, initial conditions. The methodology employed includes: (1)?the derivation of explicit solutions for the gravitational potential and field for a one-dimensional system with periodic boundary conditions (Ewald sums for one dimension); (2)?the development of a procedure for obtaining scale-free initial conditions for the growing mode in phase space for an arbitrary power-law index; (3)?the evaluation of power spectra, correlation functions, and generalized fractal dimensions at different stages of the system evolution. It is shown that a simple analytic representation of the power spectra captures the main features of the evolution, including the correct time dependence of the crossover from the linear to nonlinear regime and the transition from regular to fractal geometry. A possible physical mechanism for understanding the self-similar evolution is introduced. It is shown that hierarchical cluster formation depends both on the model and on the initial power spectrum. Under special circumstances a simple relation between the power spectrum, correlation function, and correlation dimension in the highly nonlinear regime is confirmed.

Groundwater Vulnerability and Risk Mapping Based on Residence Time Distributions: Spatial Analysis for the Estimation of Lumped Parameters

Specific vulnerability estimations for groundwater resources are usually geographic information system-based (GIS) methods that establish spatial qualitative indexes which determine the sensitivity to infiltration of surface contaminants, but with little validation of the working hypothesis. On the other hand, lumped parameter models, such as the Residence Time Distribution (RTD), are used to predict temporal water quality changes in drinking water supply, but the lumped parameters do not incorporate the spatial variability of the land cover and use. At the interface between these two approaches, a GIS tool was developed to estimate the lumped parameters from the vulnerability mapping dataset. In this method the temporal evolution of groundwater quality is linked to the vulnerability concept on the basis of equivalent lumped parameters that account for the spatially distributed hydrodynamic characteristics of the overall unsaturated and saturated flow nets feeding the drinking water supply. This vulnerability mapping method can be validated by field observations of water concentrations. A test for atrazine specific vulnerability of the Val d’Orléans karstic aquifer demonstrates the reliability of this approach for groundwater contamination assessment.

Three ways to compute accurately the probability of the referendum paradox

In a recent paper published in MSS, Wilson and Pritchard (2007) exhibit some results suggesting that the limiting probability of the referendum paradox given in Feix et al. (2004) could be wrong. After having explained the origin of this disagreement, we propose in this note some further analytical (and complementary) methods to compute the probability of this paradox.

Ewald sums for one dimension.

We derive analytic solutions for the potential and field in a one-dimensional system of masses or charges with periodic boundary conditions, in other words, Ewald sums for one dimension. We also provide a set of tools for exploring the system evolution and show that it is possible to construct an efficient algorithm for carrying out simulations. In the cosmological setting we show that two approaches for satisfying periodic boundary conditions-one overly specified and the other completely general-provide a nearly identical clustering evolution until the number of clusters becomes small, at which time the influence of any size-dependent boundary cannot be ignored. Finally, we compare the results with other recent work with the hope to provide clarification over differences these issues have induced.

Cosmology in one dimension: Vlasov dynamics

Numerical simulations of self-gravitating systems are generally based on N-body codes, which solve the equations of motion of a large number of interacting particles. This approach suffers from poor statistical sampling in regions of low density. In contrast, Vlasov codes, by meshing the entire phase space, can reach higher accuracy irrespective of the density. Here, we perform one-dimensional Vlasov simulations of a long-standing cosmological problem, namely, the fractal properties of an expanding Einstein-de Sitter universe in Newtonian gravity. The N-body results are confirmed for high-density regions and extended to regions of low matter density, where the N-body approach usually fails.

Fractal dimension computation from equal mass partitions.

Numerical methods which utilize partitions of equal-size, including the box-counting method, remain the most popular choice for computing the generalized dimension of multifractal sets. However, it is known that mass-oriented methods generate relatively good results for computing generalized dimensions for important cases where the box-counting method is known to fail. Here, we revisit two mass-oriented methods and discuss their strengths and limitations.

On the Probability to Act in the European Union

Since its foundation by Arrow in his seminal contribution (Arrow, 1963), one of the main merit of social choice theory has been to provide a coherent framework for the analysis and comparison of different voting rules. First, many normative requirements about voting rules can be expressed precisely in this framework. Then it is possible to check whether a given voting rule satisfies a given property. Ideally, this type of analysis may lead to the axiomatic characterization of a voting rule. At last the propensity of situations for which a voting rule fails to satisfy a condition can be evaluated. Peter Fishburns contributions to this research program have been extremely important. For example, he proposed many new normative conditions for the analysis of voting rules (see in particular Fishburn, 1974, 1977; Fishburn & Brams, 1983), and developed axiomatic analysis for binary voting (Fishburn, 1973) and approval voting (Fishburn, 1978). Together with Gehrlein, he launched an important research program on the probabilistic analysis of voting rules. After Guilbaulds paper (Guilbauld, 1952), the use of probability models in voting was limited to the evaluation of the majority voting paradox under the assumption that each voter would pick his preference independently from the others from a uniform distribution. This assumption, today called the Impartial Culture assumption, puts an equal weight on each profile. Fishburn and Gehrlein developed the use of probabilistic models in two directions. First, to analyze the occurrence of Condorcet cycles, they proposed in Gehrlein and Fishburn (1976) a new probability assumption, the Impartial Anonymous Culture assumption, which assumes that each anonymous profile is equally likely to appear. Secondly, they applied these two probability models to a wider range of problems, the relationships between the scoring rules and the Condorcet principle being their favorite issue (see Fishburn & Gehrlein, 1976; Gehrlein & Fishburn, 1978a, 1978b). The results we will present in this paper are clearly a continuation of this research program, as we will compare voting rules suggested for the European Union on their propensity to fulfill a given property according to different probability assumptions.

Collective Effects Triggered by Individual Effects in One‐Dimensional Plasmas

Abstract The relationship between individual and collective effects in a two‐component plasma is investigated in the case of an unstable equilibrium given by a cold two‐stream distribution. The full dynamics of this system is solved using an exact N‐body code. As the graininess parameter is large, such a cold plasma should be dominated by individual effects. Indeed, during an initial phase much longer than the plasma period, ions and electrons simply oscillate around each other forming neutral “molecules.” Subsequently, however, the system switches to a regime where collective effects are important: the two‐stream configuration becomes unstable and phase space structures appear. On a longer time scale, the streams are destroyed and the system evolves towards thermal equilibrium. The present results show that collective effects can emerge even in a plasma dominated by individual interactions, provided that the initial distribution is unstable.