arXiv subjects

Giuseppe Toscani

Publications and source records attributed to Giuseppe Toscani.

At least 55 records · Page 3Linked to original sources

arXiv2019

Multiple-interaction kinetic modelling of a virtual-item gambling economy

In recent years, there has been a proliferation of online gambling sites, which made gambling more accessible with a consequent rise in related problems, such as addiction. Hence, the analysis of the gambling behaviour at both the individual and the aggregate levels has become the object of several investigations. In this paper, resorting to classical methods of the kinetic theory, we describe the behaviour of a multi-agent system of gamblers participating in lottery-type games on a virtual-item gambling market. The comparison with previous, often empirical, results highlights the ability of the kinetic approach to explain how the simple microscopic rules of a gambling-type game produce complex collective trends, which might be difficult to interpret precisely by looking only at the available data.

physics.soc-ph↗

arXiv2019

Kinetic modeling of alcohol consumption

In most countries, alcohol consumption distributions have been shown to possess universal features. Their unimodal right-skewed shape is usually modeled in terms of the Lognormal distribution, which is easy to fit, test, and modify. However, empirical distributions often deviate considerably from the Lognormal model, and both Gamma and Weibull distributions appear to better describe the survey data. In this paper we explain the appearance of these distributions by means of classical methods of kinetic theory of multi-agent systems. The microscopic variation of alcohol consumption of agents around a universal \emph{social} accepted value of consumption, is built up introducing as main criterion for consumption a suitable value function in the spirit of the prospect theory of Kahneman and Twersky. The mathematical properties of the value function then determine the unique macroscopic equilibrium which results to be a generalized Gamma distribution. The modeling of the microscopic kinetic interaction allows to clarify the meaning of the various parameters characterizing the generalized Gamma equilibrium.

physics.soc-ph↗

arXiv2019

Hydrodynamic models of preference formation in multi-agent societies

In this paper, we discuss the passage to hydrodynamic equations for kinetic models of opinion formation. The considered kinetic models feature an opinion density depending on an additional microscopic variable, identified with the personal preference. This variable describes an opinion-driven polarisation process, leading finally to a choice among some possible options, as it happens e.g. in referendums or elections. Like in the kinetic theory of rarefied gases, the derivation of hydrodynamic equations is essentially based on the computation of the local equilibrium distribution of the opinions from the underlying kinetic model. Several numerical examples validate the resulting model, shedding light on the crucial role played by the distinction between opinion and preference formation on the choice processes in multi-agent societies.

physics.soc-ph↗

arXiv2018

The information-theoretic meaning of Gagliardo--Nirenberg type inequalities

Gagliardo--Nirenberg inequalities are interpolation inequalities which were proved independently by Gagliardo and Nirenberg in the late fifties. In recent years, their connections with theoretic aspects of information theory and nonlinear diffusion equations allowed to obtain some of them in optimal form, by recovering both the sharp constants and the explicit form of the optimizers. In this note, at the light of these recent researches, we review the main connections between Shannon-type entropies, diffusion equations and a class of these inequalities.

math.FA↗

arXiv2018

Human Behavior And Lognormal Distribution. A Kinetic Description

In recent years it has been increasing evidence that lognormal distributions are widespread in physical and biological sciences, as well as in various phenomena of economics and social sciences. In social sciences, the appearance of lognormal distribution has been noticed, among others, when looking at body weight, and at women's age at first marriage. Likewise, in economics, lognormal distribution appears when looking at consumption in a western society, at call-center service times, and others. The common feature of these situations, which describe the distribution of a certain people's hallmark, is the presence of a desired target to be reached by repeated choices. In this paper we discuss a possible explanation of lognormal distribution forming in human activities by resorting to classical methods of statistical mechanics of multi-agent systems. The microscopic variation of the hallmark around its target value, leading to a linear Fokker--Planck type equation with lognormal equilibrium density, is built up introducing as main criterion for decision a suitable value function in the spirit of the prospect theory of Kahneman and Twersky.

physics.soc-ph↗

arXiv2018

Kinetic models for optimal control of wealth inequalities

We introduce and discuss optimal control strategies for kinetic models for wealth distribution in a simple market economy, acting to minimize the variance of the wealth density among the population. Our analysis is based on a finite time horizon approximation, or model predictive control, of the corresponding control problem for the microscopic agents' dynamic and results in an alternative theoretical approach to the taxation and redistribution policy at a global level. It is shown that in general the control is able to modify the Pareto index of the stationary solution of the corresponding Boltzmann kinetic equation, and that this modification can be exactly quantified. Connections between previous Fokker-Planck based models and taxation-redistribution policies and the present approach are also discussed.

physics.soc-ph↗

arXiv2018

The size distribution of cities: a kinetic explanation

We present a kinetic approach to the formation of urban agglomerations which is based on simple rules of immigration and emigration. In most cases, the Boltzmann-type kinetic description allows to obtain, within an asymptotic procedure, a Fokker--Planck equation with variable coefficients of diffusion and drift, which describes the evolution in time of some probability density of the city size. It is shown that, in dependence of the microscopic rules of migration, the equilibrium density can follow both a power law for large values of the size variable, which contains as particular case a Zipf's law behavior, and a lognormal law for middle and low values of the size variable. In particular, connections between the value of Pareto index of the power law at equilibrium and the disposal of the population to emigration are outlined. The theoretical findings are tested with recent data of the populations of Italy and Switzerland.

physics.soc-ph↗

arXiv2018

Call center service times are lognormal. A Fokker--Planck description

Call centers are service networks in which agents provide telephone-based services. An important part of call center operations is represented by service durations. In recent statistical analysis of real data, it has been noticed that the distribution of service times reveals a remarkable fit to the lognormal distribution. In this paper we discuss a possible source of this behavior by resorting to classical methods of statistical mechanics of multi-agent systems. The microscopic service time variation leading to a linear kinetic equation with lognormal equilibrium density is built up introducing as main criterion for decision a suitable value function in the spirit of the prospect theory of Kahneman and Twersky.

physics.soc-ph↗

arXiv2017

Wealth distribution in presence of debts. A Fokker--Planck description

We consider here a Fokker--Planck equation with variable coefficient of diffusion which appears in the modeling of the wealth distribution in a multi-agent society. At difference with previous studies, to describe a society in which agents can have debts, we allow the wealth variable to be negative. It is shown that, even starting with debts, if the initial mean wealth is assumed positive, the solution of the Fokker--Planck equation is such that debts are absorbed in time, and a unique equilibrium density located in the positive part of the real axis will be reached.

q-fin.MF↗

arXiv2017

Kinetic models for goods exchange in a multi-agent market

We introduce a system of kinetic equations describing an exchange market consisting of two populations of agents (dealers and speculators) expressing the same preferences for two goods, but applying different strategies in their exchanges. We describe the trading of the goods by means of some fundamental rules in price theory, in particular by using Cobb-Douglas utility functions for the exchange. The strategy of the speculators is to recover maximal utility from the trade by suitably acting on the percentage of goods which are exchanged. This microscopic description leads to a system of linear Boltzmann-type equations for the probability distributions of the goods on the two populations, in which the post-interaction variables depend from the pre-interaction ones in terms of the mean quantities of the goods present in the market. In this case, it is shown analytically that the strategy of the speculators can drive the price of the two goods towards a zone in which there is a marked utility for their group. Also, the general system of nonlinear kinetic equations of Boltzmann type for the probability distributions of the goods on the two populations is described in details. Numerical experiments then show how the policy of speculators can modify the final price of goods in this nonlinear setting.

q-fin.GN↗

arXiv2016

Recent advances in opinion modeling: control and social influence

We survey some recent developments on the mathematical modeling of opinion dynamics. After an introduction on opinion modeling through interacting multi-agent systems described by partial differential equations of kinetic type, we focus our attention on two major advancements: optimal control of opinion formation and influence of additional social aspects, like conviction and number of connections in social networks, which modify the agents' role in the opinion exchange process.

physics.soc-ph↗

arXiv2016

Kinetic and mean field description of Gibrat's law

We introduce and analyze a linear kinetic model that describes the evolution of the probability density of the number of firms in a society, in which the microscopic rate of change obeys to the so-called law of proportional effect proposed by Gibrat. Despite its apparent simplicity, the possible mean field limits of the kinetic model are varied. In some cases, the asymptotic limit can be described by a first-order partial differential equation. In other cases, the mean field equation is a linear diffusion with a non constant diffusion coefficient that models also the geometric Brownian motion and can be studied analytically. In this case, it is shown that the large-time behavior of the solution is represented, for a large class of initial data, by a lognormal distribution with constant mean value and variance increasing exponentially in time at a precise rate. The relationship between the kinetic and the diffusion models allow to introduce an easy-to- implement expression for computing the Fourier transform of the lognormal distribution.

q-fin.GN↗

arXiv2015

Entropy inequalities for stable densities and strengthened central limit theorems

We consider the central limit theorem for stable laws in the case of the standardized sum of independent and identically distributed random variables with regular probability density function. By showing decay of different entropy functionals along the sequence we prove convergence with explicit rate in various norms to a Lévy centered density of parameter $λ>1$ . This introduces a new information-theoretic approach to the central limit theorem for stable laws, in which the main argument is shown to be the relative fractional Fisher information, recently introduced by the same author (cf. arXiv 1504.07057). In particular, it is proven that, with respect to the relative fractional Fisher information, the Lévy density satisfies an analogous of the logarithmic Sobolev inequality, which allows to pass from the monotonicity and decay to zero of the relative fractional Fisher information in the standardized sum to the decay to zero in relative entropy with an explicit decay rate.

math.PR↗

arXiv2015

Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities

This paper is devoted to improvements of functional inequalities based on scalings and written in terms of relative entropies. When scales are taken into account and second moments fixed accordingly, deficit functionals provide explicit stability measurements, i.e., bound with explicit constants distances to the manifold of optimal functions. Various results are obtained for the Gaussian logarithmic Sobolev inequality and its Euclidean counterpart, for the Gaussian generalized Poincar{é} inequalities and for the Gagliardo-Nirenberg inequalities. As a consequence, faster convergence rates in diffusion equations (fast diffusion, Ornstein-Uhlenbeck and porous medium equations) are obtained.

math.AP↗

arXiv2015

The fractional Fisher information and the central limit theorem for stable laws

A new information-theoretic approach to the central limit theorem for stable laws is presented. The main novelty is the concept of relative fractional Fisher information, which shares most of the properties of the classical one, included Blachman-Stam type inequalities. These inequalities relate the fractional Fisher information of the sum of $n$ independent random variables to the information contained in sums over subsets containing $n-1$ of the random variables. As a consequence, a simple proof of the monotonicity of the relative fractional Fisher information in central limit theorems for stable law is obtained, together with an explicit decay rate.

cs.IT↗

arXiv2015

Kinetic models of opinion formation in the presence of personal conviction

We consider a nonlinear kinetic equation of Boltzmann type which takes into account the influence of conviction during the formation of opinion in a system of agents which interact through the binary exchanges introduced in [G. Toscani, Commun. Math. Sci. 4, 481 (2006)]. The original exchange mechanism, which is based on the human tendency to compromise and change of opinion through self-thinking, is here modified in the parameters of the compromise and diffusion terms, which now are assumed to depend on the personal degree of conviction. The numerical simulations show that the presence of conviction has the potential to break symmetry, and to produce clusters of opinions. The model is partially inspired by the recent work [L. Pareschi, G. Toscani, Phil. Trans. R. Soc. A 372, 20130396 (2014)], in which the role of knowledge in the formation of wealth distribution has been investigated.

physics.soc-ph↗

arXiv2015

Nonlinear diffusions: extremal properties of Barenblatt profiles, best matching and delays

In this paper, we consider functionals based on moments and non-linear entropies which have a linear growth in time in case of source-type so-lutions to the fast diffusion or porous medium equations, that are also known as Barenblatt solutions. As functions of time, these functionals have convexity properties for generic solutions, so that their asymptotic slopes are extremal for Barenblatt profiles. The method relies on scaling properties of the evo-lution equations and provides a simple and direct proof of sharp Gagliardo-Nirenberg-Sobolev inequalities in scale invariant form. The method also gives refined estimates of the growth of the second moment and, as a consequence, establishes the monotonicity of the delay corresponding to the best matching Barenblatt solution compared to the Barenblatt solution with same initial sec-ond moment. Here the notion of best matching is defined in terms of a relative entropy.

math.AP↗

arXiv2015

Mean field dynamics of collisional processes with duplication, loss and copy

In this paper we introduce and discuss kinetic equations for the evolution of the probability distribution of the number of particles in a population subject to binary interactions. The microscopic binary law of interaction is assumed to be dependent on fixed-in-time random parameters which describe both birth and death of particles, and the migration rule. These assumptions lead to a Boltzmann-type equation that in the case in which the mean number of the population is preserved, can be fully studied, by obtaining in some case the analytic description of the steady profile. In all cases, however, a simpler kinetic description can be derived, by considering the limit of quasi-invariant interactions. This procedure allows to describe the evolution process in terms of a linear kinetic transport-type equation. Among the various processes that can be described in this way, one recognizes the Lea-Coulson model of mutation processes in bacteria, a variation of the original model proposed by Luria and Delbrück.

math.PR↗