Kinetic models of opinion formation in the presence of personal conviction

Wealth Distribution Involving Psychological Traits and Non-Maxwellian Collision Kernel

A kinetic exchange model is developed to investigate wealth distribution in a market. The model incorporates a value function that captures the agents' psychological traits, governing their wealth allocation based on behavioral responses to perceived potential losses and returns. To account for the impact of transaction frequency on wealth dynamics, a non-Maxwellian collision kernel is introduced. Applying quasi-invariant limits and Boltzmann-type equations, a Fokker-Planck equation is derived. We obtain an entropy explicit stationary solution that exhibits exponential convergence to a lognormal wealth distribution. Numerical experiments support the theoretical insights and highlight the model's significance in understanding wealth distribution.

Curvature-driven growth and interfacial noise in the voter model with self-induced zealots

We introduce a variant of the voter model in which agents may have different degrees of confidence in their opinions. Those with low confidence are normal voters whose state can change upon a single contact with a different neighboring opinion. However, confidence increases with opinion reinforcement, and above a certain threshold, these agents become zealots, irreducible agents who do not change their opinion. We show that both strategies, normal voters and zealots, may coexist (in the thermodynamical limit), leading to competition between two different kinetic mechanisms: curvature-driven growth and interfacial noise. The kinetically constrained zealots are formed well inside the clusters, away from the different opinions at the surfaces that help limit their confidence. Normal voters concentrate in a region around the interfaces, and their number, which is related to the distance between the surface and the zealotry bulk, depends on the rate at which the confidence changes. Despite this interface being rough and fragmented, typical of the voter model, the presence of zealots in the bulk of these domains induces a curvature-driven dynamics, similar to the low temperature coarsening behavior of the nonconserved Ising model after a temperature quench.

On the steady state of continuous-time stochastic opinion dynamics with power-law confidence

This paper introduces a non-linear and continuous-time opinion dynamics model with additive noise and state-dependent interaction rates between agents. The model features interaction rates which are proportional to a negative power of the opinion distances. We establish a non-local partial differential equation for the distribution of opinion distances and use Mellin transforms to provide an explicit formula for the stationary solution of the latter, when it exists. Our approach leads to new qualitative and quantitative results on this type of dynamics. To the best of our knowledge these Mellin transform results are the first quantitative results on the equilibria of opinion dynamics with distance-dependent interaction rates. The closed-form expressions for this class of dynamics are obtained for the two-agent case. However, the results can be used in mean-field models featuring several agents whose interaction rates depend on the empirical average of their opinions. The technique also applies to linear dynamics, namely with a constant interaction rate, on an interaction graph.

Network Structured Kinetic Models of Social Interactions

The aim of this paper is to study the derivation of appropriate meso- and macroscopic models for interactions as appearing in social processes. There are two main characteristics the models take into account, namely a network structure of interactions, which we treat by an appropriate mesoscopic description, and a different role of interacting agents. The latter differs from interactions treated in classical statistical mechanics in the sense that the agents do not have symmetric roles, but there is rather an active and a passive agent. We will demonstrate how a certain form of kinetic equations can be obtained to describe such interactions at a mesoscopic level and moreover obtain macroscopic models from monokinetics solutions of those. The derivation naturally leads to systems of nonlocal reaction-diffusion equations (or in a suitable limit local versions thereof), which can explain spatial phase separation phenomena found to emerge from the microscopic interactions. We will highlight the approach in three examples, namely the evolution and coarsening of dialects in human language, the construction of social norms, and the spread of an epidemic.

Data-driven discovery of emergent behaviors in collective dynamics

Particle- and agent-based systems are a ubiquitous modeling tool in many disciplines. We consider the fundamental problem of inferring the governing structure, i.e. interaction kernels, in a nonparametric fashion, from observations of agent-based dynamical systems. In particular, we are interested in collective dynamical systems exhibiting emergent behaviors with complicated interaction kernels, and for kernels which are parameterized by a single unknown parameter. This work extends the estimators introduced in Lu et al. (2019), which are based on suitably regularized least squares estimators, to these larger classes of systems. We provide extensive numerical evidence that the estimators provide faithful approximations to the interaction kernels, and provide accurate predictions for trajectories started at new initial conditions, both throughout the "training" time interval in which the observations were made, and often much beyond. We demonstrate these features on prototypical systems displaying collective behaviors, ranging from opinion dynamics, flocking dynamics, self-propelling particle dynamics, synchronized oscillator dynamics, to a gravitational system. Our experiments also suggest that our estimated systems can display the same emergent behaviors as the observed systems, including those that occur at larger timescales than those in the training data. Finally, in the case of families of systems governed by a parametric family of interaction kernels, we introduce novel estimators that estimate the parametric family of kernels, splitting it into a common interaction kernel and the action of parameters. We demonstrate this in the case of gravity, by learning both the "common component" 1/r ² and the dependency on mass, without any a priori knowledge of either one, from observations of planetary motions in our solar system.

Long route to consensus: Two-stage coarsening in a binary choice voting model

Formation of consensus, in binary yes-no type of voting, is a well-defined process. However, even in presence of clear incentives, the dynamics involved can be incredibly complex. Specifically, formations of large groups of similarly opinionated individuals could create a condition of "support-bubbles" or spontaneous polarization that renders consensus virtually unattainable (e.g., the question of the UK exiting the EU). There have been earlier attempts in capturing the dynamics of consensus formation in societies through simple Z_{2}-symmetric models hoping to capture the essential dynamics of average behavior of a large number of individuals in a statistical sense. However, in absence of external noise, they tend to reach a frozen state with fragmented and polarized states, i.e., two or more groups of similarly opinionated groups with frozen dynamics. Here we show in a kinetic exchange opinion model considered on L×L square lattices, that while such frozen states could be avoided, an exponentially slow approach to consensus is manifested. Specifically, the system could either reach consensus in a time that scales as L^{2} or a long-lived metastable state (termed a "domain-wall state") for which formation of consensus takes a time scaling as L^{3.6}. The latter behavior is comparable to some voterlike models with intermediate states studied previously. The late-time anomaly in the timescale is reflected in the persistence probability of the model. Finally, the interval of zero crossing of the average opinion, i.e., the time interval over which the average opinion does not change sign, is shown to follow a scale-free distribution, which is compared with that seen in the opinion surveys regarding Brexit and associated issues since the late 1970s. The issue of minority spreading is also addressed by calculating the exit probability.

Forecasting Efficient Risk/Return Frontier for Equity Risk with a KTAP Approach—A Case Study in Milan Stock Exchange

We introduce and discuss a dynamics of interaction of risky assets in a portfolio by resorting to methods of statistical mechanics developed to model the evolution of systems whose microscopic state may be augmented by variables which are not mechanical. Statistical methods are applied in the present paper in order to forecast the dynamics of risk/return efficient frontier for equity risk. Specifically, we adopt the methodologies of the kinetic theory for active particles (KTAP) with stochastic game-type interactions and apply the proposed model to a case study analyzing a subset of stocks traded in Milan Stock Exchange. In particular, we evaluate the efficient risk/return frontier within the mean/variance portfolio optimization theory for 13 principal components of the Milan Stock Exchange and apply the proposed kinetic model to forecast its short-term evolution (within one year). The model has the aim to pave the way to many different research perspectives and applications discussed eventually in the paper. In particular, the case of efficient frontier obtained by minimizing the Conditional Value-at-Risk (CVaR) is introduced and a preliminary result is proposed.

A critical analysis towards research perspectives: Reply to comments on "Modeling human behavior in economics and social science"

We take advantage of the challenging comments to the modeling approach we proposed in [35] to look ahead at a number of applications of the methods to the alternative questions these comments raise. In turn, our effort results in a number of interesting and valuable research perspectives. The presentation goes along three main lines. In the first line, we summarize briefly the aims and results in [35]. In the second section we give a technical the issues raised and, finally, the focus moves to the above mentioned research perspectives.

Modeling human behavior in economics and social science

The complex interactions between human behaviors and social economic sciences is critically analyzed in this paper in view of possible applications of mathematical modeling as an attainable interdisciplinary approach to understand and simulate the aforementioned dynamics. The quest is developed along three steps: Firstly an overall analysis of social and economic sciences indicates the main requirements that a contribution of mathematical modeling should bring to these sciences; subsequently the focus moves to an overview of mathematical tools and to the selection of those which appear, according to the authors bias, appropriate to the modeling; finally, a survey of applications is presented looking ahead to research perspectives.

Symmetry breaking by heating in a continuous opinion model

We study the critical behavior of a continuous opinion model, driven by kinetic exchanges in a fully connected population. Opinions range in the real interval [-1,1], representing the different shades of opinions against and for an issue under debate. Individuals' opinions evolve through pairwise interactions, with couplings that are typically positive, but a fraction p of negative ones is allowed. Moreover, a social temperature parameter T controls the tendency of the individual responses toward neutrality. Depending on p and T, different collective states emerge: symmetry broken (one side wins), symmetric (tie of opposite sides), and absorbing neutral (indecision wins). We find the critical points and exponents that characterize the phase transitions between them. The symmetry breaking transition belongs to the usual Ising mean-field universality class, but the absorbing-phase transitions, with β=0.5, are out of the paradigmatic directed percolation class. Moreover, ordered phases can emerge by increasing social temperature.