Hydrodynamic models of preference formation in multi-agent societies

TL;DR

This work develops a multi-scale framework for preference formation in multi-agent societies by coupling opinion dynamics, modeled via Boltzmann-type kinetics with diffusion, to a transport-driven evolution of personal preferences. Through an inhomogeneous Boltzmann equation and a local-equilibrium closure, it derives first- and second-order hydrodynamic models for the density $\rho(t,\xi)$ and mean opinion $m(t,\xi)$, revealing how the symmetry of the interaction kernel $P$ and the perceived social opinion $\alpha$ shape polarisation. Theoretical results are complemented by extensive numerical tests (MC and structure-preserving schemes) that validate the kinetic-to-hydrodynamic transition and show how distinct polarisation patterns emerge, including multiple preference poles, depending on $\alpha$, $\lambda=\sigma^2/\gamma$, and $\Delta$. The findings highlight the crucial distinction between opinion and preference formation and offer a tractable framework to study polarisation phenomena relevant to elections and referendums, with potential applications to strategic social influence and control.

Abstract

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.

Figures (15)

• Figure 1: Solution of \ref{['eq:micro_w']} with $N=50$ agents and $P$ given by \ref{['eq:P_BC']} for decreasing values of the confidence threshold $\Delta$. The initial opinions $w_{0,i}$ have been sampled uniformly in $[-1,\,1]$. The ODE system has been integrated numerically via a standard fourth order Runge-Kutta method.
• Figure 2: The curves $t\mapsto\xi_i(t)$ generated by the coupled system \ref{['eq:micro_xi']}-\ref{['eq:micro_w']} with $N=50$ agents and $P$, $\Phi$ like in \ref{['eq:P_BC']}, \ref{['eq:Phi_3poles']} with $\Delta=1$ and $\alpha=\pm 0.3$. The initial values $w_{0,i}$, $\xi_{0,i}$ have been sampled uniformly in the interval $[-1,\,1]$. The ODE system has been integrated numerically via a standard fourth order Runge-Kutta method.
• Figure 3: The same as in Figure \ref{['fig:xi1']} but with $\Delta=0.4$, cf. Figure \ref{['fig:BCmicro']}(b), and $\alpha=\pm 0.3$ (top row), $\alpha=\pm 0.6$ (bottom row).
• Figure 4: Asymptotic opinion distribution \ref{['eq:ginf']} with mean $m=0.25$ and four different values of the parameter $\lambda$.
• Figure 5: Top row: contours of the distribution function $g$ computed numerically for $\tau\in (0,\,T]$, $T=50$, from the Fokker-Planck equation \ref{['eq:FP.gen']} with the SP scheme. Bottom row: comparison of the numerical approximations at $\tau=T$ of the large time distribution $g_\infty$ obtained with the previous SP scheme and with the MC scheme for the Boltzmann-type equation \ref{['eq:boltzmann_strong']} with two decreasing values of the parameter $\epsilon$ simulating the quasi-invariant regime. In both rows, the confidence thresholds are $\Delta=1$ (left) and $\Delta=0.4$ (right).

Theorems & Definitions (2)

• Proposition 4.1
• Remark 5.1