Socioeconomic agents as active matter in nonequilibrium Sakoda-Schelling models

How robust are socioeconomic agent-based models with respect to the details of the agents' decision rule? We tackle this question by considering an occupation model in the spirit of the Sakoda-Schelling model, historically introduced to shed light on segregation dynamics among human groups. For a large class of utility functions and decision rules, we pinpoint the nonequilibrium nature of the agent dynamics, while recovering an equilibrium-like phase separation phenomenology. Within the mean-field approximation we show how the model can be mapped, to some extent, onto an active matter field description. Finally, we consider nonreciprocal interactions between two populations and show how they can lead to nonsteady macroscopic behavior. We believe our approach provides a unifying framework to further study geography-dependent agent-based models, notably paving the way for joint consideration of population and price dynamics within a field theoretic approach.

Figure 1

(a) Utility function $u\left(\rho \right)=-|\rho -{\rho }^{★}{|}^{\alpha }$ for ${\rho }^{★}=0.5, \alpha =\{0.5,1,2\}$. Panels (b), (c), and (d) show snapshots of the stationary state for these different utility functions, starting from the same homogeneous profile at ${\rho }_0=0.5$. Here $\mathrm{\Gamma }=100, \sigma =3$ and $L_x=L_y=100$. The stationary density ${\rho }_d$ in the dense phase is ${\rho }_d=0.575\left(5\right)$ for $\alpha =0.5$ in (b), ${\rho }_d=0.575\left(5\right)$ for $\alpha =1$ in (c), and ${\rho }_d=0.585\left(5\right)$ for $\alpha =2$ in (d). These bulk densities are all significantly higher than the density ${\rho }^{★}$ for which agents maximize their utility. Note the accumulation of agents at the edge of the empty domain in (b) and (c); see Sec. 4.

Figure 2

Loop of four configurations with $N=3$ agents on $M=5$ sites breaking Kolmogorov's criterion when the utility is nonlinear and is a function of an individual perceived density. Shaded and unshaded nodes correspond to occupied and empty sites, respectively. The dashed line indicates a possible segmentation of the system into two distinct neighborhoods.

Figure 3

(a) Typical dense domain size $L_d\left(t\right)$ during coarsening as a function of time $t$. A unit of time is defined as $N$ Monte Carlo steps, where $N$ is the number of agents. $L_d\left(t\right)$ is averaged over five independent simulations. Panels (b), (c), (d), and (e) show snapshots at different times. Starting from a disordered configuration, we quench the system at low temperature, or high rationality $\mathrm{\Gamma }$, corresponding to $T\simeq T_c/6$. Parameters: $L_x=L_y=600, {\rho }_0=0.3, \sigma =1, \alpha =3/2, T=0.01$.

Figure 4

Numerical experiments for $\sigma =1, \alpha =3/2$. (a) Binodal densities measured for $L_x=200$ and $L_y=66$ ($\ell =33$), inset showing the Binder cumulant as a function of the density and fitted (continuous line) to determine the critical density. Panels (b), (c), and (d) show the numerical measurements of the critical exponents close to the critical point $({\rho }_c,T_c)=(0.271,0.0620)$ determined using various system sizes ranging from $\ell =20$ to $\ell =40$.

Figure 5

Comparison between Monte Carlo simulations and mean-field results for $\alpha =3/2, \sigma =7$ and $L_x=200, L_y=66$ ($\ell =33$). (a) Phase diagram in the $({\rho }_0,T)$ plane. The mean-field binodal (continuous black line) is given by measuring the densities of the bulk of each plateau in a phase-separated state. The circles are the bulk averaged densities in Monte Carlo (MC) simulations. The dashed black line represents the mean-field spinodal, which is obtained analytically from the linear stability analysis [see Eq. (20)], with ${\widehat{G}}_{\sigma }\left(k_1\right)=e^{-{\sigma }^2{k}_1^2/2}$ and $k_1=2\pi /L_x$. The diamonds indicate the loss of stability of the homogeneous state in the MC simulations. The black square is the critical point for $\sigma /L\to 0$. (b) Averaged density profile $\rho \left(x\right)$ from MC simulations (continuous green line) for ${\rho }_0=0.35, T=0.05$. The dashed black line is the stationary solution of the mean-field equation Eq. (15) for the same parameters, solved on a grid of step size 1 with a Euler explicit scheme.

Figure 6

(a) Utility $u\left(\rho \right)$ (solid line), and change of variable $R\left(\rho \right)$ (dashed line) for $\alpha =3/2$ and ${\rho }^{★}=1/2$. (b) Comparison between the semianalytical prediction (dark line) and the binodal densities, both obtained via the Monte Carlo simulations (green circles) and solving numerically the mean-field Eq. (15) (green dot-dashed line). The decision rule here is the logit function, whose values in 0 are $f_{\mathrm{\Gamma }}\left(0\right)=1/2$ and $f_{\mathrm{\Gamma }}^{\prime }\left(0\right)=\mathrm{\Gamma }/4$. Inset: Double-tangent construction on $\mathrm{\Phi }\left(R\right)$ for $T=0.083$. The dense phase is given by $R_d=0.170$, yielding ${\rho }_d=0.53$, and the gaseous phase is given by $R_g=-0.69$, yielding ${\rho }_g=0.022$.

Figure 7

Snapshots of the system for two frustrated interaction parameter choices. (a) Stationary demixing in a region where LSA presents complex eigenvalues. The agent $A$ phase still contains some $B$ agents. Parameters: $c_1=-2, c_2=1, \sigma =3, {\overline{\rho }}_A=0.2, {\overline{\rho }}_B=0.5, \mathrm{\Gamma }=10$. (b) Chaotic propagation of polarized blobs in a region where LSA presents pure real eigenvalues (null imaginary part). Parameters: $c_1=-2, c_2=0.5, \sigma =7, {\overline{\rho }}_A=0.6, {\overline{\rho }}_B=0.2, \mathrm{\Gamma }=100$. For both (a) and (b), $L_x=L_y=300$. Movies are available online [74].

Figure 8

Snapshot of a Monte Carlo simulation. Green sites are occupied, black sites are empty. We draw the boxes of size $\ell =L_x/6$ that are used to measure the liquid and the gas densities. Here the system size is $L_x=200, L_y=66$. Parameters: $\alpha =3/2, {\rho }_0=0.35, T=0.05$.

Figure 9

Binder cumulant, order parameter, and compressibility close to the critical point $({\rho }_c,T_c)=(0.271,0.0620)$ computed for $\alpha =3/2$ and $\sigma =1$, as a function of the temperature $T$.

Figure 10

1D steady-state profiles of the density $\rho$ computed by solving the mean-field dynamics with local moves (solid line) or nonlocal moves (dashed line). The two density profiles superimpose almost exactly, independently of the various parameter values. It was checked that different initial configurations lead to the same final state. (a) Parameters: utility exponent $\alpha =1, {\rho }_0=0.4, \sigma =12, L=300, \mathrm{\Gamma }=9$; (b) $\alpha =3/2, {\rho }_0=0.3, \sigma =10, L=300, \mathrm{\Gamma }=15$.

Figure 11

(a) Monotonic utility $u\left(\rho \right)=\mathrm{sgn}(\rho -{\rho }^{★}){|\rho -{\rho }^{★}|}^{1/2}$ (solid line), and bijective change of variable $R\left(\rho \right)=\mathrm{sgn}(\rho -{\rho }^{★}){|\rho -{\rho }^{★}|}^{3/2}$ (dashed line) for ${\rho }^{★}=1/2$. (b) Density profile when phase separation occurs, for $\mathrm{\Gamma }=2.5, \sigma =10, L_x=300$. The plateaus of the liquid and the gas phase do not match the plateaus predicted by the generalized thermodynamic mapping because the gradient expansion is no longer valid close to $\rho =0.5$.

Figure 12

(a) Double-tangent construction (in red) on the function $\mathrm{\Phi }\left(R\right)$ and (b) Maxwell equal-area construction on $h_0\left(w\right)$, with $h_0\left(w\right)=\overline{h}$ in red. Shaded areas in (b) are equal. We have plotted these functions for $\alpha =3/2, {\rho }^{★}=1/2, f_{\mathrm{\Gamma }}\left(0\right)=1/2, f_{\mathrm{\Gamma }}^{\prime }\left(0\right)=\mathrm{\Gamma }/4$, and $\mathrm{\Gamma }=10$. We find $\overline{h}=0.0212, R_{\ell }=1.092$, and $R_g=0.327$ (or $w_{\ell }=0.915$ and $w_g=3.058$), translating into ${\rho }_{\ell }=0.508$ and ${\rho }_g=0.047$.

Figure 13

(a) Nonmonotonic price field $\psi \left(\varphi \right)$ as a function of the locally perceived density $\varphi$, given by Eq. (J1). Parameters: ${\rho }_{\mathrm{p}}^{★}=0.25$ and ${\alpha }_{\mathrm{p}}=2$. (b) Spinodal curves for parameters $\lambda =5,{\alpha }_{\mathrm{p}}=2,\alpha =3$ and densities $[{\rho }^{★}=0.2,{\rho }_{\mathrm{p}}^{★}=0.4]$ (solid line), $[{\rho }^{★}=0.45,{\rho }_{\mathrm{p}}^{★}=0.2]$ (dotted line), and $[{\rho }^{★}=0.5,{\rho }_{\mathrm{p}}^{★}=0.1]$ (dashed line).