Density and Affinity Dependent Social Segregation and Arbitrage Equilibrium in a Multi-class Schelling Game

TL;DR

The paper addresses why larger, denser urban environments can exhibit greater segregation by formulating a Schelling-like, utility-driven model within the statistical teleodynamics framework. It derives a density-dependent effective utility $h_i(\rho_i)$ with affinity, congestion, exploration, and competition terms, and a global potential $\phi$ whose maximization yields arbitrage equilibrium ($h^*$ equal across sites). The main finding is that non-monotone $h(\rho)$ induces phase separation (spinodal-like) in both one-class and two-class settings, with agent-based simulations corroborating the emergence of coexisting high- and low-density phases that share the same $h^*$. This provides a formal mathematical mechanism that aligns with observed density-dependent segregation and offers a rigorous lens for interpreting urban dynamics, while acknowledging that real-world factors may modulate these idealized outcomes.

Abstract

Contrary to the widely believed hypothesis that larger, denser cities promote socioeconomic mixing, a recent study (Nilforoshan et al. 2023) reports the opposite behavior, i.e. more segregation. Here, we present a game-theoretic model that predicts such a density-dependent segregation outcome in both one- and two-class systems. The model provides key insights into the analytical conditions that lead to such behavior. Furthermore, the arbitrage equilibrium outcome implies the equality of effective utilities among all agents. This could be interpreted as all agents being equally "happy" in their respective environments in our ideal society. We believe that our model contributes towards a deeper mathematical understanding of social dynamics and behavior, which is important as we strive to develop more harmonious societies.

Figures (13)

• Figure 1: Net benefit of a resource for $\alpha N_i - \beta {N_i}^2$ ( $\alpha=6$ , $\beta =1$)
• Figure 2: Effective Utility vs Density: $h$ vs $\rho$ for different $\alpha$. The black points are the spinodal points ($\rho_{s1} = 0.146, h_{s1} = 2.934; \rho_{s2} = 0.854, h_{s2} = 5.066$). The red points are the binodal points ($\rho_{b1} = 0.021, h_{b1} = 4.00; \rho_{b2} = 0.979, h_{b2} = 4.00$).
• Figure 3: The parametric space within the yellow region is where segregation is guaranteed to occur.
• Figure 4: This figure shows the cross-section of the solid from Figure \ref{['fig:phase_separation']}. Each line represents values of $\alpha$, showing the range of densities where segregation would occur for values of $\beta$.
• Figure 5: Equilibrium configuration after five million time steps on a $300\times 300$ grid, $M=50\times 50$ and $\beta=0$, for different $\rho_0$ and $\alpha$. Top row(A, B, C), middle row(D, E, F), and bottom row(G, H, I) correspond to an average density of $\rho_0 = 0.1, 0.25, 0.5$, respectively. Left column (A, D, G), middle column (B ,E, H), and right column (C ,F, I) correspond to $\alpha=0, 4, 8$, respectively. For single-phase systems, the final density is seen to be the same as $\rho_0$. For multiphase systems the density of the two phases are F. 0.039 and 0.988 (utility of 3.512, 3.523, respectively) I. 0.061 and 0.991 (utility of 3.223, 3.247 respectively)