Competition between collective and individual dynamics

• The city has Q blocks with H cells each, at most one agent per cell, and block density ρ_q determines the utility shared by its residents.The configuration x records each block’s density.Source: Model · p. 1
• Collective utility sums agents’ utilities across blocks, while configuration entropy S(x) quantifies the number of cell arrangements consistent with the block densities.The collective utility is defined as U(x) = H times the sum over blocks of ρ_q u(ρ_q).Source: Model · p. 1
• At each step, a randomly selected agent and vacant cell in different blocks form a proposed move, accepted with a probability that depends on its cost.Temperature T adds noise to the decision process and can represent influences omitted from the utility function.Source: Model · p. 1
• The move cost combines changes in the mover’s utility and total utility, with α weighting the collective contribution from selfish-only at α = 0 to collective-only at α = 1.Varying α continuously interpolates between these limiting behaviors.Source: Model · p. 1

Potential function

• When cost is a potential difference, detailed balance gives a stationary distribution based on F(x) = V(x) + TS(x); the linking function is constructed so that ℓ′(ρ) = u(ρ).The linking function connects individual and collective levels by matching changes in utility to changes in L.Source: Potential function · pp. 1–2
• The potential F predicts global town states analytically for arbitrary utility functions, with homogeneous states when f meets its concave hull and phase separation otherwise.The approach generalizes the free-energy function used in physics and provides an analytical alternative to numerical simulations for individualistic Schelling-type models.Source: Potential function · p. 2
• At α = 1, all blocks have density 1/2, whereas at α = 0 the town segregates into empty blocks and blocks with density ρs > 1/2.At positive temperature, a slight utility decrease can trigger departures from a block at density 1/2, creating an avalanche that empties it.Source: Potential function · p. 2
• Mixed and segregated states are separated at αc = 1/(3 − 2m); for m > 2/3, satisfaction stays constant until αt = (3m − 2)/(6 − 5m).The transition reflects competition between collective and individual dynamics; the paper interprets the unsatisfactory global state as an externality effect and reports αc = 1/3 at m = 0.Source: Potential function · p. 2
• At low temperatures, the qualitative conclusions persist; at higher temperatures, noise drives the city toward homogeneity.At low temperatures, entropic contributions modify the phase diagram only at extreme values of ρ0.Source: Potential function · p. 2

Link to Schelling’s original segregation model

• At T → 0 with equal overall concentrations, low α yields segregated blocks containing one agent type at density ρ0, while α ≥ αc yields mixed blocks with group density 0 or 1/2.The spatial constraint ρqR + ρqG ≤ 1 prevents the two-population model from reducing to two independent one-population models, though stationary states can still be computed.Source: Link to Schelling’s original… · p. 3
• The segregated-to-mixed transition is unaffected by whether neighborhoods are predefined blocks or Schelling’s eight nearest neighbors [9].The paper concludes that block neighborhoods suit this simple modeling aim of showing stylized segregation transitions.Source: Link to Schelling’s original… · p. 3
• Unlike Schelling’s original model [9], the block approach permits analytical solutions for arbitrary utility functions; the nearest-neighbor approach is limited to a modified model and a specific utility function.The paper notes that neither neighborhood definition has a decisive realism advantage for describing social neighborhoods.Source: Link to Schelling’s original… · p. 3

Conclusion

• Maximizing the linking function ℓ(ρ) captures the emergent collective state, subject to constraints in overall town density.The function depends only on the global town configuration and can be interpreted as the sum of agents’ marginal utilities as they fill the city from a zero-utility reservoir.Source: Conclusion · p. 3
• At α = 1, the stationary state maximizes average utility, whereas at α = 0 it differs from the collection of individual optima, illustrating links between micromotives and macrobehavior.Source: Conclusion · p. 3
• At stationarity, maximal potential L means no agent can increase utility by moving because ∆u = ∆L, consistent with the economists’ definition of a Nash equilibrium.Source: Conclusion · p. 3
• A link function analogous to thermodynamic state functions or game-theory potential functions extends statistical mechanics to a Schelling-like model and may enable analytical treatment of wider systems governed by individual strategies.Source: Conclusion · p. 3

Temperature effects

• Higher temperature favors homogeneous states and introduces a second homogeneous phase for ρ0 < 1/2, where homogeneity results from noise rather than utility maximization.For ρ0 > 1/2, homogeneity is the agents’ optimal choice; for small finite temperatures (roughly T < 0.2), the phase diagram changes mainly at extremal ρ0 values.Source: Figure 4 · p. 5

Neighborhood definition

• The segregated-to-mixed transition is unchanged between block-based and nearest-cell neighborhood definitions in simulations.For α = 0, configurations are segregated; for α = 1, they are mixed or homogeneous. Simulations used 10% vacant cells and T = 0.1.Source: Figure 5 · p. 5