Cities at Play: Improving Equilibria in Urban Neighbourhood Games

TL;DR

The work analyzes how urban investments interact with strategic, location-choosing agents in a Schelling-inspired bounded-neighbourhood model featuring concave, non-monotone utilities. It shows that small, carefully designed interventions—costing at most $0.81 \epsilon^2 \cdot \text{opt}$—can guarantee that every Nash equilibrium attains at least $\epsilon \cdot \text{opt}$ social welfare, thus turning potential supra-negative outcomes into supra-positive ones. The authors introduce a rudimentary strategy for improving a location and a principled method for selecting where to invest, including a two-case analysis based on the existence of a critical location. They also prove the bound is tight up to constants and illustrate the approach with a case study reversing the donut effect, highlighting practical implications for targeted urban planning and decentralized collective behavior.

Abstract

How should cities invest to improve social welfare when individuals respond strategically to local conditions? We model this question using a game-theoretic version of Schelling's bounded neighbourhood model, where agents choose neighbourhoods based on concave, non-monotonic utility functions reflecting local population. While naive improvements may worsen outcomes - analogous to Braess' paradox - we show that carefully designed, small-scale investments can reliably align individual incentives with societal goals. Specifically, modifying utilities at a total cost of at most $0.81 ε^2 \cdot \texttt{opt}$ guarantees that every resulting Nash equilibrium achieves a social welfare of at least $ε\cdot \texttt{opt}$, where $\texttt{opt}$ is the optimum social welfare. Our results formalise how targeted interventions can transform supra-negative outcomes into supra-positive returns, offering new insights into strategic urban planning and decentralised collective behaviour.

Figures (8)

• Figure 1: An instance with an equilibrium of high social welfare.
• Figure 2: An instance with an equilibrium of low social welfare.
• Figure 3: An instance with supra-negative investment returns.
• Figure 4: An instance with supra-positive investment returns.
• Figure 5: A circular city composed of one central inner-city region (red) and an equal-area suburban region (blue), the latter subdivided into ten equally sized neighbourhoods. The spatial domain is discretized on a $60 \times 60$ grid, and population density is represented by coloured filled pixels.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Theorem \ref{thm:main}
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['thm:main']}