Individual and group fairness in geographical partitioning
Ilya O. Ryzhov, John Gunnar Carlsson, Yinchu Zhu
TL;DR
The paper addresses fair geographical partitioning for heterogeneous populations by enforcing both individual fairness (assignment depends only on location) and group fairness (demographic parity per facility) while minimizing expected travel costs. It recasts the problem within a semidiscrete optimal transport framework, derives a finite-dimensional concave dual with weights $w_{k,z}$ that define a multigroup generalization of additively weighted Voronoi diagrams, and shows how to compute the solution via a simple stochastic approximation algorithm. The authors provide a complete characterization through nonlinear equilibrium conditions and demonstrate the approach on a realistic LA County case study with seven subgroups, revealing the cost of fairness and how group granularity affects district geometry and office closures. The work offers theoretical and practical insights into the cost-benefit tradeoffs of demographic parity in districting, and it provides a tractable method for policymakers to quantify fairness costs using real-world travel-cost data.
Abstract
Socioeconomic segregation often arises in school districting and other contexts, causing some groups to be over- or under-represented within a particular district. This phenomenon is closely linked with disparities in opportunities and outcomes. We formulate a new class of geographical partitioning problems in which the population is heterogeneous, and it is necessary to ensure fair representation for each group at each facility. We prove that the optimal solution is a novel generalization of the additively weighted Voronoi diagram, and we propose a simple and efficient algorithm to compute it, thus resolving an open question dating back to Dvoretzky et al. (1951). The efficacy and potential for practical insight of the approach are demonstrated in a realistic case study involving seven demographic groups and $78$ district offices.