The Increasing Gap Dynamics in a General Spatial Matching Model

TL;DR

This paper investigates a spatial matching model where servers and requests interact, revealing a persistent increasing gap dynamic that leads to inefficient equilibria, even under optimal assignment policies, with implications for real-world ride-hailing systems.

Contribution

The paper introduces the concept of Increasing Gap Dynamics (IGD) in spatial matching models and demonstrates its persistence under optimal policies through analytical and simulation results.

Findings

In 1D, the system converges to an inefficient equilibrium.

Optimal policies do not prevent the emergence of IGD.

Simulations in 2D and real data confirm IGD's impact.

Abstract

We study a representation of a problem that appears in numerous transport systems: $N$ servers distributed over a given space (e.g., cars on an urban network), receive random requests from arriving users who get assigned to the closest server, after which this server is replaced by a new one at a random location. We show that this creates a negative feedback loop, which we call \textit{Increasing Gap Dynamics} (IGD): when a server is assigned a spatial gap forms, which is more likely to attract new users that further widen the gap. The simplest version of our model is a one-dimensional circle, for which we derive analytical results showing that the system converges to an inefficient equilibrium, worse than both balanced and fully random distributions of servers. We prove that an optimal assignment policy always matches the user to one of its two neighbouring servers so that long gaps tend to widen. Hence, the IGD persists even when assigning optimally rather than greedily. In two dimensions, the appearance of the IGD is illustrated through simulations on a square region. Finally, simulations of a proper ride-hailing system using real data from Manhattan confirms that the IGD arises and that it is responsible for the appearance of the well-known Wild Goose Chase.