The Increasing Gap Dynamics in a General Spatial Matching Model

TL;DR

This work introduces Increasing Gap Dynamics (IGD), a negative feedback mechanism in spatial matching where assigning a nearby user to a server creates a larger gap that is more likely to attract future requests, further widening disparities. Using a general Markovian framework, the authors analyze a minimal one-dimensional circle model, prove that the IGD equilibrium is inefficient relative to balanced or random server distributions, and show that an optimal policy reduces to neighbor-based decisions, yet cannot prevent IGD. They extend insights to a unit square via visualisations and demonstrate the phenomenon in realistic ride-hailing scenarios with Manhattan data, linking IGD to the Wild Goose Chase. The study combines analytical results, discrete verbal arguments, and microsimulations to reveal robust IGD/WGC dynamics across settings and discuss potential mitigation via rebalancing and policy design.

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.

Figures (15)

• Figure 1: An example of the dynamic that prevents servers from remaining balanced in a spatial setting.
• Figure 2: Two interpretations of the metric space, either as a circle (left), or as the $[0,1]$ interval where 0 and 1 are identified as the same.
• Figure 3: When a new user arrives, the two intervals that surround the assigned vehicle $s_{i+1}$ get merged.
• Figure 5: Comparison of the IGD with having the drivers randomly or uniformly located.
• Figure 6: The policy $\rho$ assigns $u$ to $s_a$, so $\rho'$ assignes its neighbour $s_n$ instead.

Theorems & Definitions (14)

• Lemma 1
• Theorem 1
• Lemma 2
• Theorem 2
• Theorem 3
• proof
• Theorem 4
• proof
• Corollary 1
• Lemma 3