The Increasing Gap Dynamics in a General Spatial Matching Model

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.

💡 Research Summary

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The paper presents a unified stochastic model for a broad class of spatial matching problems that arise in modern on‑demand transport systems such as ride‑hailing, dock‑less bike‑sharing, and on‑street delivery parking. The authors consider a metric space ((X,d)) populated by (N) servers (vehicles) and a stream of users arriving according to an i.i.d. distribution (Q). When a user arrives, the closest server is assigned (greedy matching) and incurs a cost equal to the Euclidean (or graph) distance. Immediately after the assignment, the used server disappears from the idle pool and a new server appears at a location drawn independently from another distribution (P). This replacement rule keeps the number of idle servers constant but introduces a dynamic feedback loop.

The evolution of the server configuration (\mathbf S_t) is a homogeneous Markov chain on the state space (X^N). When (X) is finite (or countable) the chain is irreducible and aperiodic, guaranteeing a unique invariant distribution (\pi). By standard Markov‑Decision‑Process arguments, the expected matching cost (E