Dynamic Spatial Matching

Motivated by a variety of online matching platforms, we consider demand and supply units which are located i.i.d. in $[0,1]^d$, and each demand unit needs to be matched with a supply unit. The goal is to minimize the expected average distance between matched pairs (the "cost"). We model dynamic arrivals of one or both of demand and supply with uncertain locations of future arrivals, and characterize the scaling behavior of the achievable cost in terms of system size (number of supply units), as a function of the dimension $d$. Our achievability results are backed by concrete matching algorithms. Across cases, we find that the platform can achieve cost (nearly) as low as that achievable if the locations of future arrivals had been known beforehand. Furthermore, in all cases except one, cost nearly as low as the expected distance to the nearest neighboring supply unit is achievable, i.e., the matching constraint does not cause an increase in cost either. The aberrant case is where only demand arrivals are dynamic, and $d=1$; excess supply significantly reduces cost in this case.