Problem definition: Motivated by ride-hailing platforms such as Uber, Lyft and Didi, we study the problem of matching riders with self-interested drivers over a spatial network. We focus on the performance impact of two operational platform controls—demand-side admission control and supply-side repositioning control—considering the interplay with two practically important challenges: (i) spatial demand imbalances prevail for extended periods of time; and (ii) self-interested drivers strategically decide whether to join the network, and, if so, whether to reposition when not serving riders. Methodology/results: We develop and analyze the steady-state behavior of a novel game-theoretic fluid model of a two-location, four-route loss network. First, we fully characterize and compare the steady-state system equilibria under three control regimes, from minimal control to centralized control. Second, we provide insights on how and why platform control impacts equilibrium performance, notably with new findings on the role of admission control: the platform may find it optimal to strategically reject demand at the low-demand location even if drivers are in excess supply, to induce repositioning to the high-demand location. We provide necessary and sufficient conditions for this policy. Third, we derive upper bounds on the platform’s and drivers’ benefits caused by increased platform control; these are more significant under moderate capacity and significant cross-location demand imbalance. Managerial implications: Our results contribute important guidelines on the optimal operations of ride-hailing networks. Our model can also inform the design of driver compensation structures that support more centralized network control.