<p>We analyse a non-cooperative game between two competing ride-hailing platforms, each of which is modeled as a two-sided queueing system, where drivers (with a limited level of patience) are assumed to arrive at a fixed rate, while the arrival process of (price-sensitive) passengers is split across the two platforms based on Quality of Service (QoS) considerations. The key novelty of our formulation is that the total market share is fixed across the platforms. The game thus captures the competition between the platforms over market share, with pricing being the lever used by each platform to influence its share of the market. The platform aims to maximize the rate of revenue generated from matching drivers and passengers. We analyse the equilibria associated with this game in certain limiting regimes. We also show that these equilibria remain relevant in the more practically meaningful ‘pre-limit.’ Interestingly, we show that for a certain range of system parameters, no pure Nash equilibrium exists. Instead, we demonstrate a novel solution concept called an <i>equilibrium cycle</i>, which has interesting dynamic connotations—intuitively, the equilibrium cycle represents the limit set of oscillating game dynamics. Our results highlight the interplay between competition, passenger-side price sensitivity, and passenger/driver arrival rates.</p>