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 according to a Poisson process 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. As a benchmark, we also consider a monopolistic scenario, where each platform gets half the market share irrespective of its pricing strategy. 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 market share split is modeled via two different QoS metrics: (i) probability that an arriving passenger obtains a ride, and (ii) the average passenger pick-up time. The platform aims to maximize the rate of revenue generated from matching drivers and passengers. In each of the above settings, we analyse the equilibria associated with the 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 \textit{equilibrium cycle}, which has interesting dynamic connotations. Our results highlight the interplay between competition, passenger-side price sensitivity, and passenger/driver arrival rates.