  We analyse a non-cooperative strategic game among two ride-hailing platforms,
each of which is modeled as a two-sided queueing system, where drivers (with a
certain patience level) are assumed to arrive according to a Poisson process at
a fixed rate, while the arrival process of passengers is split across the two
providers based on QoS considerations. We also consider two monopolistic
scenarios: (i) each platform has half the market share, and (ii) the platforms
merge into a single entity, serving the entire passenger base using their
combined driver resources. The key novelty of our formulation is that the total
market share is fixed across the platforms. The game thus captures the
competition among the platforms over market share, which is modeled using two
different Quality of Service (QoS) metrics: (i) probability of driver
availability, and (ii) probability that an arriving passenger takes a ride. The
objective of the platforms is to maximize the profit generated from matching
drivers and passengers.
  In each of the above settings, we analyse the equilibria associated with the
game. Interestingly, under the second QoS metric, we show that for a certain
range of parameters, no Nash equilibrium exists. Instead, we demonstrate a new
solution concept called an equilibrium cycle. Our results highlight the
interplay between competition, cooperation, passenger-side price sensitivity,
and passenger/driver arrival rates.
