  Autonomous vehicles will be an integral part of ride-sharing services in the
future. This setting differs from traditional ride-sharing marketplaces because
of the absence of the supply side (drivers). However, it has far-reaching
consequences because in addition to pricing, players now have to make decisions
on how to distribute fleets across network locations and re-balance vehicles in
order to serve future demand. In this paper, we explore a duopoly setting in
the ride-sharing marketplace where the players have fully autonomous fleets.
Each ride-service provider (RSP)'s prices depend on the prices and the supply
of the other player. We formulate their decision-making problems using a
game-theoretic setup where each player seeks to find the optimal prices and
supplies at each node while considering the decisions of the other player. This
leads to a scenario where the players' optimization problems are coupled and it
is challenging to find the equilibrium. We characterize the types of demand
functions (e.g.: linear) for which this game admits an exact potential function
and can be solved efficiently. For other types of demand functions, we propose
an iterative algorithm to compute the equilibrium. We conclude by providing
numerical insights into how different kinds of equilibria would play out in the
market when the players are asymmetric. Our numerical evaluations also provide
insights into how the regulator needs to consider network effects while
deciding regulation in order to avoid unfavorable outcomes.
