  Autonomous vehicles (AVs) are attractive for ride service providers (RSPs) in
part because they eliminate the need to compete for human drivers. We
investigate a scenario where two RSPs with AVs compete for customers. We model
the problem as a game where the RSPs select prices for each origin-destination
pair over multiple time periods in an underlying graph representing the
customers' desired trips. Each RSP also decides the number of AVs to be
stationed at each node at each time period to serve the customers' demands. The
number of customers who avail of the service of an RSP depends on the price
selected by the RSP and its competitor. Since the strategy choices available to
an RSP depends on its competitor, we seek to compute a Generalized Nash
equilibrium (GNE). We show that there may be multiple GNEs. However, when an
RSP selects prices in order to deter its competitor when it is not serving a
source-destination pair, the game has a potential function and admits a unique
GNE. We also compare the competitive prices with a monopoly price where only
one RSP is in the market. Numerically, we show that if a network consists of
two equal-size spatial clusters of demand where the demand between clusters is
low, the RSPs may partition the market, i.e, one cluster is served by only one
RSP. Hence, the competitive price may become close to the monopoly price.
