  We consider one-way vehicle sharing systems where customers can rent a car at
one station and drop it off at another. The problem we address is to optimize
the distribution of cars, and quality of service, by pricing rentals
appropriately. We propose a bidding approach that is inspired from auctions and
takes into account the significant uncertainty inherent in the problem data
(e.g., pick-up and drop-off locations, time of requests, and duration of
trips). Specifically, in contrast to current vehicle sharing systems, the
operator does not set prices. Instead, customers submit bids and the operator
decides whether to rent or not. The operator can even accept negative bids to
motivate drivers to rebalance available cars to unpopular destinations within a
city. We model the operator's sequential decision-making problem as a
\emph{constrained Markov decision problem} (CMDP) and propose and rigorously
analyze a novel two phase $Q$-learning algorithm for its solution. Numerical
experiments are presented and discussed.
