  Optimizing shared vehicle systems (bike/scooter/car/ride-sharing) is more
challenging compared to traditional resource allocation settings due to the
presence of \emph{complex network externalities} -- changes in the
demand/supply at any location affect future supply throughout the system within
short timescales. These externalities are well captured by steady-state
Markovian models, which are therefore widely used to analyze such systems.
However, using such models to design pricing and other control policies is
computationally difficult since the resulting optimization problems are
high-dimensional and non-convex.
  To this end, we develop a \emph{rigorous approximation framework} for shared
vehicle systems, providing a unified approach for a wide range of controls
(pricing, matching, rebalancing), objective functions (throughput, revenue,
welfare), and system constraints (travel-times, welfare benchmarks,
posted-price constraints). Our approach is based on the analysis of natural
convex relaxations, and obtains as special cases existing approximate-optimal
policies for limited settings, asymptotic-optimality results, and heuristic
policies. The resulting guarantees are non-asymptotic and parametric, and
provide operational insights into the design of real-world systems. In
particular, for any shared vehicle system with $n$ stations and $m$ vehicles,
our framework obtains an approximation ratio of $1+(n-1)/m$, which is
particularly meaningful when $m/n$, the average number of vehicles per station,
is large, as is often the case in practice.
