  The increasing presence of decentralized renewable generation in the power
grid has motivated consumers to install batteries to save excess energy for
future use. The high price of energy storage calls for a shared storage system,
but careful battery management is required so that the battery is operated in a
manner that is fair to all and as efficiently as possible. In this paper, we
study the tradeoffs between efficiency and fairness in operating a shared
battery. We develop a framework based on constrained Markov decision processes
to study both regimes, namely, optimizing efficiency under a hard fairness
constraint and optimizing fairness under hard efficiency constraint. Our
results show that there are fundamental limits to efficiency under fairness and
vice-versa, and, in general, the two cannot be achieved simultaneously. We
characterize these fundamental limits via absolute bounds on these quantities,
and via the notion of price of fairness that we introduce in this paper.
