  We describe mechanisms for the allocation of a scarce resource among multiple
users in a way that is efficient, fair, and strategy-proof, but when users do
not know their resource requirements. The mechanism is repeated for multiple
rounds and a user's requirements can change on each round. At the end of each
round, users provide feedback about the allocation they received, enabling the
mechanism to learn user preferences over time. Such situations are common in
the shared usage of a compute cluster among many users in an organisation,
where all teams may not precisely know the amount of resources needed to
execute their jobs. By understating their requirements, users will receive less
than they need and consequently not achieve their goals. By overstating them,
they may siphon away precious resources that could be useful to others in the
organisation. We formalise this task of online learning in fair division via
notions of efficiency, fairness, and strategy-proofness applicable to this
setting, and study this problem under three types of feedback: when the users'
observations are deterministic, when they are stochastic and follow a
parametric model, and when they are stochastic and nonparametric. We derive
mechanisms inspired by the classical max-min fairness procedure that achieve
these requisites, and quantify the extent to which they are achieved via
asymptotic rates. We corroborate these insights with an experimental evaluation
on synthetic problems and a web-serving task.
