A simple Online Fair Division problem.

A fixed set of $n$ agents share a random object: the distribution $mu$ of the profile of utilities is IID across periods, but arbitrary across agents. We consider a class of online division rules that learn the realized utility profile, and only know from $mu$ the individual expected utilities. They have no record from past realized utilities, and do not know either if and how many new objects will appear in the future. We call such rules prior-independent. A rule is fair if each agent, ex ante, expects at least $1/n$-th of his utility for the object if it is a good, at most $1/n$-th of his disutility for it if it is a bad. Among fair prior-independent rules to divide goods (bads) we uncover those collecting the largest (lowest) total expected (dis)utility. There is exactly one fair rule for bads that is optimal in this sense. But for goods, the set of optimal fair rules is one dimensional. Both in the worst case and in the asymptotic sense, our optimal rules perform much better than the natural Proportional rule (for goods or for bads), and not much worse than the optimal fair prior-dependent rule that knows the full distribution $mu$ in addition to realized utilities.