We study a setting in which an algorithm must output a fair allocation of indivisible items while "learning on the job". More specifically, the algorithm is to output an allocation satisfying EF1, PROP1, or similar fairness notions; however, the algorithm initially has no information about the agents' valuations, and can only learn about them by (repeatedly) proposing an allocation, and obtaining feedback about a fairness violation in the allocation. Importantly, the observed fairness violation may be adversarially chosen. The algorithm's goal is to converge to a fair allocation in rounds polynomial in the number of agents and items, ideally with only polynomial computation. We prove two main results: first, when the valuations are additive, then even for mixed items (goods and chores), an allocation satisfying EF1 or PROP1 can be found in polynomial time using the corresponding feedback. These results are instantiations of a more general framework which maintains a polytope of candidate valuations consistent with all past feedback. The algorithm repeatedly constructs putative valuations and uses them to propose allocations; the observed violations then define separating hyperplanes, allowing the algorithm to emulate the ellipsoid method. When the valuations are monotone, we present an algorithm which is guaranteed to find an EF1 allocation in polynomially many iterations; however, its internal calculations are not guaranteed to be polynomial. The algorithm again maintains putative valuations, and only considers allocations in which each agent obtains an interval plus one additional item with respect to an arbitrary ordering of the items. It is known that such an EF1 allocation always exists; however, because the existence proof is non-constructive, the internal step of constructing an allocation from the putative valuations is not known to take polynomial time.

翻译：暂无翻译