Online Nash Social Welfare via Promised Utilities
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We consider the problem of allocating a set of divisible goods to N agents in an online manner over T periods, with adversarially-chosen normalized valuations in each period. Our goal is to maximize the Nash social welfare, a widely studied objective which provides a balance between fairness and efficiency. On the positive side, we provide an online algorithm that achieves a competitive ratio of O(log N) and O(log T), but also a stronger competitive ratio of O(log k) in settings where the value of any agent for her most preferred item is no more than k times her average value. We complement this by showing this bound is essentially tight: no online algorithm can achieve a competitive ratio of O(log^1-ϵ N) or O(log^1-ϵ T) for any constant ϵ>0.
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