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-\epsilon} N) or O(\log^{1-\epsilon} T) for any constant \epsilon>0.
We consider the problem of allocating a set of divisible goods to N agents in an online manner , aiming to maximize the Nash social welfare, a widely studied objective which provides a balance between fairness and efficiency. The goods arrive in a sequence of T periods and the value of each agent for a good is adversarially chosen when the good arrives. We first observe that no online algorithm can achieve a competitive ratio better than the trivial O(N), unless it is given additional information about the agents' values. Then, in line with the emerging area of "algorithms with predictions", we consider a setting where for each agent, the online algorithm is only given a prediction of her monopolist utility, i.e., her utility if all goods were given to her alone (corresponding to the sum of her values over the T periods). Our main result is an online algorithm whose competitive ratio is parameterized by the multiplicative errors in these predictions. The algorithm achieves a competitive ratio of O(\log N) and O(\log T) if the predictions are perfectly accurate. Moreover, the competitive ratio degrades smoothly with the errors in the predictions, and is surprisingly robust: the logarithmic competitive ratio holds even if the predictions are very inaccurate . We complement this positive result by showing that our bounds are essentially tight: no online algorithm , even if provided with perfectly accurate predictions, can achieve a competitive ratio of O(\log^{1-\epsilon} N) or O(\log^{1-\epsilon} T) for any constant \epsilon>0.
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over T periods, with adversarially-chosen normalized valuations in each period. Our goal is
, aiming
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On the positive side, we provide
The goods arrive in a sequence of T periods and the value of each agent for a good is adversarially chosen when the good arrives. We first observe that no online algorithm can achieve a competitive ratio better than the trivial O(N), unless it is given additional information about the agents' values. Then, in line with the emerging area of "algorithms with predictions", we consider a setting where for each agent, the online algorithm is only given a prediction of her monopolist utility, i.e., her utility if all goods were given to her alone (corresponding to the sum of her values over the T periods). Our main result is
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that
whose competitive ratio is parameterized by the multiplicative errors in these predictions. The algorithm
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, 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
if the predictions are perfectly accurate. Moreover, the competitive ratio degrades smoothly with the errors in the predictions, and is surprisingly robust: the logarithmic competitive ratio holds even if the predictions are very inaccurate
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by showing this bound is
positive result by showing that our bounds are
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, even if provided with perfectly accurate predictions,