Before revision:  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 .
After revision:  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 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 .