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:  On the positive side, we provide an online algorithm that 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 .