The paper proposes a new bandit setup that is theoretically interesting and practically relevant. The reviewers generally agree that the results are solid, although there is still room for improvement in terms of both theory and the presentation. Please do take all the reviewers' comments into account when preparing the final version. Note: one of the main complains from Reviewer #6 is that the problem seems to have been studied before, but the reviewer provided no references. To the best of the knowledge of the other reviewers and the AC, the problem is new. Reviewer #6 promised to increase the score if this is indeed the case, but was not responsive at all during the discussion. Therefore, this review was not taken into account after all.