In this paper we consider the dynamic assortment selection problem under an uncapacitated multinomial-logit (MNL) model. By carefully analyzing a revenue potential function, we show that a trisection based algorithm achieves an item-independent regret bound of Op? T log log T q, which matches information theoretical lower bounds up to iterated logarithmic terms. Our proof technique draws tools from the unimodal/convex bandit literature as well as adaptive confidence parameters in minimax multi-armed bandit problems.

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In the multi-armed bandit problem, an agent has several possible decisions, usually referred to as "arms", and chooses or "pulls" sequentially one of them at each time step. This generates a sequence of rewards and the objective is to maximize their cumulative sum.

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Instead, we seek to learn a fair distribution overthearms. Drawing onalong lineofresearch ineconomics and computer science, we use theNash social welfareas our notion of fairness.