Probability in the Engineering and Informational Sciences (2021), 1–9 doi:10.1017/S0269964821000395 RESEARCH ARTICLE A regret lower bound for assortment optimization under the capacitated MNL model with arbitrary revenue parameters Yannik Peeters1 and Arnoud V. den Boer1,2 1 Amsterdam Business School, University of Amsterdam, Plantage Muidergracht 12, 1018 TV Amsterdam, The Netherlands. E-mail:
[email protected] 2 Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 105-107, 1098 XG Amsterdam, The Netherlands. Keywords: Assortment optimization, Incomplete information, Multinomial logit model, Regret lower bound Abstract In this note, we consider dynamic assortment optimization with incomplete information under the capacitated multinomial logit choice model. Recently, it has been shown that the regret (the cumulative expected revenue loss caused√ by offering suboptimal assortments) that any decision policy endures is bounded from below by a constant times ,where denotes the number of products and denotes the time horizon. This result is shown under the assumption that the product revenues are constant, and thus leaves the question open whether a lower regret rate can be achieved for nonconstant revenue parameters. In this note, we show that this is not the case: we show that, for any vector of product revenues√ there is a positive constant such that the regret√ of any policy is bounded from below by this constant times . Our result implies that policies that achieve O( ) regret are asymptotically optimal for all product revenue parameters. 1. Introduction In this note, we consider the problem of assortment optimization under the multinomial logit (MNL) choice model with a capacity constraint on the size of the assortment and incomplete information about the model parameters.