Ranking an Assortment of Products via Sequential Submodular Optimization
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We study an optimization problem capturing a core operational question for online retailing platforms. Given models for the users' preferences over products as well as the number of items they are willing to observe before clicking on one or abandoning the search, what is the best way to rank the relevant products in response to a search query? In order to capture both popularity and diversity effects, we model the probability that a user clicks on an element from a subset of products as a monotone submodular function of this set. We also assume that the patience level of the users, or the number of items they are willing to observe before clicking on one or abandoning the search, is a given random variable. Under those assumptions, the objective functions capturing user engagement or platform revenue can be written as a new family of submodular optimization problems over a sequence of elements. We call this family of natural optimization problems sequential submodular optimization. By building on and extending the literature on submodular maximization subject to matroid constraints, we derive a (1-1/e) optimal approximation algorithm for maximizing user engagement and a bi-criteria approximation algorithm for maximizing revenue subject to a lower bound on user engagement.
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