Ranking an Assortment of Products Via Sequential Submodular Optimization

Abstract

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.

Online Learning via Offline Greedy: Applications in Market Design and Optimization

“…For the sake of presentation, assume that the desired action is clicking on an item. We consider the model proposed by Asadpour et al (2020), which is inspired by an earlier model proposed in Ferreira et al (2019). In this model, a consumer u is characterized by a patience level θ u together with a monotone non-decreasing submodular set function κ u : 2…”

“…Product ranking problem as sequential submodular maximization. A slight reformulation of the above model casts the product ranking problem as a special case of a class of optimization problems over permutations called sequential submodular maximization (Asadpour et al 2020). We define the sequential submodular maximization problem as follows.…”

“…To the best of our knowledge, the online adversarial version of this problem has not been studied before, neither under full information nor bandit setting. Asadpour et al (2020) showed the offline problem of sequential submodular maximization is NP-hard. They also proposed an optimal (1 − 1/e)-approximation algorithm, and a simple 1 2approximation greedy algorithm for this offline problem.…”

“…where f i is a monotone submodular function that takes an element of S i as input and returns a probability in [0, 1], i.e., Asadpour et al (2020) and Ferreira et al (2019), is a greedy algorithm with n subproblems, where each subproblem corresponds to a position on the ranking.…”

“…To express user engagement as a function of the ranking over the products, we use the model proposed by Asadpour et al (2020), which is a generalization of the model presented by Ferreira et al (2019). Under this model, the offline ranking problem can be written as maximizing sequential submodular functions; see Section 6.1 for the definition of the sequential submodular functions.…”