Continuous Assortment Optimization with Logit Choice Probabilities under Incomplete Information

Motivated by several practical applications, we consider assortment optimization over a continuous spectrum of products represented by the unit interval, where the seller's problem consists of determining the optimal subset of products to offer to potential customers. To describe the relation between assortment and customer choice, we propose a probabilistic choice model that forms the continuous counterpart of the widely studied discrete multinomial logit model. We consider the seller's problem under incomplete information, propose a stochastic-approximation type of policy, and show that its regret -- its performance loss compared to the optimal policy -- is only logarithmic in the time horizon. We complement this result by showing a matching lower bound on the regret of any policy, implying that our policy is asymptotically optimal. We then show that adding a capacity constraint significantly changes the structure of the problem, by constructing an instance in which the regret of any policy after $T$ time periods is bounded below by a positive constant times $T^{2/3}$. We propose a policy based on kernel-density estimation techniques, and show that its regret is bounded above by a constant times $T^{2/3}$. Numerical illustrations show that our policies outperform or are on par with alternatives based on discretizing the product space.