Research

When customer preferences are unknown and a seller must decide in each period which products to offer, every assortment decision plays a dual role: generating revenue and revealing information about demand. This interaction between decision-making and learning gives rise to a bandit problem. We consider demand governed by a multinomial logit (MNL) model, whose shared denominator makes both purchase probabilities and the information conveyed by each choice depend on the entire assortment. Under competition, this assortment also includes rival products; consequently, other sellers’ decisions affect both a seller’s revenue and what it can learn when it does not make a sale.

This thesis studies Thompson sampling policies for this problem with either a single seller or multiple sellers learning in a decentralized manner. All policies sample a plausible vector of attraction parameters and act as if it were the true vector. They differ in how they represent uncertainty: filtering the history into product-versus-outside-option comparisons to obtain independent Beta-prime updates; preserving posterior dependence through a Gamma–Exponential representation; or approximating the shared denominator to maintain a separate state for each product.

For a single seller with a catalog of N products, capacity to offer at most K of them, and a horizon of T periods, two of the proposed policies admit regret bounds. The policy that samples attractions independently achieves a bound of order Õ(√(NKT)), whereas the policy that samples exactly from the joint posterior attains Bayesian regret of order Õ(√(NT)). Under competition, each seller combines its own draw with the potential function of the MNL game. With full observation, the expected number of periods outside the pure-strategy Nash equilibrium selected by the potential is o(T); with partial observation, the expected number of periods outside the equilibrium set is also o(T), provided that decisions are identifiable. Taken together, these results show that learning demand and stabilizing a competitive market depend on the information observed, the extent to which MNL dependence is preserved, and the computational burden one is willing to bear.