Online Learning and Pricing for Multiple Products with Reference Price Effects

We consider the dynamic pricing problem of a monopolist seller who sells a set of mutually substitutable products over a finite time horizon. Customer demand is sensitive to the price of each individual product and the reference price which is formed from a comparison among the prices of all products. To maximize the total expected profit, the seller needs to determine the selling price of each product and also selects a reference product (to be displayed) that affects the consumer's reference price. However, the seller initially knows neither the demand function nor the customer's reference price, but can learn them from past observations on the fly. As such, the seller faces the classical trade-off between exploration (learning the demand function and reference price) and exploitation (using what has been learned thus far to maximize revenue). We propose a dynamic learning-and-pricing algorithm that integrates iterative least squares estimation and bandit control techniques in a seamless fashion. We show that the cumulative regret, i.e., the expected revenue loss caused by not using the optimal policy over $T$ periods, is upper bounded by $O((n^2+n)\sqrt{T} \log T)$, which is optimal up to a logarithmic factor in terms of the time horizon $T$ and polynomially scaling with the number of products $n$. We also establish the regret lower bound (for any learning policies) to be $\Omega(n^{1.5}\sqrt{T})$. We then generalize our analysis to a more general demand model. Finally, our algorithm performs consistently well numerically, outperforming an exploration-exploitation benchmark. We also identify an interesting ``loss-leader'' phenomenon in our computational study