A pricing problem with unknown arrival rate and price sensitivity

We study a pricing problem with finite inventory and semi-parametric demand uncertainty. Demand is a price-dependent Poisson process whose mean is the product of buyers’ arrival rate, which is a constant $$\lambda $$ λ , and buyers’ purchase probability $$q(p)$$ q ( p ) , where p is the price. The seller observes arrivals and sales, and knows neither $$\lambda $$ λ nor $$q$$ q . Based on a non-parametric maximum-likelihood estimator of $$(\lambda ,q)$$ ( λ , q ) , we construct an estimator of mean demand and show that as the system size and number of prices grow, it is asymptotically more efficient than the maximum likelihood estimator based only on sale data. Based on this estimator, we develop a pricing algorithm paralleling (Besbes and Zeevi in Oper Res 57:1407–1420, 2009) and study its performance in an asymptotic regime similar to theirs: the initial inventory and the arrival rate grow proportionally to a scale parameter n. If $$q$$ q and its inverse function are Lipschitz continuous, then the worst-case regret is shown to be $$O((\log n / n)^{1/4})$$ O ( ( log n / n ) 1 / 4 ) . A second model considered is the one in Besbes and Zeevi (2009, Section 4.2), where no arrivals are involved; we modify their algorithm and improve the worst-case regret to $$O((\log n / n)^{1/4})$$ O ( ( log n / n ) 1 / 4 ) . In each setting, the regret order is the best known, and is obtained by refining their proof methods. We also prove an $$\Omega (n^{-1/2})$$ Ω ( n - 1 / 2 ) lower bound on the regret. Numerical comparisons to state-of-the-art alternatives indicate the effectiveness of our arrivals-based approach.