A pricing problem with unknown arrival rate and price sensitivity
- PDF / 795,275 Bytes
- 30 Pages / 439.37 x 666.142 pts Page_size
- 84 Downloads / 183 Views
A pricing problem with unknown arrival rate and price sensitivity Athanassios N. Avramidis1 Received: 23 February 2019 / Revised: 20 January 2020 © The Author(s) 2020
Abstract 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 λ, and buyers’ purchase probability q( p), where p is the price. The seller observes arrivals and sales, and knows neither λ nor q. Based on a non-parametric maximum-likelihood estimator of (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 and its inverse function are Lipschitz continuous, then the worst-case regret is shown to be 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 ). In each setting, the regret order is the best known, and is obtained by refining their proof methods. We also prove an (n −1/2 ) lower bound on the regret. Numerical comparisons to state-of-the-art alternatives indicate the effectiveness of our arrivals-based approach. Keywords Estimation · Asymptotic efficiency · Exploration–exploitation · Regret · Asymptotic analysis Mathematics Subject Classification 60K10 · 93E35 · 90B05 · 62G20
B 1
Athanassios N. Avramidis [email protected] School of Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, UK
123
A. N. Avramidis
1 Introduction 1.1 Background Pricing and revenue management are important problems in many industries. Talluri and van Ryzin (2005) discuss instances of this problem that range over many industries, including fashion and retail, air travel, hospitality, and leisure. Early literature assumes the relationship between the mean demand and the price is known to the seller (Gallego and van Ryzin 1994). In practice, decision makers seldom have such knowledge. Pricing and demand learning is a stream of literature concerned with pricing under incomplete knowledge of the demand process, which is estimated. A standard model is that whenever the price is set at p, the demand is a Poisson process of rate ( p), which is called the demand function (Besbes and Zeevi 2009, 2012; Wang et al. 2014). Estimation methods are broadly divided into parametric and non-parametric. The former assume a certain functional form and carry mis-specification risk, while the latter make weaker assumptions and tend to alleviate this risk (Besbes and Zeevi 2009). For the single-product problem with finite inventory, prominent work
Data Loading...
