Dynamic pricing with real-time demand learning

Abstract

In many service industries, the firm adjusts the product price dynamically by taking into account the current product inventory and the future demand distribution. Because the firm can easily monitor the product inventory, the success of dynamic pricing relies on an accurate demand forecast. In this paper, we consider a situation where the firm does not have an accurate demand forecast, but can only roughly estimate the customer arrival rate before the sale begins. As the sale moves forward, the firm uses real-time sales data to fine-tune this arrival rate estimation. We show how the firm can first use this modified arrival rate estimation to forecast the future demand distribution with better precision, and then use the new information to dynamically adjust the product price in order to maximize the expected total revenue. Numerical study shows that this strategy not only is nearly optimal, but also is robust when the true customer arrival rate is much different from the original forecast. Finally, we extend the results to four situations commonly encountered in practice: unobservable lost customers, time dependent arrival rate, batch demand, and discrete set of allowable prices.

Introduction

Dynamic pricing is a business strategy that adjusts the product price in a timely fashion in order to allocate the right service to the right customer at the right time. The rationale of dynamic pricing can be understood with an example of an airline company. When an airline sells seats in the same class, it offers different fares depending on time to departure and current seat inventory. The airline has the incentive to promote sale when the departure time is approaching with a lot of vacancies on hand, because each empty seat is worth nothing after the airplane takes off. On the other hand, the airline still wants to reserve a certain number of seats for possible last-minute travelers who are willing to pay substantially more in price. As a consequence, airfare often fluctuates in its selling horizon.

Products such as airline seats are called perishable products, which have three major characteristics: (1) the quantity is fixed and reordering is not possible; (2) there is a deadline for sale; and (3) the marginal cost of selling one more item is little, so most part of revenue goes directly to profit. Because of these characteristics, perishable products are particularly suitable for dynamic pricing. Besides being used extensively in the airline industry, dynamic pricing can also be found in other travel industries––such as hotel rooms [4], rental cars, and cruise lines [11]––to incorporate seasonal fluctuation in demand. Interested readers are referred to survey papers, such as [20] and [17], for an overview of dynamic pricing and its role in revenue management.

In general, there are two major sources of randomness in demand: customer arrival rate and customer reservation price distribution. Most existing literature concerning dynamic pricing assumes that both customer arrival rate and customer reservation price distribution are well known before the sale begins. In many service industries, however, whereas the seller can use historical data to estimate the customer reservation price to a good extent, it is rather difficult to accurately forecast the customer arrival rate before the sale begins. For example in the travel industry, the demand rates for air travel services and for hotel rooms on different days may be different if an event––such as a commencement, a trade show, or a conference––takes place at the destination city. For another example in the entertainment industry, when a pop singer goes on an international tour, it is relatively easy to know how much a loyal fan is willing to pay for a ticket, but it is rather difficult to know how many fans there are in each city and how many of those fans will be aware of the event. In these cases, if the seller roughly estimates the customer arrival rate and dynamically sets the product price based on this rough estimation, he faces a significant risk. If the true customer arrival rate is much lower than the estimated rate, the seller will end up with many unsold items at the end. On the other hand, if the true arrival rate is much higher, the seller will be out of stock quickly and loses the opportunity to take advantage of the excess demand. The dynamic pricing literature does not adequately address this risk.

In this paper, we present a dynamic pricing model where customers arrive in accordance with a conditional Poisson process, whose rate is not known to the seller in advance. Instead, through preliminary pre-sale market research, the seller obtains a prior distribution of the customer arrival rate. As the sale moves forward, the seller uses real-time sales data from the realized demand to fine-tune the arrival rate estimation, and then uses the fine-tuned arrival rate estimation to better understand the demand curve in the future. Consequently, the seller updates the future demand distribution in real time, and then dynamically sets the product price to maximize the expected total revenue.

In recent years, the problem of dynamic pricing has drawn much attention. Most research on dynamic pricing assumes that the customers arrive according to a stochastic process that has independent increments; that is, the numbers of customers in disjoint time intervals are independent random variables. With this assumption, knowing the number of customers that have shown up so far provides no additional information about how many more customers will show up later on, so learning is not possible. For example, in a continuous-time setting, a common assumption is that customers arrive according to a Poisson process with a given intensity function [6], [8], [9]. In a discrete time setting, time is divided into small intervals such that in each time interval there is a small probability a customer will arrive, independent of everything else [13], [19]. With the assumption that the demand process has independent increments, the problem is often formulated as a Markov decision process. In most cases, it can be shown that the optimal product price increases in the remaining time and decreases in the current inventory level. However, because the optimal policy is difficult to derive, most research focuses on developing heuristic policies.

Learning models have been studied in the operations management literature to better forecast the future demand curve. Most work assumes that price is exogenous, while the firm decides how much inventory to replenish in each time period [2], [12], [16]. Learning models that incorporate both price and replenishment decisions include [3] and [18]. In both papers, the demand curve in each time period is a deterministic and identical function of the price, while the parameters of the function are unknown to the decision maker. Based on the realized demand in early periods, the decision maker learns about the demand curve in order to set a proper price later on. Burnetas and Smith [5] considered a similar problem except that the demands in different periods are independent and identically distributed random variables. They developed a policy with which the realized profit converges with probability one to the optimal value under complete information. These learning models are different from our model because the seller learns from repetition of identical experiments (same flight number through different days), and in our model the seller learns throughout the sales horizon of a single event.

The rest of this paper is organized as follows. In Section 2, we introduce a dynamic pricing model where customers arrive according to a conditional Poisson process. We show how the seller can improve the estimation on the customer arrival rate from the real-time sales data as the sale moves forward. Motivated by these preliminary results, we consider a surrogate dynamic pricing model and derive its optimal policy in Section 3. Then in Section 4, we use the results from this surrogate model to develop the variable-rate policy for the original problem described in Section 2. The numerical experiments show that this variable-rate policy is not only nearly optimal, but also robust even when the pre-sale estimation on the customer arrival rate is relatively poor. In Section 5, we extend the results to four settings that are often encountered in practice: (1) lost customers are not observable; (2) the customer arrival process is non-stationary; (3) each customer can request more than one item; and (4) the allowable price set is discrete. Finally we conclude the paper and discuss future research directions in Section 6.