Performance bounds on optimal fixed prices

Abstract

We consider the problem of selling a fixed stock of items over a finite horizon when the buyers arrive following a Poisson process. We obtain a general lower bound on the performance of using a fixed price rather than dynamically adjusting the price. The bound is 63.21% for one unit of inventory, and it improves as the inventory increases. For the one-unit case, we also obtain tight bounds: 89.85% for the constant-elasticity and 96.93% for the linear price-response functions.

Introduction

We consider the problem of selling a fixed stock of items over a finite horizon. Potential customers arrive following a Poisson process and decide whether to buy the product or not based on the posted price and their maximum willingness-to-pay. The seller knows the probability distribution of the customers’ maximum willingness-to-pay and adjusts the price as a function of remaining time in the horizon so as to maximize his/her expected revenue. It is assumed that each customer has a chance to buy the product only at his/her (first) arrival, i.e., the seller cannot recall a customer who already decided not to buy the product (no recalls) or a customer cannot return later to buy the product at a lower price (no strategic consumer behavior).

Kincaid and Darling  [7] were the first to study this problem, which they call “inventory pricing problem”, and to characterize the function that the seller should use to update the price over the horizon. Elfving  [3] notices that the special case of a single unit of inventory is similar to what is known as the “secretary problem” or the “best choice problem” (see  [5]), except that the arrivals (of candidates, i.e., potential buyers in our problem) follow a point process and the decision maker has distributional information about the desirability of candidates. Elfving  [3] investigates the use of a “critical curve” to govern the decision maker’s decision to accept or reject an arriving candidate. This curve essentially plays the role of price (as a function of time) in our problem; if an arriving candidate’s desirability (or offer) is larger than or equal to the value of the function at the candidate’s arrival time, the candidate is accepted and the process stops. Elfving  [3] characterizes the unique curve in the presence of discounting. Stadje  [10] derives conditions that are equivalent to those obtained in  [7] for pricing a single or multiple units of inventory.

The generalized version of this problem for multiple units has been studied in a seminal work by Gallego and van Ryzin  [6] and subsequently by Bitran and Mondschein  [2]. Since then, dynamic pricing in the presence of inventory considerations has been the subject of an extensive literature in Operations Research (see [11], [4] for reviews). An important question is whether dynamic pricing offers substantial revenue improvements over keeping the price constant over the horizon. The main finding in  [6] is that using a constant price at a level that is determined by the solution of the deterministic version problem has a bounded worst performance and is asymptotically optimal as the expected demand and starting inventory go to infinity. Note that solving the deterministic version of the problem to obtain the fixed price is a heuristic even for static pricing; one can also optimize among all possible prices. The numerical results in  [6] show that, even for smaller-size problems, optimal fixed prices lead to a very good performance. In fact, under an exponential price-response function, the worst performance reported in that article is 94.5%. The authors state that they have never observed a suboptimality gap more than 7% (p. 1009). Zhao and Zheng  [12] find that the constant-demand elasticity price-response function leads to larger gaps than the exponential price-response function. The worst performance that they report is 92.74% (the price is selected from a set of 11 alternatives in this case). The worst performance reported in an extensive numerical study in  [9] that involves linear, logit, and exponential price-response functions is 93.43%. For a discrete-time version of the problem, Bearden et al.  [1] report that the worst performance in their numerical experiments is 94%.

In this paper, we develop theoretical lower bounds for the worst performance of fixed pricing heuristics. For multiple units of starting inventory, we obtain a lower bound for the fixed pricing heuristic that is independent of the price-response function and the length of the selling season. For a single unit of inventory, we obtain tight bounds for constant-elasticity and linear price-response functions. These price-response functions are widely used in theory and practice, and they represent the two extreme forms for the potential benefits of dynamic pricing. Under the constant-elasticity price-response function, potential buyers have the same sensitivity to price changes at all price levels. This allows the seller to test higher prices in the beginning of the horizon since price reductions remain effective later in the horizon if needed. In contrast, under a linear price-response function, the elasticity decreases as the price is lowered over the course of the horizon. Confirming these intuitions, we show that dynamic pricing is most useful under a constant-elasticity price-response function and that the theoretical worst performance of optimal fixed pricing is 89.85%. On the other hand, the worst performance of optimal fixed pricing under a linear price-response function is significantly higher, at 96.93%. Our numerical results show that other parametric price-response functions have worst performances that are between these two bounds.