Omar Besbes

Dynamic Pricing Without Knowing the Demand Function: Risk Bounds and Near-Optimal Algorithms

We consider a single-product revenue management problem where, given an initial inventory, the objective is to dynamically adjust prices over a finite sales horizon to maximize expected revenues. Realized demand is observed over time, but the underlying functional relationship between price and mean demand rate that governs these observations (otherwise known as the demand function or demand curve) is not known. We consider two instances of this problem: (i) a setting where the demand function is assumed to belong to a known parametric family with unknown parameter values; and (ii) a setting where the demand function is assumed to belong to a broad class of functions that need not admit any parametric representation. In each case we develop policies that learn the demand function “on the fly,” and optimize prices based on that. The performance of these algorithms is measured in terms of the regret: the revenue loss relative to the maximal revenues that can be extracted when the demand function is known prior to the start of the selling season. We derive lower bounds on the regret that hold for any admissible pricing policy, and then show that our proposed algorithms achieve a regret that is “close” to this lower bound. The magnitude of the regret can be interpreted as the economic value of prior knowledge on the demand function, manifested as the revenue loss due to model uncertainty.

Blind Network Revenue Management

We consider a general class of network revenue management problems, where mean demand at each point in time is determined by a vector of prices, and the objective is to dynamically adjust these prices so as to maximize expected revenues over a finite sales horizon. A salient feature of our problem is that the decision maker can only observe realized demand over time but does not know the underlying demand function that maps prices into instantaneous demand rate. We introduce a family of “blind” pricing policies that are designed to balance trade-offs between exploration demand learning and exploitation pricing to optimize revenues. We derive bounds on the revenue loss incurred by said policies in comparison to the optimal dynamic pricing policy that knows the demand function a priori, and we prove that asymptotically, as the volume of sales increases, this gap shrinks to zero.

On Implications of Demand Censoring in the Newsvendor Problem

We consider a repeated newsvendor problem in which the decision maker (DM) does not have access to the underlying demand distribution. The goal of this paper is to characterize the implications of demand censoring on performance. To that end, we compare the benchmark setting in which the DM has access to demand observations to a setting in which the DM may only rely on sales data. We measure performance in terms of regret: the difference between the cumulative costs of a policy and the optimal cumulative costs with knowledge of the demand distribution. Through upper and lower bounds, we characterize the optimal magnitude of the worst-case regret for the two settings, enabling one to isolate the implications of demand censoring. In particular, the results imply that the exploration--exploitation trade-off introduced by demand censoring is fundamentally different in the continuous and discrete demand cases, and that active exploration plays a much stronger role in the latter case. We further establish that in the discrete demand case, the need for active exploration almost disappears as soon as a lost sales indicator (that records whether demand was censored or not) becomes available, in addition to the censored demand samples. This paper was accepted by Gerard P. Cachon, stochastic models and simulation.

Going Bunkers: The Joint Route Selection and Refueling Problem

Managing shipping vessel profitability is a central problem in marine transportation. We consider two commonly used types of vessels---“liners” (ships whose routes are fixed in advance) and “trampers” (ships for which future route components are selected based on available shipping jobs)---and formulate a vessel profit maximization problem as a stochastic dynamic program. For liner vessels, the profit maximization reduces to the problem of minimizing refueling costs over a given route subject to random fuel prices and limited vessel fuel capacity. Under mild assumptions about the stochastic dynamics of fuel prices at different ports, we provide a characterization of the structural properties of the optimal liner refueling policies. For trampers, the vessel profit maximization combines refueling decisions and route selection, which adds a combinatorial aspect to the problem. We characterize the optimal policy in special cases where prices are constant through time and do not differ across ports and prices are constant through time and differ across ports. The structure of the optimal policy in such special cases yields insights on the complexity of the problem and also guides the construction of heuristics for the general problem setting.

Non-Stationary Stochastic Optimization

We consider a non-stationary variant of a sequential stochastic optimization problem, in which the underlying cost functions may change along the horizon. We propose a measure, termed variation budget , that controls the extent of said change, and study how restrictions on this budget impact achievable performance. We identify sharp conditions under which it is possible to achieve long-run average optimality and more refined performance measures such as rate optimality that fully characterize the complexity of such problems. In doing so, we also establish a strong connection between two rather disparate strands of literature: (1) adversarial online convex optimization and (2) the more traditional stochastic approximation paradigm (couched in a non-stationary setting). This connection is the key to deriving well-performing policies in the latter, by leveraging structure of optimal policies in the former. Finally, tight bounds on the minimax regret allow us to quantify the “price of non-stationarity,” which mathematically captures the added complexity embedded in a temporally changing environment versus a stationary one.

Revenue Optimization for a Make-to-Order Queue in an Uncertain Market Environment

We consider a revenue-maximizing make-to-order manufacturer that serves a market of price-and delay-sensitive customers and operates in an environment in which the market size varies stochastically over time. A key feature of our analysis is that no model is assumed for the evolution of the market size. We analyze two main settings: (i) the size of the market is observable at any point in time; and (ii) the size of the market is not observable and hence cannot be used for decision making. We focus on high-volume systems that are characterized by large processing capacities and market sizes, and where the latter fluctuate on a slower timescale than that of the underlying production system dynamics. We develop an approach to tackle such problems that is based on an asymptotic analysis and that yields near-optimal policy recommendations for the original system via the solution of a stochastic fluid model.

Testing the Validity of a Demand Model: An Operations Perspective

The fields of statistics and econometrics have developed powerful methods for testing the validity (specification) of a model based on its fit to underlying data. Unlike statisticians, managers are typically more interested in the performance of a decision rather than the statistical validity of the underlying model. We propose a framework and a statistical test that incorporate decision performance into a measure of statistical validity. Under general conditions on the objective function, asymptotic behavior of our test admits a sharp and simple characterization. We develop our approach in a revenue management setting and apply the test to a data set used to optimize prices for consumer loans. We show that traditional model-based goodness-of-fit tests may consistently reject simple parametric models of consumer response (e.g., the ubiquitous logit model), while at the same time these models may “pass” the proposed performance-based test. Such situations arise when decisions derived from a postulated (and possibly incorrect) model generate results that cannot be distinguished statistically from the best achievable performance---i.e., when demand relationships are fully known.

Dynamic Pricing with Financial Milestones: Feedback-Form Policies

We study a seller that starts with an initial inventory of goods, has a target horizon over which to sell the goods, and is subject to a set of financial milestone constraints on the revenues and sales that need to be achieved at different time points along the sales horizon. We characterize the revenue maximizing dynamic pricing policy for the seller and highlight the effect of revenue and sales milestones on its structure. The optimal policy can be written in feedback form, where the price at each point in time is selected so as to track the most stringent among all future milestones. Building on that observation, we propose a discrete-review policy that aims to dynamically track the appropriate milestone constraint and show that this simple and practical policy is near optimal in settings with large initial capacity and long sales horizons even in settings with no advance demand model information. One motivating application comes from the sales of new multiunit, residential real estate developments, where intermediate milestone constraints play an important role in their financing and construction. This paper was accepted by Gerard P. Cachon, stochastic models and simulation.

Dynamic Pricing Strategies in the Presence of Demand Shifts

Many factors introduce the prospect of changes in the demand environment that a firm faces, with the specifics of such changes not necessarily known in advance. If and when realized, such changes affect the delicate balance between demand and supply and thus current prices should account for these future possibilities. We study the dynamic pricing problem of a retailer facing the prospect of a change in the demand function during a finite selling season with no inventory replenishment opportunity. In particular, the time of the change and the postchange demand function are unknown upfront, and we focus on the fundamental trade-off between collecting revenues from current demand and doing so for postchange demand, with the capacity constraint introducing the main tension. We develop a formulation that allows for isolating the role of dynamic pricing in balancing inventory consumption throughout the horizon. We establish that, in many settings, optimal pricing policies follow a monotone path up to the change in demand. We show how one may compare upfront the attractiveness of pre- and postchange demand conditions and how such a comparison depends on the problem primitives. We further analyze the impact of the model inputs on the optimal policy and its structure, ranging from the impact of model parameter changes to the impact of different representations of uncertainty about future demand.

On Information Distortions in Online Ratings

Consumer reviews and ratings of products and services have become ubiquitous on the Internet. This paper analyzes, given the sequential nature of reviews and the limited feedback of such past reviews, the information content they communicate to future customers. We consider a model with heterogeneous customers who buy a product of unknown quality and we focus on two different informational settings. In the first setting, customers observe the whole history of past reviews. In the second one they only observe the sample mean of past reviews. We examine under which conditions, in each setting, customers can recover the true quality of the product based on the feedback they observe. In the case of total monitoring, if consumers adopt a fully rational Bayesian updating paradigm, then they asymptotically learn the unknown quality. With access to only the sample mean of past reviews, inference becomes intricate for customers and it is not clear if, when, and how social learning can take place. We first analyze ...