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Author: N. Bora Keskin
Publisher:
ISBN:
Category :
Languages : en
Pages : 52
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Book Description
We consider a monopolist who sells a set of products over a time horizon of T periods. The seller initially does not know the parameters of the products' linear demand curve, but can estimate them based on demand observations. We first assume that the seller knows nothing about the parameters of the demand curve, and then consider the case where the seller knows the expected demand under an incumbent price. It is shown that the smallest achievable revenue loss in T periods, relative to a clairvoyant who knows the underlying demand model, is of order √T in the former case and of order logT in the latter case. To derive pricing policies that are practically implementable, we take as our point of departure the widely used policy called greedy iterated least squares (ILS), which combines sequential estimation and myopic price optimization. It is known that the greedy ILS policy itself suffers from incomplete learning, but we show that certain variants of greedy ILS achieve the minimum asymptotic loss rate. To highlight the essential features of well-performing pricing policies, we derive sufficient conditions for asymptotic optimality.
Author: N. Bora Keskin
Publisher:
ISBN:
Category :
Languages : en
Pages : 52
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Book Description
We consider a monopolist who sells a set of products over a time horizon of T periods. The seller initially does not know the parameters of the products' linear demand curve, but can estimate them based on demand observations. We first assume that the seller knows nothing about the parameters of the demand curve, and then consider the case where the seller knows the expected demand under an incumbent price. It is shown that the smallest achievable revenue loss in T periods, relative to a clairvoyant who knows the underlying demand model, is of order √T in the former case and of order logT in the latter case. To derive pricing policies that are practically implementable, we take as our point of departure the widely used policy called greedy iterated least squares (ILS), which combines sequential estimation and myopic price optimization. It is known that the greedy ILS policy itself suffers from incomplete learning, but we show that certain variants of greedy ILS achieve the minimum asymptotic loss rate. To highlight the essential features of well-performing pricing policies, we derive sufficient conditions for asymptotic optimality.
Author: Omar Besbes
Publisher:
ISBN:
Category :
Languages : en
Pages : 34
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Book Description
We consider a multi-period single product pricing problem with an unknown demand curve. The seller's objective is to adjust prices in each period so as to maximize cumulative expected revenues over a given finite time horizon; in doing so, the seller needs to resolve the tension between learning the unknown demand curve and maximizing earned revenues. The main question that we investigate is the following: how large of a revenue loss is incurred if the seller uses a simple parametric model which differs significantly (i.e., is misspecified) relative to the underlying demand curve. This "price of misspecification'' is expected to be significant if the parametric model is overly restrictive. Somewhat surprisingly, we show (under reasonably general conditions) that this may not be the case.
Author: Georgia Perakis
Publisher:
ISBN:
Category :
Languages : en
Pages : 50
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Book Description
We consider the dynamic pricing problem of a retailer who does not have any information on the underlying demand for a product. The retailer aims to maximize cumulative revenue collected over a finite time horizon by balancing two objectives: textit{learning} demand and textit{maximizing} revenue. The retailer also seeks to reduce the amount of price experimentation because of the potential costs associated with price changes. Existing literature solves this problem in the case where the unknown demand is parametric. We consider the pricing problem when demand is non-parametric. We construct a pricing algorithm that uses piecewise linear approximations of the unknown demand function and establish when the proposed policy achieves near-optimal rate of regret, tilde{O}( sqrt{T}), while making O( log log T) price changes. Hence, we show considerable reduction in price changes from the previously known mathcal{O}( log T) rate of price change guarantee in the literature. We also perform extensive numerical experiments to show that the algorithm substantially improves over existing methods in terms of the total price changes, with comparable performance on the cumulative regret metric.
Author: Wang Chi Cheung
Publisher:
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Languages : en
Pages : 30
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Book Description
In a dynamic pricing problem where the demand function is not known a priori, price experimentation can be used as a demand learning tool. Existing literature usually assumes no constraint on price changes, but in practice sellers often face business constraints that prevent them from conducting extensive experimentation. We consider a dynamic pricing model where the demand function is unknown but belongs to a known finite set. The seller is allowed to make at most m price changes during T periods. The objective is to minimize the worst case regret, i.e., the expected total revenue loss compared to a clairvoyant who knows the demand distribution in advance. We demonstrate a pricing policy that incurs a regret of O(log^(m) T), or m iterations of the logarithm. Furthermore, we describe an implementation at Groupon, a large e-commerce marketplace for daily deals. The field study shows significant impact on revenue and bookings.
Author: N. Bora Keskin
Publisher:
ISBN:
Category :
Languages : en
Pages : 39
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Book Description
We consider a price-setting firm that sells a product over a continuous time horizon. The firm is uncertain about the sensitivity of demand to price adjustments, and continuously updates its prior belief on an unobservable sensitivity parameter by observing the demand responses to prices. The firm's objective is to minimize the infinite-horizon discounted loss, relative to a clairvoyant that knows the unobservable sensitivity parameter. Using partial differential equations theory, we characterize the optimal pricing policy, and then derive a formula for the optimal learning premium that projects the value of learning onto prices. We compare and contrast the optimal pricing policy with the myopic pricing policy, and quantify the cost of myopically neglecting to charge a learning premium in prices. We show that the optimal learning premium for a firm that looks far into the future is the squared coefficient of variation (SCV) in the firm's posterior belief. Based on this principle, we design a simple variant of the myopic policy, namely the SCV rule, and prove that this policy is long-run average optimal.
Author: Yiwei Chen
Publisher:
ISBN:
Category :
Languages : en
Pages : 60
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We consider a firm that sells a single type product on multiple local markets over a finite horizon via dynamically adjusted prices. To prevent price discrimination, prices posted on different local markets at the same time are the same. The entire horizon consists of one or multiple change-points. Each local market's demand function linearly evolves over time between any two consecutive change-points. Each change-point is classified as either a zero-order or a first-order change-point in terms of how smooth the demand function changes at this point. At a zero-order change-point, at least one local market's demand function has an abrupt change. At a first-order change-point, all local markets' demand functions continuously evolve over time, but at least one local market's demand evolution speed has an abrupt change. The firm has no information about any parameter that modulates the demand evolution process before the start of the horizon. The firm aims at finding a pricing policy that yields as much revenue as possible. We show that the regret under any pricing policy is lower bounded by CT^{1/2} with C>0, and the lower bound becomes as worse as CT^{2/3} if at least one change-point is a first-order change-point.We propose a Joint Change-Point Detection and Time-adjusted Upper Confidence Bound (CU) algorithm. This algorithm consists of two components: the change-point detection component and the exploration-exploitation component. In the change-point detection component, the firm uniformly samples each price for one time in each batch of the time interval with the same length. She uses sales data collected at the times that she uniformly samples prices to both detect whether a change occurs and judge whether it is a zero-order or a first-order change if it occurs. In the exploration-exploitation component, the firm implements a time-adjusted upper confidence bound (UCB) algorithm between two consecutive detected change-points. Because demand dynamically evolves between two consecutive change-points, we introduce a time factor into the classical UCB algorithm to correct the bias of using historic sales data to estimate demand at present. We show that the CU algorithm achieves the regret lower bounds (up to logarithmic factors).
Author: Chen, Xi
Publisher:
ISBN:
Category :
Languages : en
Pages : 51
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Book Description
This paper studies the dynamic pricing problem under model mis-specifi cation settings. To characterize the model mis-specification, we extend the "eps-contamination model | the most fundamental model in robust statistics and machine learning, to the online setting. In particular, for a selling horizon of length T, the online "eps-contamination model assumes that the demands are realized according to a typical unknown demand function only for (1-eps)T periods. For the rest of eps T periods, an outlier purchase can happen with arbitrary demand functions. Under this model, we develop new robust dynamic pricing policies to hedge against outlier purchase behavior. For the dynamic pricing problem, there are two critical prices, the revenue-maximizing price and inventory clearance price, and the optimal price is the larger price. The challenge is that the seller has no information about which price is larger, and the revenues near these two prices behave entirely differently. To address this challenge, we propose robust online policies for both cases when the optimal price is the revenue-maximizing price and when the optimal price is the clearance price, and then develop a meta algorithm that combines these two cases. Our algorithm is a fully adaptive policy that does not require any prior knowledge of the outlier proportion parameter ". Our simulation study shows that our policy outperforms existing policies in the literature.
Author: Jinzhi Bu
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Author: Mr. Nuri Bora Keskin
Publisher:
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Category :
Languages : en
Pages :
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Pricing decisions often involve a tradeoff between learning about customer behavior to increase long-term revenues, and earning short-term revenues. In this thesis we examine that tradeoff. Whenever a firm is not certain about how its customers will respond to price changes, there is an opportunity to use price as a tool for learning about a demand curve. Most firms try to solve the tradeoff between learning and earning by managing these two goals separately. A common practice is to first estimate the parameters of the demand curve, and then choose the optimal price, assuming the parameter estimates are accurate. In this thesis we show that this conventional approach is far from being optimal, running the risk of incomplete learning--a negative statistical outcome in which the decision maker stops learning prematurely. We also propose several remedies to avoid the incomplete learning problem, and guard against poor performance. In Chapter 1, we model a learn-and-earn problem using a theoretical framework in which a seller has a prior belief about the demand curve for its product, and updates his belief upon observing customer responses to successive sales attempts. We assume that the seller's prior is a binary distribution, i.e. one of two demand curves is known to apply, although our analysis can be extended to any finite prior. In this setting, we first analyze the myopic Bayesian policy (MBP), which is a stylized representative of the estimate-and-then-optimize policies described above. Our analysis makes three contributions to the literature: first, we show that under the MBP the seller's beliefs can get stuck at a confounding value, leading to poor revenue performance. This result elucidates incomplete learning as a consequence of myopic pricing. Our second contribution is the development of a constrained variant of the MBP as a way to tweak the MBP in the binary-prior setting. By forbidding prices that are not sufficiently informative, constrained MBP (CMBP) avoids the incomplete learning problem entirely, and moreover, its expected performance gap relative to a clairvoyant who iv knows the underlying demand curve is bounded by a constant independent of the sales horizon. Finally, we generalize the CMBP family to obtain more flexible pricing policies that are suitable in case the seller has an arbitrary prior on model parameters. The incomplete learning result and the pricing policies we design have a practical significance. Because firms have no means to check whether they are suffering from incomplete learning, the myopic policies used in practice need to be modified with some kind of forced price experimentation, and our policies provide guidelines on how price experimentation can be employed to prevent incomplete learning. In Chapter 2, we consider several research questions: for example, when a seller has been charging an incumbent price for a very long time, how can he make use of the information contained in that incumbent price? Or, when a seller offers multiple products with substitutable demand, can he safely employ an independent price experimentation strategy for each product? More importantly, what if the particular pricing policies in literature are not feasible in a given business setting? To handles such cases, can we derive general principles that identify the essential ingredient of successful price experimentation policies? We address these questions using a fairly general dynamic pricing model, where a monopolist sells a set of products over a given time horizon. The expected demand for products is given by a linear curve, the parameters of which are not known by the seller. The seller's goal is to learn the parameters of the demand curve as he keeps trying to earn revenues. This chapter makes four main contributions to the learning-and-earning literature. First, we formulate an incumbent-price problem, where the seller starts out knowing one point on its demand curve, and show that the value of information contained in the incumbent price is substantial. Second, unlike previous studies that focus on a particular form of price experimentation, we derive general sufficient conditions for accumulating information in a near-optimal manner. We believe that practitioners can use these conditions as guidelines to design successful pricing policies in various settings. Third, we develop a unifying theme to obtain performance bounds in operations management problems with model uncertainty. We employ (i) the concept of Fisher information to derive natural lower bounds on regret, and (ii) martingale theory to analyze the estimation errors and generate well-performing policies. Finally, we analyze the pricing of multiple products with substitutable demand. Our analysis shows that multi-product pricing is not a straightforward repetition of single-product pricing. Learning in a high dimensional price space essentially requires sufficient "variation" in the directions of successive price vectors, which brings forth the idea of orthogonal pricing. In Chapter 3, we extend our analysis to the case where information can become obsolete. The particular dynamic pricing problem we consider includes a seller who tries to simultaneously learn about a time-varying demand curve, and earn sales revenues. We conduct a simulation study to evaluate the revenue performance of several pricing policies in this setting. Our results suggest that policies designed for static demand settings do not perform well in time-varying demand settings. Moreover, if the demand environment is not very noisy and the changes are not very frequent, a simple modification of the estimate-and-then-optimize approach, which is based on a moving time window, performs reasonably well in changing demand environments.
Author: Steffen Christ
Publisher: Springer Science & Business Media
ISBN: 3834961841
Category : Business & Economics
Languages : en
Pages : 363
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Book Description
Steffen Christ shows how theoretic optimization models can be operationalized by employing self-learning strategies to construct relevant input variables, such as latent demand and customer price sensitivity.
