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Technical Note—Personalized Dynamic Pricing of LimitedInventoriesGoker Aydin, Serhan Ziya,

To cite this article:Goker Aydin, Serhan Ziya, (2009) Technical Note—Personalized Dynamic Pricing of Limited Inventories. Operations Research57(6):1523-1531. http://dx.doi.org/10.1287/opre.1090.0701

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TECHNICAL NOTE

Personalized Dynamic Pricing ofLimited Inventories

Goker AydinDepartment of Industrial and Operations Engineering, University of Michigan,

Ann Arbor, Michigan 48109, ayding@umich.edu

Serhan ZiyaDepartment of Statistics and Operations Research, University of North Carolina,

Chapel Hill, North Carolina 27559, ziya@unc.edu

Prior work has investigated time- and inventory-level-dependent pricing of limited inventories with finite selling horizons.We consider a third dimension—in addition to time and inventory level—that the firms can use in setting their prices: theinformation that the firm has at the individual customer level. An arriving customer provides a signal to the firm, whichis an imperfect indicator of the customer’s willingness to pay, and the firm makes a personalized price offer depending onthe signal, inventory level, and time. We consider two different models: full personalization and partial personalization.In the full personalization model, the firm charges any price it wishes given the customer signal, while in the partialpersonalization model, the firm can charge one of two prices. We find that a mere correlation between the signals andcustomers’ willingness to pay is not sufficient to ensure intuitive relationships between the signal and the optimal prices.We determine a stronger condition, which leads to several structural properties, including the monotonicity of the optimalprice with respect to the signal in the full personalization model. For the partial personalization model, we show that theoptimal pricing policy is of threshold-type and that the threshold is monotonic in the inventory level and time.

Subject classifications : inventory/production: policies; marketing: dynamic pricing, personalized pricing.Area of review : Revenue Management.History : Received October 2007; revisions received April 2008, September 2008; accepted November 2008. Publishedonline in Articles in Advance August 17, 2009.

1. IntroductionFirms that sell a limited inventory of a perishable orseasonal product (e.g., airlines, apparel retailers) havelong been used to adjusting their prices over time basedon inventory levels, a practice commonly referred to asdynamic pricing. In addition to adjustments based on timeand inventory levels, a firm can also tailor the price itcharges to its customers based on information availableat the individual customer level. This practice is usuallyreferred to as personalized pricing. In this paper, our goalis to explore the interactions between personalized pric-ing and dynamic pricing. In particular, we analyze how theavailability of a product interacts with customer informa-tion to determine the price offered to a customer.Many sellers have the ability to identify individual cus-

tomers, collect information about them, and track their pur-chasing behavior (e.g., catalog retailers, online retailers,and stores that issue loyalty cards to their customers). Suchsellers can and do use personalized pricing. In 1996, DeniseKatzman found that a Victoria’s Secret catalog sent to amale colleague offered a deeper discount than the nearlyidentical catalog she received and sued the company (Weiss

and Mehrotra 2001). Victoria’s Secret was not alone inits practice of charging different prices to different cus-tomers. In fact, Blank (2001) report that catalogers engagein geographical price discrimination, that is, they use dif-ferent prices in catalogs mailed to different zip codes.Online retailers are also known to engage in personalizedpricing. For example, an online retailer charged differentprices for the same digital camera, depending on whethercustomers had previously visited a price-comparison site(Bridis 2005). According to a Forrester report, out of 30online retailers interviewed in 2000, 57% had plans to trysome form of personalized pricing (Johnson et al. 2000).Likewise, traditional grocery stores make personalized dis-count offers: More than 300 Jewel-Osco and Albertsonsstores have installed kiosks where customers insert theirmembership cards to be presented with customized dis-count offers (Desjardins 2007).A natural question is whether implementing such discrim-

inatory pricing policies is legal. With some exceptions, itis widely considered to be so. It is telling that consumerssued Victoria’s Secret for mail fraud as opposed to tryingto make the case that the company’s practice violated the

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Robinson-Patman Act.1 According to Ramasastry (2005),“The reality is that Internet price customization does exist—and, contrary to popular opinion, is typically legal.”It appears that the real challenge for the firms is to

manage customers’ perception of the practice because itcan cause customer ill-will when not framed appropriately.For example, in the summer of 2000, Amazon was caughtselling an X-Files DVD box at prices ranging from $80to $100. There was an uproar from customers who feltthat Amazon was tracking their purchase history to chargehigher prices to more loyal customers, while Amazondenied the claim and said the prices were chosen at ran-dom as part of price testing (Adamy 2000). Furthermore, asurvey by Turow et al. (2005) finds that 90% of customersdisagree with the statement that “It’s OK if a store chargesme a price based on what it knows about me.” Nonetheless,there are many cases in which customers do not appear tobe bothered by personalized pricing. In fact, it appears thatcustomers have little problem with different people pay-ing different prices for the same product, as long as thepricing scheme is perceived to be fair. For example, hardlyanyone objects to student or senior discounts, which areultimately a form of personalized pricing based on demo-graphic information. Two different customers buying twoidentical vehicles are likely to pay different prices, depend-ing on their propensity to negotiate. Airlines charge higherprices to travelers who are not willing to stay the Saturdaynight, presumably because these are business travelers whoare less price-sensitive.The use of personalized pricing requires that the seller

acquire customer-specific information. In this paper, we usea model where each customer furnishes the seller with asignal, which captures the information that the seller hasabout the individual. This signal might take many formsand there might be many ways in which the seller can col-lect such signals. For example, the signal might take one ofthree possible values indicating whether the customer car-rying the signal is a student, a senior citizen, or neither, andthis signal can be obtained by simply asking students andsenior citizens to identify themselves. In this case, the firmmight be willing to bank on the fact that students and seniorcitizens typically have lower willingness to pay and offerthem a discount. In many instances, this might be the bestaction for the firm, but offering a discount to a student (orsenior citizen) is not always the best possible action becausesurely there are students (or senior citizens) who have muchdeeper pockets and higher willingness to pay than an aver-age customer. That is, as useful as the signal might be tothe firm, it is not necessarily a perfect indicator of the cus-tomers’ willingness to pay. Firms can actually develop moresophisticated and more useful signals. For example, loy-alty programs give firms not only demographic informationabout a customer, but also possibly a way to track the pur-chases of the customer over time. In cases where loyaltyprograms are used, customer signal could be the amount thecustomer spent with the seller in the last year or the time

elapsed since the customer last made a purchase from theseller. Such signals are revealed to the seller every time thecustomer identifies herself, for example, by swiping a loy-alty card. Similarly, online retailers have access to a wealthof information about registered customers. Another exam-ple of a signal is the zip code, which is easily available toand frequently used by catalog retailers. What is commonin all these examples is that even though the signal froma customer is a useful bit of information, this informationis not a perfect indicator of the customer’s willingness topay. Hence, in this paper, we develop a model where cus-tomer signals provide only limited information about thecustomer’s reservation price.If a seller were to use personalization to its full extent,

then each unique signal would prompt a unique price fromthe seller. We consider such a model with full personal-ization. If personalization cannot be implemented to thatextent, a seller may want to use a personalization strategywhere there is an announced price and a single discountlevel that can possibly be offered to a customer, dependingon the signal from the customer. To analyze such partialpersonalization of prices, we consider a model in which arange of signals are bunched together and offered a singleprice.This paper characterizes the properties that a signal

should have for it to be an intuitive determinant of thefirm’s prices. For example, if there is a positive correlationbetween the signal and customers’ reservation prices, doesthat mean that the firm should charge higher prices to cus-tomers with higher signals? As we see in §4, the answer is“no.” Positive correlation is not sufficient. A stronger con-dition is needed. In §4, we give a precise description of thiscondition, which leads to useful relationships between thesignal and optimal prices. For example, when this “strong”correlation condition is satisfied, under full personalization,the optimal price is higher for customers with higher sig-nals. Under partial personalization, the optimal pricing pol-icy is of threshold type, meaning that customers above thethreshold pay the higher price while those below the thresh-old pay the lower price. The stronger correlation conditionalso leads to several monotonic properties for the optimalpolicies. For example, the threshold level changes mono-tonically with respect to the inventory level and time whenprices are kept at fixed levels. These are all properties thatwould make implementation of price personalization eas-ier in practice. However, none of these structural propertiesexists in the absence of the strong correlation condition.The presence of personalization may give customers

an incentive to act strategically. For example, a customermight wish to hide her identity if she feels that the sellerwill charge a lower price to an anonymous customer. A cus-tomer might even provide false information to a seller ifshe believes that such information will qualify for a lowerprice. In this paper, we do not allow such strategic behavioron the part of customers. However, in the online supple-ment, we consider an extension where some customers inthe population do not provide signals.

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The rest of this paper is organized as follows. In §2,we review the relevant literature, and in §3, we describeour base model. Sections 4 and 5 present our results forthe full personalization and partial personalization models,respectively. Section 6 gives our concluding remarks. Allthe proofs are given in the online supplement. An electroniccompanion to this paper is available as part of the onlineversion that can be found at http://or.journal.informs.org/.

2. Literature ReviewFollowing the pioneering work of Gallego and van Ryzin(1994) and Bitran and Mondschein (1997), there has been agrowing interest in dynamic pricing of perishable productsbased on inventory levels and time. Both of these papersare concerned with the pricing of a single product overa finite horizon with no replenishment opportunities, butthey use different formulations (one in continuous-time set-ting, the other in discrete-time setting) and come up withdifferent insights. There has been a significant volume ofresearch on dynamic pricing since then. For comprehensivereviews of this research pre-2003, we refer the reader toElmaghraby and Keskinocak (2003), Bitran and Caldentey(2003), and Talluri and van Ryzin (2004). Among the morerecent work, Maglaras and Meissner (2006) and Zhangand Cooper (2009) consider dynamic pricing of multipleproducts. Aviv and Pazgal (2005) consider a model wherethere is a high level of uncertainty about the demand (asa function of the price) but the firm learns more aboutthe demand throughout the sales horizon by observing cus-tomer reaction to the prices. Aviv and Pazgal (2008), Liuand van Ryzin (2005), Elmaghraby et al. (2008), and Su(2007) investigate dynamic pricing decisions when cus-tomers act strategically. Monahan et al. (2004) deal with thedecision of determining the initial inventory for productsthat are under dynamic pricing. Popescu and Wu (2004)and Ahn et al. (2007) consider models where price in agiven period influences the demand in other periods.Our work differs from the above mainly in that in our

models, the firm has the flexibility of adjusting the pricenot only depending on the inventory level and time but alsodepending on the information that the firm has at the indi-vidual customer level. Some of the recent work investigatedvery specific forms of such use of personalized pricing. Forexample, Kuo et al. (2009) investigate negotiation, whileNetessine et al. (2006) and Aydin and Ziya (2008) investi-gate upselling/cross-selling, in which case the price offeredto a customer depends on what the customer already boughtfrom the seller. All these papers provide insights that arerelevant within the specific forms of personalized pricingthey consider. Furthermore, from more of a technical pointof view, as a consequence of this focused interest in spe-cialized forms of personalized pricing, they consider sig-naling formulations that are fairly restrictive. For example,in Kuo et al. (2009) the signal is the offer made by a bar-gainer. In Aydin and Ziya (2008), the signal is whether the

customer bought another product at an advertised price. Inthis paper, we are not restricting ourselves to any particularform of personalized pricing. We use a fairly general sig-naling formulation so as to make our results relevant to alarge class of personalized pricing practices. Perhaps moreimportantly, unlike the other three papers, this paper is pri-marily focused on developing a better understanding aboutthe signal itself. For example, we provide insights on theproperties the signal should have in order for it to be amore intuitive determinant of customers’ reservation priceand thus be more useful in practice.Personalized pricing is closely related to price discrim-

ination, which has been studied extensively in economicsand marketing literature. (This body of work is fundamen-tally different from ours in that inventory considerationstypically play no role in pricing decisions made by thefirms.) For a survey of this literature and detailed bibli-ography, see Varian (1989). More recent work has con-centrated on the profitability of personalized pricing. Someresearchers found that in a competition environment per-sonalized pricing may actually hurt the firms because itintensifies competition. For example, see Thisse and Vives(1988), Shaffer and Zhang (1995), Fudenberg and Tirole(2000), and Chen and Iyer (2002). Villas-Boas (2004) andLiu and Zhang (2006) show that the practice may not beprofitable even for a monopolist (due to strategic behaviorsof the customers in the case of Villas-Boas 2004, and mis-alignment of incentives within the supply chain in the caseof Liu and Zhang 2006). On the other hand, Chen et al.(2001) find that when customer targetability is sufficientlylow, personalized pricing is profitable for two competingfirms, but for high targetability levels, firms might be worseoff because a “prisoner’s dilemma” occurs.2 Shaffer andZhang (2002) show that when two competing firms areasymmetric, one of the two competing firms can profit frompersonalized pricing. Based on these conflicting findings itappears that, perhaps not surprisingly, whether or not per-sonalized pricing is profitable is context specific. For super-markets, even when there is competition, Kumar and Rao(2006) find that using past purchase information for per-sonalized pricing is profitable. Using data from the ketchupmarket, Besanko et al. (2003) also find the practice to bea profitable strategy. For an extensive survey of market-ing literature on behavior-based price personalization, seeFudenberg and Villas-Boas (2006). For a broader survey ofresearch on personalization in marketing, see Murthi andSarkar (2003).

3. Model DescriptionConsider a firm with I units of inventory that will be sal-vaged at the end of a finite selling season. Following thestandard approach in dynamic pricing models, we assumethat the firm does not have an option to replenish inven-tory during the selling season. This is certainly true foran airline that has a fixed number of seats on a flight or

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an apparel retailer that faces long procurement lead timescoupled with short life cycles for its seasonal products. Fol-lowing the approach first used by Bitran and Mondschein(1997), we assume that the selling season is divided into Tperiods, where each period is short enough that at most onecustomer arrives in a period. Let � denote the probabilitythat a customer arrives in a given period.As we will discuss shortly, our model uses two differ-

ent forms of stochastic ordering: failure rate ordering andlikelihood ratio ordering. For two cumulative distributionfunctions (cdfs) �1 and �2 (with corresponding probabilitydensity functions (pdfs) �1 and �2), if the failure rate of�1 is less than that of �2, i.e., if �1�x�/�1 − �1�x�� ��2�x�/�1−�2�x�� ∀x, then we say �1 dominates �2 infailure rate ordering, and we write �1 �fr �2. Furthermore,if �1�x�/�2�x���1�y�/�2�y� for any x > y, then we say�1 dominates �2 in likelihood ratio ordering, and we write�1 �lr �2. If �1 and �2 are cdfs for discrete random vari-ables, the pdfs in the definition of likelihood ratio orderingare replaced by corresponding probability mass functions(pmfs). See Online Appendix B for more detailed defi-nitions of the forms of stochastic ordering used in thispaper. We refer the reader to Müller and Stoyan (2002) orShaked and Shanthikumar (2007) for more on stochasticorderings, but here it is useful to mention that likelihoodratio ordering is a stronger condition implying failure rateordering, which implies first-order stochastic dominance,which in turn implies ordering of the means. Note thatthroughout the paper we use increasing/decreasing and pos-itive/negative in the weak sense unless specifically qualifiedas strictly increasing/decreasing or nonpositive/negative.Suppose that the consumer population is divided into two

segments, one with higher willingness to pay than the other.Let qi, i= 1�2 denote the fraction of segment i customers.We assume that all customers in a given segment have inde-pendent and identically distributed (iid) reservation prices.Let Fi denote the cdf of the reservation prices of customersin segment i, and fi their pdf. Let �Fi�·� �= 1− Fi�·�. Wemake the following assumptions on the reservation pricedistributions.

Assumption (A1). Fi�·�, i = 1�2, are twice-continuously-differentiable, strictly increasing functions, and they bothhave the same nonnegative support.

Assumption (A2). Fi�·�, i = 1�2, have strictly increas-ing generalized failure rates, i.e., xfi�x�/ �Fi�x� is strictlyincreasing.

Assumption (A3). F1�·��fr F2�·�.Assumption (A2) is satisfied by a large family of dis-

tributions, including all Weibull distributions and the pos-itive part of the normal distribution. (For a comparisonof various assumptions on reservation prices used in rev-enue management problems, see Ziya et al. 2004.) Theordering stated in (A3) implies that the absolute price elas-ticity of demand is smaller for customers in segment 1;

i.e., segment 1 is less price-sensitive. A further implica-tion of (A3) is that the reservation price of a customer insegment 1 stochastically dominates that of a customer insegment 2.We assume that the customer’s segment is not directly

observable by the seller. In other words, the seller can-not say with certainty if an individual is coming from thepopulation with high or low price elasticity of demand.However, each arriving customer provides the seller witha signal, which embodies the information available aboutthe customer. (Argon and Ziya 2009 use a similar formu-lation within a queueing context, where customers’ signalsdetermine their priority levels.)Depending on the information a seller collects and uses,

the signal could be demographic information about the cus-tomer (e.g., age, gender, zip code) or information regardingtransaction history (e.g., the last time the customer madea purchase from the seller, the amount the customer spentwith the seller in the last year). This signal, while notenough in itself to determine the segment of the customer,might still be valuable for the seller in updating its beliefabout the customer’s segment. The signals from customersin segment i will be distributed over a range because a seg-ment consists of similar yet heterogenous customers. Weassume that the signals of the customers in segment i areiid random variables, denoted by the generic random vari-able Si. We allow Si to be either discrete or continuous, butnot a mixture of the two. Let Gi, i = 1�2 denote the cdfof Si, and gi denote the pdf of Si if Si is continuous andthe pmf if Si is discrete. We impose the following technicalassumption on the signal distributions.

Assumption (A4). Gi�·�, i = 1�2, are strictly increasingfunctions on the support set, and they both have the samenonnegative support.

In addition, we assume the following ordering betweenthe signals from the two segments:

Assumption (A5). G1�·��lr G2�·�.Assumption (A5) imposes a strong stochastic order

between the signals from the two segments. (Likelihoodratio ordering implies failure rate ordering, which in turnimplies first-order stochastic dominance.) In the next sec-tion, we investigate the signal’s effect on the optimal priceand illustrate the significance of Assumption (A5).

4. Dynamic Pricing with PersonalizationWe first consider a seller who does not have to commit toa price prior to the arrival of the customer. Upon arrival,the customer furnishes the seller with a signal. The sellerthen quotes a price to the customer based on the signal.One way to implement such a practice is to ask customersto identify themselves with their loyalty cards and offerthem customized discounts, as done by Jewel-Osco andAlbertsons stores. Such a practice is also technologically

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feasible for many online retailers because the retailer canpresent different users with different prices after observing,for example, the IP address of the user, or customers mightsimply choose to log in and identify themselves. Althoughthis practice might be feasible, it might not be easily imple-mentable, as evidenced by the customer complaints thatAmazon received during its random price testing.Let S denote the signal furnished by a customer, a ran-

dom variable prior to the arrival of the customer. Supposethat a customer arrives with signal S = x. After observingthis signal, the seller can update its belief about the seg-ment this particular customer belongs to. Let �qi�x� denotethe probability that a customer with signal x belongs tosegment i. Using Bayes’ rule, �qi�x� is given by

�qi�x�=qigi�x�

q1g1�x�+ q2g2�x�� i= 1�2� (1)

Suppose that the seller has y units of inventory with t peri-ods to go until the end of the season. Let Vt�y� denotethe seller’s optimal expected revenue to go. The followingoptimality equations characterize the dynamic program tobe solved by the seller:

Vt�y�=ES

[maxp

{�( �q1�S� �F1�p�+ �q2�S� �F2�p�

)(p+Vt−1�y−1�

)

+[1−�� �q1�S� �F1�p�+ �q2�S� �F2�p��

]Vt−1�y�

}]�

y>0� t=1�����T �Vt�0�=0� t=1�����T � and V0�·�=0�

In the remainder of this section, let

��x�p� �= �q1�x� �F1�p�+ �q2�x� �F2�p�� (2)

Hence, ��x�p� is the probability that a customer with asignal x is willing to purchase the product at price p.After some algebraic manipulation, one can rewrite Vt�y�as follows:

Vt�y�= Vt−1�y�+�ES

[maxp���S�p��p−�t�y���

]�

y > 0� t = 1� � � � � T �where

�t�y�= Vt−1�y�−Vt−1�y− 1�� y > 0� t = 1� � � � � T � (3)

Here, �t�y� can be interpreted as the marginal value ofinventory. Let p∗�x� y� t� denote the optimal price quotedto a customer who furnishes signal x when the seller hasy units in inventory with t periods to go until the end ofthe season. (In case there are multiple optimal prices, wedefine p∗�x� y� t� to be the smallest optimizer.)We next discuss how the optimal price p∗�x� y� t�

depends on the signal x. Suppose that a customer provides

signal x and consider �q1�x�, the probability that this cus-tomer belongs to segment 1. This probability, given by (1),can be rewritten as

�q1�x�=q1g1�x�/g2�x�

q1g1�x�/g2�x�+ q2� (4)

Notice from (4) that �q1�x� is increasing in the signal x dueto our assumption that the signal from segment 1 dominatesthe signal from segment 2 in likelihood ratio ordering. Inother words, the larger the signal from a customer, the morelikely the customer is to belong to segment 1, which is thesegment with higher willingness to pay. Hence, one wouldexpect that the optimal price p∗�x� y� t� increases with thesignal x, which is formalized in the following theorem.

Theorem 1. Given time t and inventory level y, the(smallest) optimal price, p∗�x� y� t�, is increasing in thesignal x provided by the customer.

Theorem 1 is not surprising, given that a higher signalimplies a higher probability that the customer belongs tothe segment with higher willingness to pay. What is per-haps surprising is that Theorem 1 does not hold underweaker versions of Assumption (A5). Consider the follow-ing example.

Example. Suppose that signals are discrete random vari-ables with probability mass functions shown in Table 1.In this example, the signal from the first segment doesnot dominate the signal from the second segment in like-lihood ratio ordering, but in failure rate ordering (which isweaker).3 Due to the failure rate ordering between the twosignals, the mean signal from segment 1 is larger than themean signal from segment 2. It is not difficult to checkthat, as a consequence of this ordering between the means,the signal and the reservation price of an individual arepositively correlated. Therefore, intuition may suggest thatthe larger the signal, the higher the price must be. This,however, is not true. For example, here p∗�1�1�1� 38�56,p∗�2�1�1� 44�16, p∗�3�1�1� 41�46, and p∗�4�1�1� 46�66. Hence, the optimal price for a customer with a sig-nal of 3 is smaller than the optimal price for a customerwith a signal of 2 when the inventory level is 1 and thetime remaining is 1 period. In fact, optimal prices satisfythe same relationship for many different pairs of inventory

Table 1. The probability mass functions g1�·� and g2�·�for the signals from segments 1 and 2, respec-tively.

x 1 2 3 4

g1�x� 0�1 0�3 0�2 0�4g2�x� 0�25 0�25 0�25 0�25

Notes. In addition, suppose that q1 = 0�3, q2 = 0�7, � = 0�5, F1 isWeibull with shape and scale parameters, 2 and 100, respectively,and F2 is Weibull with shape and scale parameters, 2 and 50,respectively, so that F1 �fr F2.

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level and time. This happens because a customer with asignal of 2 is more likely to belong to segment 1 than acustomer with a signal of 3.

The above example shows that positive correlationbetween signals and reservation prices does not imply thatcustomers with higher signals should be charged higherprices. In fact, the example says more: Even failure rateordering between the two signals might not be enough toguarantee an intuitive relationship between the signal andthe optimal price. For the optimal price to be increasing inthe signal, we need the likelihood ratio ordering imposedin Assumption (A5).

Remark 1. Prior work in dynamic pricing literature hasestablished the monotonicity of the optimal prices withrespect to inventory level and time under various settings.One can show that similar monotonicity results hold in ourproblem as well: The optimal price p∗�x� y� t� is increas-ing in the remaining time t and decreasing in the inventorylevel y.

In many cases, it is unlikely that a seller will delaythe pricing decision until after the revelation of the sig-nal because consumers are likely to find the resulting pricediscrimination unacceptable. However, for many retailers,be they online or traditional, it is quite possible to offerpersonalized discounts from an announced price. In effect,such a discounting strategy is similar to a retailer provid-ing customers with special discount offers. While such apractice boils down to charging different prices to differentcustomers, it somehow appears to be more palatable to con-sumers, either because the discrimination is less transparentor because the consumers find it “fair” that different indi-viduals qualify for different discounts. Therefore, one wayto implement the personalized pricing strategy discussedin this section is to announce a price at the beginning ofeach period and then provide each arriving customer witha personalized discount after seeing the customer’s signal.Provided that the announced price is high enough, sucha discounting strategy, which we refer to as personalizeddiscounting, will achieve the same results as not commit-ting to a price until after seeing the customer’s signal. Aswe discuss next, there is a natural ceiling on how high theannounced price needs to be, which makes personalizeddiscounting all the more attractive.Define p∗

i ��� �= argmaxp �p−�� �Fi�p�, i = 1�2. Notethat p∗

i ��� is the price that a seller would charge to a cus-tomer who is known to be from segment i, given that theseller’s marginal value of one unit of inventory is �. Fur-thermore, one can show that p∗

1��� � p∗2��� because seg-

ment 1 customers are less price-sensitive. In our model, aseller does not know with certainty what segment an arriv-ing customer belongs to, even after observing the signalfrom the customer. Therefore, regardless of what the cus-tomer’s signal is, a seller using personalized pricing willquote a price somewhere between p∗

1��� and p∗2���. This

observation suggests that the following discounting strategyperforms as well as not committing to a price until afterseeing the customer’s signal: Set the announced price equalto p∗

1���, and offer the optimal personalized discount afterseeing the customer’s signal. The following proposition for-malizes this result.

Proposition 1. Suppose that a seller has y units of inven-tory with t periods to go. Then:(a) p∗

1��t�y�� � p∗�x� y� t� � p∗2��t�y�� for any

signal x.(b) Suppose that a seller uses the following personal-

ized discounting strategy: At the beginning of period t,announce the price to be p∗

1��t�y�� and then offer a dis-count of p∗

1��t�y�� − p∗�x� y� t� after observing the cus-tomer’s signal x, thereby charging an effective price ofp∗�x� y� t� to this particular customer. The seller’s expectedrevenue under such a strategy is the same as the optimalexpected revenue of a seller who quotes prices only afterseeing the signal of the customer.

One caveat is that the higher announced price couldresult in some potential customers giving up on the productprematurely, even though some of those customers wouldqualify for lower prices had they made an attempt to pur-chase and revealed their signals in the process. In such acase, the potential loss of revenue must be weighed againstthe benefits from personalization.

5. Dynamic Pricing with PartialPersonalization

In the previous section, we have considered a scenariowhere the seller uses not only dynamic pricing (adjustingthe price over time based on inventory levels) but also per-sonalized pricing (adjusting the price in response to thesignal from the customer). In that model, at any given timeand inventory level, each unique signal prompts a uniqueprice from the seller. Such a pricing strategy, which usespersonalization to its full extent, might be harder to jus-tify and implement than a pricing strategy in which cus-tomers are grouped into a few classes, with each classbeing charged a different price. While customers might beuncomfortable with full personalization, it might be mucheasier to gain customer acceptance for group pricing. Inthis section, we consider such a group pricing model with“partial personalization.” Here, the seller continues to usepersonalized dynamic pricing but picks only two differentprices at the beginning of each period, before observing thecustomer’s signal. After observing the customer’s signal,the firm decides which of the two prices to offer.The optimality equations can be written as follows:

V Pt �y�=max

p1�p2ES

[max

p∈�p1�p2�{�( �q1�S� �F1�p�+�q2�S� �F2�p�

)�p+V P

t−1�y−1��

+[1−�� �q1�S� �F1�p�+ �q2�S� �F2�p��

]V Pt−1�y�

}]�

y>0� t=1�����T �V Pt �0�=0� t=1�����T � and V P

0 �·�=0�

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The outer maximization corresponds to the problem ofpicking the two prices at the beginning of period t, and theinner maximization corresponds to the problem of decidingwhat price to offer for a given signal, S. Although this is arelatively complicated optimization problem, it is possibleto prove a certain structure for the optimal policy, stated inthe following proposition.

Theorem 2. Suppose that a seller has y units of inventorywith t periods to go and charges the customers one of thetwo prices p1 and p2 with p1 > p2. Then, there exists athreshold z̄�y� t� such that if an arriving customer has asignal z� z̄�y� t�, it is optimal to offer that customer pricep1; otherwise, it is optimal to offer the customer p2.

Theorem 2 shows that, for any given prices p1 and p2,a threshold-type policy will be used to determine which ofthe two prices to offer a given customer. The likelihoodratio condition imposed in (A5) is crucial for Theorem 2.The threshold structure does not necessarily exist when theordering is not as strong, e.g., when the ordering is in fail-ure rate sense.

5.1. Fixed Prices, Dynamic Threshold Signal

To gain further insight into the choice of the threshold sig-nal z, we now focus on the problem where the prices p1 � p2are fixed exogenously at the beginning of the horizon, butthe seller picks the optimal threshold at the beginning ofeach period. (We know from Theorem 2 that a threshold-type policy is optimal when deciding which of the twoprices to offer.) Let V FP

t �y� denote the optimal expected rev-enue of the seller, given that the seller has y units of inven-tory with t periods to go. The optimality equations for theseller’s problem are as follows:

V FPt �y�

=maxz

{��q1 G1�z� �F1�p1�+q2 G2�z� �F2�p1���p1+V FP

t−1�y−1��+��q1G1�z� �F1�p2�+ q2G2�z� �F2�p2���p2+V FP

t−1�y− 1��+ "1−��q1 G1�z� �F1�p1�+ q2 G2�z� �F2�p1��

−��q1G1�z� �F1�p2�+ q2G2�z� �F2�p2��#V FPt−1�y�

}�

y > 0� t = 1� � � � � T � (5)

V FPt �0�= 0� t = 1� � � � � T � and V FP

0 �·�= 0�As before, after some algebraic manipulation, one canrewrite V FP

t �y� as follows:

V FPt �y�=V FP

t−1�y�+�maxz

{�q1 G1�z� �F1�p1�+q2 G2�z� �F2�p1��

·�p1−�FPt �y��+�q1G1�z� �F1�p2�+q2G2�z� �F2�p2��

·�p2−�FPt �y��

}� y>0� t=1�����T �

where

�FPt �y�=V FP

t−1�y�−V FPt−1�y−1�� y>0� t=1�����T � (6)

Let z∗�y� t� denote the optimal threshold signal (or thesmallest optimal threshold when there is more than oneoptimal value). The following proposition indicates that ifthe seller has more inventory or less time until the end ofthe selling season, then the seller increases the thresholdsignal, thus making it more likely that an arriving customerwill qualify for the lower price, p2.

Proposition 2. Suppose that a seller has y units of inven-tory with t periods to go.(a) If y increases, z∗�y� t� increases.(b) If t increases, z∗�y� t� decreases.

5.2. Fixed Signal Threshold, Dynamic Prices

To explore the pricing problem further, we now analyze theproblem where the firm chooses the two prices p1 and p2dynamically, depending on the time and inventory level, butfixes the threshold signal at the beginning of the horizon. Inessence, fixing the threshold signal ahead of time impliesthat the firm is designating classes of customers that willreceive a discount regardless of time and inventory level;e.g., committing to offering discounts to students. We termthe customers whose signals exceed the threshold as class-1customers and those whose signals are below the thresholdas class-2 customers. Let z denote the threshold signal cho-sen by the firm. Given the threshold z, let V FT

t �y� z� denotethe firm’s optimal expected revenue under partial person-alization when starting period t with y units of inventory.Then, V FT

t �y� z� is given by the same optimality equationsas in (5), but the maximization is over p1 and p2 insteadof z.4

Let p∗j �z� y� t� denote the optimal price quoted to a

class-j customer when the seller has y units in inventorywith t periods to go (or the smallest optimal price whenthere are multiple optimizers). The following propositiondescribes how the optimal prices depend on the thresholdsignal.

Proposition 3. Suppose that the threshold signal is z andthe seller has y units of inventory with t periods to go.Then:(a) Class-1 customers are charged a higher price than

class-2 customers; i.e., p∗1�z� y� t�� p∗

2�z� y� t�.(b) If the firm increases the threshold z to be used in

period t while keeping the threshold signal unchanged inother periods, both p∗

1�z� y� t� and p∗2�z� y� t� increase.

Proposition 3(a) is not necessarily true under some con-ditions that are weaker than the likelihood ratio ordering.Proposition 3(a) suggests a way in which this pricing strat-egy can be implemented. Given a threshold signal z, theseller could announce the price to be p∗

1�z� y� t� at thebeginning of period t, and then give class-2 customers adiscount in the amount of p∗

1�z� y� t�−p∗2�z� y� t�.

Proposition 3(b) indicates that if the firm raises thethreshold signal for a single period only, optimal pricesfor that period also increase. When the threshold is higher,

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a customer must exhibit a higher signal to be classifiedas a class-1 customer, which indicates that the customeris likely to have a higher reservation price as well. As aresult, the price charged to a class-1 customer increases. Asfor class-2 customers, as the threshold signal increases, thisclass grows to include customers with larger signals whowere earlier classified as class-1, thus growing to includecustomers who are likely to have higher reservation prices.Therefore, the higher the threshold is, the higher the pricecharged to a class-2 customer as well.

6. ConclusionArguably, the biggest challenge for firms in personalizingprices is to effectively manage their customers’ perceptionsof the practice. Amazon’s experience clearly shows that anot well-thought-out implementation might seriously upsetthe customers. As Phillips (2005) argues, however, fram-ing can make the whole difference. For example, a cus-tomer will be more satisfied by receiving a discount froma higher price as opposed to paying a premium on top ofa lower price, even if both practices resulted in the sameeffective price. Depending on the industry, customers mightalso have different attitudes toward price personalization.For example, it is reasonable to expect that in industrieswhere customers are already accustomed to paying dif-ferent prices (such as travel, hotel, or apparel industries),personalizing prices might be more acceptable. For suchindustries, time- and inventory-level-dependent pricing oflimited inventories has been well researched. In this paper,we investigated the optimal pricing policies in the pres-ence of a third dimension (in addition to time and inventorylevel) that can be used in setting prices: information thefirm has at the individual customer level.When making pricing decisions based on customer-

specific information, firms in some way need to relate theinformation they have about a particular customer withthe customer’s willingness to pay. Typically, firms useinformation that they believe is correlated with the will-ingness to pay so that they can set their prices accord-ingly. We have shown, however, that a mere correlationbetween the information used (which we call the signal)and the willingness to pay does not necessarily imply thatcustomers with higher signals should be charged higherprices. A stronger technical condition than correlationmight be needed (the likelihood ratio ordering conditionin our model) to ensure that the signal is a more intu-itive determinant of the price that needs to be charged.This suggests that firms need to carefully determine thebit of information they use when setting their prices, andit thus points to an interesting avenue for future research:How should the signal be determined, or more precisely,how should the signal be designed so that higher val-ues of the signal would indeed imply higher prices to becharged?In this paper, we mainly considered two different mod-

els for two different price personalization policies. In the

first model, we assumed that there are no limitations onthe number of different prices that the firm can charge, andwe showed that under the likelihood ratio ordering con-dition that ensures a “strong” positive correlation betweenthe signal and willingness to pay, optimal prices increasewith the signal. For the second model, considering thepossibility that firms would typically limit the number ofdifferent prices they would charge, we assumed that thefirm at any one point can only charge one of two differentprices. Here, we showed that regardless of whether pricesare dynamically set or they are set at the beginning ofthe horizon, the optimal policy is of threshold-type, i.e.,customers with signals that are above a certain level arecharged the higher price while those with lower signals arecharged the smaller price. We have also established sev-eral monotonicity properties. For example, we showed thatwhen prices do not change dynamically, the optimal thresh-old level changes monotonically with the inventory leveland time.In the online supplement we include an extension to

consider the case where the firm cannot observe the sig-nals of some of the customers. The analysis of this exten-sion shows that when signals from some of the customersare not available, the lack of a signal becomes a signalin itself, which the firm can utilize in setting prices. Infact, the firm prefers that the customers with high willing-ness to pay do not signal. When that is the case, the firmcharges a high price to the nonsignaling, high-willingness-to-pay customers and tailors the price for the signaling,low-willingness-to-pay customers.

7. Electronic CompanionAn electronic companion to this paper is available as partof the online version that can be found at http://or.journal.informs.org/.

Endnotes1. The Robinson-Patman Act of 1936 prohibits anti-competitive price discrimination, primarily in the contextof wholesale prices charged to retailers and distributors.2. Targetability is defined as “the ability to predict the pref-erences and purchase behaviors of individual consumers forthe purpose of customizing price or product offers.” Theauthors consider a model where the firm can make errorswhen classifying customers, a possibility that we also allowin this paper.3. The signal distributions assumed in Example 1 aretaken from Shaked and Shanthikumar (2007). Shaked andShanthikumar give these distributions as an example todemonstrate that it is possible that two distributions are notordered in the likelihood ratio while they are ordered infailure rate and reversed failure rate.4. Online Appendix C provides an equivalent yet moretractable formulation for the optimality equations.

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