MANAGEMENT SCIENCEVol. 60, No. 7, July 2014, pp. 1792–1811ISSN 0025-1909 (print) ó ISSN 1526-5501 (online) http://dx.doi.org/10.1287/mnsc.2013.1839

©2014 INFORMS

Optimal Multiperiod Pricing with Service Guarantees

Christian BorgsMicrosoft Research, New England Lab, Cambridge, Massachusetts 02142, borgs@microsoft.com

Ozan CandoganFuqua School of Business, Duke University, Durham, North Carolina 27708, ozan.candogan@duke.edu

Jennifer ChayesMicrosoft Research, New England Lab, Cambridge, Massachusetts 02142, jchayes@microsoft.com

Ilan LobelStern School of Business, New York University, New York, New York 10012, ilobel@stern.nyu.edu

Hamid NazerzadehMarshall School of Business, University of Southern California, Los Angeles, California 90089,

hamidnz@marshall.usc.edu

We study the multiperiod pricing problem of a service firm with capacity levels that vary over time. Cus-tomers are heterogeneous in their arrival and departure periods as well as valuations, and are fully

strategic with respect to their purchasing decisions. The firm’s problem is to set a sequence of prices that max-imizes its revenue while guaranteeing service to all paying customers. We provide a dynamic programmingbased algorithm that computes the optimal sequence of prices for this problem in polynomial time. We showthat due to the presence of strategic customers, available service capacity at a time period may bind the priceoffered at another time period. This phenomenon leads the firm to utilize the same price in multiple periods,in effect limiting the number of different prices that the service firm utilizes in optimal price policies. Also,when customers become more strategic (patient for service), the firm offers higher prices. This may lead tothe underutilization of capacity, lower revenues, and reduced customer welfare. We observe that the firm cancombat this problem if it has an ability, beyond posted prices, to direct customers to different service periods.

Keywords : dynamic pricing; strategic customers; nonlinear programming; dynamic programmingHistory : Received June 17, 2011; accepted September 3, 2013, by Dimitris Bertsimas, optimization. Published

online in Articles in Advance February 14, 2014.

1. IntroductionDynamic pricing is one of the key tools available to aservice firm trying to match time-varying supply withtime-varying demand. It is, however, a delicate toolto use in the presence of customers who strategicallytime their purchases. As customers change the timingof their purchases, not only might the firm lose rev-enue, but it might also cause the firm’s capacity to bestrained in periods where a low price is offered.We consider a general formulation of a multiperiod

pricing problem of a service firm trying to maxi-mize its revenue while selling service to strategic cus-tomers who arrive and depart over time. We assumethe firm is constrained by its time-varying servicecapacity level and that it wishes to provide serviceguarantees to its customers. More specifically, thefirm announces a sequence of prices in advance; eachcustomer chooses the period with the lowest pricebetween her arrival and departure. The sequence ofprices is chosen to maximize the revenue of the firm

while guaranteeing that each customer who is will-ing to pay the price at a given period will obtainservice. Our main contributions are to provide algo-rithms that compute the firm’s optimal pricing pol-icy and to characterize the properties of such optimalpolicy.Service guarantees are an important contract fea-

ture that are often used when the customers them-selves are businesses that rely on the service they pur-chase to run their own operations. An example ofa setting with the aforementioned properties comesfrom the cloud computing market where firms sellcomputational services to their customers.1In this market, despite demand for service vary-

ing quite significantly over time, customers typicallydemand reliability from their providers, in the sense

1 Cloud computing is a large and quickly growing business. Thecombined revenues of this market were estimated to be more than$22 billion in 2010 and are expected to reach $55 billion by 2014(see Lohr 2011).

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that they should be able to purchase service when-ever they need it and rationing is not tolerated.For instance, consider the case of Domino’s Pizza,which is a client of Microsoft’s Windows Azure cloudcomputing platform. As explained by Domino’s direc-tor of eCommerce: “We have daily peaks for din-ner rush, with Friday night being the biggest. SuperBowl, however, has a peak 50 percent larger than ourbusiest Friday night. Windows Azure allows me tofocus on customer facing functionality, and not haveto worry about whether or not I have enough hostingcapacity to support it” (Vitek 2009). For Domino’s andmany other companies, the key managerial benefit ofpurchasing cloud computing services is that it per-mits them to completely ignore the hosting capacityneeds of their online businesses. This is only the casebecause the cloud computing providers go to greatlengths to ensure that their service is always availableand, therefore, companies who rely on them need notworry about rationing risk.2The clients of cloud computing services are also

highly heterogeneouswith respect to their willingness-to-wait for service. Although some companies utilizecloud computing to run on-demand services and web-sites, and thus always need immediate service, othersuse the cloud to run simulations or solve large-scaleoptimization problems such as the ones that arise infinancial analysis, weather forecasting, and genomesequencing. Such clients will typically display morestrategic behavior in their purchasing of cloud ser-vices and wait for lower prices before purchasing ser-vice (e.g., see the AWS case study DNAnexus (AmazonWeb Services 2011)).Currently, most cloud computing services are sold

via static pricing or via a combination of long-termcontracts and static pricing (e.g., Amazon Web Ser-vices 2012, Windows Azure 2012). More specifically,the customers can purchase computation capacity(starting at around 10 cents per hour) in a pay-as-you-go model where the per-hour price is constant overtime; this hourly rate is reduced for customers whopay in advance, via yearly contracts, to reserve capac-ity. The largest players in this market have mostlyshied away from selling their higher quality ser-vices via dynamic pricing;3 this could potentially be

2 “Organizations worry about whether utility computing serviceswill have adequate availability, and this makes some wary of cloudcomputing. 0 0 0Google Search has a reputation for being highlyavailable, to the point that even a small disruption is picked upby major news sources. Users expect similar availability from newservices...” (Armbrust et al. 2010, p. 54).3 The main exception would be Amazon’s spot market (Ama-zon Web Services 2012), which is a secondary market run by Ama-zon to sell the excess capacity of its main platform. Although thespot market prices fluctuate over time, the exact manner in whichthese prices are determined is not publicly available (see Ben-Yehuda et al. 2011).

attributed to the difficulty of maintaining service levelguarantees while customers are strategically timingtheir purchases. Our work provides the firm withtechniques for using dynamic pricing in such a con-text and, therefore, giving them a tool to better man-age their resources and revenues.There are other examples of service firms that need

to set profit-maximizing prices while guaranteeingservice. For instance, the increasingly popular Uberonline taxi service offers a flat pricing scheme onmost days of the year, but utilizes dynamic pricingin high demand days (see Bilton 2012). As explainedby its CEO, Uber is “aiming to provide a reliableride to anybody who needs one, no matter how crazydemand is or what is going on in the city” (Kalanick2011). Uber is able to provide such a service guaranteeby, in times of higher demand, conserving resourcesby charging higher prices.Another interesting application of our work is in

the context of electricity markets. As smart meters(see Federal Energy Regulatory Commission 2008) arebeginning to be widely deployed, allowing customersto immediately respond to price changes, the tech-niques we develop in this paper will become increas-ingly useful since electricity markets feature many ofthe elements of our model: capacity, demand, andprices are time varying, and rationing of service ishighly undesirable.In the industries discussed above, it is mainly the

firm’s responsibility to set prices that ensure that allservice requests can be accommodated with a limitedservice capacity. This is in contrast to settings such astraditional retailing, where customers are exposed torationing risk. In a traditional retail setting, strategiccustomers consider the risk of a stock-out and thisincentivizes them to purchase the good earlier. Thisrationing risk mitigates the effect of strategic customerbehavior on the firm’s ability to set its own prices. Inour setting, the firm’s need to offer service guaran-tees places the entire burden of matching supply anddemand over time on the firm. The intuition is thatthe firm should increase its prices when demand ishigh (or capacity is low) to shift some of the demandto the time periods with enough capacity. This is simi-lar to the pricing scheme used by the major cellphonecarriers that, in order to decongest their networksduring business hours, often charge lower prices formaking calls on nights and weekends (e.g., see AT&T2012). The dynamic pricing tools we propose hold thepromise of helping firms in such industries improvetheir resource utilization by better matching supplyand demand over time.

1.1. Our Framework and ContributionsWe consider a monopolist that offers service to cus-tomers over a finite horizon. The firm faces a (possibly

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time-varying) capacity constraint at each time period.The firm’s objective is to implement a posted pric-ing scheme to maximize its revenue. At time 0, themonopolist declares—and commits to—a sequence ofprices for its service, one for each time period. Giventhose preannounced prices, customers decide whetherand when to purchase service. The firm needs to solvethe constrained optimization problem of determiningthe prices that maximize revenue while still fulfillingall customer purchase requests.Each customer is assumed to be infinitesimal and

demands an (also infinitesimal) single unit of ser-vice. The valuation of a given customer for a unitof service is drawn from a known distribution. Sheis also associated with an arrival and a departuretime. The arrival time corresponds to the time sheenters the system and the departure time representsher deadline for obtaining service. All customers arefully strategic about whether and when they purchaseservice from the firm. That is, each customer eitherrefuses to buy service (if her valuation is below anyof the prices offered while she is present) or buys ser-vice at the period when she is offered the lowest priceamong all the periods in which she is present; if twoperiods have the same low price, the customer prefersthe earlier one.First, we consider a deterministic baseline model

where the monopolist knows the total mass of cus-tomers that arrive at each given time period as wellas their departure periods. The assumption of deter-ministic demand is justified when the number of cus-tomers is large and fairly predictable, which is oftenthe case in the market for cloud computing. This mod-eling choice allows us to study the impact of strate-gic customers and time-varying demand and capacityon the optimal sequence of prices, but it deliberatelyremoves the element of uncertainty from the model.The demand being deterministic also implies the opti-mality of using a sequence of preannounced prices.We show later in the paper that many of the insightsobtained in this simple environment naturally extendto general settings that allow for uncertainty in themodel.Interestingly, even the solution of this baseline

model is far from trivial. We show that because ofthe presence of strategic customers, the set of feasi-ble price vectors is neither convex nor closed. Thismeans no off-the-shelf software can be used to solvethis problem efficiently. Despite these challenges, weare able to establish that the firm’s price optimizationproblem is a tractable one and we provide an efficientpolynomial-time algorithm for computing the optimalpricing policy. This result relies on two crucial ideas:the set of prices that the firm needs to consider is

not too large, and prices can be combined into a pol-icy via dynamic programming because strategic cus-tomers never wait past a low price to purchase serviceat a future price that is higher.We extend our results to models where the firm

does not know its service capacity levels or thenumber of arriving customers. We do so in twodistinct ways. First, we consider a robust optimiza-tion framework (see Ben-Tal and Nemirovski 2002,Bertsimas and Thiele 2006), where there is uncertaintyabout the firm’s capacity and the size of the customerpopulation at any given period, and the firm onlyknows that these parameters belong to given sets.In this setting, the firm tries to maximize its worst-case revenues while ensuring that the capacity con-straints are not violated for any realization of demandand capacities. Second, we study the model in astochastic setting where the seller knows the distribu-tion of the uncertain parameters and is able to obtainadditional capacity at a cost; the goal is to determine asequence of (preannounced) prices that maximizes theexpected profit. Additionally, this model allows forproduction costs and different value distributions forcustomers with differing patience levels. We establishthat the insights from our baseline model carry on tothese general environments. That is, using a dynamicprogramming approach similar to our baseline model,we are able to provide algorithms that compute the(near) optimal pricing policy in polynomial time.We also consider a related setting where cus-

tomers do not simply choose the earliest time periodwith the lowest price to receive service, but ratherthe firm chooses how customers should break tiesbetween time periods with equal prices. In this set-ting, customers are guaranteed to receive service atthe lowest price available to them, but the firm canmore efficiently use its capacity by scheduling cus-tomers appropriately. Interestingly, our algorithmscan be modified to account for this additional flexi-bility and solve the optimal pricing problem of thefirm.Finally, we conduct numerical studies using our

algorithms and obtain further insights on the effectof strategic customers and service guarantees on boththe firm and its customers. We show that even insettings with high volatility in service capacity anddemand, the number of price levels that optimalpricing policy employs is small. For instance, in a24-period model, the optimal price sequence includesfour different price levels on average. This shows thateven in complex multiperiod settings, the customers’strategic behavior severely constrains the firms choiceof price sequence. We also observe that if patient cus-tomers can wait longer for service, both the revenueof the firm and the aggregate customer welfare maydecrease. This occurs because the firm is forced to

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use higher prices to maintain its service guarantees,and consequently the service capacity is underuti-lized. Thus we conclude that in a phenomenon simi-lar to Braess’s paradox (Basar and Olsder 1999), whencustomers have additional freedom in choosing thetime period they purchase service, the overall perfor-mance of the system may decrease.

1.2. Related WorkIn this section, we present a brief overview of theliterature on pricing mechanisms in the presence ofcustomers who strategically time their purchases anddiscuss how the results in the literature relate to ours.There is also an extensive literature on dynamic pric-ing with myopic customers (see, e.g., Lazear 1986,Wang 1993, Gallego and Ryzin 1994, Feng and Gallego1995, Bitran and Mondschein 1997, Federgruen andHeching 1999). We do not provide a summary of thisline of literature here, but we refer the reader to excel-lent surveys by Talluri and van Ryzin (2004), Bitranand Caldentey (2003), Chan et al. (2004), Shen and Su(2007), and Aviv et al. (2009).The study of monopoly pricing in the presence of

strategic customers was pioneered by Coase (1972).Coase conjectured that in a setting in which a monop-olist sells a durable good to patient customers, if themonopolist cannot commit to a sequence of postedprices, then the prices would converge to the pro-duction cost. Later, Stokey (1979, 1981), Gul et al.(1986), and Besanko and Winston (1990) showed thata decreasing sequence of prices is optimal for sellingdurable goods when customers face the trade-off ofconsuming right away versus the possibility of pur-chasing at the lower prices in the future. They observethat customers with high valuations buy in earlierperiods and pay higher prices compared to the lowvaluation customers.In the context of revenue management, Aviv and

Pazgal (2008) study a model where a monopolist sellsmultiple items over a finite time horizon to strategiccustomers who arrive over time. The authors considertwo classes of pricing strategies: contingent postedpricing, where the firm’s prices may depend on theremaining inventory, and preannounced posted pric-ing. They observe that commitment (preannounceddiscount) can benefit the seller when customers arestrategic. Also, ignoring the strategic customer behav-ior can lead to significant loss of revenue. Elmaghrabyet al. (2008) and Dasu and Tong (2010) extend theanalysis to a setting where the seller can reduce theprices multiple times over the time horizon. Otherpapers that consider commitment to a pricing pol-icy include Arnold and Lippman (2001), Levin et al.(2010), and Cachon and Feldman (2013). These worksmainly consider markdown pricing. Su (2007) showsthat if the customers are heterogeneous regarding

their time sensitivity, then the optimal sequence ofposted prices might also be increasing.In the aforementioned works, the service provider

uses the customers’ fear of rationing to extractmore revenue from strategic customers (see Liu andvan Ryzin 2008). In contrast, in our model the firmensures the customers does not face such risks andprovides service guarantees. Su and Zhang (2009)consider the issue of rationing in the presence ofstrategic customers and find that sellers have anincentive to overinsure consumers against the risk ofstockouts, thus showing that providing service guar-antees can be in the firm’s interest.Another paper related to ours is the one by Ahn

et al. (2007), which considers joint pricing and produc-tion decisions. Unlike us, they assume that customersare myopic with respect to prices, but, similarly toour model, they assume customers stay in the systemfor a number of periods unless they make a purchase.Interestingly, their analysis also relies on the notionthat a low price effectively separates past and futurethrough what they call regeneration points.An altogether different approach to this problem

is the one taken by the dynamic mechanism designliterature. There, the firm offers a direct mechanismthat allocates its service as a function of customersreports of their private valuations, entry period, anddeparture period. See Bergemann and Said (2011) fora survey. The paper closest to this one within this lit-erature is Pai and Vohra (2013), where strategic cus-tomers arrive and depart over time, but the allocationproblem they study is quite dissimilar to the one weconsider.The model we consider here differs from many

papers in the literature in at least three key aspects:in our model, the firm guarantees service to all pay-ing customers; therefore, the customers do not facerationing risk. Second, instead of having a fixedinventory at time 0, in our model, the firm has a time-varying service capacity, which is nonstorable. Finally,in the previous work, the customers are either presentfrom the beginning of the time horizon or arrive overtime but remain until the end of the horizon (or untilthey make a purchase). In our model, buyers arriveand depart the system over time.

2. The Baseline ModelIn this section, we formulate the firm’s revenue max-imization problem, which will be studied in the sub-sequent sections. The firm sets a vector of prices overa finite horizon t = 11 0 0 0 1T . The prices, denoted byp = 4p11 0 0 0 1pT 5, are announced up front, one pricefor each period t.4 Customers arrive and depart over

4 For instance, the horizon of the problem can be chosen as a day,with periods corresponding to different hours during the day, to

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time and are infinitesimal. We denote the populationof customers that arrive at period i and depart atperiod j by ai1 j . With slight abuse of notation, we alsorepresent the mass of the population that arrives atperiod i and departs at period j by ai1 j .Each customer wants one unit of service5 from

which she obtains a (nonnegative) value, and cus-tomers are strategic with respect to the timing oftheir purchases. Given the vector of prices p, a cus-tomer from population ai1 j , with value v for the ser-vice, purchases the service at a time period with thelowest price between times i and j , if her value islarger than the lowest price, i.e., if v �min`2 i`j8p`9.If there is more than one period with the lowest pricein 8i1 0 0 0 1 j9, the customer chooses the earliest period(with the minimum price) to obtain the service.6

Given a price vector p, we can assign to each pop-ulation ai1 j a service period, denoted by èi1 j4p5. Thisperiod has the lowest price among periods in 8i1 0 0 0 1 j9and is the earliest one (in 8i1 0 0 0 1 j9) with this price.Each member of population ai1 j considers purchas-ing service at time èi1 j4p5 and will purchase serviceif her value exceeds the price at that period. We callthe mass of customers that, given prices p, considerobtaining service at period t as the potential demand attime t and denote it by êt4p5. Formally, the potentialdemand is given by

êt4p5=X

i1 j21itjT

ai1 j18t =èi1 j4p591 (1)

where 1 is an indicator function.Each customer assigns a nonnegative value for

obtaining service. The fraction of customers withvalue below v is given by F 4v5. For simplicity of pre-sentation, we assume that F is a continuous functionand v 2 60117 for all customers. We also assume thatcustomer valuations are independent of their arrivaland departure periods, an assumption that we relax in§7. Hence, given price vector p, the demand at time t,denoted by Dt4p5, is equal to Dt4p5= 41É F 4pt55êt4p50The firm’s objective is to maximize its revenue,

which is given byPT

t=1 ptDt4p5. However, the firm isconstrained by a service capacity level of ct for eacht 2 811 0 0 0 1T 9. The firm provides service guarantees toits customers, so it must set prices that ensure thatthe demand Dt4p5 does not violate the capacity ct at

capture the problem of the firm selecting prices for its next businessday. Such an approach would be reasonable when deciding day-ahead prices for cloud computing services or electricity markets.5 A customer that wants multiple units of service could be consid-ered as multiple customers in our model.6 We relax this assumption in §8.

any period t. Thus, the firm’s decision problem isgiven by

supp�0

TX

t=1

ptDt4p5

s.t. Dt4p5 ct1 for all t 2 811 0 0 0 1T 91

(OPT-1)

where p � 0 is a shorthand notation for pt � 0 forall t 2 811 0 0 0 1T 9. The above problem searches for thesupremum of the objective function instead of themaximum, since the maximum of OPT-1 does notalways exist. We demonstrate the nonexistence of anoptimal solution in §3, where we also present ourtechnique for handling this issue.If there were no capacity constraints, the firm could

use a single price p at all periods to maximize its rev-enue,7 and this would result in a revenue equal top41 É F 4p55

Pij ai1 j . Since

Pij ai1 j is a constant, we

call p41É F 4p55 the uncapacitated revenue function. Wemake the following regularity assumption to simplifyour analysis.

Assumption 1. The uncapacitated revenue functionp41 É F 4p55 is unimodal. That is, there exists some mo-nopoly price pM such that p41É F 4p55 is increasing for allp < pM and decreasing for all p > pM .

Note that this assumption implies that pM maxi-mizes p41É F 4p55, and it is satisfied for a wide rangeof distributions, including the uniform, normal, log-normal, and exponential distributions.We now show, by means of an example, that the set

of feasible prices of OPT-1 is nonconvex.

Example 1. Let the time horizon be T = 3 andassume that a single unit-mass of customers with uni-form valuations in 60117 arrive at period 1 and departat period 3. Assume that c2 = 0 and c11 c3 = 1. Then theprice vectors 401001115 and 411001105 are both feasi-ble. However, the average of these two price vectors,4005100110055, is infeasible since all customers withvaluation above 001 seek service at period 2, violat-ing the service capacity c2 = 0. Therefore, the set offeasible prices of OPT-1 is nonconvex.

The above example illustrates that OPT-1 is a non-convex optimization problem, and we cannot hopeto solve it using off-the-shelf optimization tools. Weshow in §5 that despite being nonconvex, there existsa polynomial-time algorithm that solves this opti-mization problem. The construction of this algorithmrelies on the structural properties of this pricing prob-lem that are explored in §§3 and 4.

7 Because customer valuations are independent of arrival anddeparture periods, it can be seen from OPT-1 that if there are nocapacity constraints, setting pt = argmaxp p41É F 4p55 for all t maxi-mizes revenue.

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3. Optimizing over Prices andRankings

In this section, we show that there does not alwaysexist a feasible solution achieving the supremum inthe firm’s optimization problem. To address this issue,we construct a closely related optimization problemwhere the firm tries to maximize not only prices butalso rankings of the prices. We show that this opti-mization problem always admits an optimal solu-tion that can be used to obtain feasible solutionsarbitrarily close to the supremum of the originalproblem.We start with an example that shows that the supre-

mum of OPT-1 may not be achieved by a feasibleprice vector. The main idea is that since the customersalways seek the lowest price available, the potentialdemand function êt is a discontinuous function of p;thus, the feasible set of OPT-1 is open.Example 2. Consider a two-period model with cus-

tomer valuations drawn uniformly from 60117, capac-ity levels c1 = 1

2 and c2 =à, and customer populationsa111 = a112 = 1 (and a212 = 0). Observe that solutions ofthe form 4p11p25= 4 121

12 É Ö5 are feasible for any Ö> 0:

the members of population a111 with value above 12

obtain service at time 1 and the members of popula-tion a112 with value above 1

2 É Ö are served at time 2.Hence, 4p11p25 = 4 121

12 É Ö5 yields the revenue of 1

2 ⇥12 + 4 12 ÉÖ5⇥ 4 12 +Ö5= 1

2 ÉÖ2. The revenue is decreasingin Ö and as Ö tends to 0 the revenue approaches 1

2 .The uncapacitated problem provides an upper boundon the revenue obtained, which is 1

2 . Therefore, thesupremum of OPT-1 is equal to 1

2 . However, 4p11p25=4 121

12 5 is not a feasible solution because under this

price vector, both populations will choose the firstperiod for service, and this violates the capacity con-straints. Therefore, the feasible set of price vectors isopen and the supremum of OPT-1 is not achieved bya feasible price vector.The nonexistence of an optimal solution can be

addressed by finding (feasible) solutions that are arbi-trarily close to the (infeasible) supremum. In theremainder of this section, we introduce the notion ofrankings and an alternative optimization formulationthat allow us to obtain such solutions for OPT-1 (orthe optimal solution itself in instances where the opti-mum is feasible).We refer to permutations of 811 0 0 0 1T 9 as rankings.

We use the notation Rt to denote the rank of timeperiod t under ranking R. We say that a ranking R isconsistent with a price vector p if periods with lowerrank have lower prices. More precisely, R is consis-tent with p if for all t and t0, Rt < Rt0 implies thatpt pt0 .We define the customer-preferred ranking, denoted by

RC4p5, as a ranking consistent with p, such that whenthere are multiple periods with the same price, the

earlier periods are ranked lower. Namely, if pt = p0tand t < t0 then Rt <Rt0 . It can be seen from the defini-tion of service period èi1 j4p5 (introduced in §2) that inOPT-1 for a given price vector p, each population ai1 jchooses the time period between i and j , with the low-est customer-preferred ranking to (potentially) receiveservice. Hence potential demand êt can be expressedas a function of this ranking.More formally, for any period t and ranking of

prices R we define the R-induced potential demand,denoted by êt4R5, as

êt4R5=X

ij

ai1 j1⇢Rt = min

k2 ikj8Rk9

�0 (2)

Similarly, the R-induced demand, denoted by Dt4pt1R5,is defined as

Dt4pt1R5= 41É F 4pt55êt4R50 (3)

It follows from (1), (2), and the definition ofcustomer-preferred ranking that for any price vec-tor p and customer-preferred ranking RC4p5, we haveêt4R

C4p55 = êt4p5 and Dt4pt1RC4p55 = Dt4p5. That is,

it is possible to express demand (Dt) in terms ofthe R-induced demand function (Dt) and customer-preferred ranking (RC).Suppose that in OPT-1 the firm could select not

only the vector of prices p but also any rank-ing R consistent with p (potentially different from thecustomer-preferred ranking), and customers decidedwhen to obtain service according to this ranking;i.e., each customer chooses the period with the low-est ranking between her arrival and departure time.Then, the demand at any period is given by Dt4pt1R5,and the corresponding revenue maximization prob-lem can be formulated as

maxp�01R2P4T 5

TX

t=1

ptDt4pt1R5

s.t. Dt4pt1R5 ct

for all t 2 811 0 0 0 1T 91

Rt <Rt0 ) pt pt0

for all t1 t0 2 811 0 0 0 1T 91

(OPT-2)

where P4T 5 is the set of all possible rankings of811 0 0 0 1T 9. Although this problem is different fromOPT-1, the solutions of these problems are closelyrelated, as we explain next.Since Dt4pt1R

C4p55= Dt4p5, it follows that any fea-sible solution p of OPT-1 corresponds to a feasiblesolution of OPT-2 given by 4p1RC4p55, and these solu-tions lead to the same objective values. Additionally,it can be seen from (1) and (2) that for a given price

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vector p and any ranking R consistent with p, the(potential) demand levels in OPT-1 and OPT-2 areequal except for periods where the price is equal tothe price offered at another period. Intuitively, unlikeOPT-1, in OPT-2 the firm can choose how the cus-tomers collectively break ties between time periodswith equal price, by choosing the ranking R prop-erly, and this may lead to a difference in demandlevels only at such time periods. These observationscan be used to show that OPT-2 always has an opti-mal solution, and this solution can be used to con-struct a solution of OPT-1 that is arbitrarily close tothe supremum.

Lemma 1. The following claims hold:1. The problem OPT-2 admits an optimal solution

4p?1R?5.2. Let 4p?1R?5 be an optimal solution of OPT-2. For any

Ö > 0, the price vector p? + ÖR? is a feasible solution ofOPT-1 and the revenue it obtains converges to the supre-mum of OPT-1 as Ö tends to 0.3. If p is an optimal solution of OPT-1, then 4p1RC4p55

is an optimal solution of OPT-2.

The proof of this lemma can be found in the onlineappendix (available at http://pages.stern.nyu.edu/⇠ilobel/multiperiod-pricing-online-app.pdf). The ideabehind this lemma is that the projection of the set offeasible solutions of OPT-2 onto the set of prices isthe closure of the feasible set of OPT-1. Therefore, theoptimal prices generated by OPT-2 can be perturbedin a way that maintains the ranking of prices, lead-ing to a solution of OPT-1 that is arbitrarily close tothe supremum. In the rest of the paper, we focus onthe solution of OPT-2, keeping in mind that an opti-mal solution (or a solution arbitrarily close to optimal,if the optimal solution does not exist) of OPT-1 with(almost) the same prices can be constructed using thissolution.

4. Structure of the Optimal PricesIn this section, we explore the structure of optimalprices in OPT-2. We first show that at all periodsthe monopolist has incentive to keep the prices atleast as high as the monopoly price pM . Then we usethis observation to study the optimality conditions inOPT-2. Exploiting these conditions, we construct a setof prices that contains all the possible candidate opti-mal prices. We show that the cardinality of this set ispolynomial in the time horizon T , a result we later usein §5 to obtain a polynomial-time algorithm to solveOPT-2. Proofs of the results presented in this sectioncan be found in the online appendix.To gain some intuition, we first consider the opti-

mal solution in a single period setting. By Assump-tion 1, choosing any price p < pM is suboptimal, andthe firm has incentive to increase its price to pM .

If setting the price equal to pM violates the capacityconstraints, then the firm increases its price to theminimum price that respects the capacity constraints.Since customers’ values are bounded by 1, such aprice exists. Thus, it follows that an optimal pricein 6pM117 can be found. The following propositionshows that this intuition extends to multiperiodsettings.

Proposition 1. There exists an optimal solution 4p1R5of OPT-2 such that pt � pM for all t.

To prove this result, we assume that a solutionwhere pt < pM for some t is given, and we raise pricesthat are below the monopoly price pM in a way thatmaintains the ranking of the prices. This ensures thatas the prices increase to pM , the revenue increaseswhile the demand decreases. Thus, it is possible toobtain a feasible solution that (weakly) improves rev-enues and satisfies pt � pM .Note that conditioned on prices being above the

monopoly price pM , by the assumption of unimodal-ity of the uncapacitated revenue function, the incen-tives of the firm and the customers are aligned: boththe firm and the customers prefer lower prices overhigher ones. The firm never raises prices to obtainmore revenue, only to satisfy capacity constraints.We next provide a further characterization of the

prices that are used at an optimal solution of OPT-2.This characterization significantly narrows down theset of prices that needs to be considered to find anoptimal solution.

Proposition 2. There exists an optimal solution 4p1R5of the optimization problem OPT-2 such that for eachperiod t one of the following three statements is true: pt =pM , pt = 1, or pt = pt for some t, such that ct =Dt4pt1R5and pt 2 6pM117.

The proof of this proposition follows by showingthat unless the conditions of the proposition hold,the monopolist can modify the prices in a way thatincreases its profits while maintaining the feasibilityof the capacity constraints.8 The third condition of theproposition suggests that the price at time t is eithersuch that the capacity constraint at time t is tight orthat this price is equal to the price offered at anothertime period, and the capacity constraint at this othertime period is tight. Hence, because of the presence ofstrategic customers, capacity constraints at one periodmay bind the prices at another period, but this requiresthe prices to be identical at these two periods.This proposition implies that for an optimal solu-

tion 4p1R5 of OPT-2, each entry of the price vector p

8 We note that if we strengthen Assumption 1 to impose concavityon the uncapacitated revenue function, then the proposition canbe proved using the Karush–Kuhn–Tucker conditions.

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either belongs to 8pM119 or is above pM and satisfiesthe equation

ct =Dt4p1R5= êt4R541É F 4p55 (4)

for some time period t. However, to characterize theset of all prices that may appear at an optimal solu-tion, we still need to consider all possible rankings.Although there are T ! possible rankings R, there area significantly smaller number of prices that satisfyequations of the form (4). To formalize this idea, weintroduce the notion of attraction range, which is arepresentation of all the populations that choose thesame period for service.

Definition 1 (Attraction Range). For a givenconsistent price-ranking pair 4p1R5 the attraction rangeof a time period k is defined as the largest inter-val 8 t1 0 0 0 1 t9✓ 811 0 0 0 1T 9 containing k such that Rk =min`28 t10001t9R`.

Assume that the attraction range of time period kfor a consistent price-ranking pair 4p1R5 is 8 t1 0 0 0 1 t9.Since customers choose the time period with the low-est ranking available to them when purchasing ser-vice, customers who arrive at the system betweenperiods t and k, and who can wait until time period kbut not beyond time period t, are exactly the oneswho will seek service at period k. Thus, the attractionrange concept can be used to identify customers whoare “attracted” to a particular time period for receiv-ing service (see Example 3).

Example 3 (Attraction Range). Consider a prob-lem instance with six time periods. Assume that aconsistent price-ranking pair 4p1R5 for this problemis given and the prices at different time periods areas in Figure 1. Since prices at all time periods are dif-ferent, there is a unique ranking R consistent withthese prices. The attraction range of time period 4 inthis example is 821 0 0 0 159. Thus, customers who arrivebetween time periods 2 and 4 (inclusive) and whocannot wait beyond time period 5 are the ones whoseek service at period 4.

Figure 1 Attraction Range of Period 4 Is 821 0 0 0 159 in This Six-Period

Problem Instance

Periods

Prices

1 2 3 4 5 6

2 4 5

This example suggests that attraction ranges canbe used to determine R-induced potential demandêt4R5. Assume that 4p1R5 is a consistent price-rankingpair and consider the attraction range of sometime period k 2 811 0 0 0 1T 9, denoted by 8 t4k1R51 0 0 0 1t4k1R59. As discussed earlier, customers who arriveat the system between t4k1R5 and k (inclusive) andwho can wait until time k but not beyond time t4k1R5are the only ones who can request service at time k.Thus, we obtain that êk4R5=

Pki= t4k1R5

Pt4k1R5j=k aij . From

this equation it follows that êk4R5 can immediatelybe obtained by specifying the attraction range oftime period k. By considering all the possible attrac-tion ranges 8 t1 0 0 0 1 t9 corresponding to time period kwe conclude that for any ranking R, we haveêk4R5 2 8

Pki= t

Ptj=k aij ó t k t9. Using this observa-

tion, it follows that any p satisfying (4) for some Rand êk4R5 belongs to the set

Lk4=⇢max

⇢pM1 F É1

✓1É

✓ck

Pki= t

Ptj=k aij

◆◆�����

ck kX

i= t

tX

j=k

aij1 and t k t

�0 (5)

Here the condition ck Pk

i= t

Ptj=k aij is present since

F É1 is defined over the domain 60117. The maximumwith pM is taken to make sure that all the prices in Lk

are at least equal to pM , which follows from Propo-sition 2. By construction each element of Lk corre-sponds to an attraction range 8 t1 0 0 0 1 t9. Since thereare O4T 25 attraction ranges (there are O4T 5 values tand t can take), the cardinality of Lk is O4T 25. Thus,we reach the following characterization of optimalprices, which is stated without proof because it imme-diately follows from Propositions 1 and 2 and the def-inition of Lk given in (5).

Proposition 3. Let L be defined as L4= 4

STk=1 Lk5 [

8pM 9 [ 819. There exists an optimal solution 4p1R5 ofOPT-2 such that pt 2 L for all t 2 811 0 0 0 1T 9. Moreover,the cardinality of L is O4T 35.

This proposition implies that without actually solv-ing OPT-2, it is possible to characterize a supersetof the prices that will be used at an optimal solu-tion. Moreover, this set has polynomially-many ele-ments, and it is sufficient for the monopolist to con-sider these prices when making its pricing decisions.However, finding the vector of optimal prices couldstill be a computationally intractable problem even ifL has small cardinality. In the next section, we showthat this is not case, and we develop a polynomial-time algorithm that determines the optimal sequenceof prices.

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5. A Polynomial-Time AlgorithmIn this section, we use the characterization ofthe optimal prices obtained in §4 to design apolynomial-timealgorithm for computing the optimalsequence of prices.As shown in Proposition 3, an optimal solution of

OPT-2 can be obtained by restricting attention to set ofprices L given in Proposition 3. Thus, an optimal solu-tion to OPT-2 can be obtained by restricting attentionto prices in L and solving the following optimizationproblem:

maxp2LT 1R2P4T 5

TX

t=1

ptDt4pt1R5

s.t. Dt4pt1R5 ct

for all t 2 811 0 0 0 1T 91

Rt <Rt0 ) pt pt0

for all t1 t0 2 811 0 0 0 1T 90

(OPT-3)

We next show that it is possible to find an optimalsolution of OPT-3 by recursively solving problemsthat are essentially smaller instances of itself.Consider an optimal solution of OPT-3, denoted by

4p?1R?5. Suppose time period k has the lowest rank-ing; i.e., R?

k = 1. In this case, the attraction range ofk is 811 0 0 0 1T 9, and êk4R

?5 = Pki=1

PTj=k aij . Hence, all

customers who are present in the system at time kwill seek service at time k. This implies that onlypopulations ak11k2 , 1 k1 k2 < k can receive serviceat time periods 811 0 0 0 1kÉ 19 (similarly, only popula-tions ak11k2 , k < k1 k2 T can receive service at timeperiods 8k + 11 0 0 0 1T 9). Therefore, if the monopolistknows p?k and that R?

k = 1, it can solve for optimalprices at other time periods by solving two separatesubproblems for time periods 811 0 0 0 1k É 19 and 8k +11 0 0 0 1T 9: maximize the revenue obtained from timeperiods 811 0 0 0 1kÉ19 assuming only populations ak11k2are present (with 1 k1 k2 < k) and similarly fortime periods 8k + 11 0 0 0 1T 9. Note that in the solutionof the subproblems, we need to impose the conditionthat prices are weakly larger than p?k because other-wise p?l < p?k for some l, and we obtain a contradictionto R?

k = 1.The above observation suggests that given the time

period k with the lowest ranking, the pricing problemcan be decomposed into two smaller pricing prob-lems, where the prices that can be offered are lowerbounded by the price offered at k. We next exploitthis observation and obtain a dynamic programmingalgorithm for the solution of OPT-3.Letó4i1 j1 p5 denote themaximum revenue obtained

from an instance of OPT-3 assuming (i) ak11k2 = 0 unless

i < k1 k2 < j and (ii) restricting prices to be weaklylarger than p. That is,

ó4i1j1 p5= maxp2LT 1R2P4T 5

jÉ1X

t=i+1

ptDijt 4pt1R5

s.t. Dijt 4pt1R5ct

for all t28110001T 91Rt <Rt0 )ptpt0

for all t1 t0 28110001T 91pt� p for all t28110001T 9

(6)

where Dijt is defined similarly to (3) and denotes the

demand at time t, assuming ak11k2 = 0 unless i < k1 k2 < j . Observe that the optimal objective value ofOPT-3 is equal to ó401T + 1105.For i+ 1> j É 1, we assume ó4i1 j1 p5 is equal to 0.

On the other hand, for any i1 j such that i+ 1 j É 1,we have

ó4i1 j1 p5= maxk28i+110001jÉ19

⇢max

p2L2p� p8ó4i1k1p5+É

ij

k 4p5

+ó4k1 j1p59

�1 (7)

where Éij

k 4p5 is given by

Éij

k 4p5=

8>>>>>>>>><

>>>>>>>>>:

✓ kX

l=i+1

jÉ1X

m=k

alm

◆41É F 4p55p

if✓ kX

l=i+1

jÉ1X

m=k

alm

◆41É F 4p55 ck1

Éà otherwise.

(8)

To see why the recursion in (7) holds, consider a solu-tion of (6) and assume that in this solution, k is thetime period in 8i+11 0 0 0 1 jÉ19with the lowest rankingand pk � p is the corresponding price. Then all popula-tions that are present in the system at k receive serviceat this time period. The total mass of these populationsisPk

l=i+1PjÉ1

m=k alm, since ak11k2 = 0 unless i < k1 k2 < j ,as can be seen from the definition of ó4i1 j1 p5. Since atthe optimal solution of (6), the capacity constraints aresatisfied, the revenue obtained from time period k isgiven by É

ij

k 4pk5. Since k has the lowest ranking among8i+ 11 0 0 0 1 j É 19, only populations ak11k2 such that i <k1 k2 < k can receive service before time k, and theprices offered at those time periods should be weaklylarger than pk. It follows from the definition of ó thatthe maximum revenue that can be obtained from thesepopulations (with prices weakly larger than pk) isgiven by ó4i1k1pk5. Similarly, it follows that the maxi-mum revenue that can be obtained from time periods

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after k equals to ó4k1 j1pk5. Thus, we conclude tható4i1 j1 p5=ó4i1k1pk5+ É

ij

k 4pk5+ó4k1 j1pk5. The recur-sion in (7) follows since it searches for time period kwith the lowest ranking and the corresponding pricepk that maximizes the objective of (6). Note that sinceÉij

k 4p5=Éà when a capacity constraint is violated, thesolution obtained by solving this recursion also satis-fies the capacity constraints.The following theorem shows that a solution of

OPT-2, or equivalently a solution of the alternativeformulation in OPT-3, can be obtained by solving forprices using the dynamic programming recursion in(7) and constructing a ranking vector consistent withthese prices.

Theorem 1. The optimal solution of OPT-2 can becomputed in time O4T 65.

This theorem, which is proved in the appendix,suggests that firms that provide service guaranteescan effectively implement optimal pricing policies bysolving OPT-2. In the subsequent sections, we showthat this result extends to other settings where thereis uncertainty about the problem parameters and thefirm has more general objectives.

6. A Robust OptimizationFormulation

The baseline model we considered in previous sec-tions assumes that both demand and service capacityavailable over the planning horizon can be fully antic-ipated. This assumption is motivated by our onlineservices application, where the planning horizon istypically counted in hours or, at most, days. Over thenext two sections, we show how to solve the firm’sproblem when this assumption is not valid, by eithertaking a robust or a stochastic view of the demandor capacity uncertainty. In particular, in this sectionwe introduce a robust optimization formulation of thefirm’s pricing problem. We show that when there isuncertainty about either the service capacity levels orthe size of the customer population, a variant of thealgorithm of §5 can be used to obtain a solution thatmaximizes revenue while maintaining feasibility forall possible values of uncertain parameters. Further-more, we can bound the firm’s worst-case revenueloss as a function of the uncertainty in the problemparameters. The proofs of this section can be found inthe online appendix.Suppose the firm does not know its service capac-

ity level ct at a given period but only knows thatit belongs to an interval Ct = 6cLt 1 c

Ut 7. Similarly, the

firm does not know the mass of customers in popu-lation ai1 j , but instead it knows only that ai1 j 2Ai1 j =6aLi1 j1a

Ui1 j 7. We refer to a collection of population sizes

A= 8ai1 j91i1 jT as a population matrix and represent

the set of all possible capacity levels by C = Qt Ct

and the set of all possible population matrices by A=Qi1 j Ai1 j . To make dependence of the demand (defined

in Equation (3)) on population size explicit, in thissection we denote the demand at period t, when pop-ulation matrix is given by A 2 A, and the firm usesprice pt and ranking R 2P4T 5 by Dt4pt1R1A5.The problem of selecting prices and ranking that

are feasible for all capacity levels in C and popula-tion matrices in A, and that maximize the worst-caserevenue, can be formulated as follows:

maxp�01R2P4T 51M

M

s.t. M TX

t=1

ptDt4pt1R1A5

for all A 2A1

Dt4pt1R1A5 ct

for all t1 ct 2Ct and A 2A1

Rt <Rt0 ) pt pt0

for all t1 t0 2 811 0 0 0 1T 91

(OPT-4)

Our next result shows that this robust optimizationproblem is tractable.

Proposition 4. The optimal solution of OPT-4 can becomputed in time O4T 65.

The optimization problem in OPT-4 is not a specialcase of OPT-2 because it uses the population matrixAL when computing revenue and a different popu-lation matrix AU when computing feasibility. How-ever, with minor modifications, the structural insightsand the polynomial-time algorithm from §5 still apply,leading to the result in Proposition 4.The robust optimization formulation finds a conser-

vative solution that is feasible for all possible valuesof uncertain parameters. We next quantify the poten-tial revenue loss due to the uncertainty, when thesolution obtained from this formulation is used forpricing. Assume that a solution of OPT-4 is obtainedusing uncertainty sets C and A. Let V ROB4C1A1 c1A5denote the revenue the firm achieves, using this solu-tion, when realized parameter values are c 2 C andA 2A. We denote the revenues that could be obtainedif we knew with certainty c and A by V 4c1A5. Thefollowing proposition bounds the decrease in the rev-enues when there is uncertainty, and the solution ofOPT-4 is used to obtain a robust pricing rule.

Proposition 5. Suppose cUt 41 + à5cLt for all t 2811 0 0 0 1T 9 and aUi1 j 41+ à5aLi1 j for all i1 j 2 811 0 0 0 1T 9.Then,

supc2C1A2A

4V 4c1A5ÉV ROB4C1A1 c1A55 3àX

i1 j

aUi1 j 0

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This proposition implies that when the uncertaintyin the problem parameters is small (i.e., when à issmall), the revenue loss is also small, provided that asolution of OPT-4 is used for pricing. The propositionalso suggests that if a nominal version of the prob-lem with parameters 4cNOM1ANOM5 is known, and therealized parameters are between 1Éò and 1+ò timesthe nominal ones (i.e., 1+ à = 1+ ò/41É ò5), then themaximum revenue loss due to uncertainty is equal to446Ö41+ ò55/41É ò55

Pi1 j a

NOMi1 j .

7. A General Framework and anApproximation Scheme

In this section, we generalize our baseline model byincorporating random customer arrivals and capacitylevels, production costs, and customer valuations thatare dependent on arrival and departure periods to themodel. Namely, the distribution of the size and valua-tions of each population is known in advance and thefirm determines a sequence of (preannounced) pricesto maximize its expected profit. In order for the firmto be able to satisfy its service guarantees, we considersoft capacity constraints in the sense that the firm canexceed the allowable capacity by paying a penalty orpurchasing more capacity.At this level of generality, the characterization of

the set of prices in an optimal solution, presentedin §4, does not hold. However, we can extend ouralgorithm to obtain a fully polynomial-time approx-imation scheme (FPTAS) for the general model. AnFPTAS is an approximation algorithm that for anyò> 0 obtains a solution within a factor of 1Éò of theoptimal solution and is polynomial in the size of theproblem and in 1/ò. Therefore, using an FPTAS, onecan obtain a solution arbitrarily close to the optimal.Intuitively, this problem is still tractable since, simi-

lar to our baseline model, if a time instant has the low-est price in the horizon, then all customers who arepresent in the system at this time instant prefer receiv-ing service there. Consequently, our main divide-and-conquer approach is applicable, and we can reformu-late the problem as a dynamic programming problemfollowing the approach in §5. In this section, we for-mally explain this idea. The proofs of our results arepresented in the online appendix.We start by introducing an abstract problem that is

the focus of this section:

maxp260117T 1R2P4T 5

TX

t=1

gt4pt1R5

s.t. ht4pt1R5 0

for all t 2 811 0 0 0 1T 91

Rt <Rt0 ) pt pt0

for all t1 t0 2 811 0 0 0 1T 90

(OPT-5)

We make the following assumption throughout thissection:

Assumption 2. For any ranking R and period t, thefunctions gt4·1R5 and ht4·1R5 satisfy the following:1. Each customer prefers the time period with the lowest

rank (among those during which she is present) to (poten-tially) receive service. Hence, the dependence of functionsgt2 ✓⇥P4T 5!✓ and ht2 ✓⇥P4T 5!✓ on R is throughthe attraction range of time period t. That is, there existfunctions g, h such that

gt4pt1R5= gt4pt1 bt4R51 et4R55 and

ht4pt1R5= ht4pt1 bt4R51 et4R55(9)

where 8bt4R51 0 0 0 1 et4R59 is the attraction range of timeperiod t, when ranking R is chosen.2. ht4pt1R5 is decreasing in pt , and gt4pt1R5 is Lips-

chitz continuous in pt with parameter lt .

The first part of the assumption implies that thedemand for service at period t and, therefore, theprofit earned at period t, depends only on the price ptand the attraction range generated by the ranking ofprices R. That is, if we modify price pt+1 while leav-ing the ranking of prices R intact, thus leaving theattraction range intact, this change in the price vec-tor p will have no effect on the demand at period t.This excludes customer discounting, for example, butis a generalization of the customer behavior modelassumed in earlier sections. The second part of theassumption just implies that demand in a givenperiod decreases continuously in that period’s priceif we maintain a constant ranking of prices. Observethat OPT-2 is a special case of OPT-5, when gt4pt1R5=ptDt4pt1R5 and ht4pt1R5=Dt4pt1R5Éct , assuming thatdemand Dt4pt1R5 is Lipschitz continuous in pt , for afixed ranking R.For a constant Ö 2 40115, consider the set of prices

PÖ = 8kÖ ó k 2 ⇢+1kÖ 19 and assume that we seek asolution to OPT-5 by restricting attention to the pricesthat belong to this set, i.e.,

maxp2PT

Ö 1R2P4T 5

TX

t=1

gt4pt1R5

s.t. ht4pt1R5 0

for all t 2 811 0 0 0 1T 91

Rt <Rt0 ) pt pt0

for all t1 t0 2 811 0 0 0 1T 90

(OPT-6)

Note that any feasible solution of OPT-6 is feasiblein OPT-5. We next show that for small Ö the opti-mal objective values of these problems are also close.Hence, an optimal solution of OPT-6 can be used toprovide a near-optimal solution of OPT-5.

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Lemma 2. Let the optimal solutions of OPT-5 andOPT-6 have objective values v and vÖ, respectively. Then,vÖ � vÉ Ö

PTt=1 lt .

We next show that a modified version of the algo-rithm in §5 can be used to solve OPT-6. Our approachis again based on obtaining a solution by recursivelysolving smaller instances of the problem. For this pur-pose, we first define ó4i1 j1 p5 to be the maximumutility that can be obtained assuming only popula-tions ak11k2 , i < k1 k2 < j , are present and the pricesthat can be used at periods 8t ó i < t < j9 are (weakly)larger than p. It can be seen that the optimal value ofOPT-6 is equal to ó401T + 1105.We set ó4i1 j1 p5= 0, for i+1> jÉ1. Using the same

argument given in §5 to justify the recursion in (7), itfollows that for i + 1 j É 1, the following dynamicprogramming recursion holds:

ó4i1j1 p5 = maxk28i+110001jÉ19

⇢max

p2PÖ 2p� p8ó4i1k1p5+É

ij

k 4p5

+ó4k1j1p59

�1 (10)

where Éij

k 4p5 denotes the utility obtained at time kwith price p, from all populations ak11k2 that canreceive service at this period and that satisfy i < k1 k2 < j . That is, É ij

k 4p5= gk4p1 i1 j5, if hk4p1 i1 j5 0 andÉij

k 4p5=Éà otherwise.The intuition behind (10) is similar to the intuition

of (7): to find ó4i1 j1 p5, we search for the time periodwith the lowest rank (maximization over k in (10)),and we search for the best possible price for this timeperiod (maximization over p). Since all populationsthat are present at the time period with the lowestranking (say k) receive service at this time period, thepayoff obtained from this time period can be givenby É

ij

k 4p5. We then solve for prices of subproblems fortime periods 8i + 11 0 0 0 1k É 19 and 8k + 11 0 0 0 1 j É 19.Since the time period with the lowest ranking alsohas the lowest price, we impose the prices for thesesubproblems to be weakly larger than p. Thus, thepayoffs of the subproblems are given by ó4i1k1p5 andó4k1 j1p5. Hence, we obtain the recursion in (10) forcomputing optimal prices in OPT-6.In the following lemma, we use this dynamic pro-

gram to construct optimal prices and ranking for thesolution of OPT-6 and characterize the computationalcomplexity of the solution. Note that since we aredealing with general functions gt and ht , our resultdepends on the computational complexity of evaluat-ing these functions.

Lemma 3. Assume that for any given t1p1R, compu-tation of gt4p1R5 and ht4p1R5 takes O4s4T 55 time. Anoptimal solution of OPT-6 can be found in O44T 3s4T 55/Ö25time.

Lemmas 2 and 3 imply that an approximate solu-tion to OPT-5 can be found in polynomial time pro-vided that gt4p1R5 and ht4p1R5 can be evaluated inpolynomial time.

Theorem 2. Assume that for any given t1p1R, compu-tation of gt4p1R5 and ht4p1R5 takes O4s4T 55 time. An Ö-optimal solution of OPT-5 can be found in O44T 3s4T 55/Ö25time.

The proof immediately follows from Lemmas 2 and3 and is omitted. In many of the relevant cases (such asrevenue maximization subject to capacity constraintsas introduced in §§2 and 3), for given prices and rank-ings, evaluating constraints and the objective function(ht and gt) can be completed in O415 time. In such set-tings, Theorem 2 implies that an approximate solutioncan be obtained in O4T 3/Ö25 time.We conclude this section by showing that this

general framework allows us to find approximatelyoptimal prices in polynomial time for probleminstances that involve random arrivals and capacitylevels, production costs, and a richer class of cus-tomer valuations.Correlated Valuations. Here, we relax the assumption

made before that the customers’ valuations are inde-pendent of their arrival and departure periods. LetFi1 j4v5 represent the fraction of the ai1 j population thatvalues service at most v. We assume that all valua-tions are in 60117 and that Fi1 j is differentiable, but weno longer suppose that Assumption 1 holds; i.e., thecorresponding uncapacitated revenue function neednot be single peaked. Customer demand at period tas a function of price pt and the ranking of prices Ris now given by

Dt4pt1R5=X

ij

ai1 j41ÉFi1 j4pt5518RtRk for all ik j90

By choosing gt4pt1R5 = ptDt4pt1R5 and ht4pt1R5 =Dt4pt1R5 É ct , the corresponding revenue maximiza-tion problem is an instance of OPT-5. Note that fora fixed R, denoting fi1 j4p5= dFi1 j4p5/dp and assumingfi1 j is bounded by li1 j1 we conclude����°Dt4pt1R5

°pt

���� =X

ij

ai1 j fi1 j4pt518Rt Rk for all i k j9

X

ij

ai1 j li1 j 0

Thus, it follows that when fi1 j is bounded for all i,j , Dt4·1R5 is Lipschitz continuous. This implies thatgt4·1R5 is also Lipschitz continuous for all t and R.Moreover, ht is decreasing in pt (since demand isdecreasing in pt). Furthermore, for any t1p1R eval-uating Dt4pt1R5 and in turn gt4p1R5 and ht4p1R5takes O4T 25 time. Thus, Theorem 2 applies, and weconclude that the approximate revenue maximizationproblem can be solved in O4T 5/Ö25.

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Production Costs and Soft Capacity Constraints. Wealso incorporate production costs into the model. Weassume that it costs the firm åt4d5 to provide serviceto mass d of customers at period t. We make the fol-lowing regularity assumption on the production costs:

Assumption 3. Assume that for any period t, theproduction cost åt is a nonnegative, nondecreasing,ã-Lipschitz continuous function.

Along with capturing the cost of producing the ser-vice to be delivered, the function åt can also capturea soft capacity constraint: if ct represents a capacitylevel above which any unit produced costs å, then wecan capture this by setting åt4d5 =max801 å4d É ct59.Even with soft capacity constraints, we still assumethat the firm provides service guarantees. Wheneverthe firm is incapable of providing the purchased ser-vice itself, it contracts service delivery out to a thirdparty with a unit cost of å.By letting gt4pt1R5= ptDt4pt1R5Éåt4Dt4pt1R55 and

ht4pt1R5 = Dt4pt1R5 É ct , we obtain an instance ofOPT-5. From Assumption 3, it follows that gt4pt1R5 isLipschitz continuous in pt . Since demand is decreas-ing with price, we observe that ht4pt1R5 decreaseswith price. Thus, Theorem 2 applies, and sinceDt4p1R5 (and hence gt4pt1R5, ht4pt1R5) can be eval-uated in polynomial time for any given p and R, itfollows that an approximate solution of the problemswith production costs and soft capacity constraintscan be obtained in polynomial time.Stochastic Arrival and Capacity Processes. Assume that

population sizes 8ai1 j9i1 j and capacities 8ct9t are ran-dom variables with known distributions. Let E6ai1 j 7=ai1 j and E6ct7 = ct . In this setting, if a monopolistwants to guarantee that the total service request doesnot exceed the capacity for any realization of theparameters, it can use the robust optimization frame-work in §6. On the other hand, if the firm has thecapability to contract service delivery out wheneverthe capacity is exceeded (hence it has soft capacityconstraints), it can solve the following expected rev-enue maximization problem:

maxp2PÖ1R2P4T 5

TX

t=1

E6ptDt4pt1R5Éåt4Dt4pt1R557

s.t. Rt <Rt0 ) pt pt0

for all t1 t0 2 811 0 0 0 1T 90

(OPT-7)

By choosing

ht4pt1R5= 0 and

gt4pt1R5= E6ptDt4pt1R5Éåt4Dt4pt1R5571

we obtain an instance of OPT-5. Note that if åt is Lip-schitz continuous, then so is gt4pt1R5= E6ptDt4pt1R5Éåt4Dt4pt1R557. Thus, under Assumption 3, provided

that the expectation in E6ptDt4pt1R5 É åt4Dt4pt1R557can be evaluated in polynomial time, OPT-7 can besolved using the dynamic programming recursion in(10) in polynomial time.

8. Multiperiod Pricing withCustomer Scheduling

In the earlier sections, we studied the pricing problemof a firm in a setting where the customers choose theearliest time instant with the lowest price to receiveservice. Consider an example where all customersarrive at the initial period and can wait until the endof the horizon to receive service, but service capac-ity is spread over many periods. For any pricing rule,in this example all customers receive service at thesame time instant (with the lowest price). However,this results in inefficient use of capacity and reducedrevenues.Motivated by this example, in this section, we con-

sider the pricing problem in a setting where the firmcan choose how customers should break ties betweentime instants with equal prices. That is, the customersstill receive service at a time instant with the low-est price, but if there are multiple such time instants,the firm schedules how customers should be served.Observe that in the example described above, such ascheduling of customers would avoid the inefficiencycreated by serving all customers at the same period(and wasting the capacity in the remaining periods).Note that it may not always be in the power of

the firm to schedule its customers as described abovesince at the least this requires knowledge of the arrivaland departure times (or deadlines) of each individualcustomer, which is not always available. However, inthis section we establish that if the firm has the nec-essary means to schedule its customers, then it candecide on the optimal prices and schedule by follow-ing a dynamic programming approach similar to theone discussed in §5.In this setting, the firm’s optimization problem can

be formulated as follows:

maxp2PT 1x

TX

t=1

pt

✓ X

i1 j2 itj

xti1 j

◆

s.t.X

i1 j2 itj

xti1 j ct

for all t 2 811 0 0 0 1T 91

xti1 jpt = xt

i1 j mink2 ikj

8pk9

for all t 2 811 0 0 0 1T 91X

t2 itj1 F 4pt 5<1

141É F 4pt55

xti1 j = aij

for all i1 j 2 811 0 0 0 1T 90

(OPT-8)

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Here, xti1 j corresponds to the mass of customers that

belong to population ai1 j and receive service at time t,and P is the set of prices that can be used by the firm.For simplicity, in this section we assume that P is afinite set. The first constraint suggests that the capac-ity constraint is satisfied at all time instants, and thesecond one ensures that if a fraction of population ai1 jis scheduled to receive service at time t (i.e., xt

i1 j > 0),then the price at time t should be equal to the mini-mum price offered from time i up to (and including)time j . The final constraint guarantees that all cus-tomers that belong to some population ai1 j and havevaluation larger than the lowest price between i andj will receive service.It is possible to interpret our paper as a two-

stage game where the firm moves first and selects asequence of prices and the consumers respond in thesecond stage. This two-stage game might have mul-tiple equilibria. One equilibrium is the one we callthe baseline model—consumers break ties in favor ofbuying service early. Another potential equilibrium isthe one where consumers break ties in favor of buyingat the most favorable period for the firm. The secondequilibrium is the one studied in this section.Observe that our baseline model can be viewed as

a version of OPT-8 where the firm is restricted tobreaking ties by assigning the customers to the ear-liest time instant with the lowest price. Hence, if thefirm has the ability to break the ties favorably, it mayobtain a higher revenue. We next show that in thiscase the optimal pricing policy (i.e., the solution ofOPT-8) can be obtained by using a dynamic program-ming approach that generalizes the one used for thesolution of our baseline model. In particular, givena set of prices that can be used by the seller P, thisalgorithm finds the optimal solution of OPT-8 in timepolynomial in T and size of P.To compute the optimal revenue in the OPT-8, sim-

ilar to §5, define ó4i1 j1 p5 as the maximum revenuethat can be obtained from all populations arrivingbetween periods i and j , using prices weakly greaterthan p, with the boundary condition that all cus-tomers who can wait until or after time j receive ser-vice at a later time instant. In our original model, werecursively calculated ó4i1 j1 p5 by finding a period kwhere it gets the lowest price and highest ranking(see §5). In the new model, since a population can besplit into different periods, instead of finding a sin-gle period that takes the lowest price, we find a setof such periods. Before presenting the algorithm, weneed a couple of definitions.Let S 4i1 j5 be the set of all feasible price vectors for

subproblem 4i1 j5. Namely, S 4i1 j5 is the set of all fea-sible solutions of problem OPT-8 where the mass ofall populations except those arriving between time iand j is assumed to be zero. Define S 4i1 j1 p1k5 to be

the set of price vectors in S 4i1 j5 such that all pricesare (weakly) greater than p and k is the latest periodthat has price p (i.e., all periods between time k andj have a price (strictly) larger than p). If no suchfeasible set exists, then let S 4i1 j1 p1k5=ô. Addi-tionally, we define L4i1 j1 p1k5 = 8t ó 9S 2 S 4i1 j1 p1k5and pt4S5= p9, where pt4S5 denotes the price attime t in price vector S. This definition suggests thatL4i1 j1 p1k5 is the set of all periods that take pricep for some price vector in S 4i1 j1 p1k5. Using thesedefinitions, we first provide a characterization of theoptimal pricing policy when we restrict attention toprice vectors in S 4i1 j1 p1k5.

Lemma 4. Suppose S 4i1 j1 p1k5 is nonempty and pis (weakly) larger than the monopoly price pM . Thenthere exists a price vector S? 2 S 4i1 j1 p1k5 such thatpt4S

?5 = p for all t 2 L4i1 j1 p1k5 and S? maximizes therevenue (objective of OPT-8) among all the price vectorsS 4i1 j1 p1k5.

Now, suppose S 4i1 j1p1k5 is nonempty, and S? 2S 4i1 j1p1k5. Observe that if we remove periods inL4i1 j1p1k5 from the set of periods between i and j ,we will end up with some intervals (consecutive timeperiods belong to the same interval and we may haveintervals with only one period). We denote the `thsuch interval by 6I `i1 j1p1k101 I

`i1 j1p1k117 and note that there

are fewer than 4jÉ i5 intervals. We also observe that inS?, only populations that arrive after time 4Ii1 j1p1k10É15and leave before period 4I `i1 j1p1k11+15 receive service atthe periods in 6I `i1 j1p1k101 I

`i1 j1p1k117 because the price at

periods I `i1 j1p1k10É1 and I `i1 j1p1k11+1 is equal to p. Nowwe can state the recursion for computing ó4i1 j1 p5:

ó4i1 j1 p5 = maxk2 i<k<j

maxp2p� p

⇢Éij

k 4p5

+X

`

ó4I `i1 j1p1k10 É 11 I `i1 j1p1k11 + 11p5�1 (11)

where, similar to §5, É is given by

Éij

k 4p5=

8>>>>><

>>>>>:

✓ X

l1m29 t2L4i1 j1p1k51 ltm

alm

◆41É F 4p55p

if L4i1 j1p1k5 6=ô1

Éà if L4i1 j1p1k5=ô0

(12)

Namely, Éij

k 4p5 denotes revenue obtained from theperiods in L4i1 j1p1k5, by setting their prices equal top � p, assuming only the populations between peri-ods i and j are present in the system. Following thesame argument in §5, and using Lemma 4, it is easy toshow that the recursion calculates the optimal valuefor ó4i1 j1 p5. Note that (12) suggests that if L4i1 j1p1k5

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can be found in polynomial time, then É can be com-puted efficiently. Our next result, which is proved inthe online appendix, shows that this is the case.

Lemma 5. Set L4i1 j1p1k5 can be found in polynomialtime.

Observe that we have O4T 2óPó5 subproblemsó4i1 j1 p5. Therefore, by the lemma given above andthe recursion in (11), we immediately obtain the fol-lowing theorem.

Theorem 3. The optimal solution of OPT-8 can befound in polynomial time.

9. Numerical InsightsIn this section, we consider generic instances of thefirm’s pricing problem and obtain qualitative insightsabout the optimal pricing scheme introduced in §2.We first investigate the effect of available capacityon the optimal prices used by the firm and estab-lish that the prices closely track the capacities; e.g.,decreasing capacities induce increasing prices andvice versa. Then, we focus on how patience levelof players affect the outcome and show that as cus-tomers become more patient, the firm offers higherprices that leads to underutilization of capacity andlowered revenues and customer welfare. In addition,we observe that when customers are patient, the firmends up using only a few different prices, therebyconsiderably decreasing the complexity of the pric-ing policy. Finally, we compare the pricing schemeswe introduce in this paper with static pricing thatis commonly employed in practice and establish thatit is possible to significantly improve the revenuesusing our algorithms. We conclude by testing the runtime of our algorithms and showing that for realis-tic scenarios optimal prices can be computed in a fewminutes.Unless noted otherwise, we will always consider

problem instances with 36 time periods, where wefocus on the middle 24 periods to avoid potentialboundary effects.9 We let customer valuations be uni-formly distributed between 0 and 1. We assume thatthere are two types of populations arriving at eachtime period: (i) impatient (or myopic) customers (whoare only interested in purchasing service at the periodthey arrived), and (ii) strategic (or s-patient) cus-tomers, who are willing to wait up to s periods topurchase service. This is captured by setting all ai1 jequal to 0 unless j = i (myopic customers) or j = i+ s(strategic ones). For each i, ai1 i is generated at ran-dom from a uniform distribution between 0 and m1,

9 We note that no significant changes are observed in our results inany of the subsections of this section, when the entire time horizonis used for the analysis.

whereas ai1 i+s is generated from a uniform distribu-tion between 0 and m2, where m1 and m2 are simula-tion parameters.

9.1. Effects of Capacity ConstraintsWe first consider different capacity regimes and tryto understand their impact on pricing rules. We con-sider capacity vectors that satisfy one of the followingcases: (case 1) ct = 1 for all t 2 811 0 0 0 1T 9, i.e., con-stant capacity; (case 2) ct = 1025É0054tÉ15/4T É15 forall t 2 811 0 0 0 1T 9, i.e., capacity decreasing from 1025 to0075 over the horizon; (case 3) ct = 0075+ 0054t É 15/4T É 15 for all t 2 811 0 0 0 1T 9, i.e., capacity increasingfrom 0075 to 1025 over the horizon; (case 4) ct = 1025É4t É 15/4T É 15 for all t T /2 and ct = 0075 + 4t É1 É T /25/4T É 15 for all t > T /2, i.e., capacity firstdecreasing from 1025 to 0075 and then increasing backto the original level, with a midday minimum. Thefirst three cases capture the constant, decreasing, andincreasing capacity settings, respectively. The last onecaptures a phenomenon that is typical in cloud com-puting markets: during peak business hours, part ofthe service capacity is usually unavailable because ofhigh demand on the servers due to other contractsand obligations.We next plot the average price vector over 100 ran-

domly generated problem instances (Figure 2). Weconsider three settings in which (i) the entire popu-lation is impatient (6m11m27 = 66107), (ii) half of thepopulation is patient (6m11m27= 63137), (iii) the entirepopulation is patient (6m11m27= 60167).Figure 2 shows that in all four cases (and for all

population structures), prices track service capacities.Prices are lower when service capacities are higherand vice versa (note that the ranges of the verti-cal axes in Figure 2 are not the same). Case 2 is tosome extent analogous to a typical revenue manage-ment setting. As capacity dwindles toward the endof the horizon, prices rise accordingly. An interest-ing phenomenon occurs as we move from the graphwith impatient customers, on the left, toward the onewith patient customers, on the right: Prices becomeboth smoother and higher as customers become morepatient. We explore this observation more in the nextsubsection.

9.2. Effects of Strategic BehaviorWe now investigate how strategic behavior of cus-tomers affects revenues, capacity usage, and customerwelfare as the parameters s (willingness to wait forstrategic customers) and m2/4m1+m25 (fraction of cus-tomers who are strategic) change. Our results indicatethat as customers become more patient (and strate-gically time their purchases), the monopolist usesfewer different prices that are on average higher. Thisleads to inefficient use of the available capacity and

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Figure 2 Average Price over 24 Time Periods of Interest

0 5 10 15 20 250.60

0.62

0.64

0.66

0.68

0.70

0.72

Time

Ave

rage

pric

e

Case 1Case 2Case 3Case 4

(a) Population is impatient

0 5 10 15 20 250.71

0.72

0.73

0.74

0.75

0.76

0.77

0.78

0.79

0.80

TimeA

vera

ge p

rice

(b) Half of population is patient

0 5 10 15 20 250.75

0.76

0.77

0.78

0.79

0.80

0.81

0.82

0.83

0.84

Time

Ave

rage

pric

e

(c) Entire population is patient

reduces both the revenue of the firm and the welfareof customers.We assume that willingness to wait for strategic

customers s belongs to set 80111 0 0 0 189. The capac-ities are generated independently and uniformlybetween 005 and 105 for each time period. Addi-tionally, the sizes of impatient and patient popula-tions are characterized by the following cases: (case1) 6m11m27 = 65117; (case 2) 6m11m27 = 63137; (case 3)6m11m27= 61157; (case 4) 6m11m27= 60167. That is, case1 captures the scenario, where most of the populationis impatient, whereas, case 4 captures the one whereall buyers are patient. Since parameters are generatedrandomly, we present our results by averaging themover 100 problem instances.We first consider the average price (over the hori-

zon) offered by the monopolist for different cases ands parameters. We also plot average number of dif-ferent prices used by the monopolist for the optimalsolution of the pricing problems. Proposition 2 stated

Figure 3 Average Number of Different Price Levels (a) and Average Prices (b)

0 1 2 3 4 5 6 7 82

4

6

8

10

12

14

16

18(a) (b)

Patience level Patience level

Ave

rage

num

ber

of p

rice

leve

ls

Case 1 = [5, 1]

Case 2 = [3, 3]

Case 3 = [1, 5]

Case 4 = [0, 6]

0 1 2 3 4 5 6 7 80.65

0.70

0.75

0.80

0.85

0.90

Ave

rage

pric

e

that the optimal price in a given time period musteither belong to 8pM119 or be equal to the price of atime period where the capacity is tight. This impliesthat the total number of price levels used might besmaller than the total number of periods. Our sim-ulation in Figure 3(a) shows that this is indeed thecase. In particular, it shows that the average numberof price levels over the 24-period horizon drops bothwhen a higher fraction of the population is willingto wait for service and when the customers who arewilling to wait become more patient. For example, incase 2, while roughly 14 prices are needed when cus-tomers are willing to wait only up to one period, thisnumber drops to 8 if they are willing to wait for twoperiods and 5 if they are willing to wait for three peri-ods. Moreover, this drop in the number of prices ismore significant if a larger proportion of the popula-tion is patient. We note that when an optimal solutionfor the original pricing problem OPT-1 does not exist,Lemma 1 suggests using perturbed prices 8pt + ÖRt9t

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Figure 4 Wasted Capacity, Revenue, and Customer Welfare over 24 Time Periods

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

Patience level Patience level Patience level

Per

cent

age

of w

aste

d ca

paci

ty Case 1 = [5, 1]Case 2 = [3, 3]Case 3 = [1, 5]Case 4 = [0, 6]

(a) % Wasted capacity

0 1 2 3 4 5 6 7 87

9

11

13

15

Tot

al r

even

ue0 1 2 3 4 5 6 7 8

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Tot

al c

onsu

mer

wel

fare

(c) Customer welfare(b) Revenue of the firm

for an arbitrarily small Ö to obtain solutions arbitrarilyclose to the optimal. In such cases, our results indicatethat the firm uses “essentially” few prices. Our firstconclusion in this set of simulations is that patientcustomers lead to fewer price levels.As customers become more patient, the firm

becomes more constrained in the prices it can offer.Consequently, to maintain feasibility with fewerprices and sustain its service guarantees, it may needto increase the prices at some periods. Recall that theprices were already at or above monopoly price tobegin with, so both the firm and the customers loseas the prices go up. Even a small increase in cus-tomer patience causes a fairly large increase in aver-age prices (see Figure 3(b)) and, as expected, the effectis more pronounced when a larger fraction of thepopulation is strategic. Thus, we also conclude thatpatient customers lead to higher prices.We next focus on the effect of these higher prices

on capacity usage, revenue, and customer welfare. Atfirst glance, the presence of customers that are morepatient would seem to lead to better use of resources.After all, high demand and low supply in one periodfollowed by low demand and high supply in the nextperiod could be properly matched if customers arewilling to wait. Indeed this phenomenon does showup in our numerical analysis to a small extent, whencustomers switch from being completely impatient towilling to wait for one period (see cases 1 and 2 inFigure 4(a)). However, we mainly observe the oppo-site effect. As customers become more patient (fors � 1), the firm is forced to use fewer and higherprices. These prices lead to inefficient use of the firm’sresources. The inefficiency is higher when a largerfraction of customers is impatient (Figure 4(a)). Thisphenomenon lowers the firm’s revenue and simulta-neously reduces customer welfare (i.e., total surplusof the customers who purchase the service, where sur-plus is defined as the difference between the value a

customer has for the service and her payment). In Fig-ures 4(b) and 4(c), respectively, we plot the averagerevenue and customer welfare for different cases ands parameters. We establish that as customers becomemore patient, both revenue loss and customer welfarereduction become quite significant. For instance, it canbe seen that for the case m1 = 1, m2 = 5, the revenueand welfare for s = 8 are, respectively, 35% and 75%lower compared to a scenario with s = 0. This phe-nomenon analogous to Braess’s paradox that arises intransportation problems (where opening a new roadmay lead to higher overall congestion in the trans-portation network). However, the mechanism at workhere is different from the one at Braess’s paradox sincein our setting the lower welfare is a consequence ofthe firm’s price adjustment (raising prices to maintainfeasibility of solution).

9.3. Static Pricing and Multiperiod PricingIn this subsection, we compare the performance ofour multiperiod pricing algorithms with the staticpricing that is commonly used in practice. The threescenarios considered here are the following: (case 1)monopolist can use multiple prices and selects the ser-vice time of the customers, as in §8; (case 2) monop-olist can use multiple prices, but customers receiveservice in their preferred (earliest) period, as in ourbase model; and (case 3) monopolist is restricted tousing a single price and customers receive service intheir preferred period, which is the case closest tocurrent practice in the cloud computing industry. Weassume that half the population is impatient and halfis patient with willingness-to-wait s 2 801 0 0 0 189.Figure 5 shows our result. The cases where the firm

is most flexible (case 1) and least flexible (case 3) interms of its pricing (and scheduling) policies, respec-tively, lead to the best and worst cases in terms ofcapacity management and firm profits. The efficiencygains from better pricing strategies are large, and thusthe customers are better off when the firm has the

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Figure 5 Wasted Capacity, Revenue, and Customer Welfare over 24 Time Periods of Interest

most tools in its arsenal. Interestingly, our base algo-rithm that uses only pricing to direct customers toperiods performs well for low values of patience s,but its performance degrades as s increases. If the firmhas some mechanism aside from pricing for schedul-ing customers to periods and customers are fairlypatient, then the pricing scheme with schedulingshould be used because the performance of this alter-native algorithm improves with customer patience.

9.4. Running TimeWe conclude this section by discussing the runningtime of our base algorithm. In this section, we con-sider problem instances where the horizon length isgiven by T 2 8241481969. We still assume that thecapacities are drawn from 600511057 uniformly at ran-dom. We first focus on problems where there are twopopulations impatient and s-patient, and s-patientplayers can wait for s 2 80111 0 0 0 189 time instantsto receive service. Moreover, the size of impatientand patient populations are drawn from 601m17 and601m27, respectively, uniformly at random. We thenconsider a heterogeneous setting, where all populationgroups (characterized by an arrival and departureperiod) are present and their sizes are drawn at ran-dom. For each of these randomly generated prob-lem instances, we run our simulations 100 times andreport the average running time in seconds in Table 1.Running times increase as either s or T grows,

but our results do indicate that in instances of rea-sonable size (up to T = 96), the optimal prices canbe found in a few minutes using a standard laptop

Table 1 Average Running Times for Random Problem Instances (in Seconds)

s= 0 s= 1 s= 2 s= 3 s= 4 s= 5 s= 6 s= 7 s= 8 Heterogeneous

T = 24 0.99 3.55 5.88 8.48 10.24 13.23 14.23 18.77 20.10 32.74T = 48 14.9 51.4 85.0 105.6 141.8 167.3 168.4 165.0 190.7 233.6T = 96 193 655 868 1,102 1,232 1,304 1,501 1,489 1,550 1,802

(with 3006 GHz Intel Core 2 Duo processor and 4 GB1,067 MHz DDR3 memory). Therefore, the algorithmis sufficiently efficient to be implemented in practice.

10. ConclusionsWe study a service firm’s multiperiod pricing prob-lem in the presence of time-varying capacities andheterogeneous customers that are strategic withrespect to their purchasing decisions. A distinct fea-ture of our model is the service guarantees pro-vided by the firm that ensure that any customerwilling to pay the announced service price will beable to receive service. Such guarantees are quiteappealing to customers because they allow them toignore rationing risk, tremendously simplifying theconsumers’ decision-making process. However, pro-viding such guarantees requires the firm to use pricesthat ensure the firm has sufficient service capacityin every period. We propose an efficient algorithmto compute revenue-maximizing prices while main-taining service guarantees. We show, via numericalsimulation, that in a typical instance the optimal pric-ing policy involves only a few prices and it enablesthe firm to obtain significantly more revenues thanthe static pricing schemes that are common in prac-tice. We show that such algorithms and insightsgeneralize to complex versions of the problem withrandom arrivals, departures and capacity levels, pro-duction costs, and customer valuations that dependon arrival and departure periods. We also constructan algorithm that the firm can use if it is capableof scheduling customer service times in addition to

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using dynamic pricing and demonstrate numericallythat such an algorithm could yield even higher rev-enues and better resource utilization for the firm.

AcknowledgmentsThe authors thank Ishai Menache (Microsoft Research),Georgia Perakis (MIT), Rakesh Vohra (University ofPennsylvania), and the anonymous associate editor and ref-erees for their fruitful comments and suggestions.

Appendix

Proof of Proposition 2. Note that if for a ranking R,we have êt4R5= 0, then, without loss of generality, we canlet pt = 1. Now, by Proposition 1, we can assume that 4p1R5is an optimal solution of OPT-2 such that pM pt 1 andpt = 1 if êt4R541 É F 4pt55 = 0 for all t 2 811 0 0 0 1T 9. We willprove that 4p1R5 is such that pt necessarily satisfies one ofthe conditions 1–3 of the proposition for all t 2 811 0 0 0 1T 9. Bycontradiction, assume that for 4p1R5 none of the conditions1–3 hold at time t0. Since conditions 1–2 do not hold 1 >pt0 > pM . Note that pt0 6= 1 implies that êt0

4R541É F 4pt0 55 6= 0,and hence êt0

4R5> 0. Since condition 3 does not hold, for t 2S

4= 8t 2 811 0 0 0 1T 9 ó pt = pt09 we have that ct is not tight; i.e.,

êt4R541É F 4pt55< ct for all t 2 S0 (13)

Let Ñ be a constant such that

Ñ=8<

:pt0 É pM if pt0 pk for all k 2 811 0 0 0 1T 91pt0 É max

8t1 ópt1<pt0 9pt1 otherwise.

Consider the price vector p, for which pk = pk for k y S, andpk = pk É Ö otherwise, for some 0 < Ö < Ñ. It follows fromthe definition of Ñ that if pi pj for some i1 j 2 811 0 0 0 1T 9,then pi pj . Hence, the price vector p is also consistent withranking R. Moreover, since 41É F 4p55 is a continuous func-tion, by (13) we conclude that Ö can be chosen small enoughto guarantee that for time periods t 2 S, êt4R541ÉF 4pt55< ct .Since 4p1R5 is feasible and pt = pt for t y S, it also followsthat for t y S, we have êt4R541ÉF 4pt55= êt4R541ÉF 4pt55 ct .Consequently, 4p1R5 is feasible in OPT-2. The definition ofÑ also suggests that pM < pt < pt = pt0 for t 2 S. It follows bythe definition of pM and the unimodality of the uncapaci-tated revenue function that pt41É F 4pt55 < pt41É F 4pt55 fort 2 S. Thus, since êt0

4R5 6= 0 we conclude that the revenueobtained from time periods t 2 S increases under p; i.e.,

X

t2Spt41É F 4pt55êt4R5>

X

t2Spt41É F 4pt55êt4R50 (14)

Since pt = pt for t y S, it also follows thatP

tyS pt41 ÉF 4pt55êt4R5 =

PtyS pt41 É F 4pt55êt4R5. Hence, we conclude

that the overall revenue improves when 4p1R5 is used.Therefore, we reach a contradiction and 4p1R5 has to satisfyone of the conditions 1–3 of the proposition.

Proof of Theorem 1. We first describe how the opti-mal prices and ranking in OPT-2 are obtained, and then weconsider the computational complexity of the solution. Asexplained in the text, given the set L, the optimal solution ofOPT-2 can be obtained by solving OPT-3. The solution of thelatter problem is identical to that of (6), with i= 0, j = T +1,

p= 0, and hence the optimal value is equal to ó401T +1105.Given ó4i1 j1 p5, for 0 i j T + 1, one can construct theoptimal sequence of prices in this problem using the recur-sion in (7): We say that k is the solution for ó4i1 j1 p5 if theright-hand side of Equation (7) takes its maximum at k andk is the earliest time period that achieves the maximum. Let4k⇤1pk⇤ 5 be the optimal solution of ó401T +1105 in (7). Thenthe price of time period k⇤ in the optimal solution of (6) ispk⇤ , and prices for time periods earlier and later than k⇤ canbe obtained by solving for the prices in the subproblemsó401k⇤1pk⇤ 5 and ó4k⇤1T + 11pk⇤ 5.

We assume that at each step the leftmost subproblem issolved first. We say that the time period k⇤, which solves theith subproblem, has priority i (hence the time period thatsolves ó401T + 1105 has priority 1). Using these prioritiestogether with prices, we next construct the ranking vector(consistent with the already obtained prices) that appearsin the solution of OPT-3 (or equivalently to (7) with i = 0,j = T + 1, p = 0). Consider time periods k1 and k2. If pk1 6=pk2 , it is clear how to rank them: the lower price will have asmaller rank. Now suppose pk1 = pk2 ; then the time periodwith lower priority receives lower ranking. Note that underthis ranking, the ranking vector is consistent with prices.Moreover, when there are multiple time periods with thesame price, the time period that has lower ranking is the onethat is used by the algorithm to solve an earlier subproblem.This implies that the ranking is consistent with the timeperiod each population receives service in the solution ofthe recursion (7).

We next characterize the computational complexity ofproviding a solution to OPT-2. Note that by Proposition 3,there exists an optimal solution for OPT-2 with prices thatbelong to set L. It can be seen from (5) that to computethe prices in this set we need quantities of the form zijk =Pk

t1=i

Pjt2=k at11t2 for all i j k. Note that there are O4T 35

values zijk can take, and each value takes at most O4T 25to compute. Thus, all values of zijk and the set L can becomputed in O4T 55 time (the computation time can be fur-ther reduced by exploiting the relation between differentvalues of zijk ; this is omitted because it does not affect ourfinal complexity result). Thus, in O4T 55 time we can reduceOPT-2 to OPT-3. We characterize the computational com-plexity of the latter problem.

Observe that the algorithm relies on characterizingó4i1 j1p5 for all time periods i j and p 2 L. Since cardinal-ity of L is O4T 35, there are O4T 55 values of ó4i1 j1p5 thatneed to be characterized. These can be computed using thecondition ó4i1 j1 p5= 0 if i+ 1 � j É 1 and the recursion in(7). At each step of the recursion, there are O4T 5 differentvalues k can take. On the other hand, for a given value ofk, the corresponding optimal pk can be computed in O415:Since for all p 2 L we have p � pM , it follows that É ij

k 4p5 isdecreasing in p, provided that p 2 L. Moreover, ó4i1 j1p5 isalso decreasing in p for all i1 j since larger p correspondsto tighter constraints in (6). Thus, the pk that solves (7)is the smallest p � pM that makes the capacity constraintfeasible. Therefore, it follows that pk =max8pM1 F É141É ck/Pk

l=i+1PjÉ1

m=k alm59, where the latter is the price that makesthe capacity at time k tight. Since by construction both theseprices belong to L, and elements of L were computed ear-lier, it follows that given k, pk can be constructed in O415.

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Thus, we conclude that each step of the recursion in (7)can be computed in O4T 5. Thus, the overall complexity ofcomputing all ó4i1 j1p5 is O4T 65. Finally, given all valuesof ó4i1 j1p5, the construction of the prices that solve (6)takes O4T 25 following the procedure described in the begin-ning of the proof: to solve for each pk, an instance of therecursion (7) needs to be solved. This takes O4T 5 time, andthere are O4T 5 prices to be solved for. Similarly constructingpriorities and rankings consistent with these prices takesanother O4T 5. Thus, the overall complexity of the algorithmis O4T 5 + T 6 + T 2 + T 5=O4T 65.

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