optimal dynamic pricing of perishable items by a...

Optimal Dynamic Pricing of Perishable Items by aMonopolist Facing Strategic Consumers

Yuri Levin, Jeff McGill, Mikhail NediakSchool of Business, Queen’s University, Kingston, ON, Canada K7L 3N6, [email protected], [email protected],

[email protected]

We introduce a dynamic pricing model for a monopolistic company selling a perishable product to a finite population ofstrategic consumers (customers who are aware that pricing is dynamic and may time their purchases strategically). This

problem is modeled as a stochastic dynamic game in which the company’s objective is to maximize total expected revenues,and each customer maximizes the expected present value of utility. We prove the existence of a unique subgame-perfectequilibrium pricing policy, provide equilibrium optimality conditions for both customer and seller, and prove monotonicityresults for special cases. We demonstrate through numerical examples that a company that ignores strategic consumerbehavior may receive much lower total revenues than one that uses the strategic equilibrium pricing policy. We also showthat, when the initial capacity is a decision variable, it can be used together with the appropriate pricing policy to effectivelyreduce the impact of strategic consumer behavior. The proposed model is computationally tractable for problems of realis-tic size.

Key words: customer behavior; dynamic pricing; strategic consumers; stochastic gamesHistory: Received: February 2008; Accepted: December 2008 by Costis Maglaras; after 3 revisions.

1. IntroductionThe recent growth in internet-based marketing hasstimulated widespread experimentation with dynamicpricing—the practice of varying prices for the samegoods over time or across customer classes in an at-tempt to increase total revenues for the seller. In thispaper, we examine a particular version of dynamicpricing that has been widely applied to sales ofperishable items over a finite time period. Familiarexamples include passenger transportation tickets,hotel bookings, automobile rentals, end-of-seasonstyle goods, and expiring models of electronic equip-ment or other high-value items.

Particularly notable successes of dynamic pricinghave been reported in the airline industry, which has,perhaps, the longest history of controlled dynamicpricing through their revenue management (RM) sys-tems (traditional RM is more properly viewed asinventory control across price classes; however, thedistinction between RM and dynamic pricing isbecoming increasingly blurred as the use of onlinebooking expands). There is an extensive literature onRM and related practices. For a review of this area, seePhillips (2005) and Talluri and van Ryzin (2004).

In dynamic pricing, it is crucial to take into accountkey characteristics of the market such as consumerbehavior and the availabilities of goods and services.Classical dynamic pricing models often assume thatconsumer behavior is myopic – a consumer makes apurchase as soon as the price drops below his/her

valuation for the product while ignoring future prod-uct availability and purchasing opportunities. Thisassumption leads to tractable models, significantlysimplifies analysis, and can lead to attractive struc-tural properties for pricing policies. But many real-lifeconsumers are strategic; that is, they attempt to ‘‘getmore for less’’ by delaying purchase in anticipation ofprice reductions. When supplies are limited, suchstrategic consumers must also take into account prod-uct availability and the strategic behavior of otherconsumers. Thus, a comprehensive model of strategicconsumer behavior needs to capture interactions be-tween the seller and the consumers as well as amongthe consumers.

1.1. Survey of Existing LiteratureDynamic pricing models for an infinite population ofmyopic customers include those of Feng and Xiao(2000), Gallego and van Ryzin (1994), and Zhao andZheng (2000). In their book, Talluri and van Ryzin(2004) discuss myopic vs. strategic consumer models(Section 5.1.4.1). They point out that the myopic as-sumption provides an acceptable approximationwhen customers are relatively spontaneous in mak-ing decisions (typical with low-price products) orwhen they do not have enough time or information tobehave strategically. However, for more expensiveand durable products and when information is avail-able, this assumption becomes less realistic. Theyargue that failure to account for strategic customer

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PRODUCTION AND OPERATIONS MANAGEMENTVol. 19, No. 1, January–February 2010, pp. 40–60ISSN 1059-1478|EISSN 1937-5956|10|1901|0040

POMSDOI 10.3401/poms.1080.01046

r 2009 Production and Operations Management Society

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behavior could significantly reduce expected revenuesfrom dynamic pricing.

The literature on consumer behavior issues in RMfor perishable products is growing rapidly; for a sur-vey see Shen and Su (2007). Here, we only reviewmodels that are the most relevant in terms of theirassumptions and conclusions relative to the presentwork. In general terms, any analysis of individualstrategic consumer behavior requires simplifying as-sumptions. For example, many studies in this areaassume deterministic demand, and, even in the morerealistic stochastic case, most articles restrict thedynamics of price changes to markdowns or considerproblems with only two decision periods. Each ofthese make additional specific assumptions on con-sumer behavior.

An early deterministic model of dynamic pricingwith strategic consumers is presented in Besanko andWinston (1990). They show that the subgame-perfectequilibrium pricing policy for the firm is to lowerprices over time in a manner similar to price-skimming.The selling capacity in this model is unlimited. Su(2007) considers a deterministic demand model withcustomers partitioned into four segments according totheir valuation level and waiting costs, and derivesconditions relating to when the seller should usemarkdowns or markups. The company controls boththe price and the fill rate in this work.

Aviv and Pazgal (2008) study the optimal pricingof fashion-like seasonal goods in the presence of stra-tegic customers who arrive according to a Poisson’sprocess but have deterministically declining valua-tions over time. This work considers subgame-perfectNash equilibria between a seller and strategic consum-ers for the cases of inventory-contingent discountingstrategies and announced fixed discount strategies. Thepaper finds that the seller cannot effectively avoid theadverse impact of strategic consumer behavior evenunder low levels of initial inventory, and that fixeddiscounting strategies may outperform contingent pric-ing strategies. Elmaghraby et al. (2008) study optimalmarkdown mechanisms in the presence of strategicconsumers who have fixed valuation throughout theselling season.

A number of authors consider fully dynamic pric-ing but under specific behavioral assumptions. Xuand Hopp (2004) study a continuous-time modelwith strategic customers who arrive according to aPoisson process and whose price sensitivity varieswith time. In this model, customers commit to a pur-chasing time which is determined from an open-loop(state independent) decision process. One of theirresults is that the optimal prices follow a submartin-gale if the price sensitivity is constant or decreas-ing, and supermartingale if the price sensitivity isincreasing.

Papers which study two-period pricing modelswith strategic consumers include Liu and van Ryzin(2008) who use quantity decisions (rather than pric-ing) as means to induce early purchases. Zhang andCooper (2008) also consider the effect of strategiccustomers in a two-period model with regular andclearance prices and an option of restricting productavailability in the second period. Cachon andSwinney (2007) study the value of an additional prod-uct procurement, termed ‘‘quick response,’’ in a two-period model where the second period price is alwaysa markdown. The model considers three segments ofcustomers: myopic, bargain hunting (those who onlypurchase in the second period), and strategic. Thepaper finds that ‘‘quick response’’ offers more value inthe presence of strategic consumers.

A two-channel model is studied by Caldentey andVulcano (2007) who consider a setting in which stra-tegic consumers, who arrive according to a Poissonprocess, decide between buying immediately at a(fixed) posted price and buying later by entering theirbids to a sealed-bid multi-unit uniform price auction.Two cases are studied: the single-auction channel casewhere the company only manages the auction, andthe dual-channel case where the company managesboth the auction and the fixed-price channel. Theauthors discuss the equilibrium and a simple, asymp-totically optimal, heuristic rule for its approximation.

Several articles study the effects of customer re-sponse to RM over multiple selling seasons; see, forexample, Ovchinnikov and Milner (2007). In thiswork, the market is treated in an aggregated form,and some fraction of customers learns to expect end-of-season discounts based on previous interactionswith the seller. Within each season, given the full andsale prices, the seller selects the number of unitsavailable at the end-of-season discount. The articleprovides analytical results for the explicit multi-season model. Gallego et al. (2008) investigate a sim-ilar class of models with multiple selling seasons andtwo-period pricing within each season. Customersupdate their expectations about the probability ofacquiring an item at a discount based on previousinteraction with the seller. Both of these papers showthe existence of complex dynamic patterns of con-sumer–seller interactions.

1.2. Inherently Stochastic MarketsIn a deterministic setting, the company does notexperience uncertainty in demand and, given itspricing/sales policy, can always accurately predictconsumer behavior and evaluate the performance ofthe policy. However, most markets in which dynamicpricing is employed are inherently stochastic in thesense that outcomes (sales paths) are not whollydetermined by company pricing decisions but also

Levin, McGill, and Nediak: Dynamic Pricing with Strategic ConsumersProduction and Operations Management 19(1), pp. 40–60, r 2009 Production and Operations Management Society 41

depend on other random events. In our view, cus-tomer valuations or propensities to consume can beuncertain up to the very time of consumption, partic-ularly for advance purchases. Appropriate examplesinclude discretionary hotel reservations, vacationpackages, and sports and entertainment events whosevalue may depend on uncertain weather and timeavailability for the consumer. Uncertainty in valua-tions is also present for a wide variety of services thatneed to be reserved in advance (housekeeping, gar-dening, snow removal, etc.).

An intention to purchase at a given time or pricepoint may not result in a purchase. For example, acustomer who feels that a price is acceptable may notact immediately for a number of reasons, includingprocrastination, congestion delay, or terms and con-ditions of purchase. Also, many customers exhibit‘‘wavering’’ around a decision, particularly for dis-cretionary purchases – deciding one moment that theywill purchase and the next that they really shouldn’t.It can also take more than a single inquiry before aconsumer settles on a purchase. (Uncertainty in indi-vidual choices, even under seemingly identicalconditions, has been reported in empirical studies[see, e.g., Tversky 1972]. Moreover, Xie and Shuganindicate that ‘‘buyers are nearly always uncertainabout their future valuations for most services’’ [Xieand Shugan 2001, p. 219].) All of these behaviors aredifficult to capture with fixed valuation assumptions.

Demand uncertainty often brings product availabil-ity into consideration for both the company and itscustomers. As a result, the pricing policy and con-sumer response will, in general, depend on theremaining capacity level and remaining time. Finally,implementation of a deterministic policy assumes thatthe company can credibly commit to a price path, anassumption that does not always hold in practice. Theissue of credible price commitment is not importantwhen consumers are myopic and, in that case, adeterministic policy can be asymptotically optimalGallego and van Ryzin (1994), However, as shown byAviv and Pazgal (2008), this becomes an importantissue in the presence of strategic consumers.

In this paper, we present a dynamic pricing modelfor a finite population of strategic consumers in amarket which is inherently stochastic in its operation.The seller is a monopolist firm that contingently pricesa fixed stock of items over a finite time horizon, andthe consumers exercise choice based on the strategicfactors discussed above. Our consumer choice modelis similar in spirit to that of Besanko and Winston(1990), but we extend that work by taking demanduncertainty into consideration and allowing for ca-pacity restrictions. In contrast to the deterministicmodel of Su (2007), which also treats capacity restric-tions, we do not assume that the firm has explicit

control of the fill rate. Dynamic explicit control ofthe fill rate may not be easy to implement since se-lective rejection of purchase attempts over time canalienate customers, and it may be hard to keep a cus-tomer from making another purchase attempt if theoriginal attempt is rejected. Our model also differsfrom that of Su (2007) in the description of strategicconsumer behavior, since the latter introduces explicitwaiting costs for the consumers. In contrast to manyprior stochastic models, we do not limit the dynamicsof prices to two time periods, and we do not limit themodel to decreasing price paths (markdowns). Whilemarkdown policies are appropriate in some retail set-tings, they are not universally applicable. These twoaspects allow us to study much richer dynamics ofprice and sales processes than more restricted modelspermit.

In models with deterministic valuations (or randomvaluations that are realized at the beginning of thesales season and remain fixed thereafter), homoge-neous consumers will all complete purchasessimultaneously when ‘‘conditions are right.’’ This, ofcourse, does not accord with the behavior of realmarkets. Firms are much more likely to see relativeincreases or decreases in the pace of sales as prices andinventories change. This motivates a model in whichindividual consumers respond to prices and othermarket conditions by controlling the intensity of theirdemand processes. In this way, actual completionsof sales will be distributed over time stochastically,and consumers will have to take this uncertainty intoaccount in their decision making. Such a strategic re-sponse of consumers in the form of shopping intensityis a distinctive feature of this paper.

This approach to consumer behavior removes theneed for explicit rationing controls for the firm be-cause, much like in real life, random rationing ofthe product to consumers occurs as a natural conse-quence of market operation. This model also obviatesthe assumption that the choice behavior of consumersis predetermined at the beginning of the sales season.

Such generality comes with a cost – the model re-quires an assumption that consumers re-sample theirvaluations at each decision point, independently ofprevious valuations. While this can be viewed as alimiting model for consumer ‘‘wavering,’’ a more sat-isfactory model would allow dependency across time;for example, with a random-walk valuation process.This is an interesting topic for future research.

1.3. Chief ContributionsThe model described here is a stochastic dynamicgame in which the company controls prices contin-gent on time and present market conditions tomaximize total expected revenues, and each customercontrols demand intensity to maximize the expected

Levin, McGill, and Nediak: Dynamic Pricing with Strategic Consumers42 Production and Operations Management 19(1), pp. 40–60, r 2009 Production and Operations Management Society

present value of utility. In the presence of imperfectinformation, such stochastic games are notoriouslydifficult to analyze. Therefore, we make some simpli-fying assumptions which guarantee that this is aperfect information game, and its participants can ra-tionally anticipate the behavior of others. In this aspect,our game structure is similar to that of Besanko andWinston (1990).

We prove that a unique subgame-perfect equilib-rium pricing policy exists, and that the ‘‘strategicity’’of consumers affects the choice of an optimal pricingpolicy. We also provide equilibrium optimality con-ditions for both consumer and seller, and studyproperties of the equilibrium policy. The equilibriumobtained here is a direct generalization of the optimalpricing policy for myopic consumers which would beobtained in a discrete-time, finite population versionof the classical model of Gallego and van Ryzin (1994).

Practical insights derived from this work relate tothe structure of pricing policies and their relation tocapacity decisions. We show that, when consumer ra-tionality is limited, the pricing policy exhibits naturalmonotonic properties: price decreases as remainingcapacity increases or remaining time decreases. Suchproperties are familiar from the analysis of Gallegoand van Ryzin (1994) in the myopic case. When con-sumers are fully rational and strategic, prices decreasemonotonically in time in the special case of highproduct supply. This result is also in agreement withBesanko and Winston (1990) in the deterministic case.However, these authors also find that price approachesmarginal cost as the discount factor approaches one,whereas in our case price remains strictly boundedaway from marginal cost. This difference arises be-cause of the stochastic nature of the sales process.

We also demonstrate that monotonic properties donot hold in general. This contrasts with well-knownresults in the theory of dynamic pricing for the my-opic case. In practical terms, if strategic customers areonly partially rational and are not aware of the pres-ent state of the system or cannot take it into account,then a familiar monotonic property holds: price in-creases as inventory decreases. In contrast, if the cus-tomers are rational and aware of the current state ofthe system, then the company may have to counter-act their strategic behavior with policies that deviatefrom monotonicity. For example, there must be acredible threat to increase prices even in the absenceof sales.

Numerical simulations of pricing policies in thegeneral case also indicate that price paths are mark-edly different for the cases of myopic and strategicconsumers. In the myopic case, the average level ofprices starts high and decreases over time, whereasthe average level is more constant in the strategic case.Also, for strategic consumers, pricing flexibility de-

creases significantly with an increase in supply levelwhen compared with the myopic case.

The model described here can be used as a bench-mark for the importance of strategic consumerbehavior. For example, it can be used to estimate thelosses a company can incur due to strategic consumerbehavior, both when this behavior is ignored andwhen it is optimally anticipated in the company’spricing policy. Numerical examples demonstrate thata company that ignores strategic consumer behaviormay receive much lower total revenues than one thatuses the strategic equilibrium pricing policy, and thatlosses are much more profound when supply levelsare high.

While the focus of our paper is on contingent pric-ing policies, the model can be appropriately modifiedto evaluate consumer response to pre-announced, de-terministic, pricing policies. We numerically evaluatethe effects of two-period pre-announced pricing pol-icies in comparison with fully dynamic, contingent,ones and find that the latter perform better when thesupply level is low and consumer behavior is closer tomyopic. However, contingent policies may underper-form when the capacity level is high. This agrees withthe findings of Aviv and Pazgal (2008) obtained underdifferent modelling assumptions.

Finally, the expected revenue as a function of initialcapacity provided by our model can also be used in anewsvendor-type formulation in which the initialprocurement decision is followed by dynamic pricingin a strategic consumer market. This is made possibleby the stochastic market feature of the model. Weshow that, when the initial capacity is a decision vari-able, its proper use together with dynamic pricing iscrucial for effective reduction of the effects of strategicconsumer behavior. In short, dynamic pricing may notbe able to compensate for inappropriate capacity de-cisions. Also, the appropriate capacity level is lowerfor the case of strategic consumers. On the computa-tional side, the proposed model is tractable forrealistic-size problems. In summary, the chief contri-butions of this paper are:

� pricing policies (based on a game between a mo-nopolist and strategic consumers) that generalizeclassical optimal pricing policies for the myopiccase in an unrestricted dynamic setting,

� monotonic properties of pricing policies in thepresence of strategic consumers, and

� an analysis of relations between and effectsof appropriate/inappropriate procurement deci-sions and dynamic pricing policies in the pres-ence of strategic consumers.

These results are restricted to the case of monopo-listic firms that possess knowledge of consumer


valuation distributions. Such results are much harderto obtain in more complex market settings. In a com-panion paper, Levina et al. (2009), we relax theassumption of knowledge of the valuation distribu-tion and allow for learning of consumer behavior asthe sales season unfolds. That model cannot explicitlycapture the dynamic aspect of the game between thecompany and the consumers. A game-theoretic modelof interactions between the company and strategicconsumers in the presence of consumer and companylearning is an interesting direction for future research.In Levin et al. (2009), we examine dynamic pricing inan oligopoly with strategic consumers in multiplemarket segments. The complexity of the models inboth of these papers does not permit detailed studyof monotonic properties of pricing policies and theimportance of capacity decisions.

The organization of this paper is as follows. Wepresent the notation and game-theoretic model inSection 2 and show existence of the unique subgame-perfect equilibrium in Section 3. We explore monoto-nicity results of the equilibrium pricing policies inSection 4, and show the effect of strategic consumerbehavior on the performance of pricing policies andcapacity decisions in Section 5. We conclude andindicate several directions for further research inSection 6. Additional discussions of the modeling as-sumptions, a glossary of notation, and mathematicalproofs are provided in the supporting informationAppendix S1.

2. Model DescriptionWe describe the model in three stages. First, in Section2.1, we specify general model assumptions with par-ticular attention to the demand process, which ischaracterized by the shopping intensity of consumers.In Section 2.2, we describe the consumer choice modeland define the stochastic dynamic game that capturescompany–consumer interactions. Finally, in Section2.3, we derive the objective functions for consumersand the company and the equilibrium response forconsumers.

2.1. General Modeling AssumptionsConsider a product with limited availability sold bya monopolistic company over a finite planning hori-zon [0, T]. A common model of demand from apopulation of consumers is a counting process withintensity generally dependent on time and price (see,e.g., Gallego and van Ryzin 1994). Since we want todescribe behavior of individual consumers, this as-sumption takes the following form:

Demand process:

� The population demand process is a sum ofcounting demand processes originating from a

finite number of individual consumers. The pop-ulation demand intensity is the sum of consumerdemand intensities.

� Consumers control the intensity of their demandprocesses (which we call shopping intensity) inresponse to price and other market conditions.

� There exists a common upper bound �l for theshopping intensity of each consumer.

In this interpretation of demand, consumers cannotcontrol the precise timing of their purchases, butrespond to more favorable prices and other marketconditions by increased eagerness to purchase, whichtranslates to a higher expected frequency of pur-chases. This assumption is motivated by two practicalconsiderations. First, in a real marketplace, all inter-ested consumers cannot purchase at exactly the sametime because both the capacity of the purchasingchannel and the timing of access by consumers to thatchannel are constrained. Second, as pointed out bySu (2006), real-world consumers have a tendency toprocrastinate. The total demand intensity is boundedin practice: even if the product is available for free,there will be a maximum number of customers perunit of time who can purchase the product. Since eachconsumer contributes a fraction of the total intensity,it is reasonable that the individual intensity is alsobounded. The value �l captures transaction uncertain-ties in the market as well as consumer tendencytoward procrastination, and has a precise interpreta-tion as the reciprocal of the expected time for an eagercustomer to purchase the product. It can also be in-terpreted as the intensity of shopping opportunities,and the quantity �lT is the expected number of shop-ping opportunities for a customer who is eagerthroughout the sales season.

Finally, to simplify the model we take the approachemployed in Talluri and van Ryzin (2004) and dis-cretize the time into short time steps which we calldecision periods. The number of decision periods issufficiently large that any continuous-time countingprocess in the model can be well approximated by itsdiscrete-time analogue. Under such a choice of dis-cretization, the probability of more than one eventoccurring in a decision period has to be relativelysmall, and we can assume that at most one event inany of the processes can occur per decision period.For simplicity of notation, we assume that each deci-sion period has a length of exactly one time unit. Thenthe shopping intensity of a customer is equivalent tothe probability that this particular customer purchasesan item in the next decision period. Consequently, thedemand intensity of the entire market is equivalent tothe probability that any consumer purchases an itemin the given decision period. The purchase probabil-ities retain the additive property of intensities: the


probability of a purchase occurring is the sum of pur-chase probabilities of individual customers. Wesummarize this as:

Representation of intensity as probability: Consumershopping intensity decisions are expressed as pur-chase probabilities in the decision periods of thediscrete-time model.

Consumer behavior, discussed in detail in the nextsection, can be summarized as follows. In each pe-riod t, each customer draws a valuation according to agiven distribution. Draws are independent across pe-riods and are exchangeable random variables acrosscustomers. Each individual customer then uses hisdraw to select a shopping intensity from the interval½0; �l�: To ensure that our time discretization is suffi-ciently fine, we must select the time step (time units)so that the upper bound �l on demand intensity is lessthan the reciprocal of the total maximum market size.The individual customer’s decision will generally alsodepend upon the posted price, the time remaining inthe horizon, the remaining inventory, and the cus-tomer’s assessment of the expected value of delayingpurchase. Once all customers have simultaneouslyselected their probabilities, then at most one customerwill, conditionally independent of the past, make apurchase in period t according to the selected prob-abilities.

Next, we introduce notation, describe how consum-ers decide between shopping now and waiting – theconsumer choice model – and how the company caninfluence this behavior.

2.2. The GameThe goal of our model is to capture the intertemporalbehavior of customers who strategically attempt totime their purchase to periods with lower price. Theyindirectly control timing of their purchases by mod-erating their shopping intensity relative to that ofmyopic customers. Customer shopping intensity isselected dynamically in response to the company’spricing policy, which is also dynamic. The resultingmodel belongs to the general class of stochasticdynamic games.

For convenience, the notation introduced here andthroughout the paper is summarized in a glossary inAppendix S1. Recall that there are T decision periodsin our model. The initial inventory of the product attime 0 is Y, with no replenishment possible during theplanning period. At time T, the product expires andall unsold items are lost.

The product is offered to a finite number N of cus-tomers, each of whom can purchase at most one item.To ensure that probabilities in our discrete-time modeladequately represent intensities of the demand pro-cesses, we choose time discretization so that N�l ismuch o1. Without loss of generality, we assume that

the initial number of customers N � Y (the case ofNoY can be reduced to N 5 Y since any excess itemscannot be sold and should be disregarded. The case ofN 5 Y is appropriate for a company offering a make-to-order product to strategic customers). All custom-ers are present from the beginning and can monitorthe market as well as make purchases during theentire planning horizon.

With regard to the information structure of thegame, we assume by default:

Perfect market information: All game participantshave perfect information about the market includingits parametric, distributional, and dynamic character-istics (except for random valuation realizations).

A perfect information assumption implies that theremaining inventory level y � Y and the number ofcustomers who have not yet acquired the product,n � N, are public information. Because the customerpopulation is homogeneous, it is unimportant whichparticular customer has not yet acquired the product,and the state of the system in decision periodtAf0, 1, . . ., Tg can be described by the pair y, n. How-ever, since the current inventory level can berecovered from n as y 5 Y� (N� n), we can drop itfrom the state description. A transition from state n ton� 1 represents a sale of one unit of the product toone of the customers. Given maximum shopping in-tensity �l, the value of n, together with the remainingtime, allows us to quantify the risk that a consumerwill fail to complete the purchase (rationing risk). Thisis how a randomized sales process can replace explicitrationing controls considered in some models. Weview this as an advantage of the model since, aspointed out in the introduction, explicit rationingcontrols may be hard to implement when the salesprocess is truly dynamic.

The key modeling issue is how strategic consumerscompare a current purchase with a possible purchasein the future. One of the common approaches is basedon random utility models (see Section 3.2 of Andersonet al. 1992). The use of such models is pervasive ineconomics, marketing, and other fields. In a dynamicsetting under linear random utility, a consumer valuesa purchase at price p at time tAf0, 1, . . ., T� 1g asat1et� p, where at is a known constant describingconsumer perception of quality or value of the prod-uct, and et is a random variable with mean zero(usually assumed to be continuously distributed). Thequantity p0t ¼ at þ et formally corresponds to valua-tion of the product at time t. One modeling optionwould be to assume that for each consumer the val-uation is drawn from some known distribution at thebeginning and then remains constant throughoutthe selling season. However, in reality, individual con-sumer valuations cannot be measured precisely andare unlikely to be certain. In general, an individual


consumer valuation can be modelled over time as astochastic process. For example, one may assume itis a continuous-state random walk (Markov) process.By changing the degree of statistical dependence be-tween consecutive states of this process, we get arange of possible models for valuations. At one end ofthis spectrum, we have a fixed (deterministic) model –an assumption made, for example, in deterministicmodels of Besanko and Winston (1990), Su (2007),where it is completely appropriate. In a model likeours, a finite-population model with a stochastic salesprocess, the fixed valuation model will lead to a dis-continuous total demand intensity response to postedprices (intuitively, as the price increases, customerswill switch from shopping to non-shopping one byone based on their fixed utility thresholds). Moreover,as the company prices its product, it will also haveto learn the unknown and uncertain demand inten-sity profile. Such a problem falls into the class ofincomplete information control problems. In such aformulation, the state description would have to in-clude the histories of price and sales processes, whichwould make the resulting control problem verydifficult. Consumer evaluation of a possible futurepurchase would have to depend on initial valuation aswell as price and sales history, resulting in a majorcomplication for a strategic consumer behavior model.An intermediate level of statistical dependence ofconsecutive valuations is more realistic, but it does notresolve the issue of dealing with the demand-learningproblem. At the other end of the spectrum – completeindependence between valuations at different time –the demand-learning problem can be avoided if thevaluation distribution is known. This leads to the fol-lowing:

Strategic choice model: Suppose that the current de-cision period is t, and the price announced by thecompany is p; then,

� The consumer values the purchase that can occurin the current decision period as u(p0 � p), wherep0 is a possible valuation level and u( � ) is astrictly increasing utility function with an inverseu� 1( � ). An arbitrary utility reference value ischosen so that u(0) 5 0.

� Consumer valuations are distributed according toFt(p). Valuations of different customers at timet are exchangeable random variables.

� Valuations are independent random variables fordifferent t.

� Each consumer’s value of future purchase oppor-tunities as perceived in the next decision periodis the same known function of time and state,U(t11, n). Consumers may assess this valueexactly or heuristically depending on their ratio-nality level.

� There is no customer ‘‘disutility’’ associated withfailing to purchase an item, and a customer with-out an item at time T derives a utility of zero.

� The utility of buying an item in the future is dis-counted by a factor bA[0, 1] per decision period.The customer values future opportunities as theexpected value of bU(t11, � ) over all possiblestates resulting from the current one.

The degree of strategicity of a customer is determinedby the parameter bA[0, 1]. Indeed, the value b5 0means that the customer completely disregards thepossibility of a future purchase; that is, the customeris myopic. The value b5 1 means that the customervalues the current purchase the same as a purchase atany point in the future and will exhibit fully strategicbehavior. Intermediate values of b determine howlong customers can postpone their purchase withoutexcessive loss of utility. Our use of identical charac-teristics to describe different customers: b, u( � ), Ft(p)and U( � ) implies that the pool of customers is homo-geneous (in a stochastic sense). In practice, theseparameters could be estimated from such sources aspast sales history and marketing surveys. In e-com-merce applications, where it is often possible to trackbehavior of individual customers, these parameterscan be estimated from consumer responses to pricesover time. The maximum shopping intensity param-eter �l could be estimated from the number ofindividual consumer inquiries. Some approaches tolearning of customer behavior in this way are dis-cussed in a separate paper: Levina et al. (2009).

The company’s objective is to maximize the ex-pected total revenues by selecting the pricing policy inthe class of state-feedback policies: p 5 p(t, n) for staten at time t. A customer’s objective is to maximize theexpected present value of utility of acquiring theproduct. The objectives of the company and its cus-tomers determine the payoff functions in a stochasticdynamic game which unfolds over the decision peri-ods tAf0, . . ., T� 1g. If, at time toT, the system is instate n4N�Y, the company announces the pricep 5 p(t, n) (move of the company), the customers ob-serve their valuation realization p0 (which is differentfor each consumer and unknown by other consumers),and respond by choosing their shopping intensityfrom the interval ½0; �l�. The equilibrium shopping in-tensity response function of a customer is denoted aslU(t, n, p, p0), and its average over realizations of p0 (theaverage response that the company would expect inthe present state) as lU(t, n, p). The game ends either attime T regardless of the state, or if the company sellsout prior to time T (that is, in state n 5 N�Y). At theend of the game all unsold items (if any) expire, andall customers still in the market obtain a utility of 0.


We now turn to the nature of equilibrium in thisgame.

2.3. EquilibriumTo arrive at a strategic decision concerning shoppingintensity, the customer needs to know the expectedpresent value of utility from a possible future pur-chase. This value may be computed exactly orapproximately. The exact computation requires thecustomers to know the price which will be used inevery possible state in the future. This is possible if wesuppose that customers are fully rational and cancompute the optimal pricing policy used by the com-pany or if the customers can resort to third-partyservices. An alternative is to assume partial rational-ity: customers use a given heuristic rule to estimatethe expected present value of utility from a possiblefuture purchase. In either case, we assume

Rationality: The company is fully rational andknows how customers evaluate future purchasingpossibilities. We further assume that all participants ofthe game assume the optimality (in an exact or a heu-ristic sense) of the future strategies of all opponents(prices and customer responses), and each customerassumes that the concurrent responses of other cus-tomers are exactly optimal.

The sequence of moves at each t described in theprevious subsection corresponds to a hierarchicalequilibrium structure with the company as the leaderand the customers as the followers. The equilibrium ateach t can be described as a Stackelberg equilibriumbetween the company and the customers and a Nashequilibrium with asymmetric information (since eachcustomer knows their own current valuation but notthose of others) between the customers. According toour rationality assumptions, the behavior of the com-pany is always time consistent. We can make thecustomer response time-consistent as well under fullcustomer rationality. Thus, under full rationality, wehave a subgame perfect equilibrium with asymmetricinformation. The equilibrium between the customersat each t can also be viewed as a variation of a quantalresponse equilibrium, see McKelvey and Palfrey (1995,1998), because the payoff of each customer is modu-lated by a random perturbation drawn from thevaluation distribution, and decisions of each customercan be simplified to a binary choice between shoppingnow (with the maximum possible intensity) or wait-ing (that is, shopping with intensity 0).

We now derive the objective function for a cus-tomer and the equilibrium decision in terms of theshopping intensity. This derivation is for a simplercase in which valuation draws across different cus-tomers are independent and identically distributedinstead of exchangeable (which admits some depen-dency). Under this assumption, it is easy to see that

the expected payoffs and equilibrium responses of allcustomers are identical. We present the formal treat-ment in the general case of exchangeable draws inAppendix S1.

Suppose that the current decision period is t andconsider a particular customer. A future purchase canonly occur in one of the decision periods t11,. . ., T� 1. The expected value of utility at time t11from a possible future purchase, given that the state attime t11 is n, is assessed by each customer asU(t11, n). Given strategicity parameter b, the expectedpresent utility (at time t) from a possible future pur-chase is then bU(t11, n). At the time of decision, thecustomer has also observed the posted price p andhis/her own valuation level p0. We denote the deci-sion variable as l 2 ½0; �l�. Since valuation realizationsof all other customers are identically distributed andindependent of the value observed by the customer,their shopping intensities can be estimated by thecustomer at the expected value lU(t, n, p). Givent, n, p, p0, the value function VU(t, n, p, p0) is the ex-pected present value of utility over all possible

purchase events at time t (recall that �lf and lU(t, n, p)are probabilities):

VUðt; n; p; p0Þ ¼ max0�l��l

luðp0 � pÞf þ bðn� 1ÞlUðt; n; pÞ

�Uðtþ 1; n� 1Þ þ bð1� l� ðn� 1Þ� lUðt; n; pÞÞUðtþ 1; nÞg:

The first term in the above expression refers to thecertain utility that will be realized if the customercompletes the purchase now. The other terms repre-sent the expected present value of utility when apurchase by another customer occurs (second term) ordoes not occur (third term). After collecting terms, thisexpression can be rewritten as

VUðt; n; p; p0Þ ¼ max0�l��l

lðuðp0 � pÞ � bUðtþ 1; nÞÞf g

þ bðn� 1ÞlUðt; n; pÞðUðtþ 1; n� 1Þ�Uðtþ 1; nÞÞ þUðtþ 1; nÞ:

From the linearity of the objective in l, we concludethat the customer optimal shopping intensity is givenby the expression

lUðt; n; p; p0Þ ¼ �lI uðp0 � pÞ � bUðtþ 1; nÞ½ �: ð1Þ

The interpretation of this result is straightforward:the hypothetical customer will shop with the maxi-mum possible intensity of �l whenever the utility ofthe valuation/price difference is greater than or equalto the present value of utility of a possible futurepurchase. The condition inside the indicator functioncan also be expressed as p0 � p1u� 1(bU(t11, n)),where u� 1( � ) is the inverse of u( � ).


We thus have a simple rule for how customerschoose their shopping intensities after observing theirprivate valuations, but we must also specify howshopping intensity can be assessed without knowl-edge of those valuations. This is needed by thecompany, by customers considering other customers(in the fully rational case), and by customers assessingtheir own behavior in the future. To accomplish this,we take expectations of the optimal shopping inten-sity and the corresponding objective value over thevaluation distribution Ft(p) and obtain the followingproposition (where we let VUðt; n; pÞ ¼ Ep0 ½VUðt; n;p; p0Þ�):

PROPOSITION 1. Given the evaluation of the expected utilityfrom a possible future purchase U(t11, � ), which is iden-tical for all remaining customers, the expected equilibriumshopping intensity of any customer observing price p instate n 5 1, . . ., N at time t is given by

lUðt; n; pÞ ¼ �l�Ft pþ u�1ðbUðtþ 1; nÞÞ� �

: ð2Þ

The expected present value of utility for each remainingcustomer is given by

VUðt; n; pÞ ¼�lZ þ1

0

ðuðp0 � pÞ � bUðtþ 1; nÞÞf gþdFtðp0Þ

þ bððn� 1ÞlUðt; n; pÞðUðtþ 1; n� 1Þ�Uðtþ 1; nÞÞ þUðtþ 1; nÞÞ;

ð3Þ

where, for any real x, its positive part fxg1 5 x if x � 0and fxg1 5 0 if xo0.

An important observation from this analysis is thatthe customer reaction to price given in Equation (2)depends on the state n only through his/her evalu-ation of the future given by U(t11, n). Thus, for anexogenously specified U, the result contained inEquation (2) can be generalized to the case in whichthe customer is uncertain about state information: wewould replace U(t11, n) by the expected value over n.We do not pursue this generalization here but notethat this would allow extension to a variety of limited-rationality cases.

The company can use the equilibrium consumershopping intensity in its policy optimization. Theprobability of a sale occurring in the current decisionperiod and state n is equal to nlU(t, n, p). The expres-sion for the optimal expected revenue of the companyis obtained in the same fashion as the expression forthe optimal expected customer’s utility. The com-pany’s equilibrium pricing strategy pU( � ) is chosen sothat pU(t, n) delivers the maximum expected futurerevenues:

Rðt; nÞ ¼Rðtþ 1; nÞ þmaxp

� nlUðt; n; pÞðpþ Rðtþ 1; n� 1Þ�

�Rðtþ 1; nÞÞg;n ¼ N � Yþ 1; . . . ;N; t 2 f0; . . . ;T � 1g;

ð4Þ

with the terminal conditions

RðT; nÞ ¼ 0; n ¼ N � Yþ 1; . . . ;N; and ð5Þ

Rðt;N � YÞ ¼ 0; t 2 f0; . . . ;T � 1g: ð6Þ

Given the rule U( � ) for consumer evaluation ofexpected utility from a possible future purchase, pU( � )and lU( � ) describe the game equilibrium.

One particularly interesting case results when U( � )corresponds to fully rational customers. It can beobtained by

Uðt; nÞ ¼VUðt; n; pUðt; nÞÞ; n ¼ N � Yþ 1; . . . ;N;

t 2 f0; . . . ;T � 1g

and the terminal conditions on the customer utility

UðT; nÞ ¼ 0; n ¼ N � Yþ 1; . . . ;N ð7Þ

Uðt;N � YÞ ¼ 0; t 2 f0; . . . ;T � 1g: ð8Þ

We will denote the expected utility for this case asUð�Þ and the corresponding equilibrium policies aspð�Þ and lð�Þ.

If, in addition to the pricing policy, the company cancontrol its initial capacity (procurement) decision, itcan use the optimal revenue at time 0 for fixed N butdifferent values of Y to make the optimal capacitychoice. For the time being, however, we assume thatY is fixed and fully concentrate on the pricing deci-sions of the company. We return to the question of theoptimal capacity decision in Section 5 where we nu-merically examine the effects of customer strategicity.

3. Existence and Uniqueness ofEquilibrium

A typical approach in the analysis of dynamic pricingmodels is to regard the aggregate demand rather thanthe price as a decision variable for the company.In our case, it is also convenient to change the decisionvariable of the company to a quantity closely relatedto the aggregate demand. We note that a particularprice p in state n at time t results in the followingeagerness to purchase averaged over the valuation dis-tribution:

w ¼ �Ft pþ u�1ðbUðtþ 1; nÞÞ� �

:

The purchase intensity corresponding to w is lUðt;n; pÞ ¼ �lw:

To use eagerness w as a policy variable, we require�Ftð�Þ to have an inverse function gt( � ). The value gt(w)of the inverse function specifies the quantile of the


valuation distribution at time t corresponding to theright-tail probability w. To this end, we will make thefollowing assumption regarding gt(w):

ASSUMPTION (A). Let the valuation probability densityfunction be strictly positive and continuous on a (possiblyinfinite) interval ð�p; ��pÞ, and zero outside the interval.

Then the tail distribution function �Ftð�Þ is decreas-ing and has an inverse function gt( � ), which iscontinuously differentiable and decreasing. The do-main of gt( � ) is (0, 1), and its range is ð�p; ��pÞ.

If Assumption (A) holds, then we can express p asthe following continuously differentiable functionof w:

p ¼ gtðwÞ � u�1ðbUðtþ 1; nÞÞ:

In the sequel, we use the following abbreviatednotations for the quantities marginal in n:

DRðt; nÞ ¼ Rðt; nÞ � Rðt; n� 1Þ;DUðt; nÞ ¼ Uðt; nÞ �Uðt; n� 1Þ;

Du�1ðbUðt; nÞÞ ¼ u�1ðbUðt; nÞÞ � u�1ðbUðt; n� 1ÞÞ:

The dynamic programming recursion (4) for theexpected future revenues of the company can now berewritten in terms of eagerness w as follows:

Rðt; nÞ ¼Rðtþ 1; nÞ þ n�l max0�w�1

w

� gtðwÞ � u�1ðbUðtþ 1; nÞÞ�

�DRðtþ 1; nÞ�;n ¼ N � Yþ 1; . . . ;N; t 2 f0; . . . ;T � 1g:

ð9Þ

The value of the expression under the maximum isdefined at 0 and 1 in the limiting sense. An optimalpolicy value wU(t, n) delivers the maximum in thisexpression. Note the role of the u� 1(bU(t11, n)) term.When the customers are myopic, this term is equal tozero, and the price corresponding to w is given byp 5 gt(w). The quantity u� 1(bU(t11, n)) can then beviewed as a state and time-dependent transactioncost. This is a portion of the total surplus the companyhas to give up to the consumer due to his/her stra-tegicity.

Next, we use Equation (3) and the unique customerresponse (2) to derive an expression for the equilib-rium expected present value of utility VU(t, n) in termsof wU(t, n):

VUðt; nÞ ¼VUðt; n; pUðt; nÞÞ

¼�lZ þ1

0

uðp0 � gtðwUðt; nÞÞ þ u�1ðbUðtþ 1; nÞÞÞ�

�bUðtþ 1; nÞgþdFtðp0Þ� b�lðn� 1ÞwUðt; nÞDUðtþ 1; nÞ þ bUðtþ 1; nÞ:

Defining

Htðw; vÞ ¼Z þ1

0

uðp0 � gtðwÞ þ u�1ðvÞÞ � v� �þ

dFtðp0Þ;

we obtain the following relation

VUðt; nÞ ¼�lHt wUðt; nÞ; bUðtþ 1; nÞ� �

� b�lðn� 1ÞwUðt; nÞDUðtþ 1; nÞ þ bUðtþ 1; nÞ;n ¼ N � Yþ 1; . . . ;N; t 2 f0; . . . ;T � 1g:

ð10Þ

This relation becomes recursive in the fully rationalconsumer case where Uðt; nÞ ¼ Uðt; nÞ ¼ VU ðt; nÞ forall n ¼ N � Yþ 1; . . . ;N and t ¼ 0; . . . ;T � 1, and theterminal conditions are given in (7) and (8).

Even under full rationality, the computational effort(required to find the optimal policy in our strategic-customer model in this parametrization) is only mod-estly greater than the computational effort needed in atime-discretization of the myopic-customer model ofGallego and van Ryzin (1994) with the same initialinventory level. The increase results entirely from theneed to solve recursion (10) and is modest as long asthe function Ht(w, v) can be evaluated efficiently.Thus, our strategic customer model is computation-ally tractable for problems of realistic size. This resultcould be achieved because of our reliance on time-consistency and a hierarchical structure of the equi-librium.

Using the dynamic programming recursion (9) forthe company, we arrive at the following existence anduniqueness result (the proof of this and all subsequentformal statements is provided in the Appendix S1).The result applies to both the full U ¼ U and limitedconsumer rationality cases (exogenous U). It alsogives the equilibrium pricing policy of the company.We make two additional assumptions:

ASSUMPTION (B). wgt(w) is strictly concave for all t, and

ASSUMPTION (C). limw ! 0 wgt(w) 5 0 for all t,

which correspond to the requirements of concavityand regularity of the revenue rate in Gallego and vanRyzin (1994). These assumptions are satisfied for suchcommon distributions as exponential �FtðpÞ ¼ expð�pÞ,and power �FtðpÞ ¼ ð1þ pÞ�g; g41 (in these cases,gt(w) 5� ln(w) and gt(w) 5 w� l/g� 1, respectively).

PROPOSITION 2. Suppose that the valuation distributionsatisfies assumptions (A), (B) and (C). Then there exists aunique equilibrium such that, for every (t, n), the customereagerness to purchase, wU(t, n), is

� zero if limw!0ðw g0tðwÞ þ gtðwÞÞ � u�1

ðbUðtþ 1; nÞÞ þ DRðtþ 1; nÞ,


� one if limw!1ðw g0tðwÞ þ gtðwÞÞ � u�1

ðbUðtþ 1; nÞÞ þ DRðtþ 1; nÞ,� and, otherwise, satisfies the equation

wUðt; nÞg0tðwUðt; nÞÞ þ gtðwUðt;nÞÞ¼ u�1ðbUðtþ 1; nÞÞ þ DRðtþ 1; nÞ:

ð11Þ

The corresponding optimal price is

pUðt; nÞ ¼ gtðwUðt; nÞÞ � u�1ðbUðtþ 1; nÞÞ: ð12Þ

Given the customer perception of the future spec-ified by U, the equilibrium eagerness value wU(t, n)specifies the optimal customer reaction to the equi-librium price pU(t, n). When assessing their reaction,customers generally take into account how manyitems are still available as well as competition fromother customers as summarized by the state descrip-tion n. Thus, the game developed here is a dynamicStackelberg game between the company and a homo-geneous population of individual consumers. Ifcustomers are unaware of the state, the equilibriumtakes a somewhat simpler form in which the companyplays Stackelberg games with customers who makedecisions independently from each other. However,for the company, these games are linked through thelimited inventory. To interpret (11), observe that, onthe left-hand side, wg0tðwÞ þ gtðwÞ is the marginal rateof revenue per customer while, on the right-hand side,marginal expected revenue per item DR(t11, n) canbe viewed as an opportunity cost of capacity. A stan-dard interpretation of optimality conditions indynamic pricing is that the marginal revenue rateequals the opportunity cost of capacity (see, for ex-ample, the interpretation of equation (5.13) on page203 of Talluri and van Ryzin (2004)). In our model, thisis modified by addition of the certainty equivalentfrom a possible future purchase u� 1(bU(t11, n)),which we interpreted as a cost of transaction withstrategic customers.

The following corollary immediately follows fromthe optimality conditions described in the aboveproposition and the (strictly) decreasing property ofwg0tðwÞ þ gtðwÞ in w. It relates monotonic properties ofthe optimal policy to those of u� 1(bU(t11, n))1DR(t11, n) (the sum of the certainty equivalent ofpurchasing in the future and the marginal future ex-pected revenues).

COROLLARY 1. The relation

u�1ðbUðtþ 1; nÞÞ þ DRðtþ 1; nÞ � ð�Þ� u�1ðbUðtþ 1; n� 1ÞÞ þ DRðtþ 1; n� 1Þ

holds if and only if wUðt; nÞ � ð�ÞwUðt; n� 1Þ. In addi-tion, if gt( � ) is stationary (independent of t) then therelation

u�1ðbUðtþ 1; nÞÞ þ DRðtþ 1; nÞ � ð�Þ u�1ðbUðtþ 2; nÞÞþ DRðtþ 2; nÞ

holds if and only if wUðt; nÞ � ð�ÞwUðtþ 1; nÞ.

This result sheds some light on the consequences ofconsumer rationality for monotonic properties of thecompany’s policy. One of the common monotonicityresults in dynamic pricing models is that the marginalrevenues decrease in capacity; that is, DR(t11, n)� DR(t11, n� 1). On the other hand, intuition sug-gests that, for rational consumers, inequalityU(t11, n) � U(t11, n� 1) should typically hold sincethe customer competition for remaining items be-comes more acute when there is one less itemavailable (even though there is also one less cus-tomer). Because this inequality for utility is in theopposite direction to the inequality for marginal rev-enues, we conclude that consumer rationality worksagainst typical monotonicity properties. Some excep-tions to this are possible if U(t11, n) is independent ofn. The reader will see that this intuitive conclusionis confirmed in the structural results and numericalexperiments to follow. As a final remark about Prop-osition 2, we show that certain monotonic propertiesof prices (in contrast to the result of Corollary 1 whichis in terms of eagerness) can be established under thefollowing assumption:

ASSUMPTION (D). wg0tðwÞ is increasing for all t.

This assumption is also satisfied for exponentialand power distributions that we mentioned earlier(after Assumptions (B) and (C)).

COROLLARY 2. Let Assumption (D) hold in addition to (A),(B), and (C). Then the pair of inequalities

DRðtþ 1; nÞ� ð�ÞDRðtþ 1; n� 1Þ;wUðt; nÞ� ð�ÞwUðt; n� 1Þ;

�

implies pUðt; nÞ � ð�Þ pUðt; n� 1Þ. If, moreover, gt( � ) isstationary (independent of t), then the pair of inequalities

DRðtþ 1; nÞ� ð�ÞDRðtþ 2; nÞ;wUðt; nÞ� ð�ÞwUðtþ 1; nÞ;

�

implies pUðt; nÞ � ð�Þ pUðtþ 1; nÞ.

To explain this result, we rewrite Equation (12) us-

ing Equation (11) as pUðt; nÞ ¼ DRðtþ 1; nÞ � wUðt; nÞg0tðwUðt; nÞÞ. Since, under strategic consumer behavior,wU(t, n) is generally time and state dependent, appro-priate monotonic relations on both DR(t11, n) andwU(t, n) are needed to guarantee corresponding mono-tonic relations between prices. On the other hand, inthe case of myopic consumers it suffices to have


monotonic relations on DR(t11, n). In managerialterms, an interpretation of the first claim of the cor-ollary is that, if under equilibrium policies a marginalvalue of an extra sale opportunity (one extra item plusone extra customer) decreases and the shopping in-tensity of an individual customer increases with n,then the price decreases with n. Similar interpreta-tions apply to the second claim and also if inequalitiesare reversed.

As a reality check of our model, we have also ex-amined what happens if the original problem is scaledto an arbitrarily large population size, with capacityproportionally large but still smaller than the popu-lation. Under such scaling, we also need to take thenumber of decision periods to infinity. Through thelaw of large numbers, the demand process becomesincreasingly deterministic from the point of view ofthe company, but each consumer still faces a poten-tial rationing risk because of insufficient capacity.In Appendix S1, we show that in the limit the priceoptimality conditions in our discrete-time model cor-respond to a Hamilton–Jacobi–Bellman equation for adeterministic continuous time control problem. Thus,we find that the stochastic discrete-time model has anatural deterministic continuous-time analog.

4. Monotone Properties of the OptimalPolicy

In this section, we study monotone properties of theoptimal policy. We show that the optimal price andmarginal revenues possess monotone properties intwo special cases: first, when customers have limitedrationality and evaluate the expected utility fromfuture purchases independently of the state (thisincludes myopic case), and, second, when customersare strategic but do not need to compete for the itemssince the inventory is sufficient to satisfy all custom-ers. Our results show that under these conditions themonotonic properties well-known in the infinite pop-ulation myopic case hold in our model as well. Wealso present a numerical illustration which shows thatthe price may not be monotone in general. This is incontrast with the results in dynamic pricing literaturefor the myopic consumer case. In practical terms, thissuggests that the company may have to deviate frommonotonic pricing policies to counteract strategic con-sumer behavior. All results of this section are obtainedunder Assumptions (A)–(C).

We start our discussion with basic observationswhich apply to the most general form of our model.First, note that the optimal expected future revenuesare decreasing in time: R(t, n) � R(t11, n), since themaximum of expression (28) under maximization ofthe recursive relation (9) for revenues is nonnegative.Given that U � 0, it is also straightforward to estab-

lish that the equilibrium customer utility isnonnegative: VU(t, n) � 0, since we can regroup theterms in Equation (10). For U, we can establish thenonnegativity by reverse induction from terminalconditions (7) and (8).

4.1. Special Case: State Unaware, LimitedRationality ConsumersWe analyze the case of a finite population of limitedrationality customers whose evaluation of the futureis such that bU(t, n) is independent of the state n forevery t. Note that myopic consumer behavior, whichis a special case of our model when b5 0, satisfies thisrestriction. The following proposition establishes thatthe marginal revenues (with respect to the number ofremaining customers) are decreasing with respect to nand t. These properties generally coincide with thosefound in many myopic infinite population models(see, for example, Proposition 5.2 on page 203 of Tall-uri and van Ryzin (2004)). We also remark that thismonotonicity result provides a bound on the decre-ment of DR(t, n) with respect to t:

PROPOSITION 3. Let U be such that bU(t, n) does not dependon n for each t. Then we have

DRðt; nÞ � DRðt; n� 1Þ; t ¼ 0; . . . ;T; n¼ N � Yþ 2; . . . ;N; ð13Þ

DRðt; nÞ � DRðtþ 1; nÞ þ �lwUðt; nÞ gtðwUðt; nÞÞ�

�u�1ðbUðtþ 1; nÞÞ � DRðtþ 1; nÞ�;

t ¼ 0; . . . ;T � 1; n ¼ N � Yþ 1; . . . ;N:

ð14Þ

The first inequality can also be restated as D2R(t, n)� 0, and implies concavity of the expected revenuesin n.

The following corollaries establish monotonic prop-erties of the policy variables in the state unaware,limited rationality case. The first corollary shows thatthe customer eagerness to purchase increases withrespect to n, and that prices pursue the oppositebehavior:

COROLLARY 3. If U is such that bU(t, n) does not depend onn for each t, then the following inequalities hold for allt 5 0, . . ., T� 1, and all n 5 N�Y12, . . ., N:

wUðt; nÞ � wUðt; n� 1Þ; ð15Þ

pUðt; nÞ � pUðt; n� 1Þ: ð16Þ

Recall that smaller n also means lower inventorylevel y; thus, this property is natural because, underotherwise identical market conditions, our intuitionsuggests increasing price if there are fewer items left


to sell. Our model provides an additional insight thatthe price increases in spite of the fact that the marketis also smaller after a sale.

The next corollary establishes monotonic propertiesof the pricing policy with respect to time in the case ofmyopic customers and a stationary valuation distri-bution. An additional assumption made is that theconsumer evaluation of expected utility from a pos-sible future purchase decreases in time (a naturalproperty for a heuristic utility evaluation rule).

COROLLARY 4. If U is such that bU(t, n) does not depend onn, bUðt; �Þ � bUðtþ 1; �Þ for all t, and gt( � ) 5 g( � ), thenthe following inequality holds for all t ¼ 0; . . . ;T � 2 andn ¼ N � Yþ 1; . . . ;N

wUðt; nÞ � wUðtþ 1; nÞ ð17Þ

with inequality being strict if wUðtþ 1;nÞ40: In addition,under Assumption (D) or in the myopic case of b5 0

pUðt; nÞ � pUðtþ 1; nÞ: ð18Þ

This corollary asserts that, under otherwise iden-tical market conditions, consumer eagerness to pur-chase increases when there is less time remaining. Itis much less obvious that the price is monotonic sincethere are generally two opposite effects: gðwU ðt; nÞÞ �gðwUðtþ 1; nÞÞ and �u�1ðbUðt; nÞÞ � �u�1ðbUðtþ 1;nÞÞ, and it is not clear which one dominates. Whenb5 0, the last inequality holds as an equality whichimplies monotonicity of prices.

4.2. Special Case: Fully Rational StrategicConsumers Under High-Supply MarketNext, we analyze the case of fully rational strategicconsumers in a ‘‘high product supply’’ situation,when there are enough items in inventory to satisfythe purchasing requirements of all customers. Theresults imply that the game at each time t and staten decomposes into n identical games played simul-taneously between the company and each of theremaining customers individually. This is possible be-cause customers are statistically identical.

PROPOSITION 4. In the case of N 5 Y, the following equalitieshold

wðt; nÞ ¼ wðt; n� 1Þ; t ¼ 0; . . . ;T; n¼ N � Yþ 2; . . . ;N; ð19Þ

pðt; nÞ ¼ pðt; n� 1Þ; t ¼ 0; . . . ;T; n¼ N � Yþ 2; . . . ;N; ð20Þ

DRðt;nÞ ¼DRðtþ 1;nÞ þ �lwðt; nÞ gtðwðt;nÞÞ½�u�1ðbUðtþ 1; nÞÞ�DRðtþ 1;nÞ

�;

t ¼ 0; . . . ;T � 1; n ¼ N � Yþ 1; . . . ;N;

ð21Þ

DRðt; nÞ ¼ DRðt; n� 1Þ; t ¼ 0; . . . ;T; n¼ N � Yþ 2; . . . ;N; ð22Þ

Uðt; nÞ ¼ Uðt; n� 1Þ; t ¼ 0; . . . ;T; n¼ N � Yþ 2; . . . ;N: ð23Þ

The result of this proposition is that the equilibriumprice, the consumer reaction, and the expected utilityare independent of the state, while the expected rev-enues increase linearly with the state (thus, therevenue is concave in the state here as well). Suchequilibrium structure is natural since there is no com-petition between consumers, and they only need totake into account time remaining to the end of theplanning horizon. Equation (21) also implies that themarginal expected revenues decrease in time.

Based on the above proposition, we also obtain atime monotonicity result for the equilibrium eager-ness to purchase and price in the special case of fullystrategic customers.

COROLLARY 5. In the case of b5 1 N 5 Y and gt( � ) 5 g( � ),for all t ¼ 0; . . . ;T � 2 and n ¼ 1; . . . ;N

wðt; nÞ � wðtþ 1; nÞ: ð24Þ

In addition, under Assumption (D)

pðt; nÞ � pðtþ 1; nÞ: ð25Þ

4.3. General Case: Numerical IllustrationsThe monotonicity properties discussed above for spe-cial cases do not hold for more general strategicityor inventory level conditions. We illustrate this withnumerical counterexamples below. In these examples,we use a risk-neutral utility uðp0 � pÞ ¼ p0 � p and astationary exponential tail distribution of the valua-tion �Ftðp0Þ ¼ expð�p0Þ: We solve the model forT 5 10,000 decision periods, N 5 100 customers andmaximum individual shopping intensity �l ¼ 0:005.The purpose of choosing a large value for T is pri-marily to provide a sufficiently close approximation tocontinuous time.

4.3.1 Monotonicity with Respect to Time. Our firstnumerical experiment concerns the non-monotonicityof price in time when the number of customers isfixed. The initial inventory level for this experiment isfixed at Y 5 75 items. Five values of the strategicityparameter b are examined: zero (myopic), one (fullystrategic), and three intermediate values such that thevalue of a purchase after 10,000 time steps is 0.1, 0.01,and 0.0001 of the value of a purchase if it is made


immediately. These values were chosen because of thelogarithmic effects of strategicity on the optimalpolicies, and produce graphs which fill the visualgaps between myopic and fully strategic customersquite well. Figure 1 shows the optimal price pðt; 100Þ,the optimal expected revenues R(t, 100) and the‘‘booking curve’’ for the state with 100 customersand all five values of b. The graph of the optimal pricefor the myopic case is, of course, monotone decreas-ing, in full agreement with Corollary 4. However, asthe value of b increases, we see a tendency towardsnon-monotonic behavior. For b5 1, the change indirection can be seen very clearly and takes placearound time steps t 5 6200. . .7200. A possible expla-nation of this non-monotonic behavior is that thesupply level Y/N 5 75% is relatively high. Therefore,in the beginning of the interval, the state n 5 100 willprovide a plentiful opportunity for the customer tobuy. As the expiration date gets closer, the customerswill have fewer possibilities to purchase. This allowsthe company to increase prices even if all otherparameters of the model stay the same. Note that thisis possible only because the customers are strategicand aware of the reduction in future purchaseopportunities. Fully rational behavior of the com-pany ensures that a price increase is a credible threat,thus stimulating consumers to shop earlier.

Another observation from this plot is that the op-timal price tends to decrease in b. Naturally, thesame can be expected from the optimal expectedrevenues. We can see this type of behavior ofR(t, 100) in Figure 1. Recall that the total shoppingintensity of n customers at time t is given by �lnw

ðt; nÞ: This can be viewed as a ‘‘booking curve’’ overtime. Non-monotonic behavior in time is seen here

as well although not as prominently as for pðt; 100Þ(see Figure 1). Another interesting feature observedfrom this plot is that the shopping intensity is notmonotonic in b. In the beginning of the interval, theshopping intensity is almost the same for b5 0 and1, while being slightly larger for the intermediatevalues of b. A possible explanation is that both themyopic and fully strategic (b5 1) customers inthe beginning of the planning interval do not carewhen exactly their purchases occur. On the otherhand, customers with intermediate values of b effec-tively have less time than T to make sure the item isacquired than the customer with b5 1 who can takeadvantage of the whole planning interval.

4.3.2 Monotonicity with Respect to n. The secondexperiment examines the monotonic behavior ofprices with respect to n at fixed time t. The initialinventory Y is fixed, which implies that for smallervalues of n we look at respectively smaller values ofthe remaining inventory y 5 Y� (N� n). Four values ofY were examined corresponding to the supply levelsY/N of 25%, 50%, 75%, and 100%. The strategicityparameter b is fixed at 1. The graphs of the optimalprice pðt; nÞ as functions of n are shown in Figure 2for fixed values of t 5 0 (the decision period nearwhich pðt; nÞ is decreasing in t, a natural monotonicbehavior) and t 5 7000 (where we observed somedeviations from monotonic behavior). The value of p

ðt; nÞ for the supply level of 100% remains constant inagreement with Proposition 4. For lower values of thesupply level, the value of pðt; nÞ is decreasing in n atthe beginning of the planning period (t 5 0). This isnot the case for t 5 7000 and the supply level of 75%.We recall that the price is also non-monotone in t for

11.5

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0 2000 4000 6000 8000 10 000 0 2000 4000 6000 8000 10 000

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� = 1.0�T =0.1�T =0.01�T =0.0001� = 0.0

Figure 1 The Optimal Price pðt ; 100 Þ, Future Expected Revenues R(t, 100) and the Booking Curve �k100wðt ; 100 Þ as Functions of Time for Y = 75, N = 100and Different Values of b


this supply level around t 5 7000. For supply levels of25% and 50%, the prices are monotonic for bothvalues of t.

4.3.3 Typical Price Path Realizations. In additionto monotonic properties of policies themselves, it isinteresting to examine typical realizations of thesepolicies – the price paths, since this is how policies areperceived in the market place. Therefore, we havesimulated 10,000 realizations of pricing policies formyopic and fully strategic (b5 1) consumers atsupply levels of 50% and 75%. The averages ofsimulated prices plus/minus one standard deviationfor all four combinations are shown in Figure 3. In themyopic case, increased supply level leads to reducedprice flexibility (as evidenced by a smaller standarddeviation) and significantly lower prices in the secondhalf of the planning horizon. When consumers arestrategic, the effect of increased supply level is evenmore dramatic. At s 5 50%, there is relatively steady

price level of about 2.3 with progressively larger pricefluctuations as the end of selling season gets closer. Ats 5 75%, we see a steady decline in prices. Even thestarting price is just slightly above 2 and there ispractically no price fluctuation around the averageprice path. This suggests that appropriate choice ofcapacity level is especially important in the presenceof strategic consumers.

The non-monotonicity of prices for the high supplylevel also suggests that we may discover interestingbehavior in expected revenues when the values of Yand N are close. A numerical experiment addressingthis and other issues is described in the next section.

5. Effects of Strategic ConsumerBehavior on Company Performanceand Decisions

In this section, we examine the interplay betweenstrategicity and company performance and policies.

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Figure 3 Averages of Simulated Price Paths for Myopic/Fully Strategic Consumers, Supply Levels 50%/75%, N = 100 and T = 10,000

t = 0 t = 7000

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Figure 2 The Optimal Price pðt ; nÞ as a Function of the Number of Remaining Customers n for b = 1, fixed t = 0, t = 7000 and Different Values of InitialSupply Level


We start by assuming that the initial capacity Y isspecified exogenously (for example, a flight capacity isdetermined by the aircraft following a pre-determinedschedule) and, in Section 5.1, study strategicity effectson the pricing policies, expected revenues, and utilities.In particular, we examine performance of pricing pol-icies obtained under the assumption that customers aremyopic in a situation when customers are in fact fullyrational and strategic. This examination is done bytesting the effect of company deviation from the equi-librium decisions. We assume that the customers areaware of the fact that the company is behaving non-optimally, can compute their actual expected utility inthis situation exactly, and adjust their response to theobserved prices accordingly. Major observations madein this subsection include much lower or even negativemarginal revenue in terms of extra product supply forsupply levels close to 100%, possibly high costs ofassuming that customers are myopic when they arestrategic, and a description of conditions when thesecosts are low.

In Section 5.2, we relax the assumption of exoge-nous Y and study the effects of strategicity on theinitial capacity decisions. In particular, we observeimportance of appropriate initial capacity reduction tocounteract strategic consumer behavior, possibly sig-nificant profit reduction even if strategic behavior iscorrectly taken into account, and truly disastrouseffects on profits if strategicity is ignored in procure-ment decisions. We also study contribution of pricingvs. capacity decisions to the proper company responseto strategicity.

5.1. Effects on Revenues and Pricing PoliciesIn the study of pricing decisions, we keep all param-eters of the model fixed except for the values of b (thestrategicity parameter) and Y. It is convenient to presentthe results in terms of the supply level s 5 Y/N. Wedenote by pb;sð�Þ the pricing policy that is optimal forfixed values of b and s, and by Rb;sð�Þ and Ub;sð�Þ thecorresponding (optimal) future expected revenues andexpected present values of customer utility in all states,respectively. The pricing policy which is optimal formyopic customers (under the assumption that b5 0) isp0;sð�Þ. If the company uses this policy in a situationwhen b40, the behavior of the customers will bedifferent from the one expected by the company, andthis policy will no longer be optimal. Given that thecompany applies the policy p0;sð�Þ, let the actual (exact)utility values for customers with strategicity parameterb be U0

b;sð�Þ. The eagerness to purchase in response top0;sð�Þ will be denoted by w0

b;s, and is computed viaEquation (2) where we use U ¼ U0

b;s resulting in

w0b;sðt; nÞ ¼ �Ft p0;sðt; nÞ þ u�1ðbU0

b;sðtþ 1; nÞÞ

:

The utility values are computed recursively accord-ing to

U0b;sðt; nÞ ¼�lHt w0

b;sðt; nÞ; bU0b;sðtþ 1; nÞ

� b�lðn� 1Þw0b;sðt; nÞDU0

b;sðtþ 1; nÞ þ bU0b;sðtþ 1; nÞ;

n ¼ N � Yþ 1; . . . ;N; t 2 f0; . . . ;T � 1g:ð26Þ

The expected future revenues in the situation whenthe myopic policy p0;sð�Þ is used for customers witharbitrary b is denoted by R0

b;sð�Þ, and can be computedvia

R0b;sðt;nÞ ¼R0

b;sðtþ 1; nÞ þ n�lw0b;sðt; nÞ

gtðw0b;sðt; nÞÞ � u�1 bU0

b;sðtþ 1; nÞ h

�DR0b;sðtþ 1; nÞ

i; n ¼ N � Yþ 1; . . . ;N;

t 2 f0; . . . ;T � 1g:

The derivation of these recursions is straight-forward and closely follows that of Equations (9)

and (10). The boundary conditions for U0b;sð�Þ and R0

b;s

ð�Þ are identical to those for Ub;sð�Þ and Rb;sð�Þ (equa-tions of the same form as (7), (8), and (5), (6),respectively).

For the numerical experiments described below, weuse a risk-neutral utility uðp0 � pÞ ¼ p0 � p and a sta-tionary exponential tail distribution of the valuation�Ftðp0Þ ¼ expð�p0Þ: We solve the model for N 5 100customers, the maximum individual shopping inten-sity of �l ¼ 0:001 per time step and a time horizon ofT 5 1,000,000. In addition to b5 0 and b5 1, we ex-amine two values of b which correspond to apurchase after T and T/10 time steps being valuedat 10% of utility for an immediate purchase.

5.1.1 Optimal Expected Total Revenues as aFunction Supply Level. Figure 4 shows the optimalexpected total revenues Rb;sð0; 100Þ as a function of thesupply level s 5 Y/N (as well as the ratio of theexpected total revenues R0

b;sð0; 100Þ=Rb;sð0; 100Þ, theratio of the expected present value of consumerutilities U0

b;sð0; 100Þ=Ub;sð0; 100Þ; and the optimalprice pb;sð0; 100Þ). The plot of the expected revenues(top left) contains four graphs which correspond todifferent values of b. The most prominent feature isthat, although in the case of myopic customers therevenues are monotonically increasing in s, this is notgenerally true in the presence of strategic customers.As b increases, the graph of Rb;sð0; 100Þ acquires amaximum in s. This drop in revenue after a certainpoint can be explained by a decrease in competitionbetween customers for the remaining inventory. Wealso see that the optimal revenues for all values of sare decreasing in b. Moreover, the slope of the graphs


(essentially, marginal revenues in Y) is decreasing.Therefore, marginal analysis would suggest a lowerinitial stocking level when customers are strategiccompared to the myopic case. The final observationregarding these plots is a plateau-type behavior forthe highest supply levels. This behavior occursbecause the company will be less likely to sell itsentire inventory when the customers are strategic ifthe initial inventory level is too high.

5.1.2 Relative Cost of Assuming Myopic Behavior.Next, we consider the ratio of the expected totalrevenues R0

b;sð0; 100Þ=Rb;sð0; 100Þ, which represents arelative cost to the company of its assumption that thecustomers are myopic in a situation where they arenot. The ratio is shown in the top right plot of Figure 4as a function of s. A few observations are in order.First, the loss in revenues due to ignoring the strategicnature of the customers can be very significant. For b5 1the drop is as high as 20% for the supply level near87%. Other numerical trials showed that for a timehorizon of T 5 100,000 the maximum loss is smallerand is approximately 13%, for even shorter time hori-zons the maximum loss decreases further. Therefore,taking into account the strategic nature of consumersmay not be as important if customers only have a fewshopping opportunities. Another useful observationfrom the plots is that, for the supply levels underabout 30%, the loss in revenues does not exceed 1%.We also remark that the loss in revenues becomes verysmall (well under 1%) for all supply levels if thecustomer utility of a purchase after 1/10 of theplanning period is 10% of an immediate purchase,

i.e., bT/10 5 0.1. This suggests that taking into accountthe strategic nature of consumers is unnecessary ifcustomers’ strategicity is such that they prefer to maketheir purchases within much shorter time spans than theplanning period of the company. Thus, we see that even ifthe exact value of b is unknown, this model may allowa company to assess whether or not strategic behavioris important. If company losses from disregardingstrategic behavior for a wide range of strategicityparameter values are minor, then the company maystick with a pricing policy which assumes myopicbehavior. The validity of this conclusion may beaffected if the capacity is not costless and the per-formance is measured in terms of profit. We examinethis issue further in Section 5.2.

5.1.3 Relative Boost in Revenues Resulting fromMyopic Behavior Assumption for Supply Levels near100%. An additional observation about Figure 4concerns a somewhat surprising boost in revenuesresulting from the assumption that the customers aremyopic when they are, in fact, strategic for supplylevels close to 100% (also observed for shorter timehorizons). This type of phenomenon is not unusual ingame theory in general. It occurs in this case, since thepair of company-customer strategies under consider-ation is no longer an equilibrium (indeed, the com-pany assumes in its pricing policy that the customersare myopic, while the actual strategic customers areaware of that and adjust their behavior accordingly).In the well-known prisoner’s dilemma, for example,both players gain if they simultaneously deviate fromthe equilibrium. We can confirm that the prisoner’s

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Figure 4 The Optimal Expected Total Revenues Rb;sð0 ; 100 Þ, the Ratio of the Expected Total Revenues R 0b;sð0 ; 100 Þ=Rb;sð0 ; 100 Þ, the Ratio of the Expected

Present Value of Consumer Utilities U 0b;sð0 ; 100 Þ=Ub;sð0 ; 100 Þ and the Optimal Price pb;sð0 ; 100 Þ as Functions of the Supply Level s = Y/N for

Different Values of b and T = 1,000,000


dilemma-type situation occurs here by looking at theplots of utility ratios U0

b;sð0; 100Þ=Ub;sð0; 100Þ pre-sented in the bottom left plot of Figure 4. We seethat the customers gain (at any supply level) if thecompany uses the correct value of b instead of as-suming myopic behavior (b5 0) since the ratio neverexceeds 1. To conclude, we remark that the companycan avoid this non-Pareto-optimal equilibrium, if thesupply level is a decision variable, and it is knownthat the customers are strategic. Indeed, there is nogain in revenue from bringing the supply level closeto 100%. The company will always work with smallervalues of s as suggested by the plot of the expectedrevenues.

5.1.4 Optimal Price as a Function of Supply Levelfor Different b’s. We also examine the optimal pricepb;sð0; 100Þ as a function of s in the bottom right plot ofFigure 4. The optimal price in case of strategic con-sumers is decreasing for smaller-to-medium values ofs but may exhibit an upward jump at a higher value ofs. The optimal price for higher values of b is smaller.Incidentally, this explains why the consumer utilityincreases if the company knows that the customers arestrategic – because the price goes down.

5.1.5 Comparison of Fully Dynamic Contingent andPre-announced Pricing Policies. Some businessesoperate under deterministic pricing policies ratherthan contingent ones, which are the focus of this paper.In practice, such policies are often pre-announced, as inthe case of retail chain Filene’s Basement with itsautomatic 25–50–75% price reduction scheme. How-ever, even if a deterministic policy were not pre-

announced, fully rational consumers would be able toreconstruct it from the available information. There-fore, we can assume that the deterministic policy isknown to the consumers, and substitute it into theutility recursion (a similar computation was done inEquation (26) for the pricing policy which assumesthat consumers are myopic). Since the expected con-sumer utility for each time, state, and each pre-announced policy is available, the company can usethis information to optimize pre-announced prices.We have implemented such calculation for the case oftwo prices, the time horizon of T 5 10,000, and a priceswitch at time t 5 5000. The best prices were selectedusing a search over the price grid with step 0.02. Theresulting two-price policies for the cases of myopicand fully strategic consumers are shown in the toptwo plots of Figure 5. Similarly to the fully dynamiccase, we observe that the first period prices in amarket with strategic consumers are lower. However,the company prefers to keep prices higher in thesecond period when supply levels are high. This canbe explained by the desire of the company todiscourage waiting on the part of strategic consum-ers. We also observe very little price flexibility forsupply levels above 40%. The ratio of expected reve-nues of the pre-announced policies over fully dy-namic ones is shown in the bottom left plot of Figure5. When consumers are myopic, we see that pre-announced policies always perform worse than fullydynamic ones (up to 4% for low supply levels andmore than 1% for high supply levels). For the case ofstrategic consumers, we also see that pre-announcedpricing policies underperform when supply levelsare low. However, if supply levels are higher than

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Figure 5 Prices in Preannounced Two-Price Policy and the Ratios of Expected Revenues of the Preannounced Two-Price Policies Over the Fully DynamicOnes for the Myopic and Fully Strategic Cases as Functions of the Supply Level s = Y/N for T = 10,000


approximately 50%, the pre-announced policy outper-forms the fully dynamic one (up to 9% for supplylevels above 70%). This finding is in agreement withthose of Aviv and Pazgal (2008), who also observedthat announced pricing policies can be advantageousto the seller compared with contingent pricing schemes(although this conclusion was obtained under differ-ent assumptions). A study of pre-announced pricingstrategies in a model setting similar to ours is aninteresting topic for future research.

5.2. Implications of Strategic Consumer Behavior onOptimal Initial Capacity DecisionsIn this subsection, we assume that the initial marketsize N is fixed but the company can select the amountof initial capacity Y at the variable cost c per unit. Theoptimal Y is then selected to maximize the expectedpresent value of profit at time 0:

maxY¼0;...;N

Rb;Y=Nð0;NÞ � cY� �

: ð27Þ

Different values of c will result in different optimalsolutions, which we denote as Yb(c). The correspond-ing optimal profits are denoted as pb(c). All resultsdiscussed below use the same parameter settings as inthe previous subsection.

5.2.1 Effects of Strategicity on Profits and OptimalProcurement Decision When it is Correctly Takeninto Account. The top left plot of Figure 6 shows theratios of the optimal expected profit pb(c) for differentvalues of strategicity parameter b to the optimalexpected profit p0(c) in the myopic case as functions ofcA[0, 10]. We examine the same values of b as in theprevious subsection. The main conclusion is that the

impact of consumer strategicity can be quite substantial,exceeding 25% of profits if customer behavior changesfrom myopic to fully strategic, even if the company cancorrectly take it into account. On the other hand, whencustomers are only moderately strategic (bT/10 5 0.1),then the impact is o5% of profits in the entire rangeof c. The strategicity impact is stronger for the lowervalues of unit cost c, but becomes smaller as capproaches 10. As suggested in the previous sub-section, the correct way for the company to respond is todecrease the initial capacity, thus limiting strategicconsumer behavior. We can see this from the topright plot of Figure 6, which shows the optimal initialcapacity for different values of strategicity parameterb as functions of cA[0, 10]. For lower values of c theoptimal initial capacity in the strategic case is up toapproximately 30% lower than in the myopic case.However, when c is high and the optimal Y is small tostart with, the difference between myopic andstrategic case also becomes small since customershave only limited opportunities for strategic behavior.

5.2.2 Possible Loss of Profitability if Strategicity isIgnored. If the company ignores strategic consumerbehavior both in pricing and capacity decisions, theimpact on profits can be disastrous. The bottom leftplot of Figure 6 shows the actual expected profitsR0b;Y0ðcÞ=Nð0;NÞ � cY0ðcÞ resulting from using the

pricing policy p0;sð�Þ and the initial capacity decisionY0( � ) (both optimal for the myopic case) when thecustomers are in fact strategic as functions of c fordifferent values of b. Some of the worst situations – nearzero or negative profit – are when the unit cost is in therange [3,5] and near 10. One may be interested in howmuch of this dramatic drop in profit is due to pricing.

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� = 1.0�T = 0.1�T /10 = 0.1

� = 1.0�T = 0.1�T /10 = 0.1

Figure 6 The Ratio of Optimal Expected Profits pb(c)/p0(c), the Optimal Initial Capacity yb(c), the Actual Expected Profit Resulting from Using the MyopicCase Policies when Customers are Strategic, and the Ratio of the Actual and Optimal Expected Profits p0

bðcÞ=pbðcÞ as Functions of Unit Cost c forDifferent Values of b and T = 1,000,000


5.2.3 Contribution of Capacity vs. PricingDecisions in Proper Response to Strategicity. Wealso examine the actual expected profit for the casewhen the capacity is selected optimally under therealization that the pricing policy will result inrevenues R0

b;Y=Nð0;NÞ rather than Rb;Y=Nð0;NÞ:

p0bðcÞ ¼ max

Y¼0;...;NR0b;Y=Nð0;NÞ � cY

n o:

In fact, the company would be unable to realizethat the expected revenue is R0

b;Y=Nð0;NÞ under itsincorrect myopic model of demand, but the ratio p0

bðcÞ=pbðcÞ can be used to gauge how much of the op-timal profit must be attributed to using the correctpricing policy. This ratio is shown in the bottom rightplot of Figure 6 as a function of c for different valuesof b. When the unit cost is low, the pricing policycorrectly accounting for the strategic behavior con-tributes up to around 10% of the optimal profit in thefully strategic case. When the unit cost is high, acorrect capacity decision appears to be more impor-tant since the profit ratio attained by it is close to 1.This can be explained by generally low levels of theoptimal capacity when the unit cost is high.

Overall, the results of this subsection suggest thattaking into account strategic behavior of consumersis very important in capacity decisions. Moreover, aproper capacity choice can partially compensate forconsumer strategicity.

6. Conclusions and Directions forFuture Research

The paper presents a new, inherently stochastic,game-theoretical dynamic pricing model for a finitepopulation of strategic customers. This model explic-itly incorporates the strategic nature of consumers inthe decision-making process. The model falls withinthe general class of stochastic dynamic games butgeneralizes stochastic consumer behavior in a man-ner that, to our knowledge, has not previously beenachieved in the literature. We demonstrate theexistence of a unique subgame-perfect equilibriumpricing policy under reasonable assumptions, provideoptimality conditions, and explore some of the struc-tural properties of solutions. We also discuss impli-cations of strategic consumer behavior for overallperformance as well as pricing and procurementdecisions. The proposed computational procedureallows evaluation of this model on problems of real-istic size.

The future research related to this model may in-clude the following topics:

(1) considering situations when the parameters nand/or y are not known to the company or cus-tomers exactly, and describing the situations in

which it would be beneficial for the company toreveal the information about n and/or y to thecustomers; and

(2) a dynamic game model of interactions betweenthe company and strategic consumers in thepresence of consumer and company learning.

AcknowledgmentThis research was supported by Natural Sciences andEngineering Research Council of Canada (grant numbers261512-04 and 341412-07). The authors thank the Depart-ment and Senior Editors, and the Referees for theirconstructive suggestions, which helped to better explainthe model and improve the results. The third author sendsprayers of utmost gratitude to God in Whom he soughtinspiration during this work.

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Supporting InformationAdditional supporting information may be found in theonline version of this article:

Appendix S1. Assumptions, Notation, and Proofs

Please note: Wiley-Blackwell is not responsible for thecontent or functionality of any supporting materials sup-plied by the authors. Any queries (other than missingmaterial) should be directed to the corresponding author forthe article.


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