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Vol. 29, No. 4, July–August 2010, pp. 671–689issn 0732-2399 �eissn 1526-548X �10 �2904 �0671

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doi 10.1287/mksc.1090.0547©2010 INFORMS

A Customer Management Dilemma: When Is ItProfitable to Reward One’s Own Customers?

Jiwoong Shin, K. SudhirYale School of Management, Yale University, New Haven, Connecticut 06520

{[email protected], [email protected]}

This study attempts to answer a basic customer management dilemma facing firms: when should the firm usebehavior-based pricing (BBP) to discriminate between its own and competitors’ customers in a competitive

market? If BBP is profitable, when should the firm offer a lower price to its own customers rather than to thecompetitor’s customers? This analysis considers two features of customer behavior up to now ignored in BBPliterature: heterogeneity in customer value and changing preference (i.e., customer preferences are correlated butnot fixed over time). In a model where both consumers and competing firms are forward-looking, we identifyconditions when it is optimal to reward the firm’s own or competitor’s customers and when BBP increases ordecreases profits. To the best of our knowledge, we are the first to identify conditions in which (1) it is optimal toreward one’s own customers under symmetric competition and (2) BBP can increase profits with fully strategicand forward-looking consumers.

Key words : behavior-based pricing; customer heterogeneity; stochastic preferences; forward-looking customers;forward-looking firms; customer relationship management; competitive strategy; game theory

History : Received: October 22, 2007; accepted: September 28, 2009. Published online in Articles in AdvanceFebruary 16, 2010.

1. IntroductionOver the past two decades, firms have made mas-sive investments of their organizational resources(human, technical, and financial) into building infor-mation infrastructures that store and analyze dataabout customer purchase behavior. Armed with suchdata, firms can pursue detailed microsegmentationand customer management strategies. A routinelyused strategy is behavior-based pricing (BBP) suchthat firms offer different prices to different customersbased on their past purchase behavior.1 However,firms differ in whether they offer a lower price to theirown customers or competitors’ customers. Catalogretailers for items such as apparel typically send spe-cial discount “value” catalogs to existing customers,whereas magazines and software firms often offer dis-counts to buyers of competing products in the form oflowered introductory prices. Would it be more prof-itable for a magazine to offer a better price for asubscription renewal as opposed to a new subscrip-tion? Should a wireless carrier offer lower rates to itshigh-volume customers who use more minutes or toits new customers? Should a hotel, airline, or retaileroffer better rates for high-value, loyalty card membersor to new customers it seeks to acquire? This paper

1 The literature variously refers to it as behavior-based price dis-crimination (Fudenberg and Tirole 2000) or pricing with customerrecognition (Villas-Boas 1999, 2004).

answers a basic customer management dilemma forfirms practicing BBP: when should a firm use BBP todiscriminate between its own and competitors’ cus-tomers in a competitive market? If BBP is profitable,when should the firm offer the lower price to its owncustomers rather than its competitor’s customers?Although this customer management dilemma is

ubiquitous across industries, little consensus existsabout the correct answers to these questions. The aca-demic literature differs sharply from the practitionerintuition on these questions. O’Brien and Jones (1995,p. 76) epitomize the conventional wisdom of manypractitioners: “to maximize loyalty and profitability, acompany must give its best value to its best customers.As a result, they will then become even more loyaland profitable” (emphasis our own). Essentially, prac-titioners argue that firms rewarding their own cus-tomers leads to a virtuous cycle: the firm rewardsits current customers with better value propositions,which makes it optimal for those customers to deepentheir relationship with the firm, which ultimatelyincreases firm profitability (Peppers and Rogers 2004).However, academic literature is skeptical of this

conventional wisdom. In most existing models of BBP,the optimal choice is not to offer current customersa lower price. When the firm can price discriminateon the basis of consumers’ past purchase behavior,it should charge higher prices to existing customers,who already have revealed their higher willingness

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Shin and Sudhir: A Customer Management Dilemma672 Marketing Science 29(4), pp. 671–689, © 2010 INFORMS

to pay for the product (otherwise, they would nothave purchased from it), relative to its competitor’scustomers. That is, customers’ past purchases revealtheir relative preference for each firm. Furthermore, ifconsumers recognize the possibility that they will bepenalized in the future, they can alter their behaviorto reduce the ability of firms to infer their true pref-erences. Such strategic behavior by consumers andincreased competition to poach others’ customers maymake BBP unprofitable. Fudenberg and Villas-Boas’s(2006, p. 378) succinct summary of existing literatureon BBP notes that “the seller may be better off ifit can commit to ignore information about buyer’spast decisions � � �more information will lead to moreintense competition between firms.” The overall les-son is that BBP cannot be profitable if both con-sumers and firms are rational and forward-looking;furthermore, it is never optimal to reward one’s owncustomers.In this paper, we resolve this discrepancy between

theoretical predictions and practitioner intuitions byincorporating two simple but important features ofcustomer behavior into our analytical model. Ourresults nest these two viewpoints and identify condi-tions in which either practitioner intuition or currenttheoretical results hold, even when both consumersand firms are strategic and forward-looking. We nowdescribe the two key features that we add.

1.1. Customer Value HeterogeneityNot all customers are equally valuable to firms. Somepurchase more than others or contribute more to afirm’s profits. Widespread empirical support in var-ious categories confirms the 80/20 rule—that is, theidea that a small proportion of customers contributesto most of the purchases and profit in a category(Schmittlein et al. 1993). Such customer heterogene-ity is critical for capturing the practitioner notion ofa “best” customer—a feature that generally does notappear in current analytical models.Modeling customer heterogeneity in purchase quan-

tity also allows firms to gain a new dimension of infor-mation about consumers from their purchase historydata. In addition to the usual “who they bought from”data, which provide horizontal information about therelative preferences of consumers, data pertaining to“how much they bought” provide vertical informa-tion about the relative importance of consumers tothe firm.Vertical information about quantity not only pro-

vides additional information to firms but also gen-erates an information asymmetry among otherwisesymmetric firms. With these data, a firm can identifythe “best customers” only among its own customers,but not among its competitor’s customers. Althoughthe firm knows there is a mix of high- and low-volume customers among its competitor’s customers,

it cannot know the exact identity of the customer.Existing models assume that all customers buy justone unit of the product and that the market is fullycovered, which implies that if a consumer does notbuy from one firm, it must have bought from theother, and therefore, there is no information advan-tage about one’s own customers. That is, existingmodels abstract away from a major reason that firmsinvest in customer relationship management (CRM)in practice: to obtain an information advantage overcompetitors about existing customers. This informa-tion asymmetry emerges as critical for obtaining morecomplete insights into BBP.

1.2. Stochastic PreferencesWe also recognize that consumers’ preferences can bestochastic. Consumer preference for a product maychange across purchase occasions, independent ofthe marketing mix or pricing, because their needsor wants depend on the specific purchase situation,which changes over time (Wernerfelt 1994). This isrelevant in many choice contexts. For example, forstore choice, consumers’ preferred geographic loca-tions likely vary across time; a customer may gener-ally prefer Lowe’s for purchasing home improvementproducts because a store is closer to her home andoffers superior quality offerings. However, she mayvisit the Home Depot store that is on her way homefrom work. A similar logic may apply to consumers’choices of hotels; even if a customer prefers Marriottin general, he may find that a Sheraton satisfies hisneeds better on a particular trip because of its prox-imity to a conference venue.2

Previous research in BBP (e.g., Caminal andMatutes1990, Fudenberg and Tirole 2000) recognizes the possi-bility of stochastic preference, but only for the extremecase in which customers’ preferences are completelyindependent over time. Although the independenceassumption makes the analysis tractable, it is notinnocuous in a customer management context. Withcompletely independent preferences, customers’ pastpurchases are of no use in predicting future purchases.For past purchase information to be valuable to firmsin future price-setting efforts, those preferences mustbe correlated across time. Our formulation of stochas-tic preferences will accommodate such correlation inpreferences across time, a characteristic of most real-world markets.With these two features included, we identify con-

ditions in which (1) behavior-based pricing is prof-itable in a competitive market, even when firms and

2 It is important to note that we are not assuming that consumerchoice changes over time. Consumers’ purchase choice is still anendogenous decision. For example, even if the Sheraton is prefer-able because of its proximity to a conference venue, a consumermay still stay in the Marriott if it offers a special deal or some otherincentive.

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Shin and Sudhir: A Customer Management DilemmaMarketing Science 29(4), pp. 671–689, © 2010 INFORMS 673

consumers are strategic and forward-looking; and (2)firms should offer lower prices to their own best cus-tomers or to their competitor’s customers. We findthat either sufficiently high heterogeneity in customervalue or stochasticity in preference is sufficient forBBP to increase firm profits. However, both sufficientheterogeneity in customer value and stochastic prefer-ence are required for firms to reward their own bestcustomers; if both elements do not exist, they shouldreward their competitor’s customers.The rest of this article is organized as follows: Sec-

tion 2 describes the related literature. Sections 3 and 4describe the model and the analysis, respectively. Sec-tion 5 concludes our paper.

2. Literature ReviewOur research connects several interrelated areas ofmarketing and economics. Our primary contributionapplies to BBP (for a review, see Fudenberg andVillas-Boas 2006). In terms of model setup, our workcomes closest to Fudenberg and Tirole’s (2000) anal-ysis of a two-period duopoly model with a Hotellingline and in which customers and firms are forward-looking. Villas-Boas (1999) extends Fudenberg andTirole (2000) to an infinite period model with overlap-ping generations of customers. In both studies, BBPleads to a prisoner’s dilemma that induces lower prof-its than the case in which firms credibly commit notto use past purchase information, and it is never opti-mal to reward the firm’s own customers. Pazgal andSoberman (2008) replicate these results but show thatprofits can increase if only one of the firms practicesBBP, although it is still not profitable to reward owncustomers.3

Other related literature pertains to switching costs.Firms can price discriminate between their ownlocked-in customers and customers locked-in to arival through switching costs. Chen (1997) analyzesa two-period duopoly model with heterogeneousswitching costs among customers. Similar to BBP,firms always charge the lower price to their competi-tor’s customers, and the total profits are lower thanif they could not discriminate. Taylor (2003) extendsChen’s model to competition between many firmsand continues to find that firms charge the lowestprices to new customers. Shaffer and Zhang (2000)also consider a static game, similar to the secondperiod of Fudenberg and Tirole’s (2000) two-periodmodel, but they allow switching costs to be asymmet-ric across the consumers of the two firms. With sym-metric switching costs, firms always charge a lower

3 Fudenberg and Tirole (1998) also identify conditions in which afirm selling successive generations of durable goods should rewardcurrent consumers or new customers in a monopoly setting.

price to their rival’s consumers, whereas with asym-metry, the firm with lower switching costs may offerlower prices to own customers, although it is neveroptimal for both firms to charge lower prices to theirown customers. In a comprehensive review, Farrelland Klemperer (2007, p. 1993) thus conclude thatswitching-cost literature finds it hard to explain dis-crimination in favor of own customers.Our work also aligns closely with the theoret-

ical and empirical literature on targeted pricing.4

Although targeted pricing by a monopolist alwaysleads to greater profits, the competitive implicationsin oligopoly markets are subtle. Thisse and Vives(1988) and Shaffer and Zhang (1995) show that pricediscrimination effects get overwhelmed by competi-tion effects in targeted pricing, leading to a prisoner’sdilemma. Chen et al. (2001) further note that target-ing accuracy can moderate the profitability of firms.They recognize that consumer information is noisy;hence, targeting is imperfect. At low levels of accu-racy, the positive effect of price discrimination onprofit is stronger, whereas at high levels, the nega-tive effect of competition on profit is stronger. Overall,profits are greatest at moderate levels of accuracy.Empirical literature on this topic (Rossi et al. 1996,

Besanko et al. 2003, Pancras and Sudhir 2007) indi-cates that firms can improve profits through targetedpricing if they use customers’ past purchase historiesto infer preferences. Interestingly, the probit and logitempirical models in these papers that support theprofit improvement achieved through targeted pric-ing include the two key consumer features that wemodel, namely, customer heterogeneity in categoryusage (through the brand intercept terms relative tothe outside good) and changing customer preferences(through the normal and extreme value distributionsin consumer utility).We also find connections with our work in adverse

selection models in financial markets (Sharpe 1990,Pagano and Jappelli 1993, Villas-Boas and Schmidt-Mohr 1999). In contrast with the focus on a customer’srelative preference (horizontal preference informa-tion) in BBP, this literature stream models verticalinformation about own customer types (ability torepay loans), arguing that firms use this informationasymmetry to determine future loans to customers.5

Our model combines two aspects of information

4 Unlike BBP models, targeting models are static, and firms dis-criminate among consumers on the basis of perfect or noisy infor-mation about their underlying preferences. In BBP, one explicitlymodels how past purchase behavior provides firms with informa-tion about preferences, which are then used to determine discrimi-natory prices in the future. In contrast, targeting literature does notmodel how firms obtain preference information.5 Several related theoretical papers in marketing accommodatecustomer heterogeneity in purchase quantities (Kim et al. 2001,

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about customers’ past behavior; that is, (1) horizontalpreference information, which is the focus in BBP and(2) vertical information, which represents the focus ofadverse selection literature in economics and finance.Finally, our work relates to CRM literature in mar-

keting (for a general review, see Boulding et al. 2005),which investigates whether firms should use acquisi-tion or retention strategies (e.g., Syam and Hess 2006,Musalem and Joshi 2009). A large body of empiricalresearch also uses data about customers’ purchase his-tory to estimate and show significant heterogeneity incustomer lifetime value (Blattberg and Deighton 1996,Jain and Singh 2002, Venkatesan and Kumar 2003,Gupta et al. 2004).

3. ModelWe follow Fudenberg and Tirole (2000) in consider-ing a two-period standard Hotelling model with tworetailers geographically located on the two ends of aunit line. We denote the retailer located at point 0 asretailer A and the retailer at point 1 as retailer B. Theretailers sell an identical nondurable good (e.g., icecream, gasoline), from which the consumer receives agross utility of v. We assume that v is large enoughthat in equilibrium, all consumers purchase the prod-uct from one of the retailers.6 We also assume theretailer’s marginal cost for the product is constant andnormalize it to zero without loss of generality. Themarket contains two periods, and consumers makepurchase decisions in both. We denote the pricescharged by retailers A and B in period t as pAt and p

Bt ,

respectively.We adapt the Hotelling model to capture our

two new features—customer value heterogeneity andunstable preference. First, to model customer hetero-geneity parsimoniously in terms of value to the firm(i.e., the 80/20 rule), we assume there are two types ofconsumers in the market (j ∈ �LH�): a high-type seg-ment (H ) that purchases q units of the good in eachperiod and a low-type segment (L) that purchasesonly one unit.7 The proportions of H and L types in

Kumar and Rao 2006) and costs to serve (Shin 2005). However,these studies do not address behavior-based pricing—that is, offer-ing different prices to own versus competitor’s customers.6 The assumptions of exogenous locations at the ends of theHotelling line and full-market coverage are standard (e.g.,Villas-Boas 1999, Fudenberg and Tirole 2000); they ensure no dis-continuities in the demand function, which is a general problemin Hotelling-type models in which the location choices are endoge-nous (d’Aspremont et al. 1979).7 Ideally, we would model customer lifetime value throughmultiple-period purchases. However, in a two-period framework,we abstract and capture the spirit of the 80/20 rule parsimoniouslythrough different purchase quantities (q) and keep the model ana-lytically tractable. In reality, q may arise from multiple purchasesacross each period.

the market are ��>0� and 1−�, respectively. Both H -and L-type consumers’ geographic locations or pref-erences (denoted �) are uniformly distributed alongthe Hotelling line, �∼U�01�. We normalize the sizeof the market to one, such that a j-type (j ∈ �LH�)consumer located at � receives the following utilitiesfrom purchasing the product:

Uj�pAt pBt � ��

=Qj�v− pAt �− � if purchase from retailer A

Qj�v− pBt �− �1− �� if purchase from retailer B

where QL = 1 QH = q�We treat q as exogenous; some consumers need moreof the product than others for reasons exogenous tothe model. For example, a household with two peo-ple will need two cones of ice cream rather than theone cone demanded by a single-person household; aperson who lives farther from work will buy gasolinemore frequently than will someone who lives closeto work.Second, to capture the idea of stochastic prefer-

ence, we allow the preference for the retailer tochange across periods. As we discussed previously,this change in preference may occur as a result of achange in the consumer’s geographic location (e.g.,shopping trip starts from home or office). Similar toKlemperer (1987), we allow the customer location inthe second period to change with probability � butremain the same as in the first period with probabil-ity 1−�. When the location changes in the secondperiod, the new location comes from a uniform distri-bution, �∼U�01�. Let �1 be the first-period location.Then the second-period location �2 = � with proba-bility �, or �2 = �1 with probability 1− �. Therefore,customer locations correlate over the two periods, andthe expected location of the second period for a cus-tomer is E��2� = �� + �1 − ��1. If preferences arestable (�= 0), �2 = �1 all the time. In contrast, when�= 1, there is maximum instability, because consumerlocations are completely independent.8

In the first period, the retailers A and B eachoffer a single price, pA1 and pB1 , respectively, to all

8 We consider an alternative specification for preference correla-tion: the second-period location, �2, is the weighted average of thefirst-period location and the external situational shock (�2 = �1−�� ·�1+��). The advantage of this approach is that we can capture thelocal movement of customers (i.e., the second location is a func-tion of the first location). However, this specification leads to anabundance of corner solutions, which makes the analysis extremelytedious without adding insight. Nevertheless, we do find a param-eter space in which the “reward own customers” and “increasedprofit” results hold even in this specification, which suggests thatour key results are robust to alternative specifications. An alter-native formulation more amenable to empirical work would be anormal distribution that allows for serial correlation across time.

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Figure 1 Sequence of Events

Time

Retailers A and Boffer one price(p1

A, p1B) to all

consumers

• A consumer’s relative location is revealed toboth retailers

• A consumer’s current supplier has superiorinformation relative to the competitor aboutthe customer’s type

First-periodlocations (�1)endowed to

each consumer

Consumers visitone retailer and

purchase

Retailers offerthree prices

• Price for its competitor’scustomers (p2

AO, p2BO)

• Price for its own low-typecustomers (p2

AL, p2BL)

• Price for its own high-typecustomers (p2

AH, p2BH)

1st period 2nd period

Consumer visitsone retailer andmakes purchase

Consumers learnsecond-period location�2 = � with prob. �

�2 = �1 with prob. 1–�

consumers. Consumers purchase from either retailer,depending on which choice is optimal for them. Notethat we allow consumers to be strategic and forward-looking; they can correctly anticipate how their pur-chase behavior will affect the prices they will have topay in the future, and therefore, they maymodify fromwhom they buy in the first period to avoid being takenadvantage of by the retailers in the second period.In the second period, the retailer can distinguish

among three types of customers: (1) customers whobought q units from it, (2) customers who bought oneunit from it, and (3) customers who bought no units(and therefore must have bought from the competi-tor). Based on the customer’s purchase behavior in thefirst period, retailers can infer two types of informa-tion about customers: horizontal information abouttheir relative location proximity to A or B, and verticalinformation about the type of customer, based on thequantity purchased. Although the horizontal informa-tion is symmetric to both retailers, because the mar-ket is covered, the vertical information is asymmetric,in that a firm knows this information only about itsown customers. Therefore, customer heterogeneity inpurchase quantity confers an endogenous informationadvantage on retailers about their current customersfor the second period.9 This point represents a keydeparture from existing models that exclude hetero-geneity in purchase quantity.The retailer knows that consumers who purchased

from it in the first period preferred it but also rec-ognizes that no guarantee ensures they will continue

9 Regarding the idea that firms have different amounts of infor-mation about their own customers and noncustomers, Villas-Boas(1999) incorporates information asymmetry by assuming customerscan be either new or competitor’s customers. In contrast, our infor-mation asymmetry comes from the customer type (quantity). Also,Villas-Boas and Schmidt-Mohr (1999) consider the impact of asym-metric information between firms and consumers.

to prefer it in the second period, because consumerpreferences are not fixed. However, as we show in§4.1, a consumer who purchases from a retailer in thefirst period is probabilistically more likely to preferthe same retailer in the second. Thus, retailers pos-sess useful probabilistic information about the rela-tive preferences of customers in the second period,given purchase information in the first period. Weformally analyze relative location more precisely in§4.1, but for now, it suffices to say that first-periodchoice reveals information about relative location,even when consumer locations are stochastic.In the second period, on the basis of its information

set, each retailer offers three different prices to thethree groups of customers: (1) a poaching price to thecompetitor’s customers (pAO2 , pBO2 ), (2) a price for itsown L-type customers (pAL2 , p

BL2 ), and (3) a price for its

own H -type customers (pAH2 , pBH2 ). Consumers decidewhere to purchase in the second period after observ-ing all these prices. Figure 1 summarizes the outlineof the game, and Table 1 summarizes the notation weuse herein.

4. AnalysisWe solve the proposed game using backward induc-tion, solving first for the second-period equilibriumstrategies and then for the first-period strategies.Before we solve the game, we define retailer turf andformally demonstrate the relevance of customer pur-chase history information, even in the presence ofunstable customer preference.

4.1. Retailer Turf and the Value of PurchaseHistory Information

For any pair of first-period prices (all consumers pur-chase and both retailers have positive sales), there

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Table 1 Terminology

Qj Purchase quantity of customer type j ∈ �L�H�, whereQL = 1, QH = q.

pi1 Retailer i ∈ �A�B�’s first-period price to all consumers.

piL2 Retailer i ∈ �A�B�’s second-period price to its L-type

customers.piH

2 Retailer i ∈ �A�B�’s second-period price to its H-typecustomers.

piO2 Retailer i ∈ �A�B�’s second-period price to its

competitor’s customers.� Probability that consumer location changes in period 2,

� ∈ �0�1�.� If consumer location changes, consumer location in

period 2, �∼ U�0�1�.�t Consumer’s preference or geographical location at

period t ∈ �1�2�. In particular, �2 = � withprobability � and �2 = �1 with probability 1− �.

�̃j1 First-period threshold for customer type j ∈ �L�H�, suchthat all consumers of type j with � ≤ �̃j1 purchase fromretailer A and the rest purchase from retailer B.

�̃Aj2 Second-period threshold for customer type j ∈ �L�H�,who was on retailer A’s turf in the first period, suchthat all consumers of type j with � ≤ �̃Aj2 repeatpurchase from retailer A and the rest switch to B.

�̃Bj2 Second-period threshold for customer type j ∈ �L�H�,who was on retailer B’s turf in the first period, suchthat all consumers of type j with � ≥ �̃Bj2 repeatpurchase from retailer B and the rest switch to A.

PrAj Probability of repeat purchasing in second period fromretailer A for customer type j ∈ �L�H� who was on A’sturf in the first period.

PrBj Probability of repeat purchasing in second period fromretailer B for customer type j ∈ �L�H� who was on B’sturf in the first period.

�it Profit of retailer i ∈ �A�B� in period t . Total profit for

retailer i is �i =�i1 + ��i

2.� Discount rate.

EA�Uj2 � �1 = �̃j1� Expected second-period utility that marginal customer of

type j ∈ �L�H� gets when he or she purchases fromretailer A. A marginal customer is indifferent betweenpurchasing a product from A or B in the first period.

EB�Uj2 � �1 = �̃j1� Expected second-period utility that marginal customer

gets when he or she purchases from retailer B.

is a cutoff �̃j1 (j ∈ �LH�), such that all consumersof type j whose � ≤ �̃

j1 decide to purchase from

retailer A, and the rest purchase from retailer B. Fol-lowing Fudenberg and Tirole (2000), we say that con-sumers to the left of �̃j1 lie in retailer A’s turf andthose on the right lie in retailer B’s turf to emphasizeconsumers’ relative locations on the Hotelling line.For consumers who purchased from retailer A in

the first period, retailer A offers the prices pAH2 , pAL2to H - and L-type customers, respectively, whereasretailer B offers the price pBO2 to both types at thebeginning of the second period. Similarly, among con-sumers who purchased from retailer B in the first

period, retailer B charges pBH2 , pBL2 , whereas retailer A

charges pAO2 .10

Given preference correlation in the second period(E��2� = �� + �1 − ��1, where � ∈ �01� and � ∼U�01�), the conditional probability that consumerswill locate in a certain range of the Hotelling line,given their first-period purchase choice of A or B, isas follows:

Pr��2 ≤ x � �1 ≤ �̃1�=

�1−��+�x if �̃ ≤ x

�1−�� x�̃1

+�x if �̃1 > x�(1)

Pr��2 > x � �1 > �̃1�

=

�1−��+��1− x� if �̃1 ≥ x

�1−�� 1− x1− �̃1

+��1− x� if �̃1 < x�(2)

Note that Equations (1) and (2) both indicate thegeneral conditional probabilities of the second-periodlocations, assuming that a consumer purchases fromretailer A or B in the first period. Therefore, we offerthe following lemma:

Lemma 1 (Value of Purchase History Informa-tion). When stochasticity in preference is not extreme(�≤ 1/�2�1− z��, where z is the first-period market share),a consumer who purchases from retailer j (j ∈ �AB�) inperiod 1 is more likely to stay in the same retailer’s turfrather than move to the competitor’s turf.

Proof. See the appendix.This lemma shows that the customer’s first-period

purchase is probabilistically informative of second-period preference, because she or he is more likelyto remain on the same turf unless preference isextremely stochastic. For example, consider the spe-cific case of �̃1 = 1

2 and �= 12 . A consumer on A’s turf

relocates to the same turf with a probability of 34 andrelocates to B’s turf with only a probability of 14 .

11 Fig-ure 2 illustrates the redistribution of customer loca-tions in the second period for four H -type and L-typefirst-period customers of A. Each letter represents oneH - or L-type consumer. Probabilistically, three H - andL-type consumers remain relatively close to retailer

10 When we analyze the symmetric case, “no poaching” does notarise in equilibrium, but in the asymmetric case, when �̃1 < 1/4,A’s turf is very small and consists only of consumers with a strongpreference for A, and retailer A can charge a monopoly price butnot lose any sales to retailer B.11 Lemma 1 focuses on the horizontal information about the cus-tomer’s first-period relative location, as observed from past pur-chases. Past purchases also reveal vertical information about thecustomer quantities, which is always valuable, irrespective of thedegree of preference stochasticity.

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Figure 2 Redistribution of Consumer Locations for Consumers on Retailer A’s Turf

�1

�1

0 1

H

L L L

0 1

H H H H

L L LL

Second period

First period

HHH

L

~

~

First-period firm A′s turf


A, and one of each type changes his or her locationtoward retailer B in the second period, as indicatedby the arrows in Figure 2.Note that retailers do not know the exact locations

of their own customers in the first period, but theyknow their relative location, in that these customersmust be somewhere on their turf. In the secondperiod, retailers still do not know the exact locationsof their first-period customers, but they can assess theaverage probability of the relative proximity (closer toA or B� of their first-period customers.When the turf is symmetric (�̃1 = 1

2 ), a customer isalways more likely to stay on the same turf for all � ∈�01�. When � = 1 (complete independence in pref-erences), not surprisingly, they are equally likely toappear on either turf in the second period. Thus, pastpurchases are informative of future preferences for all� ∈ �01�.

4.2. Second PeriodA consumer on retailer A’s turf purchases againfrom retailer A in the second period if and onlyif Qjv − Qjp

Aj2 − �2 ≥ Q

jv − QjpBO2 − �1 − �2� ⇔

�2 ≤ �1+ �QjpBO2 −QjpAj2 ��/2 ≡ �̃Aj2 , where j ∈ �LH�

and QL = 1, QH = q. Otherwise, the consumerswitches to retailer B. We denote the repeat pur-chase probability of the high and low types forretailer A as PrAH and PrAL, respectively. Thus, PrAH =Pr��2 ≤ �1 + q�pBO2 − pAH2 ��/2 � �1 ≤ �̃H1 � and PrAL =Pr��2 ≤ �1 + �pBO2 − pAL2 ��/2 � �1 ≤ �̃L1 �, where �̃j1 is thefirst-period cut-off for type j ∈ �LH�. Similarly, therepeat purchase probabilities for retailer B are PrBH =Pr��2 > �1+ q�pBH2 − pAO2 ��/2 � �1 > �̃H1 � and PrBL =Pr��2 > �1+ pBL2 − pAO2 �/2 � �1 > �̃L1 �.

The second-period profits of retailers A and B thenare

A2 = pAH2 �q�̃H1 PrAH + pAL2 �1−��̃L1PrAL

+ pAO2{�q�1− �̃H1 ��1−PrBH�+ �1−��1− �̃L1 ��1−PrBL�

}

B2 = pBH2 �q�1− �̃H1 �PrBH + pBL2 �1−��1− �̃L1 �PrBL

+ pBO2{�q�̃H1 �1−PrAH�+ �1−��̃L1 �1−PrAL�

}�

(3)

Each retailer’s second-period demand consists ofthree parts. For example, retailer A enjoys demandfrom (1) its own previous H -type customers (�̃H1 ),who continue to be on the retailer’s turf in the sec-ond period (with probability PrAH ) and pay a pricepAH2 ; (2) its own previous L-type customers (�̃L1 ), whocontinue to be on the retailer’s turf in the secondperiod (with probability PrAL) and pay a price pAL2 ;and (3) a mix of the competitor’s previous high- andlow-type customers (�1 − �̃H1 � + �1 − �̃L1 �) who haveshifted to the retailer’s turf (with probability 1−PrBj ,where j ∈ �LH�) and pay a price pAO2 .Using Equations (1) and (2), we can obtain the

second-period prices by solving the retailers’ first-order conditions. We only look for the pure strategyequilibrium in a symmetric game. In the first-periodanalysis, we show that the symmetric outcome, inwhich both firms charge the same price in the firstperiod, is indeed the equilibrium solution.

Proposition 1. Suppose there exists a nonzero � por-tion of H -type customers in the market (0<�< 1).(a) Reward Competitor’s Customers: When pref-

erence stochasticity is low(� < !��q�

), there exists

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Figure 3 Prices When �= 12 and q = 2

iOp2

iHp2

iLp2

iOp2

iHp2

iLp2

12( , 2)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.2

0.4

0.6

0.8

1.0

1.2Profit Profit

0.2 0.4 0.6 0.8 0.90 0.92 0.94 0.96 0.98 1.00

Reward competitors’ customers Reward own customers

��

� 12( , 2)�

a symmetric pure strategy equilibrium in second-periodprices, such that retailers charge

pAH2 = pBH2 = 12q

+ �2+��1+ �q− 1��6�2−��1+ �q2− 1��

pAL2 = pBL2 = 12+ �2+��1+ �q− 1��6�2−��1+ �q2− 1�� and

pAO2 = pBO2 = �2+��1+ �q− 1��3�2−��1+ �q2− 1��

Prices follow an ordinal relationship, piO2 ≤ piH2 ≤ piL2 , wherei ∈ �AB�; that is, the competitor’s customers receive thelowest price.(b) Reward One’s Own Best Customers: When pref-

erence stochasticity is sufficiently high ��≥ !��q�� andconsumer heterogeneity in purchase quantity is sufficientlyhigh �q > q∗�, there exists a symmetric pure-strategy equi-librium in second-period prices, such that retailers charge

pAH2 = pBH2 = 2−�2q�

+ �2+��1+ �q− 1��6��2−��1−��+ q2��

pAL2 = pBL2 = 12+ �2+��1+ �q− 1��6��2−��1−��+ q2�� and

pAO2 = pBO2 = �2+��1+ �q− 1��3��2−��1−��+ q2��

Prices follow an ordinal relationship, piH2 ≤ piO2 ≤ piL2 , wherei ∈ �AB�; that is, the retailer’s own high-type customersreceive the lowest price.12

12 The exact expressions for !��q� !��q�, and q∗ are provided inthe appendix. For any given 0<�< 1, in a small range �!�� q� <�< !��q�� of �, the pure-strategy equilibrium does not exist. Wefocus our analysis only on the pure-strategy equilibrium area.

Proof. See the appendix.Proposition 1 thus highlights the importance of

the new features that we incorporate in our model.Both customer heterogeneity in quantity and suffi-ciently high levels of preference stochasticity are nec-essary to justify rewarding the firm’s own customers.Otherwise, it is optimal to reward the competitor’scustomers (see Figure 3). Consistent with existingliterature (Fudenberg and Tirole 2000), when thereis no heterogeneity (q = 1) and preferences are sta-ble (�= 0), it is optimal to reward the competitor’scustomers.The intuition for why both customer heterogene-

ity and sufficient preference stochasticity are neededto reward the firm’s own customers is as follows: asLemma 1 shows, in a symmetric market, customersare more likely to stay with the current retailer thanthey are to switch to a rival for all � ∈ �01�. Ifcustomers are all equal in value (i.e., no customerheterogeneity in quantity), retailers take advantageof customers’ revealed preferences from the previouspurchase and charge a higher price to their own cus-tomers while they expend more effort to acquire thecompetitor’s customers through low prices. Althoughretailers understand that some existing customersswitch in the second period (they cannot identifyexactly who; they only know the aggregate proba-bility of customers changing turfs), they find it opti-mal to offer a lower price to their current customersbecause both own and competitor’s customers are allequal in value.However, if customers are heterogeneous, retailers

must assess whether selective retention (rewardingtheir own best customers) is profitable. The value ofthe firm’s own high-type customers is greater than theexpected average value of the competitor’s customers.

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Figure 4(a) Customer Switching When �≤ �� q�� ̃L1 > �̃AL2 and �̃H1 > �̃AH2

0 1

Stay A Switch to B

�2AL

�2�2AH

H-type (�)

L-type

(1–�) ~~~


Figure 4(b) Customer Switching When �≥ �� q�� ̃L1 > �̃AL2 and �̃H1 ≤ �̃AH2

10

H-type (�)

L-type

(1–�)

H1

L1

Stay A Switch to B

�2AL �2

AH�1~

~~


Furthermore, as preference stochasticity increases, themonopoly power that the retailer enjoys from hori-zontal preference weakens. Beyond a certain thresh-old of preference stochasticity, the marginal gain inprofit from retaining high-type customers throughlowered prices becomes greater than the marginalbenefit of poaching a mix of high- and low-type com-petitor’s customers. Therefore, both heterogeneity inquantities and high levels of preference stochasticityare necessary to make rewarding the firm’s own high-type customers an effective strategy.Finally, it is never optimal to offer a better value

to own low-type customers, whose value is less thanthe expected average value of the competitor’s cus-tomers. They always receive the highest price offer.This result echoes practitioners’ assertion that the firmmust reward its own best customers (not all of itscustomers!).13

We illustrate the relationship between preferencestochasticity and actual customer switching behav-ior in the second period in Figure 4. Specifically,

13 Interestingly, we also find a condition in which the average prices,pi_avg2 = �piH2 + piL2 �/2, offered to one’s own customers can be lowerthan the price for the competitor’s customers when both hetero-geneity in quantities and high levels of preference stochasticity exist(i.e., the price for the H -type is sufficiently low) and the portion ofH -type customers is sufficiently small; thus the poaching price (piO2 )increases (note that the poaching price increases as � decreases).For example, when � = 1 and q = 2, we can show that if � < 0�2,then pi_avg2 < piO2 . (See details in the electronic companion to thispaper, available as part of the online version that can be found athttp://mktsci.pubs.informs.org/.) When � = 1, q = 2, and � = 0�1,pi_avg2 = 0�798< piO2 = 0�846. We thank a reviewer and the area editorfor suggesting that we compare the average prices offered to thefirm’s own and competitor’s customers.

Figure 4(a) illustrates the case in which � is low,so retailers reward the competitor’s customers. Thesecond-period cutoff thresholds for both high andlow types (�̃AH2 and �̃AL2 ) shift to the left of thefirst-period threshold, though the shift is greater forlow-type customers who pay the highest price. Theobserved level of switching is greater for both highand low types relative to their intrinsic preferencestochasticity levels.In Figure 4(b), we present the case when � is high,

so retailers reward their own best customers. AnH -type customer (H1) who bought from retailer A inthe first period but gets a shock in the second periodthat moves him closer to retailer B still remains withretailer A. An L-type customer (L1) who bought fromretailer A and receives a shock that makes her evencloser to retailer A instead may switch in the sec-ond period. In effect, the observed level of switchingamong H -types is lower relative to their preferencestochasticity level, because they still repurchase fromthe same retailer even though they are much closerto the alternative. In contrast, the observed level ofswitching among L-type customers is greater thantheir preference stochasticity level. Even if the level ofpreference stochasticity is identical for the two types(i.e., no exogenous relationship between quantity andloyalty), firms’ BBP practice creates an endogenoustendency for the most valuable (H-type) customers tostay more loyal than do L-type customers.14

14 We thank the editor for suggesting an analysis of the interplaybetween quantity and customer loyalty. To this end, we considerthe possibility of different preference stochasticity levels across two

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4.3. First PeriodIn the first period, the forward-looking consumersolves a dynamic program that takes into accountthe probabilities of second-period location and theprices offered by both retailers, as we show in Fig-ure 5. Because these prices are the outcome of adynamic strategic game played by the retailers, solv-ing for the equilibrium retailer and consumer strate-gies requires embedding the consumer’s dynamicprogramming problem within a dynamic strategicgame that involves one price for each retailer in thefirst period and three prices (own high and low types,and competitor’s customers) for each retailer in thesecond period.Let retailer A’s first-period price be pA1 and retailer

B’s first-period price be pB1 . If the first-period priceslead to a cut-off �̃j1, the marginal consumer of type j ,where j ∈ �LH� with location �̃j1, must be indifferentbetween buying a product from A or B in period 1.In other words, the consumer compares the utility ofpurchasing from either retailer A or retailer B, recog-nizing that his or her location may change becauseof preference stochasticity. Consumers are forward-looking, so they rationally anticipate the consequenceof their first-period choice in terms of the prices theywill receive in the second period. Thus, the followingequation holds for the marginal consumer:

Qjv−QjpA1 − �̃j1+ #�EA�U j2 � �1 = �̃j1��

=Qjv−QjpB1 − �1− �̃j1�+ #�EB�U j2 � �1 = �̃j1�� (4)

where # < 1 is the discount factor, and EA�U j2 ��1= �̃j1�

=EAj2 , EB�U j2 ��1= �̃j1�=EBj2 represents the expected

second-period utility that this “marginal” consumerderives from purchasing from retailer A or B in thefirst period, respectively. Thus,

EAj2 = EA�U j

2 ��1= �̃j1�

=∫ �̃

Aj2

�2=0Pr��2 ��1= �̃j1�×�Qjv−QjpAj2 −�2�d�2

+∫ 1

�2=�̃Aj2Pr��2 ��1= �̃j1�

×�Qjv−QjpBO2 −�1−�2��d�2 (5)

segments, say � for L-type and %� for H -type. Our main result—pertaining to rewarding the firm’s own customers by offering thema lower price beyond a certain threshold level of �—remains robustas long as %� is higher than a threshold level, that is, if the prefer-ence stochasticity of the high types is not much lower than that ofthe low types. However, if %� for H -type customers is extremelylow (i.e., little preference stochasticity), the firm has no incentiveto reward its own high-type customers, who are very likely to staywith the firm. See the electronic companion for details.

EBj2 = EB�U j

2 ��1= �̃j1�

=∫ �̃

Bj2

�2=0Pr��2 ��1= �̃j1�×�Qjv−QjpAO2 −�2�d�2

+∫ 1

�2=�̃Bj2Pr��2 ��1= �̃j1�

×�Qjv−QjpBj2 −�1−�2��d�2 (6)

where �̃Aj2 and �̃Bj2 represent the second-period cut-off locations of j-type customers who purchase fromretailerA and retailer B in the first period, respectively.The expected second-period utility in Equations (5)

and (6) contains two components in each case. The firstterm in Equation (5),

∫ �̃Aj2

�2=0Pr��2 ��1= �̃j1�×�Qjv−QjpAj2 −�2�d�2

represents the case in which a second-period location(�2) is such that the first-period marginal consumerdecides to repurchase from retailer A at its repeat pur-chase price. The second term,

∫ 1

�2=�̃Aj2Pr��2 ��1= �̃j1�×�Qjv−QjpBO2 −�1−�2��d�2

represents the case in which the second-period loca-tion is such that the first-period marginal consumerhas decided to purchase from retailer B with its poach-ing price. Both the repeat purchase case and poachingcase similarly constitute Equation (6).From Equation (4), it follows that the marginal first-

period customer of type j is

�̃j1=�1+QjpB1−QjpA1 +#�EA�U j

2 ��1= �̃j1�−EB�U j2 ��1= �̃j1��

2�

(7)

Added complexity arises because pAO2 , pBO2 , pAj2 , and

pBj2 , which are embedded in E

Aj2 and EBj2 , are all func-

tions of both �̃L1 and �̃H1 . Hence, E

AL2 , E

BL2 , E

AH2 , and EBH2

are again functions of both �̃L1 and �̃H1 .

We solve for the first-period equilibrium prices andoverall profits.15 The overall net present values of

15 Because we assume a uniform distribution of consumers, onemay think that we could compute the first-order condition of Equa-tion (8) using an explicit solution strategy, which simply plugsthe primitives back into the second-period profit functions. Unfor-tunately, as we can see in Equations (5) and (6), this approachwould require us to solve the integral with the boundary and theprobability itself involving the primitives of �̃j1, which is again theimplicit function of itself and pA1 . It is therefore infeasible to employthis explicit solution strategy. We follow and extend the generalapproach, first employed by Fudenberg and Tirole (2000). Further-more, we develop an indirect method to solve this challenge, whichwe show in Lemma 2 of the appendix.

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Figure 5 Consumer Decision Tree

Retailer A Retailer B

Nature endows first-period location �1∈ [0, 1]

First-periodutility

Second-periodutility

Retailer ARetailer A Retailer B Retailer B

Second-period location may or may not change

Qjv – Qj p2AJ –�2 Qjv – Qj p2

BO–(1–�2)

Qjv – Qj p1A –�1 + �{EA[U2

j|�1]} Qjv – qj p1

B – (1–�1) + �{EB[U2j

|�1]}

Qjv –Qj p2Bj– (1–�2)Qjv – Qj p2

AO–�2

E(�2) = (1– �)�1 +��

profits for retailers A and B, as viewed from period 1,respectively, are given by

A= �pA1 �·�q�̃H1 + �̃L1 �+# A2 B= �pB1 �·�q�1− �̃H1 �+�1− �̃L1 ��+# B2

(8)

where A2 , B2 are as defined in Equation (3).

Taking first-order conditions with respect to prices,we find a symmetric pure-strategy equilibrium inwhich both retailers charge the same prices (by usingthe envelope theorem and indirect method to solve forthe first-order conditions that we develop in Lemma 2in the appendix, we can arrive at the closed-formsolutions; see the appendix for details):

pA1 =pB1= ��1−��+�q�−2#�&

H�−'�̃H1 /'pA1 �+&H�−'�̃L1/'pA1 ��2��q�−'�̃H1 /'pA1 �+�1−��−'�̃L1/'pA1 ��

(9)

where &L =' AA2'pBO2

dpBO2d�̃L1

+ ' AB2

'pBL2

dpBL2d�̃L1

+ ' AB2

'pBH2

dpBH2d�̃L1

+ ' AA2

'�̃L1+ '

AB2

'�̃L1

&H = ' AA2'pBO2

dpBO2d�̃H1

+ ' AB2

'pBL2

dpBL2d�̃H1

+ ' AB2

'pBH2

dpBH2d�̃H1

+ ' AA2

'�̃H1+ '

AB2

'�̃H1�

Therefore, the equilibrium profits are

A= B= �pA1 �·��q�̃H1 +�1−��̃L2 �+# A2=(��1−��+�q�−2#�&H�−'�̃H1 /'pA1 �+&H�−'�̃L1 /'pA1 ��

2��q�−'�̃H1 /'pA1 �+�1−��−'�̃L1 /'pA1 ��)

·(�1−��+�q

2

)+# A2 � (10)

We now compare these profits obtained by BBP witha model that does not use customer purchase infor-mation. The comparison should help us understandthe benefits, if any, of BBP.The model that does not use customer purchase

information reduces to a static pricing model. In eachperiod, the two retailers maximize their current prof-its. In this static pricing scenario, both retailers wouldcharge the static price (ps1), which maximizes the staticprofit functions in each period, such that

A1 = �pA1 �·(�q

(1−qpA1 +qpB1

2

)

+�1−��(1−pA1 +pB1

2

)) and

B1 = �pB1 �·(�q

(1−qpB1+qpA1

2

)

+�1−��(1−pB1+pA1

2

))� (11)

Taking the first-order conditions and solving forprices, the equilibrium price in the static case is givenby pNo BBP1 =pA1 =pB1 = �1+�q−1��/�1+�q2−1��, andthe per-period profit is A1 = B1= ��1+�q−1��2�/�2�1+�q2−1��. The discounted net present value ofthe total profit across two periods for both retailers is

No BBP1 = A= B=(�1+�q−1��22�1+�q2−1��

)�1+#�� (12)

We begin the profit discussion by considering twospecial cases with (1) only preference stochasticity and(2) only customer heterogeneity in purchase quantity.These cases can be obtained by setting q=1 and �=0,respectively, in the general model. We isolate the trueeffect of these features and then describe the generalprofit result with both customer heterogeneity andpreference stochasticity, to clarify how these two key

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features interact. The complete analysis of the generalcase is available in the electronic companion; we focushere on the case in which high- and low-volume cus-tomers are equal in number (�= 1

2 ) to ease the expo-sition and analysis.

4.3.1. Only Preference Stochasticity, No Cus-tomer Heterogeneity in Purchase Quantity. The casewithout customer heterogeneity can be captured bysetting q=1 in the model, such that all customers buyone unit of the product. As Proposition 1(a) states,when q=1, it is always optimal for firms to rewardthe competitor’s customers, irrespective of the levelof preference stochasticity. Figure 6 shows the totalprofits across the first and second periods with BBPand without BBP, with a discount rate #≈1. The per-period profits without BBP are close to 1

2 with #≈1,and the total profits over two periods without BBPare close to 1.Consistent with Fudenberg and Tirole’s (2000)

model (�=0 and q=1 in our model), we find thatthe profit from BBP is lower than profits without BBPwhen �=0. Total profits reveal an inverted U-shapedcurve when we extend our analysis to the case ofgeneral preference stochasticity in �∈ �01�. To under-stand the intuition behind this result, we decom-pose the total profits into profits from the first andsecond periods, as shown in Figure 6. The second-period profit increases monotonically with �, becausepreference instability softens competition. More pre-cisely, firms do not wish to compete aggressivelyfor customers who could be on their own turf nat-urally through changing preference; they do notoffer aggressive prices to customers who may pre-fer their own product anyway. This competition-softening effect, similar to the “mistargeting” effectidentified by Chen et al. (2001), raises second-periodprices and profit.

Figure 6 Profitability of BBP When �=0�5 and q=1

Π2BBP

Π1BBP

ΠBBP

ΠNo BBP

1.00.80.60.40.20.0

0.2

0.4

0.6

0.8

1.0

1.2Profit

�

The effect of preference stochasticity on first-periodprofit is more subtle, including an indirect effect as aresult of the consumer’s consideration of future price.Consumers recognize that future prices will increase(both second-period repeat purchase and poachingprices increase in �), so their choices become less sen-sitive to changes in the first-period price. The lowerprice sensitivity shifts the first-period price upwardas � increases.16

However, a countervailing direct effect exertsdownward pressure on first-period prices. As �increases, the link between customers’ choices in thefirst and second periods weakens, because consumerpreferences become less correlated, and price elastic-ity in the first period is less affected by what happensin the second period. The direct effect interacts withthe indirect effect and effectively weakens the upwardpressure of the indirect effect. At the extreme, when�=1, demand in the two periods becomes indepen-dent, and price elasticity increases to short-run elas-ticity without BBP; in turn, profits with and withoutBBP become identical.In summary, the net effect on first-period price

(upward pressure from indirect effect and down-ward pressure from direct effect) leads to an invertedU-shaped curve for first-period profits (and totalprofits). Therefore, the total equilibrium profit is max-imized at �=0�6435, increasing for �∈ �00�6435� anddecreasing for �∈ �0�64351�.4.3.2. No Stochastic Preference, Only Customer

Heterogeneity in Purchase Quantity. When thereis no changing preference (�=0), we know fromProposition 1 that retailers offer the lowest price totheir competitor’s customers; therefore, piO2 ≤piH2 ≤piL2 .However, the more surprising result indicates thatwhen heterogeneity in purchase quantities is suffi-ciently high, the total profit with BBP is greater thanprofits without BBP.

Proposition 2. Suppose there is no preference stochas-ticity (�=0). If heterogeneity in purchase quantities is suf-ficiently high (q>5), both retailers increase their profitsunder BBP.

16 More precisely, the indirect effect of consumer’s future consider-ation affects the first-period price in two ways: (1) second-periodhigh repeat purchase prices decrease price elasticity in the firstperiod, because customers know they will be ripped off by thesame firm in the second period (ratchet effect); and (2) second-period low poaching price increases price elasticity in the firstperiod, causing a downward pressure on prices. As � increases,both poaching and repeat purchase prices increase, and the cor-rect expectation of high prices makes consumers less price sen-sitive in the first period (a higher repeat purchase price furtherdecreases price elasticity; a higher poaching price weakens upwardpressure on price elasticity). The indirect effect through a con-sumer’s dynamic consideration makes first-period demand lesselastic, causing firms to increase prices and profits in the firstperiod as � increases.

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Figure 7 Profits When �=0�5 and q=2

Π2BBP

Π2BBP

Π1No BBP, Π2

No BBP

ΠNo BBPΠNo BBP

ΠBBP

ΠBBP

Π1BBP

0.8 0.90 0.92 0.94 0.96 0.98 1.00

0.2

0.4

0.6

0.8

1.0

1.2

0.2

0.4

0.6

0.8

1.0

1.2

ProfitProfit

0.60.40.20.0� �

Π1No BBP, Π2

No BBPΠ1BBP

(a) Reward competitor’s customers (b) Reward one’s own best customers

Proof. See the appendix.Thus, BBP can increase both competing firms’ prof-

its even when consumer preferences are stable if thereis sufficient heterogeneity in quantities. This resultdoes not occur simply because of heterogeneity inquantities and the ability to distinguish between high-and low-type customers. The asymmetric informationabout customer types is critical.The intuition for this result is as follows: with

asymmetric information, the poacher faces an adverseselection (lemon) problem when attracting the com-petitor’s customers. Because the high types get lowerprices from their current retailer, firms recognizethat poaching disproportionately attracts L-type rel-ative to H -type customers. Therefore, firms becomeless aggressive in attracting the competitor’s cus-tomers as customer heterogeneity becomes significant(i.e., degree of adverse selection becomes greater).In other words, asymmetric customer informationsoftens competition, so the benefit of price discrim-ination dominates the cost of increased competitiontypically associated with BBP when customer hetero-geneity becomes significant.

4.3.3. Both Stochastic Preference and CustomerHeterogeneity in Purchase Quantity. We next con-sider the full model by relaxing both the q=1 and�=0 assumptions. Specifically, we consider the q=2case to allow for customer heterogeneity, which helpsbuild the intuition for the customer heterogeneity casewith the least complexity.In Figure 7, we show the total profits with and

without BBP when �=0�5 and q=2. As anticipatedfrom Proposition 1, there are two regimes based onthe pricing strategy in period 2: the “reward com-petitor’s customers” regime and the “reward one’sown best customers” regime. As a key takeaway,

BBP can be profitable in both regimes. Specifically,profit with BBP is greater than profits without BBP inthe range �∈ �0�1150�857� in the reward competitor’scustomers regime (Figure 7(a)). Similarly in the range,�∈ �0�8940�917�, profit with BBP is greater than thatwithout BBP in the reward one’s own best customersregime (Figure 7(b)).As a summary of this discussion, we state our third

proposition.

Proposition 3 (Profitability of BBP). (a) RewardCompetitor’s Customers in Second Period: When pref-erence is stable (�<!�0�52�=0�857), the total profit withBBP is greater than profits without BBP if preferencestochasticity is in the range �∈ �0�1150�857�.(b) Reward One’s Own Best Customers in Second

Period:When customer preference is sufficiently stochastic(�≥ !�0�52�=0�894), the total profit with BBP is greaterthan profits without BBP when preference stochasticity isin the range �∈ �0�8940�917�.We see a stark reversal in profit patterns across the

two periods between the reward one’s own best andreward competitor’s customers regimes. For the lat-ter regime, we see a decreasing profit pattern overtwo periods: profits in the first period are higherthan static levels, and profits in the second period arebelow the static levels. This profit-ordering reversesin the reward one’s own best customers regime, sothat we see an increasing profit pattern over time;that is, the first-period profits are below the static lev-els, whereas second-period profits are above.The intuition for profits under the reward competi-

tor’s customer regime follows directly from the previ-ous discussion regarding the direct and indirect effectsof preference stochasticity alone and the discussion ofinformation asymmetry with only heterogeneity. On

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Table 2 Summary of Results

Preference stochasticity

Low Sufficiently high

Prices Profits Prices Profits

Heterogeneity in quantity(information advantageabout current customers)

Low Reward competitorcustomers(Proposition l(a))

Behavior-based pricingless profitable(Proposition 3(a))

Reward competitorcustomers(Proposition l(b))

Behavior-basedpricing moreprofitable(Proposition 3(a))

Sufficiently high Reward competitorcustomers(Proposition l(a))

Behavior-based pricingmore profitable(Proposition 2)

Reward current high-type customers(Proposition l(b))

Behavior-basedpricing moreprofitable unlesspreference isextremely unstable(Proposition 3(b))

the other hand, the intuition for the increasing priceand profit pattern over time under the reward one’sown best customers regime follows from the lock-inintuition, similar to switching-cost models (Klemperer1995, Farrell and Klemperer 2007). Firms compete toattract more customers in the first period to gain aninformation advantage about customer types. By usingthis information advantage, firms effectively can lockin their most valuable (and profitable) H -type cus-tomers by rewarding them in the second period.17 Thislock-in intuition explains the profit pattern over time—that is, lower first-period profit and higher second-period profit. Moreover, we find that firms can be bet-ter off with BBP under this regime.However, as preference stochasticity gets closer

to one, the benefit of lock-in weakens because of theimperfect lock-in that arises from preference stochas-ticity, which in turn lowers the second-period profits.At high levels of preference stochasticity, overall prof-its can be lower with BBP when �=0�5 and q=2. Ingeneral, the total profit can be higher with BBP evenwhen preference stochasticity is very high because ofthe same adverse selection intuition in the pure het-erogeneity case. For example, when �=0�5, BBP isalways profitable if q≥5. The threshold level of pref-erence stochasticity at which the total profit with BBPis greater than that without BBP is 0.917 when �=0�5and q=2.This threshold is in fact a function of � and q.

When the value of information asymmetry is greater(larger q), the threshold rises, and rewarding one’sown best customers is a profitable strategy for a wider

17 In our model, the high types are indeed the most valuable cus-tomers for firms because the optimal prices are such that per-periodprofits will be higher for the high type (that is, q×pAH2 >��1+q�/2�×pAO2 >pAL2 for all � and q≥2). In other words, the cost to retain hightypes does not exceed the benefit of retaining them. Furthermore,it is easy to see that second-period prices are all higher than thefirst-period price; pAH2 >pA1 .

range of preference stochasticity. For example, whenq=16, which is the case suggested by the 80/20 rulewhen �=0�5,18 the range in which BBP is profitableunder the reward one’s own best customers regimebecomes much larger, �∈ �0�7310�961�.

5. ConclusionWe address a basic but controversial question in cus-tomer management: should a firm use BBP, and ifso, whom should it reward? We summarize our keyresults in Table 2. When heterogeneity in purchasequantity and preference stochasticity are both low, it isoptimal to reward competitor’s customers, and BBP isless profitable. This result is consistent with existingtheoretical models. However, if either heterogeneityin customer quantities or preference stochasticity issufficiently high, BBP can improve profits under com-petition, even when consumers and firms are strategicand forward-looking. Rewarding the firm’s own bestcustomers is optimal only when there is both hetero-geneity in purchase quantities and preference stochas-ticity are sufficiently high.This study thus reconciles the apparent conflict

between practitioners’ optimism and academic skep-ticism about using customer’s past purchase informa-tion by nesting both existing analytical results andpractitioner intuition. Both heterogeneity in customervalue and the threat of customer switching as a resultof stochastic preference are characteristics of a widevariety of markets; thus understanding the extent ofsuch heterogeneity and preference stochasticity offerscritical information about the value of using BBP andarriving at the right balance between customer acqui-sition and retention.Certain limitations of our model suggest interesting

directions for further research. First, we do not allowconsumers to split their purchases across different

18 A q that solves for the 80/20 rule is obtained by solving 0�2q=0�8�0�2q+0�8�.

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firms or even across periods;19 thus it is harder forfirms to infer the customer’s true potential and shareof customer wallet. In our current model, the first-period purchases help firms unambiguously iden-tify H - and L-type consumers. When consumers cansplit their purchases, the inference about consumertype would be only probabilistic, even if firms pos-sess share-of-wallet information. Thus share-of-walletinformation weakens the extent of information asym-metry in the model (see Du et al. 2008 as an exampleof how a firm may infer share of wallet).Second, third parties also sell estimates of customer-

spending potential, based on the observed characteris-tics of the household (e.g., zip code, demographics), butthis information is far fromperfect.Therefore, evenwiththird-party information, theobserved purchase historystill provides the information asymmetry critical tothe profitability of BBP. We therefore believe our find-ings are robust even with the introduction of noisypotential information, although it might weaken theinformation asymmetry. A systematic analysis of howshare of wallet and potential estimates may affectinformation asymmetry, and their resulting effect onBBP, would offer an important next step from boththe empirical and theoretical perspectives.20

Third, given our focus on BBP, we have abstractedaway from modeling second-degree price discrimina-tion. Our model is a reasonable abstraction of manyreal-world markets (e.g., airlines, department stores,grocery stores) in which firms only use BBP with pastcustomer purchase history data but do not offer amenu of prices. This could be because consumers dif-fer not in the quantity they use at any one time butin the frequency of their use. Firms need to aggre-gate the flow of multiple purchases across a cer-tain time period into a “past quantity” stock. In thisscenario, menu pricing is not feasible, and BBP isappropriate. Nevertheless, further research needs toconsider how robust our results are if competitorscould offer a menu of poaching prices. The second-degree price discrimination problem in a competitivesetting with customer behavior information is a tech-nical challenge that has not yet been solved generallyand awaits further research (Stole 2007). We thereforeleave the issue of combining BBP and second-degreeprice discrimination as an important, but challenging,area of further research.

19 Consumers may strategically time their purchases. This could berelated to sales force literature where in response to quotas andbonuses, salespeople sometimes shift their effort and book salestowards the end of the year to qualify for a bonus (Steenburgh2008). We thank an anonymous reviewer for raising this issue.20 A potential formulation that accommodates the notion of share ofwallet and customer potential might be q=Potential×S��, whereS�� the share of wallet of the customer, is a function of the relativepreference between firms (i.e., based on location ��.

Fourth, our theoretical model provides empiricallytestable hypotheses for further research. We find thatBBP is most likely effective when there is suffi-cient heterogeneity in customer quantities and prefer-ence stochasticity. Empirical research across categorieswhich vary along these dimensions can help ascer-tain whether these predictions are valid. We thereforehope our model serves as an impetus for further the-oretical and empirical research into BBP.

6. Electronic CompanionAn electronic companion to this paper is available aspart of the online version that can be found at http://mktsci.pubs.informs.org/.

AcknowledgmentsThe authors thank the editor, the area editor, and fouranonymous reviewers for their very constructive anddetailed comments during the review process. The authorsare grateful to Drew Fudenberg, Duncan Simester, J. MiguelVillas-Boas, and Birger Wernerfelt for their helpful com-ments. They also thank seminar participants at Duke, KoreaUniversity, Stanford, UCLA, University of Chicago, Uni-versity of Toronto, and Yale as well as participants in theNortheast Marketing Colloquium conference at the Mas-sachusetts Institute of Technology and the Summer Institutein Competitive Strategy conference at UC Berkeley for theircomments and suggestions. The authors contributed equallyand their names are listed in alphabetical order.

AppendixProof of Lemma 1 (Value of Information). We

first address retailer A. Let z be the first-period mar-ket share. Then, Pr��2≤z ��1≤z�≥Pr��2>z ��1≤z�⇔ �1−��+�z≥��1−z�. In turn, Pr��2≤z ��1≤z�−Pr��2>z ��1≤z�≥0⇔�≤1/�2�1−z��. Therefore, it always holds when �≤1/�2�1−z��. In particular, when z≥ 1

2 , the inequality alwaysholds for ∀�∈ �01�. Next, let z̄=1−z be retailer B’s first-period market share and substitute it into Equation (2). Weget exactly the same result. Q.E.D.Proof of Proposition 1 (Reward One’s Own Cus-

tomers or Competitor’s Customers).(a) We start with �<!��q�,

where !��q�= 2��3−q��1−��+2�q2�

�3+q��1−��+4�q2and !��q�< !��q� for ∀�∈ �01�, which ensures that �̃L1 >�̃AL2 and �̃H1 >�̃

AH2 , in equilibrium. Again, using Equations (1)

and (2), we know that

PrAH=(1−��̃H1

+�)1+q�pBO2 −pAH2 �

2

PrAL=(1−��̃L1

+�)1+pBO2 −pAL2

2

PrBH=(1−�1− �̃L1

+�)1−q�pBH2 −pAO2 �

2 and

PrBL=(1−�1− �̃L1

+�)1−pBL2 +pAO2

2�

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Similar to the proof of Proposition 1, we solve the first-orderconditions. When �̃j1= 1

2 , second-period prices are

pAH2 =pBH2 = 12q

+ �2+��1+�q−1��6�2−��1+�q2−1��

pAL2 =pBL2 = 12+ �2+��1+�q−1��6�2−��1+�q2−1�� and

pAO2 =pBO2 = �2+��1+�q−1��3�2−��1+�q2−1��

which confirms that �̃L1 >�̃AL2 and �̃H1 >�̃

AH2 when

�<!�q�= 2��3−q��1−��+2�q2�

�3+q��1−��+4�q2 �

We only need to show that piO2 ≤piH2 ≤piL2 . First, we noticethat

!�q�= 2��3−q��1−��+2�q2�

�3+q��1−��+4�q2is an increasing function in �∈ �01� for any q≥1. Second,we have

piO2 ≤piH2 ⇔ �2+��1+�q−1��6�2−��1+�q2−1�� ≤

12q

⇔ �≤ 2�5�1−��−q�+6q2��

7�1−��+q�+6q2�

and2�5�1−��−q�+6q2��7�1−��+q�+6q2�

for any q≥1 and ∀�∈ �01�. Hence, the inequality alwayssatisfies for all q≥1. In addition, it is obvious that piH2 ≤piL2 ⇔1/�2q�≤ 1

2 for all q≥1. �

(b) Next, we look at the case �≥ !��q�, which ensuresthat in equilibrium, the market is �̃L1 >�̃

AL2 and �̃H1 ≤ �̃AH2 ,

where

!��q� = �2�q2−�q+6��1−��+√�2�q2−�q+6��1−��2+12�1−��4�q2+�q−3��1−��

·�4�q2+�q−3��1−��−1�Again, from Equations (1) and (2), we know that

PrAH= �1−��+�(1+q�pBO2 −pAH2 �

2

)

PrAL=(1−��̃L1

+�)1+pBO2 −pAL2

2

PrBH= �1−��+�1−q�pBH2 −pAo2 �2

and

PrBL=(1−�1− �̃L1

+�)1−pBL2 +pAO2

2�

Plugging these into Equation (3), we obtain the second-period prices by solving the retailers’ first-order conditions.Specifically, when �̃j1= 1

2 , the second-period prices are

pAH2 =pBH2 = 2−�2q�

+ �2+��1+�q−1��6��2−��1−��+q2��

pAL2 =pBL2 = 12+ �2+��1+�q−1��6��2−��1−��+q2�� and

pAO2 =pBO2 = �2+��1+�q−1��3��2−��1−��+q2��

We confirm that �̃L1 >�̃AL2 and �̃H1 ≤ �̃AH2 when q>1 and �≥

!�q�. First, we verify that

piL≥piO ⇔ piL−piO= 12− �2+��1+q�6�q2+1��2−�� ≥0

⇔ 2�3q2−q+2�≥ �3q2+q+4��Note that �3q2+q+4��≤ �3q2+q+4� for �∈ �01�. Hence, theresult that piL≥piO for all �∈ �01� and q≥1 derives directlyfrom 2�3q2−q+2�>�3q2+q+4�⇔3q�q−1�≥0. Second, weshow that piH ≤piO . Notice that

piH ≤piO ⇔ 2−�q�

≤ �2+��1+q�6�q2+1��2−��

⇔ �≥ !�q��We only need to show that there exists �∈ �01�, such that� ≥ !��q�= �2�q2−�q+6��1−��

+√�2�q2−�q+6��1−��2+12�1−��4�q2+�q−3��1−��

·�4�q2+�q−3��1−��−1�Lemma 2. For all �∈ �01�, there always exists

q>max{−1+�+√

1+46�−47�28�

9�1−��5+3�

}

such that !��q�<1.Proof. Note that

!��q�

≤ 2�2�q2−�q+6��1−��+√

12�1−��4�q2+�q−3��1−��4�q2+�q−3��1−��

<2�2�q2−�q+6��1−��+4q+√

12�q−3��1−��24�q2+�q−3��1−��

<2�2�q2−�q+6��1−��+4q+4�q−3��1−��

4�q2+�q−3��1−��

<2�2�q2−�q+6��1−��+8q

4�q2+�q−3��1−��

Let us define G��q�≡ �2�2�q2−�q+6��1−��+8q�/�4�q2+�q−3��1−��> !��q� for all �∈ �01� and q. We can easilysee that G��q� is monotonically decreasing in q because

'G��q�

'q= −6−2��6−9�+2q�5q+3�6+q��−18��

�q−3+�3+q�4q−1��2 <0

for all �∈ �01� when q>�−1+�+√1+46�−47�2�/�8��.

Also, q= �9�1−��/�5+3�� is the unique cut-off value of q,such that G��q�=1. By the monotonicity of G��q� and !��q�<G��q�≤1, we recognize that !��q�<1 whenq>max��−1+�+√

1+46�−47�2�/�8��9�1−��/�5+3��.There always exists

q>q∗=max{−1+�+√

1+46�−47�28�

9�1−��5+3�

}

such that !��q�<1. �

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Lemma 1 shows that when q is sufficiently large, thereexists �∈ �01�, such that� ≥ !��q�=

(2�q2−�q+6��1−��

+√�2�q2−�q+6��1−��2+12�1−��4�q2+�q−3��1−��

)

·�4�q2+�q−3��1−��−1� Q.E.D.

Derivation of First-Period PriceWe follow Fudenberg and Tirole’s (2000) proof strategy,based on the envelope theorem. Because we are focusingon the symmetric game, we expect that pA1 =pB1 , and �̃j1= 1

2 .In other words, marginal customers of type j in the firstperiod reside at the center, and therefore, the two retailerssplit demand equally in the first period. We subsequentlyconfirm that this symmetric outcome is an equilibrium, butthe level of the first-period equilibrium price depends onthe elasticity of demand to a change in price. Because con-sumers are forward-looking, elasticity is affected by con-sumer expectations about prices in the second period. Inaddition, the H - and L-type consumers face different pricesin the second period, so the price elasticities of the twotypes in the first period differ and the optimal first-periodprices require retailers to balance the effects of a change inprice on the demand of the two types of customers.Applying the implicit function theorem, we arrive at the

following lemma:

Lemma 3. The first-period price sensitivities for L- and H -type consumers are

d�̃L1dpA1

=− F H�H−q ·F L�HF L�LF

H�H−F L�H ·F H�L

andd�̃H1dpA1

=− q ·F L�L−F H�LF L�LF

H�H−F L�H ·F H�L

where

F L�L=2−#('EAL

'�̃L1− 'E

BL

'�̃L1

) F H�L=−#

('EAH

'�̃L1− 'E

BH

'�̃L1

)

F L�H=−#('EAL

'�̃H1− 'E

BL

'�̃H1

) F H�H=2−#

('EAH

'�̃H1− 'E

BH

'�̃H1

)�

Proof. We define the implicit functions from Equa-tion (7) as follows:

F L�pA1 �̃L1 �̃

H1 � = 2�̃L1−

{1+pB1−pA1+#�EAL��̃L1 �̃H1 �−EBL��̃L1 �̃H1 ��

}=0and

F H�pA1 �̃L1 �̃

H1 � = 2�̃H1 −{

1+qpB1−qpA1+#�EAH��̃L1 �̃H1 �−EBH��̃L1 �̃H1 ��

}=0�We differentiate both equations and rearrange them asfollows:

d�̃L1dpA1

=− �'FL/'pA1 +�'F L/'�̃H1 �·�d�̃H1 /dpA1 ��

'F L/'�̃L1 and

d�̃H1dpA1

=− �'FH/'pA1 +�'F H/'�̃L1 �·�d�̃L/dpA1 ��

'F H/'�̃H1�

Solving these two equations simultaneously, we obtain thefollowing results:

d�̃L1dpA1

=− �FLp −�F L�H/F H�H�·F Hp �

�F L�L−�F L�H ·F H�L�/F H�H�=− F

H�HF

Lp −F L�H ·F Hp

F H�HFL�L−F L�H ·F H�L

and

d�̃H1dpA1

=− �F Hp −�F H�L/F L�L�·F Lp ��F H�H−�F L�H ·F H�L�/F L�L�

=− F L�LFHp −F H�L ·F Lp

F L�LFH�H−F L�H ·F H�L

�

Using

F Lp ='F L

'pA1=1 F HP = 'F

H

'pA1=q

we recognize that

F L�L='F L

'�̃L1=2−#

('EAL

'�̃L1− 'E

BL

'�̃L1

)

F H�L='F H

'�̃L1=−#

('EAH

'�̃L1− 'E

BH

'�̃L1

) and

F H�H='F H

'�̃H1=2−#

('EAH

'�̃H1− 'E

BH

'�̃H1

)� Q.E.D.

It is convenient to rewrite Equation (10) for firm A’s over-all profits using the functions AA2 , AB2 to represent itssecond-period profit from its own previous customers andfrom retailer B’s previous customers, respectively,

A = pA1(�q�̃H1 +�1−��̃L1

)+#{ AA2 (

pAi2 �pA1 p

B1 �p

Bi2 �p

A1 p

B1 ��̃

L1 �p

A1 p

B1 ��̃

H1 �p

A1 p

B1 �)

+ AB2(pAi2 �p

A1 p

B1 �p

Bi2 �p

A1 p

B1 ��̃

L1 �p

A1 p

B1 ��̃

H1 �p

A1 p

B1 �)}

(13)

where

AA2 �pAi2 p

Bi2 �̃

L1 �̃

H1 �= �pAH2 ��q�̃H1 Pr

AH�pAH2 pBO2 �̃H1 �

+�pAL2 ��1−��̃L1 PrAL�pAL2 pBO2 �̃L1 � AB2 �p

Ai2 p

Bi2 �̃

L1 �̃

H1 �

=pAO2{�q�1− �̃H1 ��1−PrBH �pAH2 pBO2 �̃

H1 ��

+�1−��1− �̃L1 ��1−PrBL�pAL2 pBO2 �̃L1 ��}�

Note that

PrAH=Pr[�2≤

1+q�pBO2 pAH2 �

2

∣∣∣�1≤ �̃H1]

and

PrBH=Pr[�2>

1+q�pBH2 pAO2 �2

∣∣∣�1>�̃H1]

are functions of pAH2 −pBO2 �̃H1 ; in addition,

PrAL=Pr[�2≤

1+q�pBO2 −pAL22

∣∣∣�1≤ �̃L1]

and

Pr[�2>

1+�pBL2 −pAO2 �2

∣∣∣�1>�̃L1]

are functions of pAL2 pBO2 �̃

L1 .

Because retailer A’s own second-period prices are set tomaximize A’s second-period profit, we can use the envelope

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theorem (' AA2 /'pAi2 =0, ' AB2 /'pAO2 =0) to write the first-

order conditions for this maximization as

��q�̃H1 +�1−��̃L1 �+pA1(�q'�̃H1'pA1

+�1−�� '�̃L1

'pA1

)

+#[{' AA2'pBO2

dpBO2d�̃H1

+ ' AB2

'pBL2

dpBL2d�̃H1

+ ' AB2

'pBH2

dpBH2d�̃H1

+ ' AA2

'�̃H1+ '

AB2

'�̃H1

}'�̃H1'pA1

+{' AA2'pBO2

dpBO2d�̃L1

+ ' AB2

'pBL2

dpBL2d�̃L1

+ ' AB2

'pBH2

dpBH2d�̃L1

+ ' AA2

'�̃L1+ '

AB2

'�̃L1

}'�̃L1'pA1

]=0�

Then, the first-order condition of Equation (11) at �̃H1 = �̃L1 = 12

simplifies to

pA1 =pB1 =��1−��+�q�−2#�&H�−'�̃H1 /'pA1 �+&H�−'�̃L1 /'pA1 ��

2��q�−'�̃H1 /'pA1 �+�1−��−'�̃L1 /'pA1 ��

where

&L=' AA2'pBO2

dpBO2d�̃L1

+ ' AB2

'pBL2

dpBL2d�̃L1

+ ' AB2

'pBH2

dpBH2d�̃L1

+ ' AA2

'�̃L1+ '

AB2

'�̃L1

&H=' AA2'pBO2

dpBO2d�̃H1

+ ' AB2

'pBL2

dpBL2d�̃H1

+ ' AB2

'pBH2

dpBH2d�̃H1

+ ' AA2

'�̃H1+ '

AB2

'�̃H1� Q.E.D.

Proof of Proposition 2. We know that if �=0, it is opti-mal to reward competitor customers for all q>1. Hence,when �= 1

2 , Equation (3) can be rewritten as

A2 = pAH2 q

2

(1+q�pBO2 −pAH2 �

2

)+ p

AL2

2

(1+pBO2 −pAL2

2

)+ p

AO2

2

·{q

(1+q�pBH2 −pAO2 �

2− �̃H1

)+(1+pBL2 −pAO2

2− �̃L1

)}

B2 = pBH2 q

2

(1−q�pAO2 −pBH2 �

2

)+ p

BL2

2

(1−pAO2 −pBL2

2

)+ p

BO2

2

·{q

(�̃H1 − 1+q�p

BO2 −pAH2 �

2

)+(�̃L1−

1+pBO2 −pAL22

)}�

The first-order conditions for the retailers’ maximizationyield

pAH2 = 3+q�2q−1�+4q�̄16q�1+q2� pAL2 = 2+q�3q−1�+4�̄1

6�1+q2�

pAO2 = 3�1+q�−4�̄13�1+q2� pBH2 = 3�q−1�+6�1+q

2�−4q�̄16q�1+q2�

pBL2 = 3�1+q�+3�1+q2�−4�̄1

6�1+q2� and pBO2 = 4�̄1−�1+q�3�1+q2�

where �̄1= �q�̃H1 + �̃L1 �/2. In turn, firms A’s and B’s overallprofit functions can be rewritten as

A = pA12�q�̃H1 + �̃L1 �+#

[49144

+ 62q+80�̄1��̄1−�1+q��144�1+q2�

]

B2 = pB1−c2�q�1− �̃H1 �+�1− �̃L1 ��

+#[49144

+ 62q+80�̄1��̄1−�1+q��144�1+q2�

]�

By using Lemma 1 and Equation (11), we obtain pA1 =pB1 =��3+#��1+q��/�3�1+q2��. We get the profit result directlyfrom plugging prices into the equation

A− s = 41#144

+ 36�1+q2�+46q#

144�1+q2� − �1+q�2�1+#�

4�1+q2�

= �q−5��5q−1�#144�1+q2� >0 if q>5� Q.E.D.

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