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A Dynamic Model of Optimal RetargetingJ. Miguel Villas-Boas, Yunfei (Jesse) Yao

To cite this article:J. Miguel Villas-Boas, Yunfei (Jesse) Yao (2021) A Dynamic Model of Optimal Retargeting. Marketing Science

Published online in Articles in Advance 03 Feb 2021

. https://doi.org/10.1287/mksc.2020.1267

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MARKETING SCIENCEArticles in Advance, pp. 1–31

http://pubsonline.informs.org/journal/mksc ISSN 0732-2399 (print), ISSN 1526-548X (online)

A Dynamic Model of Optimal RetargetingJ. Miguel Villas-Boas,a Yunfei (Jesse) Yaoa

aHaas School of Business, University of California, Berkeley, California 94720Contact: [email protected], https://orcid.org/0000-0003-1299-4324 (JMV-B); [email protected],

https://orcid.org/0000-0001-9831-5083 (Y(J)Y)

Received: January 10, 2020Revised: May 26, 2020; August 6, 2020Accepted: August 23, 2020Published Online in Articles in Advance:February 3, 2021

https://doi.org/10.1287/mksc.2020.1267

Copyright: © 2021 INFORMS

Abstract. A consumer searching for information on a product may be indicative that theconsumer has some interest in that product but is still undecided about whether topurchase it. Some of this consumer search for information is not observable to firms, butsomemay be observable. Once a firm observes a consumer searching for information on itsproduct, the firmmay then want to try to provide further information about the product tothat consumer, a phenomenon that has been known in electronic commerce as retargeting.Firmsmay not observe all activities by a consumer in searching for information, may not beable to observe the information gained by consumers, and may not be able to observewhether a consumer stopped searching for information. A consumer could stop searchingeither because he received information of poor fit with the product, because he bought theproduct (whichmay be unobservable to the firm), or because he exogenously lost interest inthe product. This paper presents a dynamic model with these features characterizing theoptimal advertising retargeting strategy by the firm. We find that a forward-looking firmcan advertise more or less than a myopic firm to gain more information about whether theconsumer is searching for information, advertising more if the effect of advertising isrelatively high. We characterize how the optimal advertising retargeting strategy is af-fected by the ability of the firm to observe when the consumer purchases the product,when the firm is better able to observe the consumer search behavior, and by the infor-mativeness of the signal received by the consumer. We find that better tracking of con-sumer search behavior could be beneficial for consumers, because it may reduce the lengthof time when a consumer receives retargeting, but that it also enlarges the region of firm’sbeliefs where retargeting is optimal. Finally, we also find that the value of retargeting ishighest for an intermediate value of the likelihood of the consumer receiving an infor-mative signal and that retargeting may allow the firm to charge higher prices if consumersare forward-looking.

History: Yuxin Chen served as the senior editor and Anthony Dukes served as associate editor forthis article.

Keywords: retargeting • targeting • advertising • consumer search • information • dynamic strategy

1. IntroductionA consumer searching for information on a productmay be indicative that the consumer has some interestin that product but is still undecided aboutwhether topurchase it. Some of this consumer search for infor-mation is not observable to firms, but some may beobservable. Once a firm observes a consumer searchingfor information on its product, the firmmay then wantto try to provide further information about the productto that consumer, a phenomenon that has been knownin electronic commerce as retargeting. This practice ofproviding targeted information to consumers search-ing for information has been present with salesforcebehavior, but, with the development of the informa-tion technologies, has become more prominent be-cause of better tracking of consumers’ informationgathering behavior. In fact, in recent years, we haveobserved firms sending online advertising when they

see a consumer searching for information on a certainproduct. This isdone throughemails,displayadvertisingof sites checked by that consumer, or with other formsof communication.Obviously, firms may not observe all activities by

a consumer in searching for information. For exam-ple, in the electronic world, a firm may not be able toknow about the offline information gathering byconsumers; even online, the consumer may be gath-ering information from sites where the firm does nottrack information. Even if a consumer is browsing in asite that has information on the product, the con-sumermay not be processing that information. A firmmay also not know if a consumer is receiving fa-vorable or unfavorable information, when it findsthe consumer searching for information. Given theposterior search or purchase behavior by the con-sumer, the firmmay infer to some degree whether the

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information obtained by the consumer was favorableor unfavorable but does not know it firsthand.

Moreover, the firm may not know if a consumerstopped searching for information, as it cannot ob-serve all search occasions, and may not be able toobserve whether and when a consumer purchases theproduct. In fact, anecdotal evidence seems to suggestthat consumers continue to receive purchase-orientedadvertising after purchasing the product, or, moregenerally, after they stop searching for information. Afirm may not be able to observe whether a consumerpurchased the product because that is done offline, isdone on a site that is not monitored, or for which theinformation collected is not cross-checked with theretargeting information. Obviously, one can considerfirms getting better at connecting consumer pur-chases with consumer search behavior and that af-fecting the optimal retargeting behavior.

This paper considers a model of these effects,taking into account the firm’s beliefs about the like-lihood of search behavior by the consumer and therole of advertising. When the firm sees a signal that aconsumer is searching for information, the firm learnsthat the consumer is searching for information, al-though it might not know whether the informationreceived by the consumer is positive or negative orwhether the consumer decided to purchase the product.When the firm does not see a signal that a consumer issearching for information, by Bayes’ rule, it reduces itsbelief that the consumer is searching for information.Advertising is modeled as increasing the likelihoodand frequency of the consumer learning informationabout the product and increasing the ability of thefirms tracking that the consumer is searching for in-formation. The optimal strategy calls for advertisingwhen the firm has a sufficiently high belief that theconsumer is considering the product.We can evaluatehow the optimal strategy can be affected by differentmarket forces and how the optimal strategy changeswhen the firm can also observe if and when theconsumer buys the product. We also characterize thelength of time that a product is advertised withoutfurther information about the consumer’s search be-havior. The model also replicates the real-world ex-periences by consumers of receiving advertised aftersearching for information on a product, and con-tinuing to receive that information, even after be-coming disinterested in the product.

We compare forward-looking with myopic firmsand find that forward-looking can do less retargetingthan myopic firms if the informational benefits ofretargeting (more signals for the consumers, andgreater ability by thefirm to track that the consumer issearching for information) are not too large and if thelikelihood of the consumer dropping out of the searchprocess exogenously is low enough. A myopic policy

does not account for the fact that without advertisingthe consumermay still end uppurchasing the productin the future and therefore can lead to greater urgencyin advertising. This has important implications forfirms about the importance of having a longer-termperspective in their retargeting decisions, and thebiases that can occur from having toomuch of a short-term perspective. These implications can also betested empirically by looking at retargeting behaviorby firms depending on their planning horizon.We find that better tracking of consumer search

behavior could be beneficial for consumers, because itmay reduce the length of time when a consumer re-ceives retargeting. Better tracking of consumer searchbehavior allows the firm to update faster that theconsumer is not searching for information when notobserving the consumer searching for information,and therefore the firms stops doing retargeting sooner.This would suggest that with improvements in thetracking technology we could observe shorter retar-geting periods. In fact, anecdotal evidence seemsto suggest that the extremely long retargeting pe-riods that occurred a few years ago seemed to havereduced over time. This has managerial implicationsfor the optimal retargeting strategy when firms de-velop technological improvements in their tracking ofconsumer search, and also has public policy implica-tions ofmaking retargeting less problematic in terms ofadvertising annoyance.Considering the value of retargeting, we find it is

highest for an intermediate value of the informa-tiveness of the signals received by consumers. If theinformativeness of those signals is too high, con-sumers quickly get informed about the value of theproduct and having the ability to do retargeting is lessvaluable. If the informativeness of those signals is toolow, then retargeting also does not help much as ittakes too long for a consumer to learn the value of theproduct. This value of retargeting is also the highestfor some intermediate value of the likelihood of theconsumer exogenously dropping out of the searchprocess. If the likelihood of the consumer exoge-nously dropping out of the search process is too low,there is not much benefit in retargeting as the con-sumer will likely find out about the value of theproduct without retargeting, and will be able tochoose to purchase the product. On the other side, ifthe likelihood of the consumer exogenously droppingout of the search process is too high, there is not muchbenefit in retargeting as the consumer will likely dropout of the search process before finding out if theproduct is a goodfit. As the informativeness of signalschanges, this result has importance about the use-fulness of firms doing retargeting and yields empir-ical predictions about the extent of firms’ retarget-ing behavior.

Villas-Boas and Yao: Dynamic Model of Optimal Retargeting2 Marketing Science, Articles in Advance, pp. 1–31, © 2021 INFORMS

We study also what happens if the informationtechnology improves in such a way that firms areimmediately able to recognize when purchases occur.In this case, firms stop retargetingwhen the consumermakes a purchase but may continue still going ondoing retargeting after the consumer receives a nega-tive informative signal. In this case, we find that thethreshold belief for the firm to do retargeting isnow lower because, when doing retargeting, the firmknows when the consumer made a purchase, andretargeting is no longer needed, and the maximumperiod of retargeting can now be longer. In additionto the managerial implications that the developmentof these technologies of recognizing purchases canbring, this result can potentially lead to an increase inadvertising annoyance and therefore lead to potentiallygreater regulation on firms’ retargeting behavior.

We consider the consumer search for information,and purchase behavior, finding that the optimal pricecan fall with lower search costs. If consumers areforward-looking and there is no disutility of receivingadvertising, the possibility of retargeting allows thefirm to charge a higher price, because of the antici-pated benefit of the additional information fromretargeting.

Existing research has focused on understandinghow consumers past purchase behavior could affectthe future behavior of the firm toward those consumers.This has been considered in the context of advertising(Shen and Villas-Boas, 2018), pricing (Villas-Boas1999, 2004; Fudenberg and Tirole 2000; Fudenbergand Villas-Boas 2006; Shin and Sudhir 2010), andproduct design (Zhang 2011). However, in the realworld, it seems that with the development of theinformation technologies, the most important prac-tice is one of conditioning the behavior of firms on thesearch behavior of consumers rather than on the pur-chase behavior of consumers.1 There is also some re-lated recent work on the search behavior of consumersfor information (Branco et al. 2012, Ke et al. 2016,Fudenberg et al. 2018, Gardete and Antill 2019, Keand Villas-Boas 2019), but this research has notconsidered how that search behavior affects the firmstrategy.2 There has also been some research on thesignificant empirical effectiveness of retargeting ad-vertising, such asManchanda et al. (2006), Lambrechtand Tucker (2013), Li and Kannan (2014), and Hobanand Bucklin (2015).

The remainder of the paper is organized as follows.Section 2 presents the model. Section 3 considersthe case in which the firm does not observe whetherand when the consumer purchases the product.Section 4 studies the value of retargeting. Section 5includes several model extensions, including the ef-fects of a greater quality of the signals received by the

consumer when the firm does retargeting, the case inwhich the firm observes consumer purchases, and theeffects of consumers being forward-looking. Sec-tion 6 concludes.

2. The Model2.1. Consumer Search Behavior and PricingWe start by presenting a model of consumer searchbehavior and optimal firm pricing that generates anexpected profit v for the firm of a consumer receivingan informative signal that the product is a fit for theconsumer. The only important element from thissubsection for the remainder of the paper is the ex-istence of this expected profit v, and the reader lessinterested on the consumer search problem, which isnot the focus of this paper, can skip to the next section.Suppose that the consumer can be in a state in

which he knows that with equal probability he haseither zero value for a product or some value ω perperiod of using the product (Tables 1 and 2 present thenotation used in the paper.). The product is a durablegood with infinite life. By searching for information,with a process that is explained later, the consumercan determine, if he is in this state, whether the value iszero or ω. The consumer also knows that with hazardrate ψ he will switch from this state to a state in whichhe has zero value of the product forever. Working interms of present value of benefits, the consumer canthen have an overall gross benefit of getting theproduct of either zero or w � ω/ψ. Suppose that thefirm chooses a price P that cannot be customized, andcannot vary with time, and w has an ex ante cumu-lative probability distribution function F(w), withdensity f (w), with positive density on, and only on,[w,w] with w > 0. One can obtain F(w) directly fromthe cumulative probability distribution function onω.We consider the situation where the consumer onlylearns w when he finds that he has a strictly positivebenefit for the product. The case in which the con-sumer knows the value ofw, while he is still uncertainwhether the benefit of the product is either zero or w,is discussed briefly at the end of Section 5.3. Finally,letm be the marginal cost of production of the product,and let P̃ � argmaxP(P− m)[1 − F(P)], the static mo-nopoly price.3 If (w−m)f (w)> 1, the static monopolyprice is P̃ � w, and all consumerswho receive apositivefully informative signal purchase the product.A consumer can search for information at a cost of

ε dt for a period of time of length dt, where ε is as-sumed small. If the firm is not advertising, the con-sumer, if searching for information, receives a signalabout the product fit in the period dt with probabil-ity p dt. If the firm is advertising, the consumer, ifsearching for information, receives a signal aboutthe product fit in the period dt with probability p̂ dt,

Villas-Boas and Yao: Dynamic Model of Optimal RetargetingMarketing Science, Articles in Advance, pp. 1–31, © 2021 INFORMS 3

with p̂ > p.We further discuss the roles of p and p̂ later.If the consumer is not searching for information, theconsumer does not get any signal, whether the firm isadvertising.4 Given that the consumer receives asignal, the probability of it being informative is q.Finally, let S(P) be the expected consumer surplus

conditional on the consumer receiving a positiveinformative signal of the product fit and the firmcharging price P. That is, S(P) � ∫ wP (w−P)dF(w). Con-sumers are assumed to be risk neutral. Possible con-sumer discounting of the future is only consideredthrough the hazard rate ψ at which time their benefitof having the product is extinguished. We restrictattention to the case of consumers being myopicwith respect to any potential future advertising thatresults from the firm finding out that the consumer issearching for information. That is, consumers areunaware of the retargeting policy by the firm. Weconsider the case of forward-looking consumers inSection 5.3.For a consumer to search for information, it must be

that the expected benefit of searching for informationis greater than the cost of searching for information.Consider the case in which the consumer is not re-ceiving advertising. In that case, for the consumer towant to search for information, it must be the case thatq2 pS(P) ≥ ε. Define P as the price that makes this in-equality bind, S(P) � 2εpq.

Table 1. Notation

Variable Description

p Hazard rate of consumer receiving product informationif no retargeting

p̂ Hazard rate of consumer information if firm is doingretargeting

c Cost per unit of time of firm doing retargetingq Probability of consumer receiving fully informative

signal given that he receivedProduct information (with no retargeting if in

Section 5.1)q̂ Probability of consumer receiving fully informative

signal given that he receivedProduct information if firm is doing retargeting and this

probability is different than theCase of no retargeting (it only appears in Section 5.1)

φ Probability of firm observing that consumer issearching for information conditional on

Consumer receiving product information if the firm isnot doing retargeting

φ̂ Probability of firm observing that consumer issearching for information conditional on

Consumer receiving product information if firm isdoing retargeting

x Firm belief that the consumer is in the state of searchingfor information

ψ Hazard rate of consumer exogenously moving to thestate of not searching for information

x̂ Endogenous belief threshold, when firm does notrecognize when purchases occur such that firm doesretargeting for x ≥ x̂ and does not do retargeting forx < x̂

x̃ Endogenous belief threshold, when firm recognizeswhen purchases occur, such that firm doesretargeting for x ≥ x̃ and does not do retargeting forx < x̃

xm Endogenous belief threshold, when firm is myopic anddoes not recognize when purchases occur, such thatfirm does retargeting for x ≥ xm and does not doretargeting for x < xm

V(x) Value function of firm in region of beliefs of noretargeting if firm does not recognize whenpurchases occur

V̂(x) Value function of firm in region of beliefs of retargetingif firm does not recognize when purchases occur

Ṽ(x) Value function of firm in region of beliefs of noretargeting if firm recognizes when purchases occur̂̃V(x) Value function of firm in region of beliefs of retargetingif firm firm recognizes when purchases occur

v Payoff for firm if consumer purchases the productT̂ Maximum amount of time that a consumer could be

retargeted when firm does not recognize whenpurchases occur

T̃ Maximum amount of time that a consumer could beretargeted when firm recognizes when purchasesoccur

B pφ + ψ + p(1 − φ)qB̂ p̂φ̂ + ψ + p̂(1 − φ̂)qA p(φ + (1 − φ)q/2)Â p̂(φ̂ + (1 − φ̂)q/2)C, Ĉ, C̃, ̂̃C Constants in firm value functions

Table 2. Notation of Variables for Consumer Search Model

Variable Description

ω Utility per unit of time for consumer of product ifproduct is of any value to consumer

w Present value of product for consumer if product is ofany value (� ω/ψ; distributed with cumulativedistribution function F(w))

m Marginal cost of productionP Price charged by firmS(P) Expected consumer surplus given price P, conditional

on consumer receiving a positive informative signalP̃ Static monopoly price (� argmaxP(P −m)[1 − F(P)])ε Search cost per unit of time for consumerP Product price such that myopic consumer is indifferent

between searching and not searching for informationP∗ Optimal priceW(t) Expected present value of utility for consumer when

searching for information and receiving retargetingfor t time since firm observed the consumer search forinformation

Wn Expected present value of utility for consumer whensearching for information and not receivingretargeting

D Constant in consumer value functionη Disutility of receiving retargeting per unit of time


If the price is only checked each time the consumergets a signal, given that the search costs are sunk, thefirmwould choose to charge the staticmonopoly priceP̃, and the consumer would only search for infor-mation if P ≥ P̃, that is, S(P̃) ≥ 2εqp, and we obtain thatthe expected profit for the firm if the consumerreceives a positive informative signal of product fitis v � (P̃ −m)[1 − F(P̃)]. In this setting, if P < P̃, noconsumer would search for information, and the firmwould not do any retargeting.

Potentially more interesting for the context beingmodeled, price could be seen as being freely checked(or fixed over time, and learned at the first search),and then the firm can use it to provide incentives forthe consumer to search for information on product fit.The remainder of this section considers this case.

As the expected profit from a consumer receiving apositive informative signal is (P −m)[1 − F(P)], if ε issmall enough we have q2 pS(P̃) > ε, and the optimalprice to charge is then P ∗ � P̃. If ε is not small enough,then that inequality does not hold for P̃, and we havethen that the optimal price is P ∗ � P. That is, we havethat the optimal price P ∗ � min[P̃,P], which is inde-pendent of the search costs ε for small ε, and de-creasing in the search costs ε for large ε. We state thisresult in the following proposition.

Proposition 1. Suppose that consumers are not aware offuture retargeting. Then, the optimal price is independent ofsearch costs for low search costs and decreasing in searchcosts for high search costs.

Figure 1 illustrates how the optimal price P∗ evolveswith the search costs ε. In this case, we would havev � (P∗ −m)[1 − F(P∗)]. If (w −m)f (w) > 1 and ε < q2pS(w), then P∗ � w and v � w −m.As consumers that have some possible positive

value for the product are ex ante equal (ω is onlylearned when the consumer receives a fully infor-mative signal), the firm prices such that all consumersin that state search for information. If consumershave some information before searching for infor-mation on their potential value for the product, orif consumers are heterogeneous in their search costs,then the firm’s pricing decision affects which con-sumers search for information, and the value of vwould have to be adjusted by the margin obtained bythe firm on each sale.One possibility to also potentially consider is that

the consumer search costs ε are lower when theconsumer is being retargeted to than when the con-sumer is not being retargeted to. This could be, forexample, because the firm could direct signals to the

Figure 1. (Color online) Optimal Price P∗ for the Myopic Consumers’ Case as a Function of the Search Costs ε for the Case inWhich F(w) � w with Support [0, 1], and m � 0, q � .2, p � 1


consumer that are easier to process as information.In this case, if the search costs without retargetingare high, then the price isP (assuming that thefirmhasto set the same price with and without retargeting),and the lower search costs during retargeting justmeans that the consumer has a strict positive surplusof searching for information during retargeting anddoes not affect the remainder of the analysis. Asdiscussed in the next section, we can also interpret p̂ >p as the consumer having some form of lower searchcosts during retargeting.

2.2. Firm’s Information and BeliefsFrom the point of view of the firm, consider a con-sumer potentially searching for information on thepurchase of a product. A consumer can be in eitherof two states: (1) searching for information on theproduct or (2) not searching for information on theproduct. Given the optimal pricing determined inthe previous section, the state of searching for in-formation on the product is the state, in terms of theprevious section, in which the consumer knows thatwith equal probability he has either zero value for theproduct or some value ω per period of using theproduct. The state of not searching for information isthe state in which the consumer has zero value for theproduct forever. There could be several reasons thatthe consumer is not searching for information on theproduct: the consumer already bought the product, orthe consumer received information that the productis a poor fit for his preferences, the consumer realized

that he no longer needs this type of product, or theconsumer was never aware of this product.The firm does not knowwhich state the consumer is

in, but, occasionally, if the consumer is searching forinformation, the firm learns that, and at that momentthe consumer is in the state of searching for infor-mation. For now, we will assume that the firm doesnot know whether the consumer bought the product,and in Section 5.2 we will allow for that possibility.Time is continuous, and, to simplify, we assume no

discounting, as the real-world phenomena consid-ered typically last only a few weeks at most. We nowpresent how the consumer evolves from the searchingstate to the no searching state at each moment in timeand describe how the firm’s beliefs about the con-sumer being in the searching state evolve over time.Figure 2 illustrates the different possibilities.Consumers in the state of not searching for infor-

mation are not useful for the firm, as going forwardthey will not purchase the product, and we assumethat there is no activity that the firm can do thatmakesconsumers switch from no search to search. Consideralso that the probability of a consumer switching fromthe no searching state to the searching state is so low,that, if a firm knew that a consumer was in the nosearching state, it would never be profitable to advertiseto that consumer as long as the firm does not see that theconsumer is searching for information.5

Consumers in the searching for information statecan either remain there or move to the not searchingfor information state, by either purchasing the product,

Figure 2. (Color online) Illustration of How Consumers Can Evolve from the Searching State to the No Searching StateDepending on the Signals Received

Notes. Retargeting signals are received at the hazard p̂ instead of p, and the probability of the firm observing the signal is φ̂ instead of φ. Dashedlines represent observations, beliefs, and actions by the firm.


receiving definitive information about the product anddeciding that the product is a poor fit, or just losinginterest in the product. The hazard rate of the con-sumer receiving a signal, given that he is searching, isp > 0, as noted in Section 2.1. In the period of time dt,the probability of the consumer receiving a signal isp dt. This probability is greater if the firm is adver-tising to the consumer,which can be done at a cost c dt,with c > 0. This is one of the twomodeledmain effectsof retargeting. As noted in Section 2.1, p̂ is the hazardrate of the consumer receiving a signal if he is beingadvertised to, with p̂ > p.6 This allows for a cleardistinction between retargeting and not retargeting.This greater probability of receiving signals duringretargeting could also be seen as reducing some formof consumer search costs during retargeting7 andtherefore increasing the rate at which the consumerreceives information. We note that retargeting (andthe advertising cost incurred) is done at the indi-vidual level—whether to send an advertising mes-sage at cost c dt, is done at the individual level, for anindividual who was observed searching for infor-mation at some point in the past.We also consider thatretargeting affects the likelihood with which the firmis able to observe the consumer searching for infor-mation and the likelihood of the consumer receiving afully informative signal, whose role we now discuss.

If the consumer receives a signal, with probabilityq ∈ (0, 1), that signal is fully informative about the fitof the product for the consumer, as noted previously.8

We can think of q as small. As we assumed, good orpoor fit are equally likely; therefore, given that theconsumer received a signal, with probability q/2 theconsumer receives a positive fully informative signal,and delivers an expected profit for the firm of v,9

moves to the no searching for information state, andwith probability q/2, the consumer receives a negativefully informative signal, delivers zero payoff for thefirm, and also moves to the no searching for infor-mation state. For now, we assume that the firm doesnot observe whether the consumer buys the product.

When the consumer receives definitive informationabout the product fit, the firm does not necessarily seethat the consumer received this signal. In fact, the firmonly sees that the consumer has seen some form ofinformation with probability φ ∈ (0, 1), given that itoccurred. Given that the firm observes the consumersearching for information, the belief by the firm xt thatthe consumer is in the searching for information stateat time t is 1 − q, aswe know thatwith probability q theconsumer receives a fully informative signal, andmoves to the no searching for information state.Asnotedpreviously, this probability φ is greater if the firm isdoing retargeting to the consumer, which we denoteas φ̂ > φ. This is the second of the two modeled maineffects of retargeting. A firmmay have a better chance

at observing when the consumer receives a signal whiledoing retargeting. For example, this could be becausethe firm becomes more active in monitoring the con-sumer search behavior or because retargeting is morelikely to increase the consumer clicking on the firm’sadvertisements, which are easier to track by the firm.We distinguish below the unique effects of p and φon retargeting.Finally, as noted previously, when in the searching

for information state, the consumer can also moveexogenously to the no searching for information statebecause of losing interest in the product or category,and this occurs with hazard rate ψ > 0. This capturesthe idea that a consumer can lose interest in a cate-gory for some factors exogenous to the problem. Forexample, a consumer could be in the market to pur-chase a new car, but then because loss of income, otheractivities that require the consumer’s attention, or theconsumer realizing that public transportation couldbe a better alternative, the consumer drops out ofbeing in the market for a car.Unconditional on any information, this structure

yields a càdlàg process on beliefs (right continuouswith left limits). Given this structure, we can useBayes’ rule to construct how thefirm’s belief about theconsumer’s search state evolves over time twhen thefirm does not receive any information that the con-sumer received a signal about product fit.For example, consider that a firm observes a con-

sumer searching for information and then does notobserve that consumer searching for information forsome period. During that period, and as time passes,the firm puts more weight on the possibility that theconsumer either received a fully informative signalor decided to stop searching exogenously. That is, astime passes without observing the consumer searchingfor information, the firm has lower and lower beliefsthat the consumer is still searching for information.Formally, let xt be the firm’s beliefs that the con-

sumer is in the searching state at time t. Letting xt+ dt �xt + dxtdt dt be the firm’s beliefs that the consumer isin the searching state conditional on not receivingany information during the period dt, we want toexpress a relation between xt and xt+ dt.10 The un-conditional expected beliefs (unconditional on in-formation obtained during period dt), which we de-note by x̃t+ dt, have to fall in expected value by theprobability of the consumer exogenously droppingout of the search process, ψ xt dt, and the probabil-ity of the consumer receiving an informative sig-nal, p q xt dt. That is, E( x̃t+dt) � xt − (ψ + pq)xt dt. Inother words, the beliefs are a supermartingale fall-ing in expected value, by (ψ + pq)xt dt in period dt.This yields a relation between the beliefs at time t,and what can occur and the beliefs at time t + dt.


Second, With likelihood pφ xt dt the firm knows forsure that the consumer is in the searching state and thebelief that the consumer is searching jumps to 1 − q.Third, with probability (1 − pφ xt dt) the firm doesnot receive any information, and the belief that theconsumer is searching goes to xt+dt � xt + dxtdt dt. Thatis, E( x̃t+dt) � xt+dt(1− pφxt dt) + (1− q)pφxt dt. Puttingthis all together yields

xt � xt + dxtdt dt( )

1 − pφ xt dt( ) + 1 − q( )pφ xt dt+ ψ + pq( )xt dt. (1)

Dividing by dt and making dt go to zero, we canthen obtain11

dxtdt

� −pφxt 1 − xt( ) − ψxt − p 1 − φ( )qxt. (2)With this information on how the firm beliefs evolveover time, we can set up the dynamic programmingproblem of when the firm should and should notretarget. Once we have the optimal retargeting strategy,we can investigate how it is affected by the differentmarket forces.

Before studying formally the optimal retargetingpolicy and setting up the dynamic programming prob-lem, let us discuss the role of the different parameters.

The hazard rate of receiving a signal p (withoutadvertising) and p̂ (with advertising) measure theextent to which the consumer receives a signal aboutthe quality of the product. When these hazard ratesget infinitely large, the consumer is receiving signalsalmost all the time. This then makes the firm’s beliefthat the consumer in in the searching state to declinevery steeply without further information, but at thesame time allows the firm to get more frequent in-formation that the consumer is searching for infor-mation. These hazard rates being infinitely large alsolead the consumer to make a decision about the productfit almost immediately. If, alternatively, these hazardrates are close to zero, then the beliefs of the firm re-garding the consumer being in the searching statedecline less steeply, but the consumer could be in thesearching state for a long time and almost never get asignal, and the firm would rarely observe that theconsumer is searching for information. The benefit ofretargeting would be almost nonexistent if p̂ → p.Obviously, when the difference p̂ − p increases, thebenefit of retargeting is greater.

An increase in the cost of advertising, c, also makesretargeting less appealing. On the other hand, when cgoes to zero, it becomes optimal to do retargetingfor almost all levels of the firm’s belief regardingwhether the consumer is in the searching state, andthe threshold belief above which the firm chooses

to do retargeting, x̂, goes to zero. Note that c > 0throughout the paper (even when we consider thecase of c approaching zero), such that there is atradeoff of whether to do retargeting.12

Consider now the probability of a signal beinginformative given that it is received by the consumer,q. If this probability goes to one, almost every timethat a consumer gets a signal, the consumer decideswhether to buy the product, and there is therefore noincentive for a firm to do retargeting. If this probabilityis zero, then the consumer never gets true informationabout the value of the product, so the consumer wouldbe in the searching state forever, and there would alsobe no incentive for a firm to do retargeting. In whatfollows, we can think of q as small, but nonzero, suchthat informative signals are not too frequent.The role of φ (without retargeting) and φ̂ (with

retargeting) is to measure the extent to which the firmis able to observe when the consumer is receiving asignal.Whenφ or φ̂ approaches one, the firm is able toobserve almost all search occasions, and thereforeeach time the consumer searches, the firm’s belief thatthe consumer is searching keeps going to 1 − q. Whenφ � 0, the firm never has information if the consumeris searching. As the information technology im-proves and the firm is better able to track the con-sumer search behavior, wewould expect thatφ and φ̂would increase.The role of ψ is to allow for the possibility of the

consumer dropping out of the search process exog-enously. Given no discounting, ψ creates an incentivefor the firm to advertise to the consumer when thebelief that the consumer is in the searching state issufficiently high, in order not to lose the consumer,and to accelerate the possibility of the consumerlearning about the product fit. In this sense, ψ > 0plays the role of discounting in the model, such thatthere is an advantage in converting the consumersooner rather than later. If ψ � 0, the firm does notgain by converting the consumer sooner and justchooses not to advertise, that is, not to do retargeting,for any beliefs. The greater is ψ, for some levels of ψ,the more important it may be to advertise now.Note also that ψ captures the effect that firms wantto advertise to consumers when they are searchingfor information and in the market, because of the riskof the consumer exogenously losing interest in theproduct and not because of discounting the futurepayoffs. A firm may obtain an estimate about ψ frommarket research or historical data analysis, and thatestimate may vary across product categories.The current modeling of retargeting is one of pro-

viding informative advertising. Alternatively, we couldconsider the possibility of retargeting providing per-suasive advertising. In that case, we could interpret q asthe probability of the advertising being fully persuasive,


andwewouldnever have thepossibility of the consumerleaving the searching state when receiving a signal. Thebelief dynamics would be exactly as above, and, in thecomputation of the expectedpayoff, thefirmwould get vwith probability q when the consumer receives asignal, whereas in the case of informative advertising,the expected payoff for the firm would be v/2 withprobability q when the consumer receives a signal.

3. Optimal RetargetingTo compute the optimal retargeting strategy, we firststudy the form of the optimal policy.

Let π(t) be the expected future payoff for the firm,conditional on the consumer being in the search state,if the firm chooses to do retargeting for a period oftime t if the firm does not observe the consumersearching for information before t (in that contin-gency, the firm would go back to the optimal policy,after just observing that the consumer is searching forinformation). Note that π(t) includes the possibility ofthe consumer dropping out of the search processbefore t (either because of the exogenous drop rate ψor because the consumer receives a negative infor-mative signal on the product fit), and the retargetingcosts c that are incurred. The expected payoff for thefirm under this strategy, given that the consumer isactually not in the search state, is −ct. Let t∗(x) be theoptimal t when the firm has a belief x that the con-sumer is in the search state.

Then, for it to be optimal for the firm to do retar-geting when it has belief x, we must have that

xπ t∗ x( )( ) + 1 − x( ) −ct∗ x( )( ) ≥ xπ 0( ). (3)Consider now the belief x′ > x, and suppose that thefirm with that belief chooses the suboptimal strategyt∗(x), as themaximum length of time that thefirmdoesretargeting if the firm does not observe that theconsumer is searching for information prior to t∗(x).The expected future payoff for the firmwould then be

x′π t∗ x( )( ) + 1 − x′( ) −ct∗ x( )( )> xπ t∗ x( )( ) + 1 − x( ) −ct∗ x( )( ) ≥ xπ 0( ). (4)

This yields that at state x′, it is optimal for the firm todo retargeting. As this was obtained for a generalx′ > x, we then have that there is a threshold x̂ suchthe firm does retargeting for x ≥ x̂ and does not doretargeting for x < x̂. The following proposition statesthe result.

Proposition 2. There is a belief threshold x̂ such that thefirm advertises for x ≥ x̂ and does not advertise for x < x̂.

Given the evolution of the firm’s beliefs (with andwithout retargeting) presented in the previous sec-tion, the format of this optimal retargeting strategy

determines t∗(x) uniquely for x > x̂, a one-to-one map-ping between x and time to continue retargeting, andt∗(x) � 0 for x ≤ x̂.From this property of the optimal retargeting strategy,

we can then fully characterize the optimal retargetingstrategy. Let V(x) be the expected present value ofthe firm’s profits if thefirm has the belief x < x̂; let V̂(x)be the expected present value of profits of the firmif the firm has the belief x > x̂; and we already havethat v is the expected profit for the firm when theconsumer receives a positive informative signal.13 Werestrict attention to the case in which it is optimal todo retargeting if the firm observes the consumersearching for information, x̂ < 1 − q, which holds if thecost of retargeting c is low enough (presented in (A.3)in the appendix). To derive the optimal retargetingpolicy, let us first consider the present value of profitswhen the belief of the firm is sufficiently low, suchthat the firm does not advertise, x < x̂. The Bellmanequation of this problem can be written as

V x( ) � p q2vx dt + pφx dt V̂ 1 − q( ) + 1 − pφx dt( )

× V x( ) + V′ x( ) dxdt

dt[ ]

, (5)

as with probability pφx dt, the firm’s beliefs jump to(1 − q) at which point the firm moves to a retargetingregion,with an expected present value of profits equalto V̂(1 − q), with probability p q2 x dt the consumer re-ceives an informative signal that the product is a goodfit, and in that case the firm gets a payoff of v, andwithprobability (1 − pφx dt), the firm does not receive anynew information, and it then gets an expected payoffof V(xt+ dt), which can be approximated with a Tay-lor’s expansion to V(xt) + V′(xt) dxdt dt.14 Dividing by dtand substituting for dxdt from (2) one can obtain thedifferential equation

pφ 1 − x( ) + ψ + pq 1 − φ( )[ ]V′ x( ) + pφV x( )� pφV̂ 1 − q( ) + p q

2v, (6)

which can be solved to obtain

V x( ) � V̂ 1 − q( ) + qv2φ

+ C B − pφx[ ], (7)where B � pφ + ψ + pq(1 − φ), C is a constant to bedetermined later, and V̂(1 − q) is the present value ofprofits when the belief that the consumer is searchingis (1 − q), which is determined by V(0) � 0.When the firm is advertising, the Bellman equa-

tion becomes

V̂ x( ) � −c dt + p̂ q2vx dt + p̂φ̂x dt V̂ 1 − q( )

+ 1 − p̂φ̂x dt( )

V̂ x( ) + V̂′ x( ) dxdt

dt[ ]

, (8)


from which one can obtain along the same linesas previously,

V̂ x( ) � V̂ 1 − q( ) + qv2φ̂

+ Ĉ B̂ − p̂φ̂x[ ]

− cB̂

+ c B̂ − p̂φ̂xB̂2

logB̂ − p̂φ̂x

x, (9)

where B̂ � p̂φ̂ + ψ + p̂q(1 − φ̂), and Ĉ is a constantwhich is determined by (9) when x � 1 − q.15 To obtainC and x̂, one then makes V(̂x) � V̂(̂x), value matchingat x̂, and V′(̂x) � V̂′(̂x), smooth pasting at x̂. Note thatC < 0. We can obtain x̂ implicitly by

2φ̂ − qB̂ vc+ 2φ̂ψ + p̂q

B̂ln

B̂ − p̂φ̂x̂( )

1 − q( )ψ + p̂q( )̂x

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

× p̂φ̂ − pφ( )

ψx̂ + 2φ̂B ψ + p̂q( ) + vcpq̂xB̂ ψ + p̂q( )

× φ̂ − φ( )

� 0.(10)

We wrote the value function as a function of the be-liefs of the firm regarding whether the consumeris searching for information. As these beliefs areuniquely determined by the extent of time because thefirm observed the consumer searching for informa-tion, we could alternatively have used time sinceobservation of the consumer searching for informa-tion instead of xt in the value function.

From (10) we can obtain that the threshold x̂ is adecreasing function of v/c, which can be seen as ex-pected as the firm wants to do more retargeting ifthere is a greater profit from a sale, greater v, or ad-vertising is less costly, lower c.Wemay consider that vis large for higher priced products such as an auto-mobile or a house, and therefore, for such products,we may expect that firms do retargeting for a longerperiod of time after a consumer makes a decision.Some anecdotal evidence seems to suggest that this isthe case for consumers on the market for an auto-mobile or for real estate. However, another factor toconsider is that firms with such high vmay bid up thecost of retargeting c, such that v/c may end up notbeing too large for such high priced items.

From (10) we can also obtain that for retargeting tobe optimal, it requires a greater propensity for theconsumer to receive signals, p̂ > p, but it does notrequire that the firm better tracks that the con-sumer is searching for information, φ̂ > φ. That is,when retargeting does not lead to a greater haz-ard rate of the consumer receiving signals, p̂ � p, thenit is optimal for the firm not to do retargeting withφ̂ > φ. The ability to better track the consumer searchdoes not lead by itself for retargeting to be optimal.In that case, by doing retargeting the firm wouldjust incur the retargeting costs and not accelerate in

any way the possibility of the consumer purchasingthe product.To obtain some further insights, we concentrate on

the case inwhich the cost of advertising, c, approacheszero, c → 0. One may expect that these costs of ad-vertising are relatively small per consumer in digitalretargeting, a central motivation for this work. Fur-thermore, our numerical analyses, as discussed later,for the case when the costs of advertising are largerreplicate the analytical results obtained for the case ofadvertising costs converging to zero.When c → 0, the firm wants to advertise for almost

all beliefs, that is, x̂ → 0. We can then obtain a mea-sure of x̂ in comparison with c as

limc→0

x̂c� 2φ̂ p̂q + ψ

qv ψφ̂ p̂ − p( ) − p̂pq φ̂ − φ( )[ ]× pφ + ψ + pq 1 − φ

( )p̂φ̂ + ψ + p̂q 1 − φ̂

( ) . (11)From (11) we can obtain the following comparativestatics, in addition to the comparative statics withrespect to v and c discussed previously.

Proposition 3. Suppose that the firm does not detect whenthe consumer makes a purchase and that c → 0. Then, theregion of beliefs for optimal retargeting, x > x̂, is increasingin the propensity for the consumer to receive signals withretargeting, p̂, the likelihood of the firm observing duringretargeting that the consumer is searching, φ̂, the likelihoodof the consumer receiving an informative signal given that hereceives a signal, q, and the likelihood of the consumer de-ciding to stop searching exogenously, ψ, for small ψ, anddecreasing in the propensity of the consumer to receivesignals without retargeting, p, and in the likelihood of thefirm observing that the consumer is searching withoutretargeting, φ.

Some of these results can be seen as one wouldexpect: The firm wants to advertise more if adver-tising generates a higher chance of the consumergetting a signal ( p̂ ), if there is a greater likelihood ofsignals being informative (q), and if the chance of aconsumer receiving a signal without advertising islower (p).More interestingly, the firm wants to advertise

more if it is better able to track that the consumer issearching for information when retargeting (φ̂). Asthe firm is better able to track that the consumer issearching for information when retargeting, the firmadvertises in a greater region of the belief space. Onthe other hand, if the ability for the firm to trackconsumer search without retargeting increases, thefirm has less of a benefit from retargeting andtherefore advertises in a smaller region of the be-lief space.


One could also consider the possibility of the abilityto track consumers increasing equally for when thefirm does retargeting and does not do retargeting (anincrease in φ̂ and in φ in the same amount). In thatcase, one could potentially consider that with betteroverall tracking a firmwould think that not observingthe consumer searching for information might indi-cate that the consumer stopped being on the market.In fact, when the firm has better overall tracking, thefirm realizes that when it advertises, it will be bet-ter able to discern whether the consumer is searchingfor information and therefore chooses to still advertisefor lower beliefs of the consumer being in the searchingfor information state.

Also interestingly, when the likelihood of the con-sumer exogenously stopping the search process isgreater (ψ), if ψ is small, the firm wants to advertisemore. In this case, the firm realizes that in the futurethe consumer’s interest may disappear and wants toadvertisemore. If ( p̂− p)[(̂pφ̂− pφ)(1− q) − pq] > p̂pq(1−φ/φ̂), we can have the firm wanting to advertise in asmaller region of the beliefs when ψ increases, if ψ issufficiently large. In this case, which can occur, forexample, if the probability of the consumer receivingan informative signal is sufficiently small, if ψ issufficiently large, the firm realizes that the potentialbenefits of advertising leading to an informativesignal are not too high, and the firm chooses to ad-vertise in a smaller region of the belief space.

Given that we have x̂, we can compute the maxi-mum amount of time that a consumer could beretargeted after being identified as searching for in-formation. This can be obtained to be (with derivationpresented in the appendix):

T̂ � 1B̂log

1 − qψ + p̂q

B̂ − p̂φ̂x̂x̂

[ ]. (12)

One interesting comparative statics on T̂ is thatit can be decreasing in φ̂, which we state in thenext proposition.

Proposition 4. Suppose that the cost of retargeting c is closeto zero. Then, the maximum amount of time that a con-sumer could be retargeted after being identified as searchingfor information, T̂, is decreasing in the firm’s ability to trackconsumer search when retargeting, φ̂.

This proposition shows that the consumer couldpotentially benefit from the firm having a greaterability to track consumer search, considering theconsumer has disutility in receiving retargeting. Whenthe firm has a greater ability to track consumer search,it updates faster that the consumer may not besearching for information, when it does not observethe consumer searching for information. Then, thefirm may prefer to stop doing retargeting sooner, as

its beliefs that the consumer is searching for infor-mation fall now more steeply.Figures 3–10 illustrate how the optimal threshold x̂

varies with the different parameters. As shown pre-viously, as the cost of advertising c increases, the firmadvertises less, and the maximum time of a consumerreceiving advertising after having bought the productdecreases. Figure 4 illustrates that the firm has a lowerthreshold x̂ to advertise when the consumer is morelikely to get informative advertising, greater q, asshown in Proposition 3. More interestingly, the maxi-mum time receiving advertising after purchases is notmonotonic on q, increasing when q is small, as the firmis willing to advertise longer, and decreasing when qis large, as then the firm realizes that not observing theconsumer search for a long time ismore likely tomeanthat the consumer is not searching (the posteriorbeliefs that the consumer is searching for informationdecrease faster over time).Figure 5 illustrates that the effect of the ability of the

firm to track consumerswhen not retargeting (φ) has amonotonic effect on the optimal firm advertisingstrategy for c > 0. Increasing the ability to track if theconsumer is searching for information when notretargetingmakes the firmwant to advertise less (higherx̂), and naturally the time during which the consumerreceives advertising after purchase decreases.Figure 6 illustrates that the effect of the ability of the

firm to track consumers when retargeting (φ̂) hasalso a monotonic effect on the optimal firm adver-tising strategy for c > 0. Increasing the ability to trackif the consumer is searching for information whenretargeting makes the firm want to advertise more(lower x̂), and the time receiving advertising afterpurchase decreases. As shown in Proposition 4, agreater φ̂ has an effect on the firm’s beliefs decliningfaster, and this has a bigger impact on T̂ than the lowerthreshold x̂. As discussed previously, this illustratesthat improvements in information technologies leadingto an increase in the tracking ability byfirms of consumersearch could actually be beneficial to consumers in re-ducing the length of time that consumers receive ad-vertising. This could also potentially provide an incen-tive for consumers to credibly disclose to what extentthey are searching for information.As shown previously, the firm advertisesmore, and

the consumer ends up spending more time receivingadvertising after purchase, if the profit for thefirmof asale, v, is greater (Figure 7). Also as expected, and asillustrated in Figure 8, as the rate of receiving infor-mation without advertising increases, the firm ad-vertises less, and the consumer spends less time re-ceiving advertising after purchase. As illustrated inFigure 9, and as expected, when the rate at which theconsumer receives information when being adver-tised to ( p̂ ) increases, thefirm is interested in advertising


more for the same beliefs ( x̂ is decreasing in p̂ ). Moreinterestingly, when p̂ increases, the length of time forwhich a consumer continues to receive advertisingafter a purchase (T̂) first increases and then decreases.It increases for low p̂ because the firm now wants toadvertise more. It decreases for high p̂ because in thatcase the firm updates more quickly that the consumermay not be in the searching for information state, andso it reaches the belief x̂ faster.

Finally, as illustrated in Figure 10, when the ex-ogenous rate of the consumer dropping out of thesearch process, ψ, increases, and ψ is small, the firmwants to advertise more (̂x is decreasing in ψ), as itwants to take advantage of the consumer searchingfor information. For some parameter values, as notedpreviously (e.g., if the signal informativeness q is lowenough), we can have x̂ increasing in ψ for large ψ, asthe potential benefits of advertising are now weaker(the case with solid lines in Figure 10). More inter-estingly, when ψ increases, the length of time forwhich a consumer continues to receive advertisingafter a purchase (T̂) first increases and then decreases.It increases for low ψ because the firm now wants toadvertise more. It decreases for high ψ because the

firm’s beliefs that the consumer is searching now fallmore steeply, which yields T̂ to fall.The optimal policy accounts for the future effects of

what the firm learns based onwhether it advertises. Itis interesting to compare this with the optimal policyif the firm were myopic. This myopic policy would beto advertise if the expected current benefit of ad-vertising, (̂p − p) q2 v xt dt, is greater than the cost ofadvertising, c dt. This would give a threshold of beliefon searching of xm � 2cvq(p̂−p). Neither the ability of thefirm to track whether consumers are searching forinformation, represented by φ̂ and φ, nor the exog-enous rate of the consumer dropping out of the searchprocess, ψ, affects the optimal myopic policy, as thebenefits of the ability to track consumers and the lossof the consumer dropping out of the search processoccur in the future.Comparing the myopic policy with the optimal

dynamic policy for the case of small c, one can get thatit could be larger or smaller. In fact, the optimaldynamic policy is to advertise more than in the my-opic policy if the product of the likelihood of trackingconsumer search and the probability of the consumerreceiving signals is high enough with retargeting,

Figure 3. (Color online) Evolution of x̂ and theMaximumTime Receiving AdvertisingWithout Search, T̂, as a Function of c forv � 2, p � .4, p̂ � 1, q � .1,φ � .5, φ̂ � .8, and ψ � .1


( p̂φ̂ − pφ) high enough, and the probability of theconsumer exogenously dropping out of the searchprocess, ψ, is high enough.

The future benefits of retargeting are greater if theproduct of the likelihood of tracking consumer searchand the probability of the consumer receiving signalsis high enough, and therefore, that makes the optimaldynamic retargeting policy to be to advertise in alarger region of the belief space. This is possible evenif the likelihood of tracking consumer search is thesame with retargeting and without retargeting. In-terestingly, the future effects of retargeting now arealso not too large if the probability of the consumerexogenously dropping out of the search process is toolow, because then the firm might as well wait for theconsumer to find out about the product fit withoutretargeting. Therefore, the probability of the con-sumer exogenously dropping out of the search pro-cess needs to be high enough for the optimal dynamicretargeting policy to prescribe a greater region of thebelief space for advertising than the myopic policy.

The myopic policy can prescribe a greater regionof the belief space than the optimal dynamic policy,

because it does not account for the fact that evenwithout advertising the consumer may find out in thefuture that the product is a good fit and purchase theproduct. Therefore, once the future potential payoffs areaccounted for, it may be optimal for the dynamic policyto prescribe less advertising than the myopic policy.We summarize these results in the following

proposition.

Proposition 5. The optimal retargeting policy prescribesadvertising for lower beliefs of consumer searching than themyopic policy if the effect of advertising is relatively high onthe interaction of the likelihood of tracking consumer searchand the probability of the consumer receiving signals (highp̂φ̂ − pφ) and the probability of the consumer exogenouslydropping out of the search process is not too low (ψ nottoo low).16

4. The Value of RetargetingIt is also interesting to evaluate the value for the firmof having the possibility of doing retargeting. This canbe seen as the amount a firm iswilling to invest to havethepossibility of doing retargeting. In evaluatingapolicy

Figure 4. (Color online) Evolution of x̂ and theMaximumTime Receiving AdvertisingWithout Search, T̂, as a Function of q forv � 2, p � .4, p̂ � 1, c � .01,φ � .5, φ̂ � .8, and ψ � .1


of making retargeting unlawful, this is the payoff for thefirm that is lost. The payoff for the consumer is evaluatedin Section 5.3.

Consider this value of retargeting evaluated at atimewhen the firm observes a consumer searching forinformation. The belief that the consumer is searchingfor information is then (1 − q) at that time. The ex-pected present value of profits with retargeting at thatmoment is then V̂(1 − q), which can be obtained from(7) as V̂(1 − q) � − qvφ − CB, given that V(0) � 0.

The maximum expected present value of profitswith the possibility of retargeting can be obtainedwith c → 0, which can be obtained to be

limc→0

p̂q 1 − q( )v2 ψ + p̂q( ) . (13)

This limit is independent of φ and φ̂ as with c → 0; thefirm is retargeting forever in the limit, which meansthat the likelihood of the firm observing the consumersearching for information becomes irrelevant. This isnot to mean that the expected present value of profitswith the possibility of retargeting does not depend on

φ or φ̂ for c > 0. In fact, this expected present value ofprofits has a component that depends on φ and φ̂ forc > 0, but that component goes to zero as c ap-proaches zero.To obtain the value of retargeting, we have to still

compute the expected present value of profits whenthere is no possibility of retargeting and the belief thatthe consumer is searching for information is (1 − q),which can be obtained to be V(1 − q) � pq(1−q)v2(ψ+pq) .We can then obtain the value of retargeting when

c → 0 to be

limc→0 V̂ 1 − q

( ) − V 1 − q( ) � q 1 − q( )vψ p̂ − p( )2 ψ + p̂q( ) ψ + pq( ) . (14)

This yields the following comparative statics.

Proposition 6. Suppose that the cost of retargeting is small.Then the value of retargeting is increasing in the propensityfor the consumer to receive signals with retargeting, p̂, theprofit earned if the consumer purchases the product, v, thelikelihood of the firm observing that the consumer is searchingwhen retargeting, φ̂, decreasing in the propensity of theconsumer to receive signals without retargeting, p, and the

Figure 5. (Color online) Evolution of x̂ and theMaximumTime ReceivingAdvertisingWithout Search, T̂, as a Function ofφ forv � 2, p � .4, p̂ � 1, c � .01, q � .1, φ̂ � .8, and ψ � .1


likelihood of the firm observing that the consumer is searchingwhen not retargeting, φ. When φ̂ � φ, the value of retar-geting is increasing in φ. The value of retargeting isincreasing (decreasing) in the likelihood of the consumerreceiving an informative signal given that he receives asignal, q, for q < (>) q∗ for a q∗ ∈ (0, 1/2), and in thelikelihood of the consumer deciding to stop searching ex-ogenously, ψ, for small ψ < (>) p̂q.

The more interesting comparative statics are theones on the probability of the consumer receiving aninformative signal given that he receives a signal, q,and on the likelihood of the consumer deciding to stopsearching exogenously, ψ. As discussed previously,when the probability of receiving an informativesignal is close to zero, there is not much gain inretargeting, as the consumer is not likely to receive aninformative signal. At the same time, if the probabilityof receiving an informative signal is close to one, thereis also not much of a benefit in retargeting, as whenthe firm observes the consumer searching for infor-mation, the consumermost likely also received a fullyinformative signal. The proposition shows that thevalue for retargeting is the highest for an intermediatevalue of q, which is less than 1/2.

Also as discussed previously, if the likelihood of theconsumer deciding to stop searching exogenously isclose to zero, there is not much gain in retargeting, asthe consumer will end up receiving a fully informa-tive signal even if the consumer is not being retar-geted to. At the same time, if the likelihood of theconsumer deciding to stop searching exogenously isvery high, there is not much gain in retargeting as theconsumer is likely to stop searching before receiving afully informative signal, even when being retargetedto. We find that the ψ that maximizes the value ofretargeting is ψ � p̂q.This value of retargeting is not the cost of retargeting c

such that the firm chooses not to do any retargeting.That threshold cost of retargeting is presented in (A.3)in the appendix. The greater the c, the less retargetingthat the firms do, and the lower the value of retargeting.One could also ask what is the value of information

for the firm of learning for sure if a consumer is or isnot searching for information. If a consumer is searchingfor information for sure, the expected presented valueof profits is V̂(1). If a consumer is not searching forinformation for sure, the expected value of profitsis V(0) � 0. The value of perfect information is then

Figure 6. (Color online) Evolution of x̂ and theMaximumTime ReceivingAdvertisingWithout Search, T̂, as a Function of φ̂ forv � 2, p � .4, p̂ � 1, c � .01, q � .1,φ � .5, and ψ � .1


xV̂(1) − V̂(x), which is positive as V̂(x) is strictlyconvex for x > x̂.17 Given the convexity of the valuefunction, we have V̂(1) − V̂′(1) < 0 and V̂(1) > V′(0) �V̂′(̂x), which means that the value of perfect infor-mation is maximized at some x∗ ∈ (̂x, 1). That is, as timepasses since the firms knows for sure that the consumeris searching for information, the value of perfect infor-mation for the firmfirst increases, until the belief reachesx∗, and then decreases.We can also get that, for c → 0,we have V(1) < V̂′(1 − q) such that x∗ ∈ ( x̂, 1 − q). Thatis, as time passes since the firm actually observes theconsumer searching for information, the value of perfectinformation for thefirmfirst increases and thendecreases.

5. Model ExtensionsThis section considers several model extensions, in-cluding the possibility of the effect of retargeting onthe quality of the signals, the possibility of the firmrecognizing when purchases occur, and the possi-bility of the consumer being forward-looking withrespect to the firm’s retargeting strategy.

5.1. Retargeting and Quality of SignalsIn this section we consider the possibility that retar-geting may not only increase the hazard rate of in-formation received by the consumers and the likeli-hood of the firmobservingwhen the consumers receivea signal but can increase the likelihood of the con-sumers receiving a fully informative signal conditionalon receiving a signal, q, aswell. For example, this couldbe because the retargeting of the firm could be moreinformative than the usual consumer informationsources. In terms of the previous model, this wouldmean that the parameter qwould be greater when thefirm is retargeting, q̂ > q.In terms of the previous analysis, considering this

case could be done by replacing q with q̂ in (9), withB̂ � p̂φ̂ + ψ + p̂̂q(1 − φ̂). The analysis would then leadto obtaining x̂ implicitly by (A.25), which is presentedin the appendix.We can obtain that, even when retargeting does not

lead to a greater hazard rate of the consumer receiv-ing signals, p̂ � p, it is still optimal for the firm to do

Figure 7. (Color online) Evolution of x̂ and theMaximumTime Receiving AdvertisingWithout Search, T̂, as a Function of v forφ � .5, p � .4, p̂ � 1, c � .01, q � .1, φ̂ � .8, and ψ � .1


retargeting with q̂ > q. That is, the ability by itself to getthe consumers to receive more informative signals withretargeting leads retargeting to be optimal. In that case,by doing retargeting, the firm accelerates the possibilityof the consumer purchasing the product before theconsumers drops exogenously out of the search process.

When the costs of retargeting approach zero we canobtain a measure of the ratio x̂/c as

limc→0

x̂c� 2φ̂ p̂̂q+ψ

v ψφ̂ p̂ q̂−pq( )+ p̂p q̂{ φ 1− φ̂( )̂q− φ̂ 1−φ( )q[ ]}× pφ+ψ+pq 1−φ

( )p̂φ̂+ψ+ p̂ q̂ 1− φ̂

( ) .(15)

From this, we can obtain that a greater informativenessof signals during retargeting leads to greater retargeting.That is, the firm prefers to increase the region of beliefswhere retargeting occurs if it leads to greater signalinformativeness, because of the acceleration in con-sumers deciding whether to purchase the product.

5.2. Recognizing PurchasesIn the previous sections, it was assumed that theseller does not know when the consumer makes thepurchase, and therefore, we have that the firm maycontinue to advertise even after a purchase. Withimprovements in tracking technologies, firms mighthave the ability to detect when consumers purchasethe product and therefore do not send further retar-geting advertising. This section considers this caseand compares the optimal strategy with the case inSection 3 in which the seller does not detect consumerpurchases. We consider the case in which if theconsumer receives a positive informative signal theconsumer always purchases the product.18

In this case, if a firm observes a consumer search forinformation (which happens with probability pφ dtwhen not advertising given that the consumer issearching for information), with probability q/2 thefirm sees the consumer purchase the product. Afterobserving the consumer search for information andnot observing any purchase, the posterior belief thatthe consumer is still searching for information is(1 − q2−q), as the probability of no purchase given

Figure 8. (Color online) Evolution of x̂ and theMaximumTime Receiving AdvertisingWithout Search, T̂, as a Function of p forφ � .5, v � 2, p̂ � 1, c � .01, q � .1, φ̂ � .8, and ψ � .1


search is (1 − q/2), and the probability of continuing tosearch for information given that a signal was re-ceived is (1 − q).

The belief updating when the firm does not observethe consumer searching for information is also nowdifferent because the firm observes a product pur-chase which occurs with probability p(1 − φ) q2 xt dt, ifthe firm does not observe consumer search, and giventhat the consumer is searching for information. As inSection 3, the beliefs at time t have to be consistentwith what can occur in the future. As in Section 3, thebeliefs are again a supermartingale, falling in ex-pected value by (ψ + pq)xt dt in period dt, and wecan obtain

xt � xt + dxdt dt( )

1 − φ p xt dt − 1 − φ( ) q2 p xt dt[ ]+ φ 1 − q

2

( )1 − q

2 − q( )

p xt dt + ψ xt dt+ q p xt dt. (16)

We can then obtain that (2) changes in this case to

dxtdt

� −p φ + 1 − φ( ) q2

[ ]xt 1 − xt( ) − ψxt − p

× 1 − φ( ) q2xt. (17)

Let x̃ be the threshold belief of the firm such that thefirm only advertises for x > x̃. With similar analysis asin Section 3, we can obtain that, when the firm is notadvertising, x < x̃, the present value of profits of thefirm can be obtained as

Ṽ x( ) � C̃ B − Ax[ ]

+qv/2 + φ 1 − q/2( )̂̃V 1 − q2−q( )

φ + 1 − φ( )q/2 , (18)where A � p(φ + (1 − φ) q2), C̃ is a constant to be de-termined later, and B is, as defined previously, B �pφ+ p(1 − φ)q + ψ.

Figure 9. (Color online) Evolution of x̂ and theMaximumTime Receiving AdvertisingWithout Search, T̂, as a Function of p̂ forφ � .5, v � 2, p � .4, c � .01, q � .1, φ̂ � .8, and ψ � .1


For the region of beliefs where the firm advertises,x > x̃, we can obtain, along the same lines as in Section 3,the present value of profits as

̂̃V x( ) � ̂̃C B̂ − Âx[ ] + c B̂ − ÂxB̂2

logB̂ − Âx

x− cB̂

+qv/2 + φ̂ 1 − q/2( )̂̃V 1 − q2−q( )

φ̂ + 1 − φ̂( )

q/2, (19)

where Â � p̂[φ̂ + (1 − φ̂)q/2], ̂̃C is a constant to bedetermined, and B̂ is, as defined previously, B̂ � p̂φ̂ +p̂(1 − φ̂)q + ψ. With the conditions Ṽ(0)�0,Ṽ(̃x)� ̂̃V(̃x),and Ṽ′(̃x) � ̂̃V′(̃x), and evaluating (19) at x � 1 − q2−q,one can obtain the value of x̃ and of the constantsC̃ and ̂̃C. The optimal x̃ is determined implicitly by(A.35), presented in the appendix.

For c → 0, we can obtain x̃ → 0, x̃ < x̂ and that x̃cconverges to

limc→0

x̃c� 2 p̂q+ψ

qvpφ+ψ+ pq 1−φ( )p̂φ̂+ψ+ p̂q 1− φ̂

( )× φ̂ 2− q

( )+ qψ 2− q( ) p̂φ̂− pφ( )+ q p̂− p( )[ ] + qp̂p φ̂−φ( ) .

(20)Proposition 7. When the firm is able to recognize purchaseswhen they occur, and the costs of doing retargeting are small,the firm has a threshold of beliefs that the consumer issearching for information to do retargeting, which is lowerthan the threshold in the case in which the firm does notrecognize purchases immediately.

In the case of recognizing purchases, the firmknows that if the firm waits longer without doingretargeting, the firm may learn that retargeting is not

Figure 10. (Color online) Evolution of x̂ and the Maximum Time Receiving Advertising Without Search, T̂, as a Function of ψfor φ � .5, φ̂ � .8, v � 2, p̂ � 1, c � .01, and q � .1

Note. The solid lines have p � .4, and the dashed lines have p � .8.


needed because the consumer purchased the product.On the other hand, when doing retargeting, the firmknows when the consumer made a purchase, andretargeting is no longer needed. It turns out that,when the costs of doing retargeting are small, thelatter effect dominates, and the threshold belief to doretargeting is lower when purchases are recognized.That is, the region of beliefs for which the firm doesretargeting is greater when the firm recognizes pur-chases than when the firm does not recognize pur-chases. We could not find parameter values for whichthis result did not hold.

Figures 11–15 illustrate how x̂ and x̃ evolve forthe different parameters for the case of c > 0, show-ing that for the parameters considered we alwayshave x̂ > x̃.

It is also interesting to evaluate how the beliefsevolve in this case where purchases are recognizedand compare them with the case above when pur-chases are not recognized. In this case of purchasesbeing recognized, we can obtain that the beliefs thatthe consumer is searching for informationasa functionof

time, because the time at which the firm recognizes thatthe consumer is actually searching for information, as

xt � 2 1 − q( )

B̂

p̂ 1 − q( ) φ̂ 2 − q( ) + q[ ] + ψ 2 − q( ) + p̂q[ ]êBt . (21)Comparing this with the previous case when pur-chases are not recognized, one can obtain that, withoutfurther information, the beliefs while the firm doesretargeting without purchase recognition are alwaysbelow the beliefs under purchase recognition. However,as in the case with purchase recognition the firm doesretargeting for longer (asdiscussed later),wehave that inthe case with purchase recognition, the beliefs can atsome point be lower in the retargeting phase than thelowest possible beliefs in the case without purchaserecognition in the retargeting phase (̂x). Figure 16 il-lustrates how the beliefs evolve over time under thetwo different conditions.As in Section 3, in this case when purchases are

recognized, we can also compute the maximum amountof time that a consumer could be retargeted after being

Figure 11. (Color online) Evolution of x̂ and x̃ as a Function of q for c � .01, v � 2, p � .4, p̂ � 1,φ � .5, φ̂ � .8, and ψ � .1


identified as searching for information, which we candenote as T̃, and is presented in the appendix. For cclose to zero, one can obtain that this length of time isgreater than the maximum amount of time of retar-geting in the case when purchases are not recognized.Recognizing purchasesmakes the extent of time that aconsumer can receive retargeting without being in thestate of search for information to be greater. This is be-cause thefirmnowhas a greater belief that the consumeris still searching for information because the firm wouldobserve whether a purchase had occurred.

5.3. Forward-Looking ConsumersThe analysis in the previous sections considered thatthe consumers were not aware that, if they wereobserved searching for information, they would re-ceive advertising later. Consider now the case offorward-looking consumers, where consumers arenow aware that if they are observed searching forinformation, they will receive retargeting advertisingthat will provide more information about the productfit. Suppose that consumers are infinitely patient, butgiven that the consumer also knows that with hazardrateψ, hewill switch from a state of having a potentialpositive benefit for the product to a state in which he

has zero benefit of the product forever, the con-sumer has bounded benefit of having the product.The existence of hazard rate ψ works as discountingon the consumer benefits. This possibility of retar-geting, given the assumed no hassle costs of receivingadvertising, makes then the consumermorewilling tosearch for information than in the case of myopicconsumers, potentially allowing the firm to increaseits price. To focus on the critical market forces, andsimplify the analysis, let us also assume that theconsumer knows when retargeting is occurring.To analyze this situation, let Wn be the expected

present value of benefits for the consumer when thefirm is not retargeting, and let W(t) be the expectedpresent value of benefits for the consumer when thefirm has been retargeting for a t period of time. FromSection 3, we also have the maximum extent of timethat a consumer is retargeted to without purchase as afunction of v as T̂. The expected payoff for the con-sumer when searching for information without beingretargeted to can be derived as

Wn � q2 S P( ) p dt − ε dt + 1 − B dt( )Wn + φ 1 − q( )

×W 0( ) p dt. (22)

Figure 12. (Color online) Evolution of x̂ and x̃ as a Function of φ for c � .01, v � 2, p � .4, p̂ � 1, q � .1, φ̂ � .8, and ψ � .1


To obtain (22), with probability q2 p dt, the consumergets an expected consumer surplus S(P), and duringthe period dt, the consumer pays search costs ε dt.Furthermore, with probability (1 − Bdt) the consumerdoes not get any full informative signal or drop ex-ogenously out of the search process, nor does the firmobserve the consumer searching for information; inthis case, the consumer gets the expectedpresent value ofbenefits when the firm is not retargeting, Wn.19 Finally,with probability φ(1 − q)p dt, the firm observes theconsumer searching for information, but the con-sumer does not receive a fully informative signal, andin that case, the consumer gets the expected presentvalue of benefits when the firm is just starting theretargeting phase, W(0).

From (22) one can obtain

BWn � q2 pS P( ) − ε + φ 1 − q( )

pW 0( ). (23)

Similarly, if the consumer is searching for informa-tion and is being retargeted to for a t period of time,

we can now derive his expected present value ofpayoffs as

W t( ) � q2S P( ) p̂ dt − ε dt + 1 − B̂ dt

[ ]W t( )[

+W′ t( ) dt] + φ̂ 1 − q( )W 0( ) p̂ dt. (24)To obtain (24), with probability q2 p̂ dt, the consumergets an expected consumer surplus S(P), and duringthe period dt, the consumer pays search costs ε dt.Furthermore, with probability (1 − B̂ dt), the consumerneither gets any full informative signal nor dropsexogenously out of the search process, and the firmdoes not observe the consumer searching for infor-mation; in this case, the consumer gets the expectedpresent value of benefits when the firm is retargetingfor an additional period dt, W(t + dt). Finally, withprobability φ̂(1 − q)̂p dt, the firm observes the con-sumer searching for information, but the consumerdoes not receive a fully informative signal, and in thatcase, the consumer gets the expected present value of

Figure 13. (Color online) Evolution of x̂ and x̃ as a Function of φ̂ for c � .01, v � 2, p � .4, p̂ � 1, q � .1,φ � .5, and ψ � .1


benefits when the firm is just starting the retargetingphase, W(0).

From (24) we can obtain

W t( ) � DêBt + 1B̂

q2p̂S P( ) − ε + φ 1 − q( )̂pW 0( )[ ], (25)

where D is a constant to be determined, and fromwhich we can obtain

W 0( ) � DB̂ +q2 p̂S P( ) − εq̂p + ψ . (26)

As the consumer’s expected value of payoffs is con-tinuous when the firm switches to stopping adver-tising,we haveWn � W(T̂), fromwhichwe can obtain,using (23), (25), and (26), the value of D, which ispresented in the appendix. We can obtain that D < 0,which yields that W(t) is decreasing in t at an in-creasing rate. That is, the consumer is better off thelonger he is likely to be receiving retargeting, and thisbenefit falls quickly when the maximum future timeis reduced.

For search costs that are not too small, the optimalpolicy of the firm is to price such that a consumer not

receiving retargeting is indifferent between searchingand not searching for information, Wn � 0. As W(t) isdecreasing in t and W(T̂) � Wn, we then have thatW(t) > 0 for t ∈ [0, T̂). From this we can obtain from(26) that q2 p̂S(P) − ε > 0; that is, when the period ofretargeting starts, the consumer has a strictly posi-tive current expected benefit of searching for infor-mation. This confirms that during the period ofretargeting the consumer continues to want to searchfor information.From (23) and W(0) > 0, we can also obtain that

q2 pS(P) − ε < 0; that is, when the consumer is not re-ceiving retargeting, the consumer has a strictly nega-tive current expected benefit of searching for infor-mation. This is the positive effect of the consumer’ssearch for information because the consumer is for-ward-looking.To obtain further insights, for c → 0, we have

T̂ → ∞, from which we can obtain

W 0( ) →q2 p̂S P( ) − εq̂p + ψ , (27)

Wn → 1Bq2pS P( ) − ε + φ 1 − q

( )p

q̂p + ψq2p̂S P( ) − ε

[ ]{ }. (28)

Figure 14. (Color online) Evolution of x̂ and x̃ as a Function of ψ for c � .01, v � 2, p � .4, p̂ � 1, q � .1,φ � .5, and φ̂ � .8


Setting Wn � 0 to obtain the optimal price P∗, we getthat when c → 0, the optimal price when ε is small isobtained by

S P∗( ) � 2εqp

B

B̂, (29)

which leads to a higher price P∗ than in the myopicconsumers case. From this, one can also obtain that, asthe search costs ε increase, P∗ has to decrease, re-ducing then v � (P∗ −m)[1 − F(P∗)]. That is, as searchcosts increase, the firm reduces the price to inducesearch, and earns a lower expected profit per con-sumer who receives a positive informative signal.

For general costs of retargeting c, P�

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