ISSN (Print) 0473-453X

Discussion Paper No. 1219

ISSN (Online) 2435-0982

STRATEGIC ANONYMITYAND BEHAVIOR-BASED PRICING

Stefano Colombo
Paolo G. Garella
Noriaki Matsushima

November 2023

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Strategic anonymity and behavior-based pricing *Stefano Colombo †Università Cattolica del Sacro Cuore
Paolo G. Garella ‡Università di Milano
Noriaki Matsushima §Osaka University

Abstract
In a model of behavior-based price discrimination (BBPD), we argue that sellers may havediscretionary power to let buyers decide whether to be identified (e.g., creating an account) or
remain anonymous (no account creation). The price equilibria generate a more fragmented
market segmentation than under the standard BBPD. Firms might prefer a policy where theyleave buyers the decision to remain or not be anonymous, breaking the standard BBPD result.
Furthermore, firms can realize higher profits than under uniform pricing, contrary to thestandard BBPD. Also, firms may adopt asymmetric policies concerning the account creation
requirement.
JEL codes: D43, L13.
Keywords: strategic anonymity; behavior-based pricing; privacy.

We gratefully acknowledge financial support from JSPS KAKENHI Grant Nos. JP20H05631,
JP21H00702, JP21K01452, JP21K18430, and JP23H00818, the International Joint Research
Promotion Program at Osaka University, and the program of the Joint Usage/Research Centerfor `Behavioral Economics' at ISER Osaka University. Although Matsushima serves as amember of the Competition Policy Research Center (CPRC) at the Japan Fair Trade Commission(JFTC), our paper does not necessarily reflect the views of the JFTC. The usual disclaimer
applies.
† Dipartimento di Economia e finanza, Università Cattolica del Sacro Cuore, Largo A. Gemelli 1,
20123, Milano, Italy. E-mail: stefano.colombo@unicatt.it.
‡ Dipartimento di Economia, Management e Metodi Quantitativi. Università di Milano, Via
Conservatorio 7, 20122 Milano, Italy. E-mail: paolo.garella@unimi.it§ Institute of Social and Economic Research, Osaka University, Mihogaoka 6-1, Ibaraki, Osaka,567-0047, Japan. E-mail: nmatsush@iser.osaka-u.ac.jp.
*

11. INTRODUCTION
The interaction between sellers and buyers over the Internet has recast many
pre-Internet era characteristics. The search for products and sellers before
purchase and the assistance after a sale—covering product usage guidance,
repairs, maintenance, and the purchase of additional complementary goods orservices—all go through within a web-based or web-augmented environment.
Sellers play a significant role in shaping this digital space, although there arelimitations on how users’ communities can organize themselves.
In this respect, as buyers display much of the pre-Internet era heterogeneity,
now they exhibit heterogeneity of preferences over websites in terms of thesites' organization, product information display, web tools, order placement
methods, and payment systems. In fact, heterogeneity in the readiness to accept
web-based self-service technologies, as related to different tastes and attitudes,
is widely recognized in the retail management literature (Liljander et al. 2006,
Ding et al. 2007). Therefore, buyers choose websites in a way that resembles the
choice between differentiated products in the physical world.
In general, however, when buyers buy from a website, they leave a trackrecord that purchasers in physical stores do not normally leave, inspiring the
research on behavior-based price discrimination (BBPD), starting with Caminal
and Matutes (1990) and Fudenberg and Tirole (2000). The literature hasdeveloped detailed analyses of the possibility to price discriminate between
returning buyers and new customers (Chen 1997, Choe et al. 2018, Laussel andResende 2022). The underlying assumption in BBPD is that firms can invariably
identify purchasers when purchasing products or services, as with mobile
phone service subscriptions.

2Data protection laws and regulations, however, tend to limit data collection
without the consent of the data owner when these are not necessary to thetransaction. While exogenous (random) imperfect recognition is the object ofsome work in the BBPD area (Colombo 2016, Esteves 2014), the level of
information accuracy is largely determined endogenously by consumers and/or
firms’ strategies. For instance, Amazon clients can purchase items only after
creating a personal account (thus renouncing anonymity). On the opposite,
eBay allows consumers to purchase after creating an account or evenanonymously, and among eBay’s clients, some choose the first option (accountcreation), whereas others choose the second one (anonymous purchase).
Endogenous imperfect recognition has been so far neglected by the literature.
Hence, the aim of the present paper is to explore endogenous imperfectrecognition (generated by strategic anonymity by both buyers and sellers).
We posit that sellers possess discretionary power in deciding whether
buyers can control the decision of being identified or remaining anonymous –we shall be a little more specific in the sequel. This discretionary power granted
to the firms then translates into an added discretionary power to buyers when
they behave strategically, and the resulting identification or anonymity of abuyer becomes entirely endogenous.
Our model, based on these considerations, simplifies the story by assumingthat sellers can identify a buyer only if the purchase occurs after creating an
account at the selling website. Sellers can, however, also allow purchases
without creating an account. The decision by the seller in our model generates
two distinct strategies: a compulsory account creation (implying identificationof the buyer) or a menu choice for the consumers that might decide whether topurchase without an account (“anonymous” transactions) or with an account.
These choices then shape the ensuing competition environment. Next, we also
3compare them with the case where firms allow only purchases without an
account (“anonymous” transactions only), which amounts to a uniform pricing
regime.
From the viewpoint of buyers, creating an account in a BBPD-like story
implies the self-hurting consequence of being banned from future discounts
that are practiced for new customers. Clearly, for a strategic consumer (asopposed to a “naïve” one), there must be a tradeoff in order to create an account.
Account creation is the way for sellers to provide a set of opportunities in web
search, post-sale services, ancillary services, advice about complementary
products, user guides, and similar services. Those opportunities ameliorate theplatform-related experience, reducing the transaction costs incurred by thebuyer. １,２ In brief, if an account is created, the buyer enters a group of identified
customers, in sequel interactions, as “returning customers”; if an account is notcreated, the customer remains “anonymous.” ３
We model the following game. In the first stage, firms noncooperatively
choose whether to (i) leave no choice to buyers and require every buyer toIn the end, creating an account implies entering a relationship with the seller that provides
some advantages while relinquishing anonymity. Strategic consumers will choose whether tocreate an account or remain anonymous based on balancing the value of the advantages against
giving up access to future price discounts reserved for new customers.
２ Casadesus-Masanell and Hervas-Drane (2015) assume there is a concave relationship between
the amount of information that a consumer provides to a firm and the quality of the purchased
item – and a negative relationship with the degree of disclosure to third parties. Similarly, Contiand Reverberi (2021) assume that opting-in for access to private information leads to higher
quality of consumption. In a similar spirit, Conitzer et al. (2012) assume that hiding informationfrom a monopolist is costly to consumers but prevents price discrimination.
３ Fake accounts are possible in the real world, and a customer may create a false identity, evenwhen account creation is mandatory for completing a transaction – with the aim to return as a
new customer in the future, under a new and maybe again false identity. Consider, however,that a shipping address is needed for a large class of purchases, severely limiting the possibilityof remaining unidentified. Furthermore, our analysis also reveals that some consumers
strategically decide to be identified if they are given the choice (so that they will not create fakeaccounts even if they could). On the other side, sellers may circumvent the buyer’s decision toremain anonymous by collecting data without the buyer’s consent: we exclude this behavior by
the sellers – possibly because of fear of intervention by regulatory agencies or of class actions.
１4

create an account or (ii) allow buyers to buy with or without an account. Then,firms compete in prices over two periods. Incidentally, at the beginning of thesecond period, buyers can create an account at the seller they will buy from,had they not done so in the first period.
We then obtain the following results. If both consumers and firms are
perfectly foresighted, we might observe either a symmetric equilibrium where
both firms offer a menu choice regarding account creation or a symmetricequilibrium where both firms adopt a no-choice policy that forces buyers tocreate an account. Multiple symmetric equilibria are also possible. Notably,
when offering a menu choice is the unique equilibrium, it is not a prisoner
dilemma for the firms. When comparing this last equilibrium with acompulsory anonymity regime (or standard uniform pricing regime), we showthat allowing consumers to choose whether or not to be anonymous might
dominate the compulsory anonymity regime in terms of profits. In other words,
behavior-based price discrimination when consumers are free to hide or nottheir identity might outperform uniform pricing, in contrast with the typicalresult when standard BBPD and uniform pricing are compared (see Fudenberg
and Tirole 2000, Villas-Boas, 1999 and 2004). Lastly, when allowing for not
perfectly foresighted consumers and firms, we obtain that asymmetric accountpolicy equilibria are possible, where one firm offers the menu choice regarding
account creation to buyers, whereas the other does not. In particular, theoutcome happens when the discount factor is not large. This helps explain theexistence of competing firms adopting different privacy policies.
While BBPD has been investigated from several perspectives (see Esteves2009 and Acquisti et al. 2016 for extended reviews of this literature), there arevery few papers trying to incorporate some sort of information endogeneity inthe model. Conitzer et al. (2012) focus on a monopolistic environment where5consumers decide whether to leave track of their previous purchases to the firmor not, and the monopolistic firm adopts BBPD. Hence, this paper does notconsider competition between firms. Ali et al. (2023) and Anderson et al. (2023)
extend to oligopoly, but they consider personalized pricing rather than thirddegree price discrimination. ４ Closer to our work, Heiny et al. (2022) consider
two competing firms. Each consumer can reveal her identity (like our “accountcreation” decision) so that a firm can kept track of their past purchases.
However, there is no heterogeneity among consumers regarding thecost/benefit of anonymity; hence, unlike our paper, the co-existence of buyersthat choose to reveal their identity and buyers that prefer remaining
anonymous is never observed in equilibrium. Moreover, we depart from Heinyet al. (2022) by investigating strategic anonymity from the firms’ perspective.
Namely, as stressed above, sellers might strategically decide whether or not toallow consumers to purchase without revealing their identity. This aspect is outof scope in Heiny et al. (2022).
The rest of the paper runs as follows. In Section 2, we introduce the model.
In Section 3, we characterize the equilibrium outcomes for the different accountpolicies of the firms, and the main results are derived. In Section 4 welfare
implications are discussed. Section 5 extends the main model to the case ofdifferent discount factors between sellers and buyers. Section 6 concludes. The
proofs of the propositions are in the Appendix.

2. THE MODEL
There are two selling websites (or firms), say A and B, located at theendpoints of a Hotelling (1929) segment of length 1 ranging from 0 to 1, that
Cong and Matsushima (2023) also consider consumer data management in a two-market
model where firms collect data in a market and apply them in another market. In their paper,
data management policy is exogenous.

４6differ in the way described in the Introduction. Firm A is located at 0, whereas
Firm B is located at 1.
Buyers display different satisfaction/abilities when surfing a website. ５
Because the characteristics of the websites are different, consumers view thosewebsites as horizontally differentiated. For example, buyers with high curiosity
or education appreciate websites that provide detailed information about theirproducts, while less knowledgeable buyers will rather rely on reviews by othercustomers (as in Tripadvisor). When a web seller has built its product/website
configuration, any buyer is defined by a “degree of mismatch” with the seller.
Consumers are uniformly distributed along the Hotelling segment, with 𝑥𝑥 ∈[0,1] indicating the “location” of the consumer on the segment. Along thesegment, proximity to a website is related to low mismatch (transportation)

costs. The usual unit transportation cost in the Hotelling model is normalized to
1. Hence, the mismatch costs of a consumer at 𝑥𝑥 with firms A and B are

respectively 1 × 𝑥𝑥 = 𝑥𝑥 and 1 × (1 − 𝑥𝑥) = 1 − 𝑥𝑥. As explained below, consumers
can reduce the unit transportation costs.

Creating an account reduces the transaction cost by reducing the per-length
costs from 1 to 𝜏𝜏<1 because it allows the consumer to obtain desirable ancillary
services that are not accessible otherwise. When creating the account, aconsumer sustains (once and forever) a cost equal to c > 0. For instance, aconsumer at x buying from firm A has transaction costs 𝑥𝑥 or 𝜏𝜏𝜏𝜏 + 𝑐𝑐, according towhether she buys anonymously or creates an account first. Therefore, if a buyer

is very close to a web seller (if she has a very low mismatch level, to begin with),
the advantage of further reducing the mismatch is likely overridden by the costof account creation. Not so for more distant buyers, for which the additional
benefit of account creation is larger.
５See, for instance, Gadalla et al. (2013).

7

There are two periods, 1 and 2. Buyers purchase in both periods. In period 1,
before purchasing, each consumer visits the websites and decides whether or
not to create an account. Even if the consumer does not create the account inperiod 1, she can create it in period 2. If a buyer creates an account in the first
period and purchases from the same firm in both periods, she is recognized bythat firm and cannot buy at a discount as a newcomer in period 2. In all theother cases, she will be treated as a newcomer in period 2 – in particular, abuyer who is a newcomer to a firm in period 2 and creates an account only inthe second period buys at a price reserved for newcomers, so that we can groupthose buyers with the “anonymous” ones for that firm. The discount factor forboth consumers and firms is 𝛿𝛿 ∈ (0,1]. In a separate extension, we consider thecase of distinct discount factors for firms and consumers.

Since in period 1, no firm is able to identify any buyer, each firm sets a
uniform price, 𝑞𝑞𝑗𝑗 (j=A,B). In period 2, Firm j recognizes those consumers(“identified consumers”) that have purchased from j in period 1 if and only ifthey have an account for website j (in both periods); it, therefore, charges themwith a price that is different from a price offered to all the other consumers(“unidentified consumers”), thus engaging in BBPD. We denote by 𝑝𝑝𝑗𝑗 the priceto its own identified consumers and by 𝑝𝑝�𝑗𝑗 the price to the unidentified

consumers.

3. ACCOUNT POLICIES
3.1 Both firms provide two options: creating an account and no creation
Suppose that each firm offers the option to buy in the first and second
periods with or without an account (strategy F, as “free choice”). We refer tothis situation as case FF. We tentatively assume that the market configuration iswith buyers near the firms’ location choosing not to set up an account. Then, we
8check if it is sustainable as an equilibrium outcome. Indeed, consistent withsecond-period utilities, the consumers located close to the firms, namely close to
the endpoints, are less interested in revealing their identity since they have low
mismatch costs to start with and low benefits from the account. The conjectured
market structure is represented in Figure 1.
In Figure 1, the consumers at 𝑤𝑤𝐴𝐴 , who buy from Firm A, are indifferent

between creating an account and buying anonymously; the buyers to the left of
𝑤𝑤𝐴𝐴 never create an account, thus never sustaining the cost c and always paying

a unit transportation cost equal to 1. The consumers between 𝑤𝑤𝐴𝐴 and 𝑘𝑘𝐴𝐴 do notcreate any account in period 1, but they create the account for Firm A in period

2, sustaining the cost c, so that they can purchase at price 𝑝𝑝�𝐴𝐴 even if buying

from Firm A in both periods (as hinted above, they are not recognized because
they have not created an account in the first period; hence, they are treated as
new customers). The consumers between 𝑘𝑘𝐴𝐴 and 𝑥𝑥𝐴𝐴 in period 1 create theaccount with Firm A, sustaining the cost c and purchasing from Firm A; inperiod 2, they purchase again from Firm A at price 𝑝𝑝𝐴𝐴 , because Firm A

identifies them as returning buyers. The consumers between 𝑥𝑥𝐴𝐴 and k in period

1 create the account with Firm A sustaining the cost c and purchase from Firm
A; in period 2, however, they create the account for Firm B sustaining the cost cagain and can purchase as new buyers from B at price 𝑝𝑝�𝐵𝐵 (Firm B treats them as“unidentified”). A similar market structure holds for consumers from k to 1. ６

It is worth mentioning that ex-ante asymmetric locations of the indifferent consumers areallowed; Figure 1 is symmetric for convenience since the equilibrium is expected to besymmetric. The same holds for Figure 2 (see later).

６

9Figure 1: market structure in case FF

The set of consumers who created accounts in period 1 is the interval[𝑘𝑘𝐴𝐴 ,𝑘𝑘𝐵𝐵 ]; hence, in period 2, the extreme points of this interval are taken as given.

The dividing points 𝑤𝑤𝐴𝐴 and 𝑤𝑤𝐵𝐵 depend upon the choices to create or notaccounts in the second period but are independent of the prices, as to be seen
below since they only depend upon c and τ.
The indifferent second-period consumers 𝑥𝑥𝐴𝐴 and 𝑥𝑥𝐵𝐵 are respectively givenby 𝑣𝑣 − 𝑝𝑝𝐴𝐴 − 𝜏𝜏𝑥𝑥𝐴𝐴 = 𝑣𝑣 − 𝑐𝑐 − 𝑝𝑝�𝐵𝐵 − 𝜏𝜏(1 − 𝑥𝑥𝐴𝐴 ) and 𝑣𝑣 − 𝑝𝑝𝐵𝐵 − 𝜏𝜏(1 − 𝑥𝑥𝐵𝐵 ) = 𝑣𝑣 − 𝑐𝑐 − 𝑝𝑝�𝐴𝐴 −𝜏𝜏𝑥𝑥𝐵𝐵 ). Hence:
𝑥𝑥𝐴𝐴 =

𝑝𝑝�𝐵𝐵 −𝑝𝑝𝐴𝐴 +𝑐𝑐+𝜏𝜏2𝜏𝜏

; 𝑥𝑥𝐵𝐵 =𝑝𝑝𝐵𝐵 −𝑝𝑝�𝐴𝐴 −𝑐𝑐+𝜏𝜏
2𝜏𝜏

(1)

.

The indifferent second-period consumers 𝑤𝑤𝐴𝐴 and 𝑤𝑤𝐵𝐵 are respectively givenby ７

𝑣𝑣 − 𝑝𝑝�𝐴𝐴 − 𝑤𝑤𝐴𝐴 = 𝑣𝑣 − 𝑐𝑐 − 𝑝𝑝�𝐴𝐴 − 𝜏𝜏𝜏𝜏𝐴𝐴 and 𝑣𝑣 − 𝑝𝑝�𝐵𝐵 − (1 − 𝑤𝑤𝐵𝐵 ) = 𝑣𝑣 − 𝑐𝑐 − 𝑝𝑝�𝐵𝐵 −𝜏𝜏(1 − 𝑤𝑤𝐵𝐵 ). Hence:
𝑤𝑤𝐴𝐴 =𝑐𝑐1−𝜏𝜏; 𝑤𝑤𝐵𝐵 = 1 −

𝑐𝑐

1−𝜏𝜏

(2)

.

The second-period profits functions are:𝜋𝜋𝐴𝐴,2 = 𝑝𝑝𝐴𝐴 (𝑥𝑥𝐴𝐴 − 𝑘𝑘𝐴𝐴 ) + 𝑝𝑝�𝐴𝐴 (𝑥𝑥𝐵𝐵 − 𝑘𝑘 + 𝑘𝑘𝐴𝐴 ),𝜋𝜋𝐵𝐵,2 = 𝑝𝑝𝐵𝐵 (𝑘𝑘𝐵𝐵 − 𝑥𝑥𝐵𝐵 ) + 𝑝𝑝�𝐴𝐴 (𝑘𝑘 − 𝑥𝑥𝐴𝐴 + 1 − 𝑘𝑘𝐵𝐵 ).

(3)
(4)

By maximizing, we get the second-period equilibrium prices, that is:
Note that 𝑤𝑤𝐴𝐴 and 𝑤𝑤𝐵𝐵 do not enter the profit functions because they do not depend on prices.
However, they contribute to characterizing the admissible parameter set.

７

10

[(3+2𝑘𝑘−4𝑘𝑘𝐴𝐴 −2𝑘𝑘𝐵𝐵 )𝜏𝜏+𝑐𝑐]

𝑝𝑝𝐴𝐴 =

3; 𝑝𝑝�𝐴𝐴 =(−1−2𝑘𝑘+2𝑘𝑘𝐴𝐴 +4𝑘𝑘𝐵𝐵 )𝜏𝜏+𝑐𝑐

𝑝𝑝𝐵𝐵 =

3(1−4𝑘𝑘+4𝑘𝑘𝐴𝐴 +2𝑘𝑘𝐵𝐵 )𝜏𝜏−𝑐𝑐; 𝑝𝑝�𝐵𝐵 =3(3+4𝑘𝑘−2𝑘𝑘𝐴𝐴 −4𝑘𝑘𝐵𝐵 )𝜏𝜏−𝑐𝑐
3

;

.

(5)

When moving to period 1, the first-period indifferent consumer k is given bythecondition:

𝑣𝑣 − 𝑐𝑐 − 𝑞𝑞𝐴𝐴 − 𝜏𝜏𝜏𝜏 + 𝛿𝛿[𝑣𝑣 − 𝑐𝑐 − 𝑝𝑝�𝐵𝐵 − 𝜏𝜏(1 − 𝑘𝑘)] = 𝑣𝑣 − 𝑐𝑐 − 𝑞𝑞𝐵𝐵 −𝜏𝜏(1 − 𝑘𝑘) + 𝛿𝛿[𝑣𝑣 − 𝑐𝑐 − 𝑝𝑝�𝐴𝐴 − 𝜏𝜏𝜏𝜏], that is:𝑘𝑘 =3(𝑞𝑞𝐵𝐵 −𝑞𝑞𝐴𝐴 )+[3−(5−6𝑘𝑘𝐴𝐴 −6𝑘𝑘𝐵𝐵 )𝛿𝛿]𝜏𝜏
2(3+𝛿𝛿)𝜏𝜏

(6).

This is similar to the market boundary defined in standard BBPD models, as
it does not depend on c. The overall profits functions are:(7)

𝜋𝜋𝐴𝐴 = 𝑞𝑞𝐴𝐴 𝑘𝑘 + 𝛿𝛿𝜋𝜋𝐴𝐴,2 ,𝜋𝜋𝐵𝐵 = 𝑞𝑞𝐵𝐵 (1 − 𝑘𝑘) + 𝛿𝛿𝜋𝜋𝐵𝐵,2 .

(8)

By maximizing, we get the first-period equilibrium prices:
𝑞𝑞𝐴𝐴 =
𝑞𝑞𝐵𝐵 =(3+𝛿𝛿+4𝛿𝛿𝑘𝑘𝐴𝐴 )𝜏𝜏−2𝛿𝛿𝛿𝛿

2(1−𝑘𝑘𝐴𝐴 −𝑘𝑘𝐵𝐵 )(27−𝛿𝛿)𝛿𝛿

3

−

3

+

(3+5𝛿𝛿−4𝛿𝛿𝑘𝑘𝐵𝐵 )𝜏𝜏−2𝛿𝛿𝛿𝛿

3(27−11𝛿𝛿)𝜏𝜏,3(27−11𝛿𝛿)𝜏𝜏.

2(1−𝑘𝑘𝐴𝐴 −𝑘𝑘𝐵𝐵 )(27−𝛿𝛿)𝛿𝛿

(9)
(10)

Now, we find the indifferent consumers 𝑘𝑘𝐴𝐴 and 𝑘𝑘𝐵𝐵 . They are obtained bysolving simultaneously the following equations: 𝑣𝑣 − 𝑐𝑐 − 𝑞𝑞𝐴𝐴 − 𝜏𝜏𝑘𝑘𝐴𝐴 + 𝛿𝛿 (𝑣𝑣 − 𝑝𝑝𝐴𝐴 −and𝜏𝜏𝑘𝑘𝐴𝐴 ) = 𝑣𝑣 − 𝑞𝑞𝐴𝐴 − 𝑘𝑘𝐴𝐴 + 𝛿𝛿 (𝑣𝑣 − 𝑐𝑐 − 𝑝𝑝�𝐴𝐴 − 𝜏𝜏𝑘𝑘𝐴𝐴 )𝑣𝑣 − 𝑐𝑐 − 𝑞𝑞𝐵𝐵 − 𝜏𝜏(1 − 𝑘𝑘𝐵𝐵 ) + 𝛿𝛿 [𝑣𝑣 − 𝑝𝑝𝐵𝐵 −𝜏𝜏(1 − 𝑘𝑘𝐵𝐵 )] = 𝑣𝑣 − 𝑞𝑞𝐵𝐵 − (1 − 𝑘𝑘𝐵𝐵 ) + 𝛿𝛿 [𝑣𝑣 − 𝑐𝑐 − 𝑝𝑝�𝐵𝐵 − 𝜏𝜏(1 − 𝑘𝑘𝐵𝐵 )], yielding:𝑘𝑘𝐴𝐴𝐹𝐹𝐹𝐹∗ =𝛿𝛿𝛿𝛿+𝑐𝑐(3−𝛿𝛿)

3(1−𝜏𝜏)+4𝛿𝛿𝛿𝛿; 𝑘𝑘𝐵𝐵𝐹𝐹𝐹𝐹∗ = 1 −𝛿𝛿𝛿𝛿+𝑐𝑐(3−𝛿𝛿)

3(1−𝜏𝜏)+4𝛿𝛿𝛿𝛿

.

(11)

The equilibrium profits, indicated by 𝜋𝜋 𝐹𝐹𝐹𝐹 , are reported in the Appendix. The

conjectured market structure – that is 1 > 𝑤𝑤𝐵𝐵 > 𝑘𝑘𝐵𝐵 > 𝑥𝑥𝐵𝐵 > 𝑘𝑘 > 𝑥𝑥𝐴𝐴 > 𝑘𝑘𝐴𝐴 > 𝑤𝑤𝐴𝐴 >0 – is verified in equilibrium when 𝑐𝑐 ≤𝜏𝜏(1−𝜏𝜏)
1+3𝜏𝜏

11. Concretely, those three variables

11

are: 𝑘𝑘 𝐹𝐹𝐹𝐹∗ = , 𝑥𝑥𝐴𝐴𝐹𝐹𝐹𝐹∗ = +2134𝛿𝛿𝜏𝜏2 +3(1+3𝜏𝜏)𝑐𝑐6𝜏𝜏(3−3𝜏𝜏+4𝛿𝛿𝛿𝛿)22and 𝑥𝑥𝐵𝐵𝐹𝐹𝐹𝐹∗ = −34𝛿𝛿𝜏𝜏2 +3(1+3𝜏𝜏)𝑐𝑐6𝜏𝜏(3−3𝜏𝜏+4𝛿𝛿𝛿𝛿). Note that

𝑥𝑥𝐴𝐴𝐹𝐹𝐹𝐹∗ > (and, symmetrically, 𝑥𝑥𝐵𝐵𝐹𝐹𝐹𝐹∗ < ), so that the “poaching” areas are lower
3

3

than 1/6, which is the case in the traditional BBPD model because the existenceof unidentified consumers closer to each firm reduces the incentives of firms tooffer low poaching prices.
3.2 Both firms require consumers to create accounts
Suppose that each firm allows only to purchase with an account (strategy N,
as “no choice”): this situation is indicated as case NN. Then, only buyers who
switch providers will pay the price to new buyers. This assumption is the usual
BBPD market configuration. Figure 2 shows how the firms segment the market
in case NN.

Figure 2: market structure in case NN

The indifferent second-period consumer 𝑥𝑥𝐴𝐴 and 𝑥𝑥𝐵𝐵 are given by (1). The

second-period profits functions are:(12)

𝜋𝜋𝐴𝐴,2 = 𝑝𝑝𝐴𝐴 𝑥𝑥𝐴𝐴 + 𝑝𝑝�𝐴𝐴 (𝑥𝑥𝐵𝐵 − 𝑘𝑘),(13)

𝜋𝜋𝐵𝐵,2 = 𝑝𝑝𝐵𝐵 (1 − 𝑥𝑥𝐵𝐵 ) + 𝑝𝑝�𝐴𝐴 (𝑘𝑘 − 𝑥𝑥𝐴𝐴 ).

By maximizing, we get the second-period equilibrium prices, that is:
𝑝𝑝𝐴𝐴 =

𝑝𝑝𝐵𝐵 =(1+2𝑘𝑘)𝜏𝜏+𝑐𝑐
3; 𝑝𝑝�𝐴𝐴 =(3−2𝑘𝑘)𝜏𝜏+𝑐𝑐
3; 𝑝𝑝�𝐵𝐵 =(3−4𝑘𝑘)𝜏𝜏−𝑐𝑐3,(−1+4𝑘𝑘)𝜏𝜏−𝑐𝑐
3

.

(14)
12

When moving to period 1, the first-period indifferent consumer k is given bythecondition:

𝑣𝑣 − 𝑐𝑐 − 𝑞𝑞𝐴𝐴 − 𝜏𝜏𝜏𝜏 + 𝛿𝛿[𝑣𝑣 − 𝑐𝑐 − 𝑝𝑝�𝐵𝐵 − 𝜏𝜏(1 − 𝑘𝑘)] = 𝑣𝑣 − 𝑐𝑐 − 𝑞𝑞𝐵𝐵 −𝜏𝜏(1 − 𝑘𝑘) + 𝛿𝛿[𝑣𝑣 − 𝑐𝑐 − 𝑝𝑝�𝐴𝐴 − 𝜏𝜏𝜏𝜏], that is:𝑘𝑘 =3(𝑞𝑞𝐵𝐵 −𝑞𝑞𝐴𝐴 )+(3+𝛿𝛿)𝜏𝜏

(15)

2(3+𝛿𝛿)𝜏𝜏

The overall profits functions are as (7) and (8). By maximizing, we get thefirst-period equilibrium prices:
𝑞𝑞𝐴𝐴𝑁𝑁𝑁𝑁∗ = 𝑞𝑞𝐵𝐵𝑁𝑁𝑁𝑁∗ =𝜏𝜏(3+𝛿𝛿)−2𝛿𝛿𝛿𝛿

(16)

3

The equilibrium profits, indicated by 𝜋𝜋 𝑁𝑁𝑁𝑁 , are reported in the Appendix.

The conjectured market structure – that is, 1 > 𝑥𝑥𝐵𝐵 > 𝑘𝑘 > 𝑥𝑥𝐴𝐴 > 0 – is verified in1equilibrium when 𝑐𝑐 ≤ 𝜏𝜏. Concretely, those three variables are 𝑘𝑘 𝑁𝑁𝑁𝑁∗ = , 𝑥𝑥𝐴𝐴𝑁𝑁𝑁𝑁∗ =
1
3

+

𝑐𝑐

6𝜏𝜏

2

, and 𝑥𝑥𝐵𝐵𝑁𝑁𝑁𝑁∗ = −
3

𝑐𝑐

6𝜏𝜏

2

. Note that when 𝑐𝑐 = 0, we are back to the equilibrium

market configuration in Fudenberg and Tirole (2000) because consumers do notneed to incur cost c to switch from one firm to the other.

3.3 Asymmetric account creation policies
Suppose that only Firm A allows its buyers to create an account or not(strategy F). Consumers of Firm B instead must create an account if theypurchase from Firm B (strategy N). This is denoted as case FN. Clearly, case NF,
where only Firm B’s consumers are free whether to create the account or not,can be treated symmetrically. Figure 3 shows how the firms segment the market
in case FN.

13

Figure 3: market structure in case FN

Here, poaching by Firm B is akin to that under standard BBPD (the segment[𝑥𝑥𝐴𝐴 ,k]), while Firm A retains a richer market segmentation, with consumers nearits location choosing not to open accounts in the first period. The indifferent
second-period consumer 𝑥𝑥𝐴𝐴 and 𝑥𝑥𝐵𝐵 are given by (1), whereas the indifferent

second-period consumer 𝑤𝑤𝐴𝐴 is given by (2). The second-period profits functionof Firm A is:(17)

𝜋𝜋𝐴𝐴,2 = 𝑝𝑝𝐴𝐴 (𝑥𝑥𝐴𝐴 − 𝑘𝑘𝐴𝐴 ) + 𝑝𝑝�𝐴𝐴 (𝑥𝑥𝐵𝐵 − 𝑘𝑘 + 𝑘𝑘𝐴𝐴 ),whereas that of Firm B is as (13). By maximizing, we get the second-period
equilibrium prices, that is:
𝑝𝑝𝐴𝐴 =

𝑝𝑝𝐵𝐵 =(1+2𝑘𝑘−4𝑘𝑘𝐴𝐴 )𝜏𝜏+𝑐𝑐
3(3−2𝑘𝑘+2𝑘𝑘𝐴𝐴 )𝜏𝜏+𝑐𝑐
3; 𝑝𝑝�𝐴𝐴 =; 𝑝𝑝�𝐵𝐵 =(3−4𝑘𝑘+4𝑘𝑘𝐴𝐴 )𝜏𝜏−𝑐𝑐3,(−1+4𝑘𝑘−2𝑘𝑘𝐴𝐴 )𝜏𝜏−𝑐𝑐
3

.

(18)

Let us consider the difference between the poaching prices when 𝑘𝑘 is close

to 1/2 . It can be observed that (𝑝𝑝�𝐴𝐴 − 𝑝𝑝�𝐵𝐵 )|𝑘𝑘=1⁄2 > 0 . Indeed, Firm A is lessaggressive with its poaching price since it also applies this price to its own
unidentified consumers.
When moving to period 1, the first-period indifferent consumer k is given bythecondition:

𝑣𝑣 − 𝑐𝑐 − 𝑞𝑞𝐴𝐴 − 𝜏𝜏𝜏𝜏 + 𝛿𝛿[𝑣𝑣 − 𝑐𝑐 − 𝑝𝑝�𝐵𝐵 − 𝜏𝜏(1 − 𝑘𝑘)] = 𝑣𝑣 − 𝑐𝑐 − 𝑞𝑞𝐵𝐵 −𝜏𝜏(1 − 𝑘𝑘) + 𝛿𝛿[𝑣𝑣 − 𝑐𝑐 − 𝑝𝑝�𝐴𝐴 − 𝜏𝜏𝜏𝜏], that is:

14

𝑘𝑘 =

3(𝑞𝑞𝐵𝐵 −𝑞𝑞𝐴𝐴 )+[3+(1+6𝑘𝑘𝐴𝐴 )𝛿𝛿]𝜏𝜏
2(3+𝛿𝛿)𝜏𝜏

.

(19)

If 𝑞𝑞𝐵𝐵 = 𝑞𝑞𝐴𝐴 , 𝑘𝑘 is higher than 1/2 unless 𝑘𝑘𝐴𝐴 = 0 . Thus, Firm A has anadvantage over Firm B in period 1 because 𝑝𝑝�𝐴𝐴 is more likely to be higher than𝑝𝑝�𝐵𝐵 , inducing the first-period indifferent consumers, who will switch in period 2,
to prefer Firm A more. ８

The overall profits functions are as (7) and (8). By maximizing, we get thefirst-period equilibrium prices:
𝑞𝑞𝐴𝐴 =
𝑞𝑞𝐵𝐵 =(3+𝛿𝛿)𝜏𝜏−2𝛿𝛿𝛿𝛿
3

+

3

−

(3+𝛿𝛿)𝜏𝜏−2𝑐𝑐

2(81−23𝛿𝛿)𝛿𝛿𝑘𝑘𝐴𝐴 𝜏𝜏
3(27−11𝛿𝛿)2(27−𝛿𝛿)𝛿𝛿𝑘𝑘𝐴𝐴 𝜏𝜏
3(27−11𝛿𝛿)

(20)

,

(21)

.

Now, we find the indifferent consumer 𝑘𝑘𝐴𝐴 . It is obtained by solving 𝑣𝑣 − 𝑐𝑐 −𝑞𝑞𝐴𝐴 − 𝜏𝜏𝑘𝑘𝐴𝐴 + 𝛿𝛿(𝑣𝑣 − 𝑝𝑝𝐴𝐴 − 𝜏𝜏𝑘𝑘𝐴𝐴 ) = 𝑣𝑣 − 𝑞𝑞𝐴𝐴 − 𝑘𝑘𝐴𝐴 + 𝛿𝛿(𝑣𝑣 − 𝑐𝑐 − 𝑝𝑝�𝐴𝐴 − 𝜏𝜏𝑘𝑘𝐴𝐴 ), that yields:
𝑘𝑘𝐴𝐴𝐹𝐹𝐹𝐹∗ =[(3−𝛿𝛿)𝑐𝑐+𝛿𝛿𝛿𝛿](27−11𝛿𝛿)3(27−11𝛿𝛿)−(81−249𝛿𝛿+34𝛿𝛿 2 )𝜏𝜏

.

(22)

The equilibrium profits, indicated by 𝜋𝜋𝐴𝐴𝐹𝐹𝐹𝐹 and 𝜋𝜋𝐵𝐵𝐹𝐹𝐹𝐹 , are reported in the

Appendix. The conjectured market structure – that is 1 > 𝑥𝑥𝐵𝐵 > 𝑘𝑘 > 𝑥𝑥𝐴𝐴 > 𝑘𝑘𝐴𝐴 >𝑤𝑤𝐴𝐴 > 0 – is verified in equilibrium when 𝑐𝑐 ≤ 𝑐𝑐̂ (𝜏𝜏, 𝛿𝛿) ≡1

Concretely, those variables except for 𝑘𝑘𝐴𝐴∗ are: 𝑘𝑘 𝐹𝐹𝐹𝐹∗ = −1

𝑥𝑥𝐴𝐴𝐹𝐹𝐹𝐹∗ = +
3

𝑐𝑐

6𝜏𝜏

+

𝐹𝐹𝐹𝐹∗
(27−20𝛿𝛿)𝑘𝑘𝐴𝐴

3(27−11𝛿𝛿)2, and 𝑥𝑥𝐵𝐵𝐹𝐹𝐹𝐹∗ = −
3

𝑐𝑐

6𝜏𝜏

Furthermore, note that 𝑞𝑞𝐴𝐴𝐹𝐹𝐹𝐹∗ − 𝑞𝑞𝐵𝐵𝐹𝐹𝐹𝐹∗ =−2𝜏𝜏(1−𝜏𝜏)(27−11𝛿𝛿)

27−11𝛿𝛿+189𝜏𝜏−23𝜏𝜏𝜏𝜏

9𝛿𝛿�(3−𝛿𝛿)𝑐𝑐+𝛿𝛿𝛿𝛿�

3(27−11𝛿𝛿)−(81−249𝛿𝛿+34𝛿𝛿 2 )𝜏𝜏

𝐹𝐹𝐹𝐹∗(27−2𝛿𝛿)𝑘𝑘𝐴𝐴

3(27−11𝛿𝛿)

.

,

.

8𝛿𝛿𝛿𝛿(9−2𝛿𝛿)[(3−𝛿𝛿)𝑐𝑐+𝛿𝛿𝛿𝛿]𝛿𝛿(33−249𝜏𝜏)−81(1−𝜏𝜏)+34𝛿𝛿 2 𝜏𝜏

> 0. Indeed,

Firm B has a stronger incentive to expand the first-period demand in order toenlarge its own no-poaching area in the second period. This logic does not

However, as shown later, the first-period price of Firm A is greater than that of Firm B in
equilibrium.

８15

apply to Firm A because Firm A needs to use the “poaching” price also to its
own unidentified consumers.
3.4 Equilibrium policies
In the first stage, each firm chooses F or N. Their binary decisions can be
represented as a 2x2 normal form game based on the ensuing equilibrium
intertemporal profits. The resulting matrix is represented for convenience.

FN

𝜋𝜋 𝐹𝐹𝐹𝐹 , 𝜋𝜋 𝐹𝐹𝐹𝐹

𝜋𝜋 𝐹𝐹𝐹𝐹 , 𝜋𝜋 𝑁𝑁𝑁𝑁

𝜋𝜋 𝑁𝑁𝑁𝑁 , 𝜋𝜋 𝐹𝐹𝐹𝐹

N
First, since 𝑐𝑐̂ ≤F𝜏𝜏(1−𝜏𝜏)
1+3𝜏𝜏

Table 1: the payoff matrix

𝜋𝜋 𝑁𝑁𝑁𝑁 , 𝜋𝜋 𝑁𝑁𝑁𝑁

and 𝑐𝑐̂ ≤ 𝜏𝜏, the relevant parameter constraint is 𝑐𝑐 ≤ 𝑐𝑐̂ .

Given the payoff structure, therefore, in order to characterize the equilibrium
policy, we have to compare 𝜋𝜋 𝑁𝑁𝑁𝑁 with 𝜋𝜋 𝐹𝐹𝐹𝐹 , and 𝜋𝜋 𝑁𝑁𝑁𝑁 with 𝜋𝜋 𝐹𝐹𝐹𝐹 , under 𝑐𝑐 ≤ 𝑐𝑐̂ . First,

we consider the case of perfectly foresighted consumers and firms (𝛿𝛿 = 1). It is

possible to characterize two critical thresholds for variable c, say 𝑐𝑐1 (𝜏𝜏) =𝜏𝜏(48+3375𝜏𝜏+14034𝜏𝜏2 +7399𝜏𝜏3 )3072+15534𝜏𝜏+13580𝜏𝜏2 −3274𝜏𝜏3

following holds:•••

and 𝑐𝑐2 (𝜏𝜏) =𝜏𝜏(48+2749𝜏𝜏)3072+3346𝜏𝜏

with 𝑐𝑐1 (𝜏𝜏) > 𝑐𝑐2 (𝜏𝜏), such that theIf 𝑐𝑐 ∈ [𝑐𝑐1 (𝜏𝜏), 𝑐𝑐̂ (𝜏𝜏)], NN is the unique equilibrium;

If 𝑐𝑐 ∈ [𝑐𝑐2 (𝜏𝜏), 𝑐𝑐1 (𝜏𝜏)], both NN and FF are equilibria;

If 𝑐𝑐 ∈ [0, 𝑐𝑐2 (𝜏𝜏)], FF is the unique equilibrium.

It is then possible to state the following result:

PROPOSITION 1. For a symmetric discount factor 𝛿𝛿 = 1 common to firms andbuyers, (a) If the reduction in transaction costs due to an account creation is below a
threshold, FF is the unique Nash equilibrium; (b) if the cost of account creation is large,
NN is more likely to emerge in equilibrium; (c) in an intermediate region the two

16equilibria FF and NN coexist; (d) when FF is the unique equilibrium, it is not a
prisoner dilemma.
Figure 4 illustrates Proposition 1:

Figure 4: equilibrium policies when 𝛿𝛿 = 1

Proposition 1 shows that for a preponderant configuration of parameters,
firms give the option to buy with or without an account (equilibrium FF).
Therefore, it is by no means clear that firms will always try to identify theircustomers by obliging them to leave their credentials at the firm. It is alsointeresting to note that even if 𝑐𝑐 = 0, not all consumers choose to reveal theiridentity at the first period– as it is apparent that neither 𝑘𝑘𝐴𝐴 nor 1 − 𝑘𝑘𝐵𝐵 vanishes

in equations (11) above if 𝑐𝑐 = 0.

Considering the impact of τ and c, we can obtain the following observations.

When τ increases, the horizontal differentiation between the firms increases aswell. Suppose that Firm A chooses F. When 𝜏𝜏 is small (large), Firm A is more(less) likely to apply its poaching price 𝑝𝑝�𝐴𝐴 to the rival’s consumers because
poaching is easier (more difficult). The pricing implies that Firm A is more (less)likely to abandon high profits from the highly profitable consumers locatedclose by Firm A. The low (high) profits from those consumers are less (more)17

likely to induce Firm A to choose F. Therefore, all else being equal, FF is more
likely when τ is large (Figure 1).
The impact of greater c is to enlarge the area of anonymous consumers, fromwhich the firm cannot extract so much surplus. Therefore, the incentive tochoose F is weaker when c is large. Also, when c goes up, the incentive tochoose N increases because protecting the turf becomes easier (indeed, understrategy N, c is a proxy of vertical differentiation between the firms). ９ , １０
Therefore, all else being equal, NN is more likely when c is larger (Figure 1).
Finally, the identification of own customers allows poaching from the rival,
which is a “best reply” – when τ is large and/or c is small – against the rival
who does not poach; however, as is also well known in the BBPD literature, thisends up being a reciprocal dumping strategy, that goes to the advantage ofbuyers, and it is detrimental to firms. Hence, leaving the consumers thepossibility of remaining anonymous does not generate a prisoner dilemma(Proposition 1, point (d)).
PROPOSITION 2. For a symmetric discount factor 𝛿𝛿 < 1, asymmetric account policy
equilibria can emerge in addition to the cases in Proposition 1, provided 𝛿𝛿 is not large

enough. As the value for 𝛿𝛿 is lowered, the parameter area of such equilibria widens; theparameter area of multiple symmetric equilibria shrinks, and then it disappears when 𝛿𝛿reaches a threshold value.

Indeed, consider the indifferent consumers 𝑥𝑥𝐴𝐴 and 𝑥𝑥𝐵𝐵 . A consumer willing to repeat thepurchase from the same firm does not sustain the cost c in the second period; instead, if she
wants to shift to the other firm, she opens the account and sustains the cost c. Therefore, thepoaching firm is a sort of “low-quality” firm from the consumer’s perspective, and c measuresthe degree of vertical differentiation.
１０ A reinforcing argument is that when c increases, the number of buyers willing to create an
account dwindles, and choosing the N strategy is a way to force them to do so, enlarging the“recognized turf” to be protected.
９18

The message of Proposition 2 is that, when consumers and firms partially
discount second-period gains, we can observe more diverse identification
policies, with firms also choosing asymmetric strategies in equilibrium. So that
our model does not necessarily predict a uniform type of selling policy.
We explain the benefit/cost of choosing F (N) to understand the mechanism
behind Proposition 2. The benefit (cost) of choosing F (N) is that the firm can(cannot) commit not to poach the rival's customers aggressively inducing astrategic effect on the rival’s first-period price because its price for new
customers is available not only to those rival's customers but also to its previous
customers who remained “anonymous”. On the other hand, the cost (benefit) of
choosing F (N) is that the firm cannot (can) extract surpluses from unidentified
consumers who strongly prefer it to the rival because it cannot offer a price forold customers to those unidentified consumers. The net gain from choosing F
emerges in period 1 and that from choosing N does in period 2. Therefore, thenet gain from choosing F outperforms that from choosing N if 𝛿𝛿 is not large.
Given that the net gain from choosing F is relatively important if 𝛿𝛿 is not large,

the parameter area of NN is substituted by the area of asymmetric equilibria NF
and FN or that of FF if 𝛿𝛿 is not large.

In what follows, we consider the implications of strategic anonymity

compared with a standard uniform pricing framework. It is well known that
BBPD reduces firms’ profits with respect to the standard Hotelling Nashequilibrium, increasing competition on contended consumers (i.e., those thatare poached in equilibrium). Therefore, we have also made a comparison to thestandard Hotelling game where no buyer is ever identified: this corresponds toa situation in which both firms voluntarily at the first stage disallow all accountcreation (or firms allow anonymous transactions only). For simplicity, we
consider perfectly foresighted consumers and firms (𝛿𝛿 = 1). In this case, the19equilibrium profits are simply 𝜋𝜋 𝑈𝑈𝑈𝑈 = 1. First, we observe that 𝜋𝜋 𝑈𝑈𝑈𝑈 > 𝜋𝜋 𝑁𝑁𝑁𝑁 : this is

in line with the standard result in the BBPD literature, where it is shown thatbehavior-based pricing (with no anonymity), tends to decrease profits withrespect to uniform pricing. However, and quite surprisingly, we observe thatallowing buyers to choose whether to be identified or not strategically may
result in higher profits than under no BBPD, that is 𝜋𝜋 𝐹𝐹𝐹𝐹 > (<) 𝜋𝜋 𝑈𝑈𝑈𝑈 if 𝜏𝜏 is high(low) and c is low (high). This result is summarized in the next proposition:

PROPOSITION 3. The standard no-poaching equilibrium profits (uniform pricing)
can be dominated by the FF equilibrium, where firms allow consumers the choice toremain anonymous or create an account.
As mentioned previously, when 𝜏𝜏 is high, letting consumers choose whether

or not to create the account implies that the firm can extract more surplus fromthe closer consumer by means of the “poaching” price (which, however, in thiscase, is used not to poach rival’s consumers, but rather to extract surplus fromcloser and anonymous consumers). On the other hand, when c is high, theprofits under FF are lower. Indeed, in order to exploit the second-period lock-in
effect, the firms compete fiercely in the first period, which is detrimental toprofits. １１ Therefore, the FF equilibrium profits are more likely to outweigh thestandard no poaching uniform pricing equilibrium profits when 𝜏𝜏 is high and c

is low. In other words, behavior-based pricing might generate larger profits

than uniform pricing when buyers are free to choose whether to remain
anonymous or not. １２

１１

It can be easily observed that

𝜕𝜕𝜋𝜋𝐹𝐹𝐹𝐹𝜕𝜕𝜕𝜕> 0 and𝜕𝜕𝜋𝜋𝐹𝐹𝐹𝐹
𝜕𝜕𝜕𝜕

< 0.

We have not considered the possibility for buyers to delete their account in period 2 after
creating it in period 1. Indeed, it is easy to show that this strategy is always dominated by thestrategy we considered in the analysis: not creating the account in period 1 and then creating
the account in period 2. Consider a consumer buying from Firm A. Under both strategies, shepays 𝑝𝑝�𝐴𝐴 in the second period. Under the deleting strategy, the cost of the buyer (net of the price)

１２20

4. WELFARE
In this section, we consider welfare implications. Since the market iscovered and each consumer purchases one unit of good, prices are irrelevant for
welfare (they only redistribute surplus from consumers to firms). Therefore,
welfare is given by １３
𝑤𝑤 𝐹𝐹𝐹𝐹∗𝑘𝑘 𝐹𝐹𝐹𝐹∗𝐴𝐴
𝑊𝑊 𝐹𝐹𝐹𝐹 = 2{𝑣𝑣(1 + 𝛿𝛿) − ∫0 𝐴𝐴 (𝑥𝑥 + 𝛿𝛿𝛿𝛿)𝑑𝑑𝑑𝑑 − ∫𝑤𝑤 𝐹𝐹𝐹𝐹∗�𝑥𝑥 + 𝛿𝛿(𝑐𝑐 + 𝜏𝜏𝜏𝜏)�𝑑𝑑𝑑𝑑
𝐴𝐴

𝐹𝐹𝐹𝐹∗
1�
𝑥𝑥𝐴𝐴
2
(𝑐𝑐 + 𝜏𝜏𝜏𝜏 + 𝛿𝛿𝛿𝛿𝛿𝛿)𝑑𝑑𝑑𝑑 − ∫𝑥𝑥 𝐹𝐹𝐹𝐹∗− ∫𝑘𝑘 𝐹𝐹𝐹𝐹∗
�𝑐𝑐 + 𝜏𝜏𝜏𝜏 + 𝛿𝛿�𝑐𝑐 + 𝜏𝜏(1 − 𝑥𝑥)�� 𝑑𝑑𝑑𝑑}
𝐴𝐴
𝐴𝐴

(23)

in the case of free choice (FF) and by
𝑥𝑥 𝑁𝑁𝑁𝑁∗

𝑊𝑊 𝑁𝑁𝑁𝑁 = 2{𝑣𝑣 (1 + 𝛿𝛿 ) − ∫0 𝐴𝐴 (𝑐𝑐 + 𝜏𝜏𝜏𝜏 + 𝛿𝛿𝛿𝛿𝛿𝛿 )𝑑𝑑𝑑𝑑
1�

2− ∫𝑥𝑥 𝑁𝑁𝑁𝑁∗(𝑐𝑐 + 𝜏𝜏𝜏𝜏 + 𝛿𝛿(𝑐𝑐 + 𝜏𝜏(1 − 𝑥𝑥)))𝑑𝑑𝑑𝑑}

(24)

𝐴𝐴

in the case of no choice (NN). By comparing (23) with (24), we can state thefollowing proposition:
PROPOSITION 4. Welfare is greater when consumers can decide whether to create an
account or not (FF) than when consumers are forced to create an account (NN).
The explanation of proposition 4 is the following. Welfare depends on threefactors:i) total account creation costs;ii) unit transportation costs;iii)number of consumers that switch in the second period.

is 𝑐𝑐 + 𝜏𝜏𝜏𝜏 in period 1, and 𝑠𝑠 + 𝑥𝑥 in period 2, where 𝑠𝑠 > 0 is the cost of account deletion. In thealternative strategy, the cost of the buyer is 𝑥𝑥 in period 1 and 𝑐𝑐 + 𝜏𝜏𝜏𝜏 in period 2. Therefore, thelatter strategy always outperforms the deletion strategy.
１３ The equilibrium welfare equations are reported in the Appendix.

21

Regarding total account creation costs (i), it is obvious that when someconsumers decide not to create the account (case FF), the total account creation
costs are lower. On the other hand, forcing all consumers to create the account(case NN) reduces the unit transportation costs – which is 𝜏𝜏 rather than 1 – forall consumers (ii): this benefits welfare. Finally, in case FF, poaching is lesseffective than in case NN (see the discussion in Section 3). Therefore, in case FF,
fewer consumers switch in the second period to patronize the most distant firm(indeed, 𝑥𝑥𝐴𝐴𝐹𝐹𝐹𝐹∗ > 𝑥𝑥𝐴𝐴𝑁𝑁𝑁𝑁∗ ): this benefits welfare. In sum, as i) and iii) outweigh ii),

welfare is greater when consumers are free to choose whether to create an
account or not.

5. EXTENSIONIn this section, we assume that consumers and firms have different discountfactors, 𝛿𝛿𝑐𝑐 and 𝛿𝛿𝑓𝑓 , respectively. We replace the discount factor 𝛿𝛿 in (19) with 𝛿𝛿𝑐𝑐

and use the profits’ functions 𝜋𝜋𝐴𝐴 = 𝑞𝑞𝐴𝐴 𝑘𝑘 + 𝛿𝛿𝑓𝑓 𝜋𝜋𝐴𝐴,2 and 𝜋𝜋𝐵𝐵 = 𝑞𝑞𝐵𝐵 (1 − 𝑘𝑘) + 𝛿𝛿𝑓𝑓 𝜋𝜋𝐵𝐵,2 .
Due to the analytical complexity, we adopt numerical simulations, １４ whichyield the following result: １５
PROPOSITION 5. Suppose 𝛿𝛿𝑐𝑐 = 1 and 𝛿𝛿𝑓𝑓 < 1. There is a unique equilibrium where

both firms offer a menu choice in terms of account creation (FF) if 𝜏𝜏 is larger than athreshold value. Suppose 𝛿𝛿𝑓𝑓 = 1 and 𝛿𝛿𝑐𝑐 < 1. There is an equilibrium where both firms
adopt a no-choice policy that forces buyers to create an account (NN) if 0.1 ≤ 𝛿𝛿𝑐𝑐 ≤ 0.8,

if c is lower than a threshold value 𝑐𝑐𝑇𝑇 for 𝛿𝛿𝑐𝑐 < 0.1, or if c is higher than the thresholdvalue 𝑐𝑐𝑇𝑇 for 𝛿𝛿𝑐𝑐 > 0.8; in addition to NN, FF is also sustainable as an equilibrium if 𝜏𝜏 islarger than a threshold value for each 𝛿𝛿𝑐𝑐 and c.
１４
１５

The details of the numerical simulations are available on request.
The threshold value of 𝑐𝑐, 𝑐𝑐𝑇𝑇 , in Proposition 5 is
𝑐𝑐𝑇𝑇 ≡

𝜏𝜏{3(7+9𝛿𝛿𝑐𝑐 )�−44+18𝛿𝛿𝑐𝑐 +27𝛿𝛿𝑐𝑐2 �+�924−4718𝛿𝛿𝑐𝑐 +1791𝛿𝛿𝑐𝑐2 +4023𝛿𝛿𝑐𝑐3 +729𝛿𝛿𝑐𝑐4 �𝜏𝜏}12(7+9𝛿𝛿𝑐𝑐 )2 −�5652−2716𝛿𝛿𝑐𝑐 −6120𝛿𝛿𝑐𝑐2 +81𝛿𝛿𝑐𝑐3 −243𝛿𝛿𝑐𝑐4 �𝜏𝜏

22.

Proposition 5 shows that when consumers are perfectly patient (𝛿𝛿𝑐𝑐 = 1) andfirms are not perfectly patient (𝛿𝛿𝑓𝑓 < 1), equilibrium FF becomes more likely (itis the unique equilibrium when 𝛿𝛿𝑓𝑓 is low). Indeed, as argued in Section 3, thegain from choosing F is prevalent in period 1 and that from choosing N inperiod 2; hence, when 𝛿𝛿𝑓𝑓 is low, F is the dominant strategy. At the same time,

Proposition 5 shows that when consumers are not perfectly patient (𝛿𝛿𝑐𝑐 < 1) andfirms are perfectly patient (𝛿𝛿𝑓𝑓 = 1), FF ceases to be a unique equilibrium (itmight emerge only together with the NN equilibrium). In other words, when
the rival chooses N, the best reply always consists in choosing N too. Indeed,
when consumers are myopic, the first-period demand elasticity is greater, thus
lowering the first-period price. １６ Hence, the second-period profits become
more important, boosting the incentive of firms to choose N.

6. CONCLUSIONSThe question addressed in this paper is whether two competing firms will
always want to identify their buyers in order to go into behavior-based pricediscrimination in the second period. As we are in an increasingly sophisticatedmarketing environment where buyers and sellers behave strategically and areaware of the consequences of the information they release during thetransactions, exploring strategic anonymity by buyers and sellers is worthwhile.
While previous works have considered the possibility of imperfect consumer
recognition due to technology imperfections (exogenous imperfect consumer
recognition), in this paper, we allow firms to give buyers the option to remain
anonymous if they so decide or to be recognized, thus endogenizing the degreeof consumer recognition imperfectness.

It is well-known (Fudenberg and Tirole, 2000) that the higher the discount factor of
consumers, the lower the demand elasticity in the first period because consumers anticipate thepoaching strategy of the firms.
１６23

Our results indicate that sellers, for a wide range of parameters, will provide
buyers with the option to remain anonymous or to buy with an account. Whenthe discount factor is not so large, we also observe multiple asymmetricequilibria, where one firm gives free choice to consumers, whereas the otherimposes account creation on its own buyers. The interpretation of this result is
twofold. First, it shows that firms can improve upon a rigid account policy that
only allows buying with being recognized. Second, it rationalizes theobservation that different sellers may use different policies for account creation(as in the Amazon/eBay example discussed in the Introduction). Indirectly, italso shows that (some) buyers, if allowed to conceal their identity, will instead
reveal it - provided this allows them to access ancillary services connected withthe account opening. We also show that, when allowing consumers to maintaintheir anonymity, behavior-based pricing might outperform uniform pricing:
this contrasts with several findings in the BBPD literature (see Fudenberg andTirole 2000, and Villas-Boas 1999 and 2004, among the others).
APPENDIX
Equilibrium profits
𝜋𝜋 𝐹𝐹𝐹𝐹
6𝑐𝑐 2 𝛿𝛿[1 − (3 − 2𝛿𝛿)𝜏𝜏]2 − 12𝑐𝑐𝑐𝑐𝑐𝑐[1 − 4𝜏𝜏(1 − 𝛿𝛿) + (3 − 8𝛿𝛿 + 4𝛿𝛿 2 )𝜏𝜏 2 ] + 𝜏𝜏 2 [27(1 − 𝜏𝜏)2 + 24𝛿𝛿 2 𝜏𝜏(3 − 𝜏𝜏) + 56𝛿𝛿 3 𝜏𝜏 2 + 24𝛿𝛿(1 + 𝜏𝜏 − 2𝜏𝜏 2 )]=6𝜏𝜏[3 − (3 − 4𝛿𝛿)𝜏𝜏]2

𝜋𝜋 𝑁𝑁𝑁𝑁 =𝜋𝜋 𝐹𝐹𝐹𝐹2

2𝑐𝑐𝑐𝑐(𝑐𝑐 − 2𝜏𝜏) + (9 + 8𝛿𝛿)𝜏𝜏
18𝜏𝜏
2

4

2 5

2

2

2

3

𝜏𝜏 [19683(1 − 𝜏𝜏) + 24𝜏𝜏𝛿𝛿 (297 − 2371𝜏𝜏) + 5336𝜏𝜏 𝛿𝛿 + 1458𝛿𝛿(1 + 70𝜏𝜏 − 71𝜏𝜏 ) − 27𝛿𝛿 (407 − 2182𝜏𝜏 − 3409𝜏𝜏 ) + 24𝛿𝛿 (121 +
3

2 4

2

2

−2324𝜏𝜏 + 6199𝜏𝜏 )] − 12𝑐𝑐𝑐𝑐𝑐𝑐[2𝜏𝜏𝛿𝛿 (253 − 3027𝜏𝜏) + 572𝜏𝜏 𝛿𝛿 + 729𝛿𝛿(1 − 3𝜏𝜏 + 2𝜏𝜏 ) − 54𝛿𝛿(11 − 135𝜏𝜏 + 232𝜏𝜏 ) +
2

=

2

2

2

2

3

2 4

2

+𝛿𝛿 (121 − 3849𝜏𝜏 + 18308𝜏𝜏 )] + 6𝑐𝑐 𝛿𝛿[729(1 − 5𝜏𝜏) + 2𝛿𝛿 𝜏𝜏(286 − 4049𝜏𝜏) + 640𝜏𝜏 𝛿𝛿 − 54𝛿𝛿(11 − 218𝜏𝜏 + 788𝜏𝜏 ) +22+𝛿𝛿 (121 − 4990𝜏𝜏 − 32293𝜏𝜏 )]6𝜏𝜏[81(1 − 𝜏𝜏) − 𝛿𝛿(33 − 249𝜏𝜏) − 34𝜏𝜏𝛿𝛿2 ]24

2

𝜋𝜋 𝑁𝑁𝑁𝑁
2

2

4

2 5

2

2

2

3

𝜏𝜏 [19683(1 − 𝜏𝜏) + 48𝜏𝜏𝛿𝛿 (121 − 879𝜏𝜏) + 3008𝜏𝜏 𝛿𝛿 + 1458𝛿𝛿(1 + 70𝜏𝜏 − 71𝜏𝜏 ) − 27𝛿𝛿 (407 − 2182𝜏𝜏 − 3409𝜏𝜏 ) + 24𝛿𝛿 (121 +
2

3

2 4

2

2

−2189𝜏𝜏 + 5821𝜏𝜏 )] − 12𝑐𝑐𝑐𝑐𝑐𝑐[2𝜏𝜏𝛿𝛿 (77 − 585𝜏𝜏) + 48𝜏𝜏 𝛿𝛿 + 729𝛿𝛿(1 − 3𝜏𝜏 + 2𝜏𝜏 ) − 54𝛿𝛿(11 − 93𝜏𝜏 + 136𝜏𝜏 ) +
2

2

2

2

3

2 4

2

+𝛿𝛿 (121 − 2061𝜏𝜏 + 7124𝜏𝜏 )] + 6𝑐𝑐 𝛿𝛿[729(1 − 𝜏𝜏) − 2𝛿𝛿 𝜏𝜏44 + 17𝜏𝜏 − 20𝜏𝜏 𝛿𝛿 − 54𝛿𝛿(11 − 32𝜏𝜏 − 70𝜏𝜏 ) +=Proof of Proposition 1 and 222+𝛿𝛿 (121 − 1162𝜏𝜏 + 337𝜏𝜏 )]6𝜏𝜏[81(1 − 𝜏𝜏) − 𝛿𝛿(33 − 249𝜏𝜏) − 34𝜏𝜏𝛿𝛿2 ]2

The conditions for (FF) to be an equilibrium are:𝜋𝜋 𝐹𝐹𝐹𝐹 ≤ (≥) 𝜋𝜋 𝑁𝑁𝑁𝑁 if 𝑐𝑐 ≥ (≤)𝑐𝑐1 (𝜏𝜏, 𝛿𝛿)𝜋𝜋 𝐹𝐹𝐹𝐹 ≤ (≥) 𝜋𝜋 𝑁𝑁𝑁𝑁 if 𝑐𝑐 ≥ (≤)𝑐𝑐2 (𝜏𝜏, 𝛿𝛿)

where

𝑐𝑐1 (𝜏𝜏, 𝛿𝛿) ≡𝜏𝜏�2187(1−𝜏𝜏)3 −81(1−𝜏𝜏)2 (37−253𝜏𝜏)𝛿𝛿+�858−19665𝜏𝜏+83412𝜏𝜏2 −64605𝜏𝜏3 �𝛿𝛿 2 �

𝐻𝐻𝜏𝜏[18𝜏𝜏�173−1798𝜏𝜏+3965𝜏𝜏2 �𝛿𝛿 3 +48𝜏𝜏2 (76−397𝜏𝜏)𝛿𝛿 4 +1384𝜏𝜏3 𝛿𝛿 5 ]𝐻𝐻, with+𝐻𝐻 ≡ 8748(1 − 𝜏𝜏)3 − 81(88 − 939𝜏𝜏 + 1758𝜏𝜏 2 − 907𝜏𝜏 3 )𝛿𝛿
+ 3(484 − 13197𝜏𝜏 + 63102𝜏𝜏 2 − 58165𝜏𝜏 3 )𝛿𝛿 2+ 2𝜏𝜏(2655 − 32954𝜏𝜏 + 68963𝜏𝜏 2 )𝛿𝛿 3 + 8𝜏𝜏 2 (792 − 4237𝜏𝜏)𝛿𝛿 4+ 2472𝜏𝜏 3 𝛿𝛿 5 ,

𝑐𝑐2 (𝜏𝜏, 𝛿𝛿) ≡𝜏𝜏[2187(1−𝜏𝜏)−81(37−181𝜏𝜏)𝛿𝛿+3(286−3805𝜏𝜏)𝛿𝛿 2 +1690𝜏𝜏𝛿𝛿 3 ] １７.
8748(1−3𝜏𝜏)−81(88−597𝜏𝜏)𝛿𝛿+3(484−7023𝜏𝜏)𝛿𝛿 2 +2302𝜏𝜏𝛿𝛿 3

Therefore,•••

If 𝑐𝑐 ∈ [max[𝑐𝑐1 (𝜏𝜏, 𝛿𝛿), 𝑐𝑐2 (𝜏𝜏, 𝛿𝛿)] , 𝑐𝑐̂ (𝜏𝜏, 𝛿𝛿)], NN is the unique equilibrium

If 𝑐𝑐 ∈ [0, min[𝑐𝑐1 (𝜏𝜏, 𝛿𝛿), 𝑐𝑐2 (𝜏𝜏, 𝛿𝛿)]], FF is the unique equilibrium

If 𝑐𝑐 ∈ [max[0, 𝑐𝑐1 (𝜏𝜏, 𝛿𝛿)] , min[𝑐𝑐2 (𝜏𝜏, 𝛿𝛿), 𝑐𝑐̂ (𝜏𝜏, 𝛿𝛿)] , both FN and NF are

equilibria•

If 𝑐𝑐 ∈ [max[0, 𝑐𝑐2 (𝜏𝜏, 𝛿𝛿)] , min[𝑐𝑐1 (𝜏𝜏, 𝛿𝛿), 𝑐𝑐̂ (𝜏𝜏, 𝛿𝛿)] , both NN and FF are
equilibria

Then we observe the following:

１７

Note that when 𝛿𝛿 = 1, we have

𝜕𝜕𝑐𝑐1
𝜕𝜕𝜕𝜕

> 0 and

𝜕𝜕𝑐𝑐2
𝜕𝜕𝜕𝜕

25

> 0.

-

When δ is high (say greater than 0.95), we have these possibilities: only
NN; both NN and FF; only FF. In other words, multiple asymmetricequilibria never arise.

-

When δ is intermediate (say about 0.9), we have these possibilities: only
NN; both NN and FF; both FN and NF; only FF. In other words, all cases-

are possible, depending on c and 𝜏𝜏.

When δ is low (say lower than 0.85), we have these possibilities: only
NN; both FN and NF; only FF. In other words, multiple symmetricequilibria never arise.

Proof of Proposition 3
By comparing 𝜋𝜋 𝑈𝑈𝑈𝑈 with 𝜋𝜋 𝑁𝑁𝑁𝑁 , 𝜋𝜋 𝑈𝑈𝑈𝑈 with 𝜋𝜋 𝐹𝐹𝐹𝐹 , we observe the following: 𝜋𝜋 𝑈𝑈𝑈𝑈 >𝜋𝜋 𝑁𝑁𝑁𝑁 and 𝜋𝜋 𝑈𝑈𝑈𝑈 < (>) 𝜋𝜋 𝐹𝐹𝐹𝐹 if 𝑐𝑐 < (>) 𝑐𝑐3 (𝜏𝜏) where 𝑐𝑐3 (𝜏𝜏) =𝜏𝜏)�𝜏𝜏(6−5𝜏𝜏)
6

𝑐𝑐 = 𝑐𝑐3 (𝜏𝜏).

𝜏𝜏(1+𝜏𝜏)1−𝜏𝜏

− (1 − 𝜏𝜏)(3 +, or if 𝜏𝜏 ≥ (≤) 𝜏𝜏3 (𝑐𝑐) , where 𝜏𝜏3 (𝑐𝑐) = 𝑐𝑐3 (𝜏𝜏)−1 , and 𝜋𝜋 𝑈𝑈𝑈𝑈 = 𝜋𝜋 𝐹𝐹𝐹𝐹 if

Welfare𝑊𝑊 𝐹𝐹𝐹𝐹= 𝑣𝑣(1 + 𝛿𝛿)−18(1 − 𝜏𝜏)2 + 16𝜏𝜏 2 𝛿𝛿 3 (1 + 𝜏𝜏) + 2𝜏𝜏𝛿𝛿 2 (9 − 10𝜏𝜏 + 𝜏𝜏 2 ) +�−𝛿𝛿(5 − 17𝜏𝜏 + 51𝜏𝜏 2 − 39𝜏𝜏 3 )
2
2 3
2 (7
2 )]
𝜏𝜏[9(1 − 𝜏𝜏) + 16𝜏𝜏 𝛿𝛿 + 4𝜏𝜏𝛿𝛿− 3𝜏𝜏) + 𝛿𝛿(11 + 2𝜏𝜏 − 13𝜏𝜏4(1 − 𝜏𝜏)[3 − (3 − 4𝛿𝛿)𝜏𝜏]2

4𝑐𝑐𝑐𝑐(1 − 𝜏𝜏)[9(1 − 𝜏𝜏)2 + 8𝜏𝜏𝛿𝛿 2 (1 + 𝜏𝜏𝛿𝛿 2 ) + 2𝛿𝛿(1 + 8𝜏𝜏 − 9𝜏𝜏 2 )] − 2𝑐𝑐 2 �

𝑊𝑊

𝑁𝑁𝑁𝑁

4𝑐𝑐𝑐𝑐(9 + 2𝛿𝛿) − 10𝑐𝑐 2 𝛿𝛿 + 𝜏𝜏 2 (9 + 11𝛿𝛿)= 𝑣𝑣(1 + 𝛿𝛿) −72𝜏𝜏

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