Softening Competition through
Unilateral Sharing of Customer Data*
Chongwoo Choe, Jiajia Cong,

Chengsi Wang§

October 20, 2022

Abstract
We study how a data-rich firm can benefit by unilaterally sharing its customer
data with a data-poor competitor when the data can be used for price discrimination. By sharing data on the segment of market that is more loyal to thecompetitor while keeping the data on the competitor’s most loyal segment toitself, the firm can induce the competitor to raise its price for consumers it
does not have data on. Such data sharing is an example of a fat-cat strategy
as it softens price competition that follows data sharing. Although consumer
surplus decreases as a result of data sharing, total surplus can increase when
the sharing firm concedes its market share to the competitor, which improves
the quality of consumer-firm matching.
Keywords: customer data sharing, price discrimination
JEL Classification Number: D43, L13

1

Introduction

In 2017, The Economist famously declared that data is the new oil in the digital economy.1 Consumer-generated data analyzed with powerful machine-learning* We thank the Associate Editor and three anonymous reviewers for many valuable comments andsuggestions. We are also thankful to Zhijun Chen, Jay Pil Choi, Noriaki Matsushima, Vai-Lam Mui,
Andrew Rhodes, Nicolas Schutz, Julian Wright, and Xiaojian Zhao for helpful comments. Choe
and Wang gratefully acknowledge financial support from the Australian Research Council (grant
number DP210102015). Cong would like to acknowledge the financial support from NSFC (Projects
72003035, 72192845) and Shanghai Pujiang Program (2019PJC007). The usual disclaimer applies.
 Department of Economics, Monash University, chongwoo.choe@monash.edu.
School of Management, Fudan University, jjcong@fudan.edu.cn.
§ The corresponding author. Department of Economics, Monash University, Clayton, Victoria
3800, Australia. chengsiwang@gmail.com.
1“The world’s most valuable resource is no longer oil, but data”, The Economist, May 16, 2017.

1

tools can facilitate data-enabled learning, which can lead to new or improved products, more target-oriented business models, effective defense against competition, etc.
(Hagiu and Wright, 2020). Consumer data can also allow firms to sharpen their pricing tools and extract more surplus from consumers through price discrimination. For
example, finer-grained analysis utilizing big data has made personalized pricing closer
to becoming a reality in some industries. The flip side is that competition can become
more intense when it is based on finer levels of customer data that can be used forprice discrimination (Thisse and Vives, 1988; Fudenberg and Tirole, 2000; Choe etal., 2018).
Can a firm benefit by unilaterally sharing its customer data with a competitor
when the data can be used for price discrimination? As alluded to in the previous
paragraph, it would seem that the sharing firm cannot benefit because data sharing will improve the competitor’s pricing capabilities and intensify competition byfacilitating poaching by the competitor.2 The purpose of this paper is to provide amodel where an informed firm can choose to share customer data with an uninformed
competitor and, given the optimally chosen amount of shared data, both firms are
better off than without data sharing. The key driver of this result is that the informed
firm strategically selects a subset of data to share with a competitor, which induces
the competitor to raise its price for consumers it does not have data on. Therefore,
the unilateral data sharing considered in this paper is a soft commitment when thesubsequent competition is in strategic complements, hence is an example of a fat-cat
strategy (Fudenberg and Tirole, 1984).3 We sketch below our main argument.
Consider a Hotelling model with two firms located at each end. Firm 1 has dataon all consumers’ locations and can set a personalized price for each consumer. Firm
2 does not have any data to start with and chooses only a uniform price. Given the
maximal differentiation, firm 1’s informational advantage does not extend to the entire
market, implying that there are some consumers served by firm 2 even without data
sharing. We call this firm 2’s customer base. Since prices are strategic complements,
2

See, for example, Kim and Choi (2010) or Chen et al. (2022). A number of studies reviewed
later provide various models where mutual, rather than unilateral, data sharing can increase industry
profits.
3
In Fudenberg and Tirole (1984), a fat-cat strategy refers to overinvestment by an incumbent
that accommodates entry by committing to play less aggressively post-entry. Subsequently, theterm has come to be used more generally in two-stage games to refer to any strategic commitmenta firm makes in the first stage that softens competition in the second stage when the second-stage
competition is in strategic complements.

2

firm 1 can benefit if it can induce firm 2 to increase its uniform price.
Suppose now firm 1 shares data on some consumers in firm 2’s customer base
except those who are very close to firm 2’s location. Then firm 2 makes pricing
decisions separately for consumers it has data on and the rest: for the former whom
we call targeted consumers, firm 2 chooses personalized prices and, for the latter,it chooses a uniform price. In the baseline model, we assume firm 2 can engage
in search discrimination, which prevents its targeted consumers from choosing its
uniform price.4 Then firm 1 can choose data sharing in such a way that leads firm
2 to set its uniform price to serve only consumers who are very close to firm 2’s
location.5 Consequently, firm 2 will raise its uniform price above the level it would
choose in the absence of data sharing. This benefits firm 1 by allowing it to increase
its personalized prices. But firm 1 does not concede any additional consumers tofirm 2 because it shares data only for some consumers in firm 2’s customer base.6
Put together, firm 1 can benefit from data sharing through higher personalized prices
at no cost of reduced market share. A clear implication from our analysis can besummarized as this: share data on consumers who are more loyal to the competitor
than to yourself; but keep the data on the competitor’s most loyal consumers to
yourself.
The welfare effects of data sharing depend on market conditions and the natureof competition that follows data sharing. We can decompose the welfare effects into
two parts. The first one is how data sharing enables firms to better extract consumer
surplus, which we call the surplus-extraction effect. Clearly, the surplus-extraction
effect impacts consumer surplus negatively. The second one is how data sharing affects
the matching between consumers and firms, called the quality-of-matching effect.
Data sharing may have a positive or negative quality-of-matching effect depending
on how market shares change after data sharing. Given our assumption of full market
coverage, the surplus extraction effect is relevant only to consumer surplus, while the4

In Section 4.1, we consider the case where firm 2 cannot engage in search discrimination. Firm
1 can still benefit from data sharing although the mechanism is now different. Firm 2’s inability tosearch discriminate puts a cap on its personalized prices it can charge to targeted consumers. This
leads firm 2 to increase its (off-the-path) uniform price, which benefits firm 1.
5
Of course, the segment of firm 2’s customer base with shared data needs to be chosen suitably
so that firm 2 does not have incentives to set its uniform price to serve some consumers outside its
customer base. In Section 3.2, we spell out the precise condition for this.
6This is an illustrative example of how firm 1 can benefit from sharing data with firm 2 without
losing its market share. Firm 1’s optimal data sharing may involve conceding some market share to
firm 2 if the benefit of doing so more than offsets the reduced market share.

3

quality-of-matching effect is relevant to both consumer surplus and total surplus.
If firm 1 increases its market share further as a result of data sharing, then bothconsumer surplus and total surplus decrease. In this case, not only does the surplusextraction effect hurt consumers, but also the quality-of-matching effect is negative.
However, if data sharing results in firm 1 giving away some of its market share to
firm 2, then total surplus increases thanks to the positive quality-of-matching effect.
We study several variations of the baseline model and examine how our main
insight extends to other economic environments. In Section 4.1, firm 2 is assumed to
be unable to use search discrimination, i.e., it cannot prevent its targeted consumers
from choosing its uniform price. We show that firm 1 continues to benefit from data
sharing, but even more than when search discrimination is possible. In Section 4.2,
we analyze the case where firm 2’s data anaytics limits use of the shared data to
third-degree price discrimination rather than personalized pricing. We show that our
main insight continues to hold, but the dampened surplus-extraction effect in thiscase implies that consumers are better off than when the shared data is used for
personalized pricing. In Section 4.3, we discuss the extent to which customer data
can be used to deter firm 2’s entry to the market. As data sharing always occurs once
entry has been accommodated, firm 2’s entry becomes more likely when data sharing
is possible. Thus the possibility of data sharing mitigates the extent to which data
plays a role as an entry barrier. In Section 4.4, we consider the case where firm 1
can be considered a dual-mode platform that charges a fixed sales fee to firm 2. Weshow that the only effect the sales fee has is to soften competition and raise all prices
by the amount of the sales fee, and our results on the optimal data sharing remainrobust. Several additional extensions of the baseline model are also discussed in theremainder of Section 4.
Contributions to the literature
Two strands of literature are directly relevant to our work. The first one relates
to information sharing among firms and the second one is on competition based on
personalized pricing.
Information sharing among firms exists in many industries such as airline, banking and finance, tourism and hospitality, etc. See, for example, Feasey and de Streel(2020) for a comprehensive discussion on information sharing in practice and therelated regulatory issues. The earlier academic literature has been focused on shar4

ing information on cost or demand conditions that can facilitate output or pricing
decisions (e.g., Gal-Or, 1985; Armantier and Richard, 2003). More recent studies
identify various conditions under which mutual information sharing, or informationexchange between firms, can increase industry profits. These conditions pertain tofirms’ asymmetric and imperfect abilities to target customers (Chen et al. 2001),
consumer switching costs (Shy and Stenbacka, 2013), or two-dimensional customer
information (Jentzsch et al., 2013).7 In contrast to these studies, our focus is on
unilateral sharing of customer information that can be used for price discrimination.
Chen et al. (2001) and Liu and Serfes (2006) provide models where unilateral
information sharing subject to side payments can increase industry profits. For example, the main mechanism in Liu and Serfes (2006) is that the firm with a small
loyal customer base but a large amount of customer data can share its data with acompetitor, thereby allowing the competitor to better extract surplus from the customers whose data is shared. Although this reduces the sharing firm’s profit, thereduced profit can be compensated by the side payment. Consequently, unilateral
information sharing does not arise in the absence of side payment. Somewhat related
studies consider the case where a monopoly data broker sells data to competing firms
to maximize its profit (Montes et al., 2019; Bounie et al., 2021). A general insight isthat data is sold to only one firm in a way to soften competition, hence data sharing
can never take place. In contrast, we show that unilateral information sharing is
individually rational even in the absence of side payments. In addition, these studies
typically assume consumer preferences are uniformly distributed whereas we consider
general distributions.
There is a large and growing body of literature on personalized pricing. Consumer
data is shown to intensify competition when it is used for personalized pricing in astatic model of horizontal differentiation (Thisse and Vives, 1988) or vertical differentiation (Choudhary et al., 2005).8 This insight is extended to a dynamic model ofbehavior-based pricing by Choe et al. (2018) and product personalization by Zhang(2011). Chen and Iyer (2002) studies the firm’s decision to invest in customer ad7

Choe et al. (2022) considers a standard model of behavior-based price discrimination, but
adds a stage when firms can agree to share customer information before price competition begins.
They show that information sharing obtains in equilibrium because it softens competition when
information is gathered, although it intensifies competition when shared information is used insubsequent competition.
8
These results are based on the case where the market is fully covered. When it is not, Rhodes
and Zhou (2022) shows the results are reversed.

5dressability which enables personalized pricing. Shaffer and Zhang (2002) allows firm
heterogeneity and Chen et al. (2020) considers consumers’ identity management thatcan bypass firms’ attempt to price discriminate. Fudenberg and Villas-Boas (2012)provides a comprehensive survey of the earlier literature, and Ezrachi and Stucke(2016) provide various examples of personalized pricing in practice. Our work contributes to this literature by analyzing the implications of unilateral data sharing
when it is used for personalized pricing, which is not formally studied in the existing
literature.
The rest of the paper is organized as follows. The baseline model is described in
Section 2 and analyzed in Section 3. Section 4 studies several extensions and variations of the baseline model explained previously. In Section 5, we provide evidence
on B2B data data sharing in practice and the implications for management that ouranalysis sheds light on. Section 6 concludes the paper with discussions on the directions for future research. Appendix contains deferred proofs while the online appendix
includes proofs of the results in Section 4 as well as some additional discussions.

2

The model

We adopt a standard location model with a general distribution for consumer
preferences. There is a continuum of consumers on [0, 1]. A consumer’s preference
is indexed by her location x ∈ [0, 1] where x has a smooth, cumulative distributionfunction F and a strictly positive density f . We assume F satisfies the monotonehazard rate condition, i.e., h(x) := f (x)/(1 − F (x)) is strictly increasing in x. We
call the consumer located at x simply consumer x. There are two firms. Firm 1 islocated at 0 and has full information about consumers’ preferences. Firm 2 is locatedat 1 and initially has no information about consumers’ preferences.
A consumer derives a gross value v from a product from either firm and incurs
a transportation cost di , the distance between her location and firm i’s location.
So if a consumer buys a product from firm i at price pi , then she obtains utilityv − pi − di . We assume v ≥ 2 so that the market is fully covered.9 Firm 1 cancostlessly share data with firm 2. When a consumer’s data is shared, firm 2 knows
the exact location of that consumer, called a targeted consumer, based on which toexercise price discrimination.
9

In Section 4.7, we discuss the case where the market is not fully covered.

6

We allow firm 1 to choose any market segment for data sharing. For example,
it may choose to share data on consumers in several disjoint segments. However, as
we show later in Lemma 1, firm 1 weakly prefers sharing data on an interval to any
other forms of data sharing. This leads us to a simple form of data sharing: firm 1chooses an interval [a, b] ⊆ [0, 1] for data sharing. For clarity of exposition, we do notconsider any monetary payment between the two firms in the baseline model.10 With
its full information, firm 1 sets personalized price p1 (x) for each consumer x ∈ [0, 1].
Without data sharing, firm 2 can choose only a uniform price, denoted by p2 . Given
data sharing on [a, b], firm 2 chooses personalized price p2 (x) for x ∈ [a, b] and uniform
price p2 for the rest of the consumers. To simplify notation, we normalize the cost of
production to zero.11
The timing of the game is as follows. First, firm 1 chooses [a, b]. Second, firm
2 decides whether or not to accept the shared data. If the shared data is rejected,
then we have the status quo. If firm 2 accepts the shared data, then the subsequent
pricing game proceeds as follows. Firm 2 moves first by choosing its uniform pricewhere relevant. Following this, the two firms simultaneously make private offers of
personalized prices to consumers they have data on. The sequential timing in priceoffers is standard in the literature on personalized pricing (Thisse and Vives, 1988;
Shaffer and Zhang, 2002; Choe et al., 2018). This not only reflects the flexibility
in choosing personalized prices, but also allows us to solve for the subgame perfectNash equilibrium in pure strategies.12 We assume that firm 2 can prevent consumers
who are given its private offers from choosing its uniform price. This can be done, forexample, by redirecting the product search by a consumer, a practice called steering or
search discrimination. Relaxing this assumption does not change our general insightand further benefits firm 1, as we show in Section 4.1.
Remark 1. Our assumption that firm 1 has full information and firm 2 has no
information is made for simplicity. As will become clear, our main insight continues
to hold if firm 1 has information on [0, m] with m ∈ (z N , 1) where z N is the location of
10

In Section 4.4, we extend our model to the case where firm 1 can charge a fixed sales fee to firm
2, a practice often adopted by dual-mode platforms. In Section 4.5, we discuss the case where firm
1 can sell data to firm 2. We show that our main insight remains robust to these extensions.
11
We can modify the model to allow for cost differences between the two firms. For example,
firm i has a constant marginal cost ci with c1 ̸= c2 . This will not change our main insight insofar asboth firms are active in equilibrium with or without data sharing.
12
Following Chen et al. (2020, Section 5.5), we can show that all equilibria in our model are inmixed strategies when firms simultaneously choose uniform and personalized prices.

7marginal consumer in the benchmark without data sharing. Then, firm 1 can choose
to share data on a subset of [z N , m], which can benefit both firms. Likewise, mutually
beneficial data sharing exists even when firm 2 has information on some consumers
close to its location, say, [n, 1] where n > z N . In this case, firm 2’s pricing decisions
on [n, 1] are delinked from its pricing decisions on [0, n]. Thus, firm 1 can choose
mutually beneficial data sharing on a subset of [z N , n′ ] for some n′ < n.
Remark 2. In our baseline model, the shared data is used for personalized pricing.
In Section 4.2, we analyze the case where the shared data is used for third-degree
price discrimination. Although we do not consider the general information design
problem where firm 1 can choose the informativeness of shared data by choosing
a disclosure strategy, which is a mapping from the Hotelling line to a distribution
over message space,13 we conjecture that our main insight may hold for general data
sharing strategies. It is because our results hold for the two polar cases, the mostinformative data sharing analyzed in Section 3 and the least informative data sharing
analyzed in Section 4.2. Analysis of the two polar cases can be useful in studying
more general cases.

3

Analysis

3.1

Benchmarks

We start with two benchmark cases. First, without data sharing, the marginal
consumer z satisfies p1 (z) + z = p2 + (1 − z). Since firm 1 can lower p1 (z) down
to zero, we have z = (1 + p2 )/2. Firm 2 chooses p2 = 2z − 1 to maximize profit
π2 = p2 (1−F (z)). From the first-order condition,14 we obtain the following equations
for firm 2’s optimal price denoted by pN
2 and the marginal consumer’s location denotedNby z :
1
1
2
N
,
z
:=
+
.
(1)
pN
2 :=
h(z N )
2 h(z N )Since firm 2 can serve the segment [z N , 1] even without data sharing, we call it firm 2’s
customer base. Given pN
2 , firm 1 chooses personalized price that leaves all consumersN
on [0, z N ] indifferent between choosing either firm, i.e., pN
1 (x) = max{p2 + (1 −13
14See, for example, Bergemann and Morris (2019).
The second-order condition is satisfied due to the monotone hazard rate condition.

82x), 0}.15 In the benchmark without data sharing, firms earn profits given by
π1N :=

Z zN

N
N
N
(pN
2 + (1 − 2x))dF (x), π2 := p2 (1 − F (z )).

(2)0Second, if firm 1 shares data on all consumers, then we have the outcome inThisse and Vives (1988): firm 1 serves [0, 1/2] with p1 (x) = 1 − 2x and firm 2serves the rest with p2 (x) = 2x − 1. Clearly, firm 1 is worse off compared to thecase without data sharing. Not only firm 1’s market share shrinks from [0, z N ] to[0, 1/2] but also personalized prices firm 1 charges to consumers it serves decreaseNfrom pN
1 (x) = p2 + (1 − 2x) to p1 (x) = 1 − 2x. Firm 2 is better off compared to thebenchmark without data sharing. It is because firm 2 now serves additional consumers
on [1/2, z N ] while serving all consumers on [z N , 1] at personalized prices higher thanN
its uniform price in the absence of data sharing, i.e., p2 (x) = 2x − 1 ≥ pN
2 = 2z − 1
for all x ∈ [z N , 1].

3.2

Mutually beneficial data sharing

We now turn to the full game and analyze when data sharing can benefit both
firms, hence is accepted by firm 2. Let us start with the following observations. First,
given data sharing on [a, b], firm 2’s problem of choosing its uniform price can bedelinked from its choice of personalized prices, due to our assumption that firm 2 canexercise search discrimination. This intensifies competition on [a, b]. Consequently,
firm 1’s profit from the segment [a, b] cannot be higher after data sharing. Second,firm 1 will set its personalized price that will leave each consumer it serves indifferent
between choosing either firm. That is, given firm 2’s uniform price p2 , firm 1 will
set p1 (x) = max{p2 + (1 − 2x), 0}. Thus, higher p2 benefits firm 1 by allowing it toraise its personalized prices. Put together, these observations imply that firm 1 canbenefit from data sharing only if it softens competition for consumers whose data
is not shared and induces firm 2 to raise its uniform price above pN
2 . From theseobservations, we can show that it is sufficient to focus on the case in which firm 1
shares data on an interval.
Lemma 1 Firm 1 weakly prefers sharing data on an interval to any other forms ofdata sharing.
15

In case of indifference, we assume consumers choose firm 1.

9

Proof: See the appendix.
Given that [z N , 1] is firm 2’s customer base, it is easy to see that, for data sharing
on [a, b] to benefit firm 1, we must have a > 1/2 and b > z N . The reasoning is asfollows. First, sharing data on [a, b] with b ≤ z N intensifies competition on this segment without the benefit of raising firm 2’s uniform price above pN
2 . Second, sharing
data on [a, b] with a ≤ 1/2 is dominated by sharing data on only [1/2, b] because theformer intensifies competition on [a, 1/2] without increasing firm 2’s uniform price.
Henceforth, we restrict analysis to the following set of (a, b).
S := {(a, b) | 1/2 < a < b, z N < b}(3)

Following data sharing, firm 2’s choice of uniform price depends on which segmentof consumers it intends to serve with uniform price. There are two possibilities. First,
if firm 2 serves only [b, 1], then it chooses p21 = 2b − 1 and earns profit
π21 (b) = (2b − 1)(1 − F (b)).

(4)

Second, if it serves [z ′ , a] ∪ [b, 1] for some z ′ ≤ a, then it chooses p22 = 2z ′ − 1 andearns profit (2z ′ − 1)(1 − F (b) + F (a) − F (z ′ )). Maximizing this leads top22 =1 1 − F (b) + F (a) − F (z ′ )1
2 (1 − F (b) + F (a) − F (z ′ ))
′
,
z
=
+
> .
′
′
f (z )2f (z )
2(5)

Denote the resulting profit by π22 (a, b). The next lemma shows that, for firm 1 tobenefit from sharing data on [a, b], firm 2’s uniform price must be p21 = 2b − 1.
Lemma 2 For data sharing on [a, b] to be profitable for firm 1, it should necessarily
induce firm 2 to choose its uniform price to serve consumers on [b, 1] only.
Proof: See the appendix.
We provide below the intuition for Lemma 2. Given a > 1/2, data sharing on [a, b]
implies that firm 2 will serve the entire segment [a, b]. Therefore, the only way firm 1
can benefit from data sharing is to induce firm 2 to raise its uniform price above pN
2 .
This is possible when firm 2 chooses its uniform price to serve consumers on [b, 1] only.
However, if firm 2 chooses its uniform price to serve additional consumers on [z ′ , a],10then firm 2’s uniform price is lower than pN
2 . It is because F satisfies the monotone′hazard rate condition, which implies z < z N as shown in the proof of Lemma 2. In
this case, not only does firm 1 concede its market share to firm 2, but its personalized
prices also decrease due to a lower uniform price chosen by firm 2. To summarize,
Lemma 2 implies firm 2’s incentive compatibility constraint that any data sharing byfirm 1 needs to satisfy:(IC) π21 (b) ≥ π22 (a, b).
(6)
In addition, data sharing needs to make firm 2 weakly better off than in thebenchmark without data sharing; or else, firm 2 will ignore the shared data. Whenfirm 2 serves consumers on [a, b] with personalized prices p2 (x) = 2x − 1 and those on
Rb[b, 1] with uniform price p21 = 2b − 1, its profit is π2 (a, b) = a (2x − 1)dF (x) + π21 (b).
Thus we have the following individual rationality constraint for firm 2:
Z b(IR) π2 (a, b) =(2x − 1)dF (x) + π21 (b) ≥ π2N .

(7)aWhen data sharing on [a, b] satisfies (IC) and (IR), firm 1 serves all consumers on[0, a] with personalized prices p1 (x) = p21 + (1 − 2x) = 2(b − x), while firm 2 serves
the rest of the market. Thus, firm 1’s profit is given by
Z a

Z a(p21 + (1 − 2x))dF (x) = 2bF (a) − 2

π1 (a, b) =0

xdF (x).

(8)0Differentiating π1 (a, b) in (8), we find ∂π1 (a, b)/∂a > 0 and ∂π1 (a, b)/∂b > 0 for
all b > a > 1/2. Thus, firm 1’s profit increases in a because it shares less data, andit increases in b because higher b leads to a higher uniform price by firm 2. Likewise,
differentiating π2 (a, b) in (7), we find ∂π2 (a, b)/∂a < 0 and ∂π2 (a, b)/∂b > 0 for allb > a > 1/2. Thus, firm 2’s profit decreases in a because of less data sharing, and it
increases in b because of more data sharing. Moreover, in the proof of Proposition 1
below, we show π1 (a, b) > π1N and π2 (a, b) > π2N for all b > a = z N . This implies that,
insofar as firm 2’s (IC) is satisfied, there exists data sharing [z N , b′ ] with b′ ∈ (z N , 1),
under which both firms have higher profits than in the benchmark without data
sharing.
Proposition 1 For any distribution F that satisfies the monotone hazard rate condition, we have π1 (a, b) > π1N and π2 (a, b) > π2N for all b > a = z N . In addition, there
11exists b′ ∈ (z N , 1) such that firm 2’s (IC) is satisfied under data sharing on [z N , b′ ],hence sharing data on [z N , b′ ] makes both firms better off compared to the benchmark
without data sharing.
Proof: See the appendix.
The intuition for the above proposition is as follows. Firm 1 can benefit fromsharing data on consumers who are in firm 2’s customer base, whom firm 1 cannot
serve even without data sharing. That is, sharing data on [z N , b′ ] does not cost firm 1
any market share. But firm 1 should keep data on consumers who have high loyalty
to firm 2. By keeping data on [b′ , 1] to itself, firm 1 can induce firm 2 to raise its
uniform price to p21 = 2b′ − 1 > pN
2 .
Sharing data in this way softens competition as firm 2 will try to extract surplus
from the consumers with high loyalty by charging a high uniform price. This allows
firm 1 to raise its own personalized prices for consumers whom firm 1 continues toserve after data sharing. Thus firm 1 benefits from data sharing through higher
personalized prices but at no cost of reduced market share. Firm 2 also benefits fromsuch data sharing because it serves the same set of consumers but at higher prices
for all of them than without data sharing.

3.3

Optimal data sharing

In the previous section, we showed that mutually beneficial data sharing exists forany distribution F . We now turn to the discussion of the optimal data sharing thatfirm 1 will choose. Firm 1’s problem can be stated as follows:
max π1 (a, b) subject to (IC) and (IR).

(9)(a,b)∈S

We already know that the following conditions must be met for the optimal data
sharing. First, data sharing has to be on an interval [a, b]. Second, we must have
1/2 < a and z N < b < 1. Third, data sharing should induce firm 2 to choose a
uniform price 2b − 1, which is accepted by consumers on [b, 1] only. These necessary
conditions characterize the optimal data sharing in reasonable details, and they do
not depend on the underlying distribution F . But one remaining issue is whether
a < z N or a ≥ z N under the optimal data sharing, which depends on F as we discuss
below.
12

When a ≥ z N , data sharing does not entail any cost of reduced market share forfirm 1. When a < z N , firm 1 sacrifices some market share if doing so increases firm 2’s
uniform price, which more than compensates for the reduced market share. Clearly,
whether a is above or below z N depends on the shape of distribution F . Recall thatfirm 2 chooses between a high uniform price p21 to serve [b, 1] and a low uniform pricep22 to serve [z ′ , a] ∪ [b, 1]. Firm 2’s (IC) requires that the former is more profitablefor firm 2 than the latter. Given that b > z N > 1/2, this is likely to be the case when
F is left-skewed with a concentrated density around [z N , 1]. Moreover, firm 1’s profit
strictly increases in b when firm 2 chooses the high uniform price p21 . Therefore, when
F is left-skewed, firm 1 can maintain large b, hence induce the high uniform price p21 ,without losing market share by setting a < z N .
Firm 1 may concede additional consumers to firm 2 by setting a < z N if F is
not highly left-skewed. Many distributions are not highly left-skewed, including all
distributions symmetric around 1/2 and hence neither left- nor right-skewed. Firm
1 then optimally reduces a below z N (i.e., losing the marginal consumer) in order tomaintain a large b (i.e., a high profit margin on all inframarginal consumers) without
violating firm 2’s (IC). To show this argument more formally, we first fully characterize
the optimal data sharing under the uniform distribution, showing that the optimal
a is indeed set strictly below z N . We then numerically solve for the optimal data
sharing for a family of distribution functions to further support our main insight.
Suppose F is a uniform distribution, hence symmetric around 1/2. Then, F (x) =x, and it is straightforward to calculate the equilibrium outcome in the benchmark
NN
without data sharing: z N = 3/4, pN
2 = 1/2, π1 = 9/16, and π2 = 1/8. Given data
sharing on [a, b], we have π21 (b) = (2b − 1)(1 − b), π22 (a, b) = (1 + 2a − 2b)2 /8, and
π1 = 2ab − a2 . Based on the above, we can solve firm 1’s problem in (9), leading to
the following equilibrium outcome for the data sharing game.
Proposition 2 If F is a uniform distribution, then firm 1’s optimal choice is givenby [a∗ , b∗ ] ≈ [0.71, 0.97].
 Firm 1 serves consumers on [0, a∗ ] with personalized price p∗1 (x) = 2b∗ − 2x andearns profit π1∗ ≈ 0.87 > 9/16 = π1N .
 Firm 2 serves consumers on [a∗ , b∗ ] with personalized price p∗2 (x) = 2x − 1,

consumers on [b∗ , 1] with uniform price p∗2 = 2b∗ − 1, and earns profit π2∗ ≈0.21 > 1/8 = π2N .
13

Proof: See the appendix.
Notice that we have a∗ < z N when F is uniform. This confirms our earlier
observation that the optimal data sharing involves choosing a < z N . That is, firm
1 sacrifices its market share by sharing data on some consumers outside firm 2’s
customer base. But firm 1’s loss from giving away market share [a∗ , z N ] is smaller
than the gain from inducing a higher uniform price from firm 2. This is becausea∗ > 1/2 and firm 1 can set higher personalized prices for all consumers on [0, a∗ ] but
its loss is limited to [a∗ , z N ].
We now consider a family of distribution functions, which all satisfy the monotonehazard rate condition, to demonstrate that a∗ ≥ z N unless F is too highly left-skewed,
and a∗ < z N , otherwise. Specifically, we consider the beta distribution with shape parameters (α, β), i.e., x ∼ Beta(α, β). The probability density function of Beta(α, β)can take on various shapes and it nests many continuous distributions as special or
limiting cases, including uniform, exponential, power, and normal distributions. In
Table 1, we present the numerical solution (a∗ , b∗ ) to firm 1’s optimal data sharing
problem for five examples of beta distribution. As shown in Table 1, the optimal data
sharing involves a∗ < z N in all cases except when (α, β) = (5, 1), the latter being thecase with an increasing density function, or a highly left-skewed distribution. This
supports our claim that a∗ ≥ z N is likely when F is highly left-skewed. To examine
this further, we plot a∗ and z N in Figure 1 for the case where β = 1 and α increases
from 1 to 5. The solid line labeled z N and the dotted line labeled a∗ cross once whenα ≈ 2.1. Thus, as α increases, hence F becomes more left-skewed, we have a∗ > z N
for all values of α greater than 2.1.16— Insert Table 1 and Figure 1 about here. —3.4

Welfare implications

How does data sharing affect consumer surplus and total surplus? Consider first
total surplus. Given full market coverage, total surplus can be proxied by the average
distance travelled by a consumer, which is minimized when the marginal consumer
When F is a beta distribution, we can numerically show (i) a∗ < z N for all α = β, i.e., F issymmetric around 1/2, (ii) a∗ > z N if s < −1 where s is the skewness of F , (iii) a∗ monotonicallyincreases as α increases holding β fixed, or as β decreases holding α fixed, i.e., F becomes moreleft-skewed. The details are available from the authors.
1614

is at 1/2 because firms are located at 0 and 1.17 It then follows that total surplus
increases if 1/2 < a∗ < z N and decreases if a∗ > z N . In the former case, the reduced
preference mismatch by firms and consumers is a salutary effect of data sharing.
If a∗ < z N , then data sharing has a positive quality-of-matching effect by allowing
consumers on [a∗ , z N ] to be matched to firm 2, their preferred choice. If a∗ > z N ,then data sharing has a negative quality-of-matching effect by leading consumers on[z N , a∗ ] to be matched to firm 1, their less preferred choice. The effect of data sharing
on total surplus is shown in the last column in Table 1 for various beta distributions.
Consider next consumer surplus. Observe that all consumers staying with thesame firm after data sharing are worse off because they incur the same transportation
cost, but pay higher price after data sharing. For these consumers, the quality-ofmatching effect stays the same but the surplus-extraction effect works against them.
But consumers who switch firms may or may not be worse off depending on different
equilibria and how the quality-of-matching and surplus-extraction effects are balanced
against each other.
Consider the case where a∗ ≥ z N . Then, consumer x ∈ [z N , a∗ ] switches to firm
1 after data sharing. Thus, for consumer x ∈ [z N , a∗ ], the quality-of-matching effectis given by (1 − x) − x = 1 − 2x < 0, while the surplus-extraction effect is given byp∗1 (x) − pN
2 , which may or may not be negative. But the overall effect is negative sinceN
her total cost is p∗1 (x)+x = 2b∗ −x after data sharing and pN
2 +(1−x) = 2z −x before
data sharing. The former is larger because b∗ > z N , hence all switching consumers
are worse off after data sharing. Put together, data sharing reduces consumer surplus
if a∗ ≥ z N .
If a∗ < z N , then consumer x ∈ [a∗ , z N ] switches to firm 2 after data sharing,so incurs a total cost p∗2 (x) + (1 − x) = x. Before data sharing, her total cost isN
pN− x. The latter is larger because x ≤ z N , hence all switching
1 (x) + x = 2z
consumers are better off after data sharing. The effect on consumer surplus thendepends on comparing the losses for consumers who stay with the same firm and thegains for those who switch firms. We show in the proof of Proposition 3 that thelosses outweigh the gains so that data sharing also reduces consumer surplus whena∗ < z N . For example, when F is a uniform distribution, we have a∗ < z N as shownin Proposition 2, which implies that the optimal data sharing decreases consumer
17

Let z beR zthe location
R 1of the marginal consumer. Then the average distance travelled by aconsumer is 0 xdF (x) + z (1 − x)dF (x), which is minimized when z = 1/2.

15

surplus but increases total surplus. We summarize the above discussions below.
Proposition 3 Suppose firm 1’s optimal data sharing is given by [a∗ , b∗ ]. Then,for any distribution F that satisfies the monotone hazard rate condition, consumer
surplus decreases unambiguously, and total surplus decreases if and only if a∗ ≥ z N .
Proof: See the appendix.
Our analysis offers clear implications for competition policy in regards to data
sharing between competitors. While data sharing unambiguously hurts consumers,18
it has a salutary effect of increasing total surplus if a dominant, data-rich firm concedes
some of its market share to its competitor as a result of sharing data. The increase
in total surplus in this case is due to reduced preference mismatch: holding prices
equal, firm 2 is a preferred choice for consumers who switch from firm 1 to firm 2
after data sharing. However, if data sharing enables the dominant, data-rich firm tofurther increase its market share, i.e., a∗ ≥ z N , then both consumer surplus and total
surplus are adversely affected.19

4

Extensions

4.1

No search discrimination

In this section, we discuss the case where firm 2 cannot engage in search discrimination. That is, firm 2 cannot prevent a consumer who is offered its personalized
price from choosing its uniform price. Given data sharing on [a, b], consumer x ∈ [a, b]then chooses the lower of the two prices offered by firm 2, i.e., min{p2 (x), p2 }. Thus,
firm 2’s uniform price acts as a cap on its personalized prices.
Suppose firm 1 chooses to share data on [a, b]. It is easy to see that, when firm 2
cannot search discriminate, it does not have incentives to choose a low uniform priceto serve consumers on [z ′ , a]. If it does, then all consumers on [a, b] will choose the18

Using the competition-in-utility-space approach, de Cornière and Taylor (2020) obtains a similar
result that giving firms more data can be anti-competitive as it makes surplus extraction more
efficient.
19
Somewhat related, Armstrong and Zhou (2022) shows that more data can reduce preference
mismatch but soften competition by amplifying perceived product differentiation. In our model,
the effect of data sharing on match quality is driven entirely by price competition because productdifferentiation is exogenously given.

16lower of the uniform price and personalized prices. As a result, firm 2’s profit is at
most (2z ′ − 1)(1 − F (z ′ )). But this is smaller than π2N = (2z N − 1)(1 − F (z N )) by
the definition of z N , which firm 2 can guarantee by ignoring the shared data. Thisimplies that, when firm 2 cannot search discriminate, its optimal pricing decision
automatically satisfies (IC). Consequently, if firm 2 accepts the shared data on [a, b],then it will choose a uniform price p2 = 2b−1 for consumers on [b, 1] and personalized
prices p2 (x) = 2x − 1 for consumers on [a, b]. In sum, although firm 2’s inability
to search discriminate gives consumers on [a, b] an extra option of purchasing at
uniform price, firm 2 instead charges a high uniform price, rendering the extra option
unattractive. This leads to firm 2’s profit given byZ b(2x − 1)dF (x) + (2b − 1)(1 − F (b)).

π2 (a, b) =a

Because firm 1 benefits when p2 = 2b−1 increases, the solution to firm 1’s problem
is then (â, b̂) such that b̂ is as large as possible and π2 (â, b̂) = π2N . This gives us b̂ = 1,
and â such that π2 (â, 1) = π2N . On the equilibrium path, firm 2 uses only personalized
prices. But the equilibrium is supported by its off-the-path uniform price p2 ≥ 1; if
p2 < 1, then consumers on [(p2 + 1)/2, 1] will deviate and choose firm 2’s uniform
price. Because p2 ≥ p∗2 = 2b∗ − 1, firm 1’s personalized prices increase for all its
consumers. Moreover, we can show â > a∗ from π2 (â, 1) = π2N . Thus, firm 1 isstrictly better off in the absence of search discrimination.
Compared to the case without data sharing, it is straightforward to check that
all consumers are worse off. The total surplus is also lower simply because â > z N ,implying that there is more preference mismatch. Compared to the case with data
sharing and search discrimination, consumer surplus is lower because prices rise forall consumers and total surplus is also lower because â > a∗ . To summarize, banningsearch discrimination can enable the dominant, data-rich firm to fine-tune its data
sharing strategy, which is then used by its competitor to further soften competition
than when search discrimination is allowed. This reduces both consumer surplus andtotal surplus.
Proposition 4 Suppose firm 2 cannot engage in search discrimination.
 Firm 1’s optimal data sharing is given by [â, 1] where â solves

π2N , and â > max{z N , a∗ }.
17

R1
â

(2x−1)dF (x) = Firm 1 serves consumers on [0, â] with personalized prices p̂1 (x) = p̂2 +(1−2x) ≥2 − 2x and earns strictly higher profit than the case with search discrimination,
where p̂2 ≥ 1 is firm 2’s uniform price.
 Firm 2 serves consumers on [â, 1] with personalized prices p̂2 (x) = 2x − 1, andearns the same profit as in the benchmark without data sharing.
 Both consumer surplus and total surplus are lower compared to the case without

data sharing, as well as the case with data sharing and search discrimination.
Proof: See the appendix.

4.2

Data sharing for third-degree price discrimination

Even when firm 1 shares data, firm 2 may not have capabilities such as data
analytics that may be necessary to process the shared data for personalized pricing.
This reduces the value of data for firm 2 and may allow it to exercise only thirddegree price discrimination, choosing one uniform price for the segment with shareddata and another uniform price for the rest of the market. But firm 1 continues touse personalized pricing for all consumers. We analyze this case in this section.
As discussed in the previous section, firm 1 can benefit from data sharing only
when it induces firm 2 to increase uniform price above pN
2 . Given data sharing on[a, b] with a ≥ 1/2, denote firm 2’s uniform price for the segment [a, b] by q2l , andthat for the rest of the market by q2h . On [a, b], the marginal consumer’s location,
denoted by zl , satisfies p1 (zl ) + zl = q2l + 1 − zl with p1 (zl ) = 0, implying q2l = 2zl − 1.
Suppose b < z N . Then we have q2l = 2zl − 1 < 2z N − 1 = pN
2 . In addition, firm 2NNNchooses q2h = p2 and serves [z , 1] by the definition of p2 . Thus, when b < z N , firm
1 cannot induce firm 2 to increase its price above pN
2 . In what follows, we focus onN
the case with b ≥ z .
Consider the segment [a, b]. Firm 2 chooses q2l to maximize profit from thissegment given by π2l = q2l (F (b) − F (zl )). From the first-order condition for aninterior solution, we obtainq2l =

1 F (b) − F (zl )2 (F (b) − F (zl ))where zl = +∈ (a, b).
f (zl )2f (zl )

Otherwise, we have a boundary solution, q2l = 2a − 1 and zl = a, hence π2l =18q2l (F (b) − F (zl )). Clearly, it is not optimal for firm 2 to charge q2l < 2a − 1, in whichcase it serves the same set of consumers but at a lower price. Likewise, firm 2 nevercharges price q2l ≥ 2b − 1 because it leads to π2l = 0.
For the rest of the market, firm 2 chooses q2h . As before, firm 2’s choice of q2h
depends on whether it serves only [b, 1], hence q2h = 2b − 1 with corresponding profit
π21 (b), or it serves additional consumers on [z ′ , a] for some z ′ < a with corresponding
profit π22 (a, b). The following lemma shows that firm 1’s optimal data sharing shouldnecessarily satisfy firm 2’s (IC): π21 (a) ≥ π22 (a, b). In addition, it is not optimal
for firm 1 to share data on [a, b] and then serve some consumers on that segment.
It is because firm 1’s personalized prices on [a, b] depend on firm 2’s low uniform
price q2l ; firm 1 is better off by not sharing data on these consumers so that it cancharge personalized prices that depend on firm 2’s high uniform price q2h . Lemma 3
summarizes these observations.
Lemma 3 Firm 1’s optimal data sharing on [a, b] should necessarily satisfy firm 2’s(IC) π21 (b) ≥ π22 (a, b), and induce firm 2 to serve the entire segment [a, b] withuniform price q2l = 2a − 1.
Proof: See the online appendix.
By Lemma 3, we can write firm 2’s profit under the optimal data sharing as
π2 (a, b) = (2a − 1) (F (b) − F (a)) + π21 (b). Thus we have the following individual
rationality constraint for firm 2:(IR) π2 (a, b) = (2a − 1) (F (b) − F (a)) + (2b − 1) (1 − F (b)) ≥ π2N .

(10)Then, firm 1’s profit under the optimal data sharing isZ a(2b − 2x)dF (x).

π1 (a, b) =0

Firm 1’s problem can be stated as follows:
max π1 (a, b) subject to b ≥ z N , π21 (b) ≥ π22 (a, b), and (10).

(11)

(a,b)

Denote the solution to (11) by (a∗∗ , b∗∗ ), and the solution to firm 1’s problem
under personalized pricing stated in (9) by (a∗ , b∗ ). Clearly, firm 1’s profit function19

is the same in both (9) and (11). Firm 2’s (IC) is also the same. The only differenceis that firm 2’s (IR) is harder to satisfy in (11) than in (9). It is because third-degree
price discrimination is a less effective tool than personalized pricing for firm 2 toextract surplus from targeted consumers: firm 2’s profit from [a, b] is smaller when itchooses q2l = 2a − 1 to serve the entire segment than when it can choose personalized
price p2 (x) = 2x − 1 for each targeted consumer. Therefore, an immediate conclusion
is that, if (a∗ , b∗ ) satisfies (10), then we must have (a∗∗ , b∗∗ ) = (a∗ , b∗ ). In this case,neither firm 1’s profit nor total surplus changes from those in Section 3.3, but firm
2’s profit is smaller, which implies that consumer surplus is larger.
Proposition 5 Suppose firm 2 uses the shared data for third-degree price discrimination and (10) holds under the optimal data sharing under personalized pricing. Then,compared to the case where firm 2 can use the shared data for personalized pricing, firm 1’s optimal data sharing does not change,
 firm 1’s profit under the optimal data sharing does not change but firm 2’s profit

decreases; total surplus does not change but consumer surplus increases.

We can show that condition (10) is satisfied at (a∗ , b∗ ) for many distributions satisfying the monotone hazard rate condition. For example, (10) holds for the uniform
distribution, power distribution, triangular distribution when the peak point of f (x)is to the right of x = 0.56, beta distribution with increasing f (x), U-shaped f (x),and unimodal f (x) when α − β is positive and relatively large. Thus the following
corollary is immediate.
Corollary Suppose F is a uniform distribution and firm 2 uses the shared data forthird-degree price discrimination. Then, firm 1’s optimal choice of [a, b] is the sameas in Proposition 2, with the only difference that consumer surplus is higher underthird-degree price discrimination.
However, (10) may not hold at (a∗ , b∗ ) when f (x) has a very thin right tail. Evenin this case, we can still show that there exists b ∈ (z N , 1) such that sharing dataon [z N , b] is mutually beneficial, similar to Proposition 1. But we would need tosolve numerically for the optimal data sharing for specific distribution functions, as
20

in Section 3.2. In Figure 2, we plot the optimal data sharing under third-degree pricediscrimination for various beta distributions with α = 2, which we label (a∗∗ , b∗∗ ).
Given α = 2, condition (10) is satisfied at (a∗ , b∗ ) for all β ≤ 1.9. Thus, we have(a∗∗ , b∗∗ ) = (a∗ , b∗ ) for α = 2 and β ≤ 1.9. But when β > 1.9, condition (10) failsat (a∗ , b∗ ) and, as shown in Figure 2, we now have a∗ < a∗∗ < z N < b∗∗ < b∗ . Thus,
compared to the case with personalized pricing, firm 1 shares less data.
— Insert Figure 2 about here. —4.3

Customer data as an entry barrier

Customer data can be a source of a firm’s competitive advantage if it is utilized
for data-enabled learning. Moreover, customer data is often an important input indigital markets and, therefore, can act as a barrier to entry. A data-rich incumbent
can cement its position by offering better-targeted products thanks to data-enabledlearning, which attracts more customers and more customer data, hence creating aself-enforcing loop. This has led to active policy discussions related to data mobility
and data openness.20 Our analysis sheds light on this by clarifying to what extent
customer data can work as an entry barrier.
We modify our model as follows. The game is played over two periods, indexedby t = 1, 2. Firm 1, an incumbent, has data on all consumers in both periods whilefirm 2, a potential entrant, has access to customer data depending on how the gameevolves. Firm 1’s location is fixed at point zero in both periods and firm 2’s location
upon entry is fixed at point one.21 In t = 1, firm 2 makes an entry decision, followedby firm 1’s data sharing decision.22 There is a fixed entry cost E > 0. If firm 2enters, then the two firms compete as in our baseline model. At the end of t = 1,
firm 2 collects data on customers it serves in addition to the data shared by firm 1,
if any. In t = 2, following firm 2’s entry, the two firms compete using the customer
data they have for price discrimination. We focus on the case where F is a uniform
20

See Crémer et al. (“EU Report”, 2019), Furman et al. (“Furman Report”, 2019), or the
ACCESS Act in the US that passed through the House Committee on June 24, 2021.
21
This is the same as setup as in Gehrig et al. (2011), whose focus is on firm 2’s entry decision
when firm 1 can price discriminate and there are consumer switching costs. Our focus is on how
data sharing affects firm 2’s entry decision.
22
One may consider an alternative timeline in which firm 1 makes and announces its data sharing
decision first, after which firm 2 makes an entry decision. This timeline can lead to time inconsistency
whereby firm 1’s announcement not to share data becomes not credible once firm 2 enters.

21

distribution, which facilitates clear discussions. To simplify notation, we assume awaytime discount. In what follows, we provide heuristic discussions only with the detailed
analysis shown in the proof of Proposition 6.
To begin with, suppose firm 1 does not have any customer data. Then, upon firm
2’s entry, the two firms compete as in standard behavior-based price discrimination
with personalized pricing (Choe et al. 2018): they compete à la Hotelling in t = 1
and, in t = 2, customer data collected in t = 1 is used for price discrimination. As
shown by Choe et al. (2018), the game has two pure-strategy equilibria, one being amirror image of the other, from which we select the one that favors firm 1.23 In thatequilibrium, firm 1’s total profit over two periods is Π1 = 109/169 ≈ 0.645 and firm
2’s total profit is Π2 = 383/676 ≈ 0.572. Thus firm 2 enters if E < 0.572.
Suppose now firm 1 has data on all consumers. Without data sharing, the two
firms compete in t = 1 as in our baseline model, leading to the marginal consumer z N .
In t = 2, firm 2 chooses a uniform price on [0, z N ] and personalized prices on [z N , 1].
Solving the two-period model gives us z N = 7/10, Π1 = 17/20, Π2 = 7/20. Recall
that, in the benchmark one-period model without data sharing, we have z N = 3/4.
Thus, firm 2’s market share in t = 1 is larger in the two-period model. It is because
firm 2 becomes more aggressive in t = 1 when it anticipates subsequent competition
in t = 2: securing a larger market share in t = 1 enables firm 2 to extract more
surplus through personalized pricing in t = 2. Indeed, without data sharing, firm 2’s
uniform price in t = 1 is 1/2 in the one-period model, but it is 2/5 in the two-period
model. Compared to the case where firm 1 does not have any customer data, we cansay that data plays a role as an entry barrier when 0.35 < E < 0.572. But this ignores
firm 1’s incentives to share data once firm 2 enters. The possibility of beneficial data
sharing mitigates the extent to which customer data can be an effective entry barrier.
We discuss this below.
Suppose firm 2 enters and firm 1 chooses to share data on [a, b]. Then, in t = 2,
firm 2 has access to data on [a, b] as well as on customers it serves in t = 1. Solving
the two-period model shows that the optimal data sharing is [a∗ , b∗ ] = [0.710, 0.943]
with each firm’s total profit given by Π1 = 1.201, Π2 = 0.431. Recall that, in the oneperiod model where F is a uniform distribution, we have [a∗ , b∗ ] = [0.71, 0.97]. Thus,
firm 1 shares less data in the two-period model, in particular, by withholding data on
consumers who are most loyal to firm 2. It is because dynamic considerations make23

For example, firm 1 may choose its uniform price in t = 1 anticipating firm 2’s entry.

22

firm 2 more aggressive in t = 1 as discussed above, which firm 1 counters by sharing
less data. Nonetheless, firm 1’s optimal data sharing is qualitatively the same as in
the one-period model in that the data on firm 2’s most loyal customers is withheld.
Both firms benefit from the optimal data sharing and firm 2 enters if E < 0.431. Note
that, without firm 1’s optimal data sharing, firm 2 enters if E < 0.35. We summarize
these discussions below.
Proposition 6 Suppose firm 2 makes a decision to enter the market at a fixed entry
cost E, after which firm 1 makes a decision to share data with firm 2.
 If firm 1 does not have customer data, then firm 2 enters if E < 0.572.
 If firm 1 is fully informed but data sharing is not possible, then firm 2 enters if

E < 0.35. Thus, data acts as an entry barrier when 0.35 < E < 0.572.
 If firm 1 is fully informed and data sharing is possible, then firm 2 enters if

E < 0.431 and firm 1 shares data on [0.710, 0.943]. Thus, data acts as an entry
barrier when 0.431 < E < 0.572, and data sharing mitigates data’s role as anentry barrier when 0.35 < E < 0.431.
Proof: See the online appendix.

4.4

When firm 1 charges a fee for sales

Many data-rich firms such as Alibaba or Amazon are digital platforms that operate
in a dual mode: they run marketplaces for third-party sellers while selling their own
products on the marketplaces.24 The platform typically charges a fee for sales tothird-party sellers. We extend our baseline model to consider a simple case wherefirm 1 charges a fixed per-unit sales fee to firm 2.25
Suppose firm 1 charges a per-unit fee k ≥ 0. This induces firm 1 to price lessaggressively for consumers loyal to firm 2. In the absence of sales fee, Bertrand
competition for each consumer implies that firm 1 will choose zero personalized pricefor consumers more loyal to firm 2. Instead, firm 1 can cede these consumers to firm
24

See Hagiu et al. (2022) for a general analysis of dual-mode platforms as well as many examples.
Many platforms use a two-part fee structure that consists of fixed fee per item sold plus variable
fee that is proportional to the sale amount. Examples include Amazon, eBay, Shopify, etc. Incorporating such a fee structure to our model is beyond the scope of current paper, which we leave for
future work.
25

23

2 and earn k per each consumer. Given this, firm 2 will raise its price by shifting thesales fee entirely onto its customers. It is because consumers have inelastic demand
insofar as firm 1 does not undercut firm 2. Since firm 1 can collect k from firm 2, it
has no incentive to undercut firm 2 when firm 2 shifts entire k onto its customers.
In addition, firm 2’s higher uniform price allows firm 1 to increase its personalized
prices. In short, the sales fee softens competition and raises all prices by k, which isthen collected by firm 1. Other than that, the sales fee does not change equilibrium
market shares, nor firm 1’s optimal data sharing. We formalize this below.
Let us start with the case without data sharing. The marginal consumer z is givenby p1 (z) + z = p2 + (1 − z). Since firm 1 should be indifferent between earning p1 (z)from serving consumer z, or ceding consumer z to firm 2 and earning a fee k instead,
we must have p1 (z) = k. From these two equalities, we obtain p2 = 2z − 1 + k.
Plugging this into firm 2’s profit, we have
π2 = (p2 − k) (1 − F (z)) = (2z − 1) (1 − F (z)) .
Observe that the above profit is exactly the same as firm 2’s profit in our baselinemodel without data sharing. Thus, z N remains the same, hence so does each firm’s
market share and firm 2’s profit. The only change for firm 2 is that its uniform priceincreases by k to 2z N − 1 + k. But firm 1 is better off because it can charge higher
personalized prices to all customers it serves and also earns a sales fee k(1 − F (z N )).
Specifically, firm 1’s personalized price is p1 (x) = p2 + (1 − 2x) = 2z N − 2x + k, hence
increases by k compared to the case without the sales fee. Thus, firm 1’s profit is
Z zN

N

N

Z zN

(2z − 1 + k)dF (x) + k(1 − F (z )) =π1 =0(2z N − 2x)dF (x) + k.

(12)

0

In the right hand side of the second equality in (12), the first term is firm 1’s profit in
the baseline model without data sharing. Thus, the sales fee has the effect of charging
a price premium k to every consumer, which becomes additional profit for firm 1.
Suppose now firm 1 shares data on [a, b]. It is easy to see that beneficial data
sharing should continue to have a > 1/2 and b > z N . For consumer x ∈ [a, b],firm 1’s indifference condition is p1 (x) = k and consumer x’s indifference condition is
p1 (x) = p2 (x) + (1 − 2x). From these two, we obtain p2 (x) = 2x − 1 + k, and firm24

2 serves all consumers on [a, b].26 For consumers outside [a, b], firm 2’s uniform pricecan be chosen to serve only [b, 1] or additional consumers to the left of a. As in ourbaseline model, firm 1’s optimal data sharing hinges on which uniform price firm 2should be induced to choose.
First, if firm 2 chooses uniform price to serve only [b, 1], then its uniform price is
p21 = 2b − 1 + k. In this case, firm 1’s profit, denoted by π11 , is
Z a

Z a

(2b − 2x)dF (x) + k,

(2b − 2x + k)dF (x) + k(1 − F (a)) =

π11 =

0

0

where the first term on the right hand side of the second equality is exactly thesame as firm 1’s profit given in (8). Second, if firm 2 chooses uniform price to serve[z ′ , a] ∪ [b, 1] for some z ′ ≤ a, then its uniform price is p22 = 2z ′ − 1 + k. Then firm 1’s
R z′
profit is π12 = 0 (2z ′ − 2x)dF (x) + k. Subtracting k from each, we have exactly thesame profit for each case as in Section 3.2. In addition, one can easily check that firm
2’s profit in each case is also exactly the same as that in Section 3.2. Thus, Lemma
2 applies and firm 2’s (IC) is the same as before. Since the addition of constant termk to firm 1’s profit does not change its optimization problem, it follows that firm 1’s
optimal data sharing remains the same as in our baseline model. We summarize theresults in Proposition 7.
Proposition 7 Suppose firm 1 charges firm 2 a per-unit fee for sales k ≥ 0.
 In the equilibrium with or without data sharing, each firm’s market share andfirm 2’s profit are the same as those without the sales fee, but all prices andfirm 1’s profit increase by k.
 Firm 1’s optimal data sharing is the same with or without the sales fee.

4.5

Data sharing with side-payment

We have so far focused on firm 1’s incentive to voluntarily share data. We abstracted away the possibility of side-payment associated with data sharing in orderto highlight the strategic benefits of data sharing. However, one may wonder how theoutcome may change if firm 1 can charge for its shared data. We consider this casein this section. For simplicity, we assume that firm 1 makes a take-it-or-leave-it offer
26

We assume consumers on [a, b] choose firm 2 in case of indifference.

25

of a lump-sum, non-negative fee for the data, denoted by Φ. The timing is as follows:
firm 1 chooses Φ and [a, b]; firm 2 accepts or rejects firm 1’s offer; if firm 2 accepts
firm 1’s offer, then firm 2 pays Φ to firm 1 in exchange for the data on [a, b]; the two
firms compete in price. The rest of the model remains the same as before, including
the use of data for personalized pricing.
An immediate observation is that any mutually beneficial data sharing in theabsence of side-payments continues to be mutually beneficial when side-payments areallowed. Firm 1 can simply share the same amount of data and use Φ to extract full
additional profit from firm 2. That is, if sharing data on [a, b] is mutually beneficial
with corresponding profits π1 (a, b) ≥ π1N and π2 (a, b) ≥ π2N , then firm 1 can continue
to share data on [a, b] and choose Φ = π2 (a, b) − π2N . Thus, we expect firm 1 to bebetter off when side-payments are allowed.
A natural question is then whether firm 1 shares more data when side-payments
are allowed. Intuition tells us it should because, given the extra tool to extract
surplus from firm 2, firm 1 would be less stringent in sharing data. Since firm 1 canuse Φ to extract any profit from firm 2, i.e., Φ = π2 (a, b) − π2N , we can focus on data
sharing that maximizes industry profit. That is, firm 1 chooses (a, b) to maximize
π1 (a, b) + π2 (a, b) subject to Φ = π2 (a, b) − π2N ≥ 0.
Suppose the optimal data sharing without side-payment is given by [a∗ , b∗ ]. Suppose now firm 1 shares additional data on [a′ , a∗ ] when side-payment is possible wherea′ = a∗ − ϵ for arbitrarily small ϵ. Note that firm 2 chooses its uniform price to serve
only [b∗ , 1] given data sharing on [a∗ , b∗ ]. Since a′ < a∗ , it continues to do so when
additional data on [a′ , a∗ ] is shared. Thus, the only change in industry profit is from[a′ , a∗ ]. If it increases, then firm 1 shares more data when side-payment is allowed.
In the absence of side-payment, consumer a′ is served by firm 1 for profit 2b∗ − 2a′ .
After additional data sharing, she is served by firm 2 for profit 2a′ −1. Thus, industry
profit from [a′ , a∗ ] increases if a′ > (2b∗ + 1)/4 or, given that ϵ is arbitrarily small,
if a∗ > (2b∗ + 1)/4. In sum, if a∗ > (2b∗ + 1)/4 at the optimal data sharing [a∗ , b∗ ]without side-payment, then firm 1 can increase its profit by sharing more data on[a′ , a∗ ] when side-payment is allowed. When a∗ ≤ (2b∗ + 1)/4, additional data sharing
on [a′ , a∗ ] needs to be accompanied by additional data sharing on [b∗ , b′ ] so that firm
2’s uniform price increases. In this case, the optimal data sharing can be derived
only numerically. For example, when F is uniform, the optimal data sharing withside-payment is [a, b] = [0.66, 0.984]. Note that, when F is uniform, the optimal data
26

sharing without side-payment is [a∗ , b∗ ] = [0.71, 0.97]. Thus, we have a∗ ≤ (2b∗ + 1)/4
so that both a∗ and b∗ need to be adjusted in a way firm 1 shares more data whenside-payment is allowed. To summarize, firm 1 shares more data and enjoys larger
profit when side-payment is allowed.

4.6

Data broker

A data-poor firm may be able to obtain data from sources other than the data-richfirm, for example, from data brokers. Data brokers’ optimal strategy of selling data tocompeting firms has been the focus of several recent studies (e.g., Montes et al., 2019;
Bounie et al., 2021). Since firm 1 in our model is fully informed, the data broker cansell data only to firm 2, but this can potentially change firm 1’s data-sharing strategy.
We discuss this below, assuming F is a uniform distribution.
Suppose that a data broker can sell the full set of customer data to firm 2 at
a fixed price P . If firm 2 purchases data from the data broker, then the two firms
compete in personalized prices as in Thisse and Vives (1988), resulting in the grossprofit of 1/4 for each firm. Firm 2’s net profit is 1/4 − P . Recall that, in thebaseline model with the optimal data sharing, the equilibrium profits are π1∗ = 0.87
and π2∗ = 0.21. Thus, the optimal data sharing in the baseline model continues to be
optimal if P ≥ 0.25 − 0.21 = 0.04. However, if P < 0.04, then firm 2 has incentives
to purchase data from the data broker, which decreases firm 1’s profit from 0.87 to
0.25. Thus, firm 1 needs to adjust its data-sharing strategy to leave more surplus tofirm 2 in order to avoid the tough competition in personalized prices. This leads toan expansion of [a∗ , b∗ ] relative to the baseline model. For each value of P < 0.04, theoptimal data sharing can be solved numerically. For the case where F is a uniform
distribution, one can check that, as P decreases from 0.04 to 0, firm 1 shares moredata by decreasing a∗ and increasing b∗ .27

4.7

Partially-covered markets

In the baseline model, we assumed v is large enough so that the market is fully
covered. We now discuss the case where the market is not fully covered even when
it is served by both firms, which can lead to quite different outcomes when firms
compete in personalized pricing (Rhodes and Zhou, 2022). Suppose v < z N where
27

The calculation is available from the authors.

27

z N is the marginal consumer in the benchmark without data sharing. Then, withoutNdata sharing, firm 2 serves [z N , 1] with uniform price pN
2 = 2/h(z ) as shown in(1), and firm 1 serves [0, v] with personalized price p1 (x) = v − x for all x ∈ [0, v].
Consequently, consumers on [v, z N ] are not served by either firm, and the market is
not fully covered without data sharing if v < z N .
In this case, the uncovered segment [v, z N ] separates the two firms, which makes
them local monopolies. Thus, they do not compete with each other and their prices
are no longer strategic complements. It then follows that firm 1 does not benefit whenfirm 2 increases its uniform price, implying that mutually beneficial data sharing does
not exist. Nonetheless, data sharing can benefit firm 2 and increase total surplus byenabling firm 2 to serve consumers who would not be served without data sharing.
For example, suppose v = 1/2 and F is uniform so that z N = 3/4. Then, when firm 1
shares data on [1/2, 3/4], firm 2 can profitably serve all consumers on [1/2, 3/4] withpersonalized price p2 (x) = x − 1/2 ≥ 0. This increases firm 2’s profit but does notchange consumer surplus, hence total surplus increases. Although firm 1’s profit does
not change in this case, firm 1 can also benefit from data sharing if it can charge firm
2 for data sharing.

4.8

When firm 1 cannot price discriminate

The main insight that unilateral data sharing can be mutually beneficial hinges
on firm 2’s ability to price discriminate after data sharing. It does not depend on
firm 1’s ability to price discriminate. We now show that the same insight continues tohold even when firm 1 charges a uniform price. Data sharing allows firm 2 to delink
its pricing decisions on the segment with shared data and the rest of the market, thelatter in competition with firm 1. As in our baseline model, suitably chosen data
sharing can soften competition on the second segment, which allows firm 1 to raise
its uniform price without losing its market share. For simplicity, our discussion below
is based on the case where F is uniform.
Suppose firm 1 is fully informed but cannot price discriminate, for example, due
to lack of data analytics necessary for price discrimination. Firm 2 does not have any
data but has access to data analytics, and can exercise personalized pricing after data
sharing. In the benchmark without data sharing, the symmetric Hotelling equilibrium
prices are p1 = p2 = 1, firm 1 serves [0, 1/2], and each firm earns profit 1/2.

28

Suppose now firm 1 shares data on [1/2, 3/5]. We will show that this benefits
both firms. Denote firm 1’s uniform price by p′1 and firm 2’s uniform price p′2 . Afterdata sharing, firm 2 will serve [1/2, 3/5] with personalized prices, hence Hotellingcompetition is only on [0, 1/2]∪[3/5, 1]. Suppose p′1 = 1+ϵ for sufficiently small ϵ > 0.
Firm 2 can undercut firm 1 slightly by choosing p′2 < 1 + ϵ to serve [γ, 1/2] ∪ [3/5, 1]
where γ is sufficiently close to 1/2 given that ϵ is sufficiently small. Or it can choosep′2 = p′1 + 1/5 = 6/5 + ϵ which makes consumer x = 3/5 indifferent, hence firm 2serves only [3/5, 1]. Given that ϵ is sufficiently small, firm 2’s profit from its uniform
price is higher from the second choice. In addition, firm 2 can charge personalized
price p2 (x) = p′1 + (2x − 1) > 1 for all consumers on [1/2, 3/5]. Thus, firm 2 is better
off after data sharing. Firm 1 is also better off because it serves the same set of
consumers but charges a higher price.28 The discussion above shows that our main
insight continues to hold even when firm 1 cannot price discriminate; what matters
is firm 2’s ability to price discriminate after data sharing.

5

Discussions and Implications for Management

5.1

B2B data sharing in practice

Although data sharing is mandated in a few industries such as banking and energy,29 the B2B data sharing considered in this paper is growing in many other industries. Based on a survey conducted during 2017-2018 in 31 countries in the European
Economic Area, Arnaut et al. (2018a) reports examples of B2B data sharing in various industries such as automotive, transport and logistics, agriculture, telecom, etc.
The survey finds that data sharing can take different forms ranging from unilateral
to more collaborate approaches, and that most B2B data sharing occurs within theirown business sector.30 Particularly relevant to our model, the survey finds that amajority of participating companies choose to share a small proportion of data they28

In the online appendix, we provide a more formal argument for when data sharing on [a, b] canbe mutually beneficial by solving for equilibrium prices given [a, b].
29
In Europe, sector-specific data sharing is imposed in banking and finance, automotive, energy,
electronic communications and postal services. See Feasey and de Streel (2020) for details.
30
The type and frequency of data shared and the mechanism for data sharing vary depending on
businesses and industries. For example, Green Energy Options (geo), a UK-based supplier of energy
monitoring devices, shares data on energy use and efficiency on a real-time basis, by relying on FTP
technologies or online data repositories. For more specific examples, see Arnaut et al. (2018b).

29

generate, rather than their complete datasets, depending on their business strategies.
But the survey does not distinguish between non-personal data and personal data, thelatter being the main type of data that can be used to facilitate price discrimination.
More specific examples of unilateral data sharing with competitors can be foundin the practices of dual-mode online platforms such as Alibaba, Amazon, JD.com,
Target, Walmart, etc. They run a marketplace for third-party sellers but also sell
their own in-house products on the marketplace, thus competing with third-party
sellers (Hagiu et al. 2022). These platforms use their advantage in collecting andanalyzing consumer data to help third-party sellers to personalize their offers to customers. For example, Amazon provides third-party sellers with some raw data on
consumer identification and consumers’ transaction details, together with the aggregated data on business performance, consumer behaviour and market trends (PPMI,
2020). Moreover, Amazon helps third-party sellers on its platform, some of whom arecompetitors to Amazon’s in-house brands, to target their coupons to specific groups
such as Amazon Prime customers, mothers, students, customers who have viewed orpurchased from certain Amazon Standard Identification Numbers, etc. As we show
in this paper, this type of unilateral data sharing can benefit both Amazon and its
third-party sellers, irrespective of how third-party sellers personalize their offers using
the shared data and whether Amazon charges fees for their platform service.
Chinese e-commerce giants, Alibaba and JD.com recently started offering free data
analytics services to sellers on their platforms through their initiatives, called the A100
program and the Zu Chongzhi platform, respectively. They profile customers by collecting and analyzing their data about gender, age, occupation, location, purchasehistory, keywords used in search, payment method, brand and product preferences,
etc. Alibaba leverages its customer data to help its online sellers to design products
and marketing strategies in order to target specific consumer segments, but goes onfurther to provide a promotional tool that can be used to implement price discrimination (Zhang et al., 2020). In this sense, Alibaba shares data with sellers on its
platform not only directly but also indirectly through data analytics and other promotional tools. Some sellers can even improve the targetability of their marketing
campaign in their brick-and-mortar shops. For instance, Bestore, a Chinese snack
food chain, encourages shoppers to save their facial data when they pay with Alibaba’s face-scanning payment tablets in its stores, and crafts campaigns to target

30

specific niches of customers by utilizing data shared by Alibaba.31 At the same time,
Alibaba and JD.com compete with these sellers in the offline market by investingheavily in their own brick-and-mortar stores in recent years. Although more details
about the data shared by these platforms are not available, it is conceivable that theseplatforms choose what to share and how in a way that is mutually beneficial so thatsellers choose to remain on their platforms.

5.2

Implications for management

The main takeaway from our analysis is that strategic use of customer data can be
an example of a fat-cat strategy. A data-rich firm can choose a proportion of customer
data to share with a data-poor competitor, which can then soften price competition.
In doing this, two general conditions need to be met. First, shared data should be on
the segment of the market that is likely to be more loyal to the competitor, but thedata on the competitor’s most loyal segment needs to be withheld. This condition
does not seem hard to satisfy given the prevalent use of various market segmentation
strategies. Second, the primary use of the data is for price discrimination. Because
our logic holds whether shared data is used for personalized pricing or third-degree
price discrimination, the latter ubiquitous in practice, we believe the second condition
is not hard to satisfy either.
While the strategic benefits of data sharing described above are clear, the question
remains whether such data sharing may not infringe competition laws. For example,
Article 101(1) of the Treaty on the Functioning of the European Union (TFEU)
can limit data sharing among competitors if it restricts competition. An exception
justifiable under Article 101(3) is when such data sharing can have beneficial effects
that outweigh anticompetitive effects (Feasey and de Streel, 2020, pp. 40-41). As ouranalysis has shown, the welfare effects of data sharing depend on market conditions
and the nature of competition. First, if the data-rich firm increases its market sharefurther as a result of data sharing, then both consumer surplus and total surplus
decrease. This is true whether competition is in personalized pricing or in third-degree
price discrimination. Second, if data sharing results in the data-rich firm concedingsome of its market share to the competitor, then total surplus increases thanks to
the improved quality of consumer-firm matching. The effect on consumer surplus
31

https://www.reuters.com/article/us-china-tech-retail-idUSKCN1TL09D

31

depends on the nature of competition in this case: consumer surplus decreases whencompetition is in personalized pricing, but the effect on consumer surplus is ambiguouswhen competition is in third-degree price discrimination, depending further on market
conditions. In view of these implications, one may say that regulators do not have a
prima facie case against the kind of data sharing studied in this paper.
A necessary condition for implementing the data sharing strategy described inthis paper is the availability of data analytics. Possessing a large amount of raw,
unstructured data in itself would not enable the data-rich firm to identify and selectwhich data to share with its competitors and how. The firm also needs capabilities toprocess and analyze the data. Thus, the investment costs in data analytics may work
as a technology barrier in implementing the optimal data sharing strategy studied inthis paper. However, as rapid advancements in digital technologies and computational
sciences reduce the costs significantly and bring down the technology barrier, thestrategic use of data sharing identified in this paper will gain more practical relevance.

6

Conclusion

We have shown when a data-rich firm can benefit by unilaterally sharing its customer data with a competitor when the data can be used for price discrimination.
Our main point is that the firm can soften competition by sharing data on consumers
who are more loyal to the competitor than to the firm itself, but withholding the dataon the competitor’s most loyal consumers. This induces the competitor to choose ahigh non-discriminatory price for its most loyal consumers, which allows the sharing
firm to raise its own prices. Given the growing prevalence of data sharing amongfirms, our analysis offers an important insight into the strategic use of data sharing,and its implications for competition policy.
We discuss below some directions for future research. First, we can examine if
our insight continues to hold more generally. Although we have employed a stylized Hotelling model for expositional clarity, we expect our key insight to be valid
in oligopoly models with product differentiation where one firm has sufficient dataadvantage over its competitors. But a more general setup will allow us to study additional issues such as the number and identity of the competitors with which thedominant firm would like to share data.32 Second, given that unilateral data sharing
32

For example, the most straightforward extension of our model to the case with three firms is a32

is often used by dual-mode online platforms, we have considered the case where theplatform charges a fixed fee for sales. It would be interesting to extend our model tostudy dual-mode platforms by incorporating a more general fee structure platforms
use, and its impact on data sharing. Third, data does not only flow from platforms
to sellers, but the other way around. For example, Amazon is reported to have useddata from third-party sellers to develop its own products.33 A key in designing a
successful online marketplace is to gain a better understanding of the interaction ofthese data flows that go in opposite directions. Finally, the extent and effectivenessof data-sharing are constrained by consumers’ privacy choice such as opt-in decisions,which can be endogenized in further research.

Appendix
Proof of Lemma 1
Suppose firm 1 chooses to share data on [a′ , b′ ] ∪ [a, b] with a′ < b′ < a < b. We
prove that firm 1 prefers sharing data on [a, b] only, or [a′ , b′ ] only. The same argument
can be applied to any pairwise comparison and show that firm 1 prefers sharing dataon n intervals to sharing data on n + 1 intervals. Then, by transitivity, firm 1 prefers
sharing data on a single interval to any other types of data sharing.
Suppose firm 1 shares data on [a′ , b′ ] ∪ [a, b]. We start with two observations.
First, for firm 1 to prefer sharing data on [a′ , b′ ] ∪ [a, b] to that on [a, b], a necessary
condition is b′ > 1/2. If b′ ≤ 1/2, then sharing data on [a′ , b′ ] in addition to [a, b] does
not affect firm 2’s uniform price because [0, 1/2] is not contestable by firm 2, but it
decreases firm 1’s profit from [a′ , b′ ]. Second, without data sharing, firm 2’s profit is(2z − 1)(1 − F (z)) where z is the marginal consumer’s location, which is maximized
at z N . This implies that data sharing with b ≤ z N does not affect firm 2’s choice of
uniform price, hence cannot benefit firm 1. These observations lead us to focus on
the cases where b′ > 1/2 and b > z N .
unit-length linear city where fully-informed firm 1 is located at 1/2 and uninformed firm 2 is at 0,
and uninformed firm 3 is at 1. It is easy to see that our main insight continues to be valid in thiscase. But it is less obvious in other cases, for example, a circular city where each firm has a loyal
customer base on each side of its location. In the online appendix, we offer some discussions on data
sharing among three firms on a Salop circle.
33
https://www.wsj.com/articles/amazon-scooped-up-data-from-its-own-sellers-to-launchcompeting-products-11587650015

33

Let z ∗ be the marginal consumer’s location that maximizes firm 2’s profit, hence
firm 2’s uniform price is p∗2 = 2z ∗ − 1. There are four possibilities: (i) z ∗ = z1 = b; (ii)z ∗ = z2 ∈ (b′ , a); (iii) z ∗ = z3 = b′ ; (iv) z ∗ = z4 ≤ a′ . Note that, when data is shared
only on [a, b], the profit-maximizing marginal consumer’s location is either z1 or z2 .
In the first two cases, i.e., z ∗ = z1 or z2 , the same z ∗ is profit-maximizing forfirm 2 when only [a, b] is chosen for data sharing, leading to the same uniform price.
Then firm 1 is better off sharing data on only [a, b] because additional data sharing
on [a′ , b′ ] reduces its profit from that segment. In the fourth case, z ∗ = z4 , we havez4 < min{z1 , z2 }, hence firm 2’s uniform price is lower than when only [a, b] is chosen
for data sharing (i.e., 2z4 − 1 < min{2z1 − 1, 2z2 − 1}). Once again, firm 1 is betteroff sharing data on only [a, b]. This leaves us the third case where z ∗ = z3 = b′ . Givenb′ > 1/2, there are two possibilities. First, if b′ ∈ (1/2, z N ), then, as shown previously,
firm 2 will set the marginal consumer’s location at z N when data is shared only on[a′ , b′ ]. Thus firm 1 is better off sharing data only on [a′ , b′ ] because firm 2’s priceincreases from 2b′ − 1 to 2z N − 1. Second, if b′ ≥ z N , then firm 2 will continue to
set the marginal consumer’s location at b′ when data is shared only on [a′ , b′ ]. In thiscase, firm 1 is indifferent between sharing data on [a′ , b′ ] ∪ [a, b] and only [a′ , b′ ] asfirm 1 cannot sell to consumers on [a, b] in either case.
Proof of Lemma 2
It suffices to show that firm 1 cannot benefit from data sharing that induces firm
2 to choose p22 . For this, we will show z ′ < z N , hence p22 < pN
2 , which proves thatfirm 1 is worse off with data sharing in this case.
From the definition of z N , we have (1 − 2z N )h(z N ) + 2 = 0. Define g(x) :=2 + (1 − 2x)h(x). Then, since h′ (x) > 0 by the monotone hazard rate condition, we
have g ′ (x) = −2h(x) + (1 − 2x)h′ (x) < 0 for all x ≥ 1/2. Rewrite z ′ asz′ =1
F (b) − F (a)
1
1
1
+
−
<
+
.
2 h(z ′ )f (z ′ )2 h(z ′ )Thus we have g(z ′ ) = 2 + (1 − 2z ′ )h(z ′ ) > 0. Since g(z N ) = 0, g ′ (x) < 0 for allx ≥ 1/2, and z ′ > 1/2, we have z ′ < z N .
Proof of Proposition 1
Consider data sharing on [a, b] with b > a, which is at no loss of generality thanks
to Lemma 1. By Lemma 2, firm 1 can benefit from data sharing only if firm 2 uses34

its uniform price to serve only consumers on [b, 1] where b < 1. Given this, π1 (a, b)
is as given in (8) and π2 (a, b) is as given in (7). We first show π1 (a, b) > π1N andNN
π2 (a, b) > π2N for all 1 > b > a = z N . Since pN
2 = 2z − 1, we can rewrite π1 asRNz
π1N = 2z N F (z N ) − 2 0 xdF (x). Then, we have
π1 (a, b) − π1N = 2 bF (a) − z N F (z N ) − 2

!

Z zN

Z a
xdF (x) −0

xdF (x)0= 2(b − z N )F (z N ) > 0 for all b > a = z N .

For firm 2, we haveZ b

(2x − 1)dF (x) + (2b − 1)(1 − F (b)) − (2z N − 1)(1 − F (z N ))
a
Z b
Z 1
Z 1
=
(2x − 1)dF (x) +(2b − 1)dF (x) −(2z N − 1)dF (x) if a = z N
N
N
z
z

Zb b
Z 1
Z 1
N
N
(2z − 1)dF (x) −(2z − 1)dF (x)>(2b − 1)dF (x) +
zN
zN
b
Z 1
(2b − 2z N )dF (x) > 0 for all b > z N .
=π2 (a, b) − π2N =b

We now show there exists b′ ∈ (z N , 1) such that data sharing on [z N , b′ ] satisfies
firm 2’s (IC). This, together with what we showed above, proves that sharing data on[z N , b′ ] benefits both firms. Fix a = z N so that we express π22 (z N , b) as a function of b
only, denoted by π̂22 (b) := π22 (z N , b). It is straightforward to check π21 (z N ) = π̂22 (z N )and π21 (1) < π̂22 (1). So mutually beneficial data sharing, if exists, must be for someb < 1. Recall that π21 (b) is maximized at b = z N by the definition of z N , hence
π21 (z N ) > π21 (z) for all z ̸= z N . But we also have π21 (z ′ ) = (2z ′ − 1)(1 − F (z ′ )) ≥π̂22 (b) = (2z ′ − 1)(1 − F (b) + F (z N ) − F (z ′ )). Put together, we haveπ̂22 (z N ) = π21 (z N ) > π21 (z ′ ) > π̂22 (b).
′Since π̂22 (z N ) > π̂22 (b) for any b > z N , we have π̂22(z N ) < 0. By the definition of z N ,′′we have π21(z N ) = 0. Since π̂22(z N ) < 0 and π21 (z N ) = π̂22 (z N ), there exists b′ > z N

35

such that π21 (b′ ) > π̂22 (b′ )., i.e., data sharing on [z N , b′ ] satisfies firm 2’s (IC).
Proof of Proposition 2
Given that z N = 3/4, we have S = {(a, b) | 1/2 < a < b, 3/4 < b}. It is easy toderive firm 2’s (IC) as follows:(2b − 1)(1 − b) ≥(1 + 2a − 2b)2.
8Since firm 2’s profit from setting personalized prices on [a, b] isa(1 − a) − b(1 − b), we can write firm 2’s (IR) as

(13)
Rb
a

(2x − 1)dx =1(2b − 1)(1 − b) + a(1 − a) − b(1 − b) ≥ π2N = .
8(14)Given the above, firm 1 serves consumers on [0, a] with personalized prices p1 (x) =Ra(2b − 1) + (1 − 2x) = 2(b − x), hence its profit is 0 2(b − x)dx = 2ab − a2 . Thus, firm
1’s problem can be stated as follows:
max 2ab − a2 , subject to (a, b) ∈ S, (13), and (14).
a,b

We consider the following two cases.
Case (i): a < 3/4. In this case, firm 1’s problem ismax 2ab − a2 , subject toa,b3
1< a < < b ≤ 1, (13), and (14).
2
4

We first solve the relaxed problem by considering only constraint (13), after which weverify that the other constraints are satisfied at the solution to the relaxed problem.
The Lagrangian is then
(1 + 2a − 2b)2
L = 2ab − a + λ (2b − 1)(1 − b) −
.
8
2

36

The first-order conditions areλ∂L= −a(λ + 2) + b(λ + 2) − = 0,
∂a

 2
∂L
7
= a(λ + 2) +− 5b λ = 0,∂b2
(1 + 2a − 2b)2λ (2b − 1)(1 − b) −= 0.
8
The above equation system yieldsa∗ ≃ 0.7077, b∗ ≃ 0.9699, λ∗ ≃ 0.2052,(15)

and the associated profit is π1∗ ≃ 0.8720. Clearly, the above solution satisfies bothconstraints, 21 < a∗ < 43 < b∗ ≤ 1 and (14). Therefore, a∗ and b∗ in (15) are indeed
the optimal solution for the case where a < 3/4.
Case (ii): 3/4 ≤ a. Firm 1’s problem in this case ismax 2ab − a2 , subject toa,b3≤ a < b ≤ 1, (13), and (14).
4Once again, we solve the relaxed problem by considering only the constraints 3/4 ≤ aand (13). We will verify the remaining constraints are satisfied at the solution to therelaxed problem. Then the Lagrangian is(1 + 2a − 2b)2
L = 2ab − a + λ (2b − 1)(1 − b) −
8
2






+ λ1

The first-order conditions are∂Lλ= −a(λ + 2) + b(λ + 2) − + λ1 ,
∂a

 2
∂L
7
= a(λ + 2) +− 5b λ,∂b2
(1 + 2a − 2b)2λ (2b − 1)(1 − b) −
= 0,
8


3
λ1 a −
= 0.
4

37

3
a−
4


.

The above equation system leads to319a∗ = , b∗ = , λ∗ = 3, λ∗1 = 0.5
4
20

(16)

and the associated profit is π1∗ ≃ 0.8625. Moreover, firm 2’s profit is (2b∗ −1)(1−b∗ )+37a∗ (1 − a∗ ) − b∗ (1 − b∗ ) = 200> 1/8, and all the remaining constraints are satisfied.
Comparing the two cases, we can conclude that firm 1 will choose a∗ ≃ 0.7077,b∗ ≃ 0.9699 and obtain profit π1∗ ≃ 0.8720. Firm 2’s profit is then π2∗ = 0.2059.
Proof of Proposition 3
It remains to show that consumer surplus decreases when a∗ < z N . We compare
the total costs consumers incur before and after data sharing. The total costs are thesum of prices paid and transportation costs. Thus the total costs for all consumers
before data sharing are
Z zN

π1N + π2N +Z 1(1 − x)dF (x).

(17)(1 − x)dF (x).

(18)

xdF (x) +zN

0After data sharing, the total costs are
π1∗ + π2∗ +

Z a∗

Z 1
xdF (x) +a∗0

Subtracting (17) from (18) and noting that a∗ < z N , we obtain(π1∗ − π1N ) + (π2∗ − π2N ) +Z zN(1 − 2x)dF (x)
Z 1
Z 1
Z b∗
∗
∗
N
(2z N − 1)dF (x) > 0.
= (π1 − π1 ) +(2x − 1)dF (x) +(2b − 1)dF (x) −
zN

a∗

b∗

zN

R1
R b∗
In the above, the equality follows from π2N = zN (2z N − 1)dF (x) and π2∗ = a∗ (2x −R1
R b∗1)dF (x) + b∗ (2b∗ − 1)dF (x), and the inequality holds because π1∗ ≥ π1N and zN (2x −R1
R11)dF (x)+ b∗ (2b∗ −1)dF (x) > zN (2z N −1)dF (x). This proves that consumer surplus
decreases after data sharing when a∗ < z N .

38

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42

Table 1: Comparison of outcomes with and without data sharing
PDF(α, β)zN[a∗ , b∗ ]π1∗ -π1N π2∗ -π2N CS ∗ -CS N T S ∗ -T S N
unimodalU-shapeddecreasing
increasing
uniform(2, 4)(0.9, 0.9)
(0.9, 2)(5, 1)(1, 1)

0.61
0.76
0.67
0.81
0.75

[0.50, 0.98][0.72, 0.97][0.50, 0.99][0.83, 0.96][0.71, 0.97]

+
+
+
+
+

+
+
+
+
+

–
–
–
–
–

+
+
+
–
+

Note: + indicates positive; − indicates negative; CS denotes consumer surplus; T S
denotes total surplus.

=1
0.95

b*

0.90
0.85

a*

0.80

zN

0.75
0.70

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Figure 1: Optimal [a∗ , b∗ ] under beta distribution

43

=2
1.0

b*
b* *

0.9
0.8

zN

0.7
0.6

a* *

a*
0.5
1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

Figure 2: Optimal [a, b] under beta distribution and third-degree price discrimination

44

Online Appendix
Proof of Proposition 4
We have already shown that firm 2’s (IC) is satisfied in the absence of search
discrimination. Thus, firm 2’s optimal pricing strategy is to choose p2 = 2b − 1 for
consumers on [b, 1] and p2 (x) = 2x − 1 for consumers on [a, b]. Given this, firm 1Rachooses [a, b] to maximize its profit π1 (a, b) = 0 (2b − 2x)dF (x) subject to firm 2’s
participation constraint:
Z b
π2 (a, b) = (2b − 1) (1 − F (b)) +(2x − 1)dF (x) ≥ π2N .

(A1)

a

We claim that b = 1 in the solution to firm 1’s problem. One can show that both
π1 (a, b) and π2 (a, b) are increasing in b. This implies that firm 1 can increase profit
by increasing b while satisfying (A1). Given b = 1, firm 1 then chooses a to satisfy
R1(A1) with equality: â solves â (2x − 1)dF (x) = π2N , hence â > z N .
Next, we show â > a∗ . Suppose a∗ ≥ â. Then we have
Z a∗

Z 1
(2x − 1)dF (x) =âZ b∗(2x − 1)dF (x) +Z 1(2x − 1)dF (x) +a∗â(2x − 1)dF (x)
b∗

Z b∗

(2x − 1)dF (x) + (2b∗ − 1) (1 − F (b∗ ))a∗≥ (2z N − 1) 1 − F (z N ) = π2N .
>This contradicts the definition of â.
Finally, we show that no search discrimination reduces consumer surplus and total
surplus. The effect on total surplus is clear because â > max{z N , a∗ }. Consider now
consumer surplus compared to the case without data sharing. All consumers whostay with the same firm are worse off because prices are higher. Consumer x ∈ [z N , â]switches to firm 1 after data sharing, hence incurs total cost p̂1 (x)+x ≥ 2−x; withoutNdata sharing, her total cost is pN
2 + (1 − x) = 2z − x. Because the former is larger,
she is worse off. Next is the case with data sharing and search discrimination. There
are four possible cases: (i) a∗ < z N < b∗ and z N < â < b∗ ; (ii) a∗ < z N < b∗ andb∗ < â; (iii) z N < a∗ < b∗ and a∗ < â < b∗ ; and (iv) z N < a∗ < b∗ and b∗ < â. We
prove our claim for case (i) only, as the other cases are similar. When a∗ < z N < b∗and z N < â < b∗ , all consumers on [0, a∗ ] ∪ [b∗ , 1] stay with the same firm, and pay
45higher prices under no search discrimination. So they are all worse off. Consumerx ∈ [a∗ , â] switches to firm 1 under no search discrimination. So her total cost changes
from p∗2 (x) + (1 − x) = 2x − 1 + (1 − x) = x to p̂1 (x) + x ≥ 2 − x. The latter is larger,
so she is worse off. Last, consumer x ∈ [â, b∗ ] stays with firm 2 and pays the same
personalized price.
Proof of Lemma 3
Claim 1: Under the optimal data sharing, we have q2l = 2a − 1, hence zl = a.
Denote firm 2’s profit from [a, b] by π̃21 when it serves the entire segment [a, b], henceπ̃21 = (2a − 1)(F (b) − F (a)), and by π̃22 when it serves a subset [zl , b] ⊂ [a, b], henceπ̃22 = (2zl − 1)(F (b) − F (zl )). Given b > z N , we need to consider only the following
two cases: (i) z N ≤ a < b and (ii) a < z N < b.
(i) Suppose z N ≤ a < b. Define G(x) := (2x − 1)(1 − F (x)). By the definitionof z N , we have G′ (z N ) = 0 and G is strictly concave due to the monotone hazard
rate condition. Thus we have G′ (x) ≤ 0 for all x ≥ z N . Define H(x) := (2x −1)(F (b) − F (x)), hence π̃21 = H(a) and π̃22 = H(zl ) where a ≤ zl . To show zl = a,it suffices to show H(a) ≥ H(zl ). It is easy to see H ′ (x) < G′ (x) for all x ∈ [z N , 1).
Since G′ (x) ≤ 0 for all x ≥ z N , we have H ′ (x) < 0 for all x ∈ [z N , 1). This proves
H(a) ≥ H(zl ), and hence zl = a.
(ii) Suppose a < z N < b. We will show that it is not optimal for firm 1 to sharedata that leads to zl > a. Suppose firm 2’s profit H(x) = (2x − 1)(F (b) − F (x)) ismaximized at zl > a. The analysis in case (i) above shows that H(x) is decreasing
for all x ≥ z N . Thus, zl > a implies that we must have zl < z N . Firm 1 serves
consumers on [a, zl ] with personalized prices p1 (x) = 2zl − 2x in this case. But firm
1 can increase its profit by sharing data on [zl , b] instead of [a, b]. By doing this, firm
1’s profit from [0, a] does not change, but its profit from [a, zl ] increases because its
personalized price on this segment increases from p1 (x) = 2zl − 2x to p1 (x) = 2b − 2x.
Thus, we must have zl = a.
Claim 2: Firm 1’s optimal data sharing should satisfy π21 (b) ≥ π22 (a, b).
There are two possible cases, z N ≤ a < b or a < z N < b. Since the proof is very
similar in both cases, we prove the claim only for the first case. Suppose z N ≤ a < b.
Firm 2’s problem on [a, b] has either a boundary solution where zl = a, or an interior
solution where a < zl < b. By Claim 1, we only need to consider the case wherezl = a. Suppose zl = a so that firm 2 serves all consumers on [a, b] with price46

q2l = 2a − 1 > pN
2 . If q2h = 2b − 1, then firm 1’s profit is π1 =′q2h = 2z − 1 for some z ′ < a, then firm 1’s profit isZ z′

0(2b − 2x)dF (x). If

Z a′(2b − 2x)dF (x).

(2z − 2x)dF (x) <

π1 =

Ra

0

0

Thus, firm 1 is better off when firm 2’s chooses q2h = 2b − 1 to serve [b, 1] only, i.e.,
firm 2’s (IC) needs to be satisfied.
Proof of Proposition 6(i) Case where firm 1 does not have customer data
We start with the case where firm 1 does not have any customer data. Suppose firm
2 enters the market. In t = 1, the marginal consumer is z = 1/2 + (p2 − p1 )/2 wherepi (i = 1, 2) is firm i’s uniform price in t = 1. At the end of t = 1, firm 1 collects dataon [0, z], called firm 1’s target segment, and firm 2 collects data on [z, 1], called firm
2’s target segment. In t = 2, firms charge personalized prices for targeted consumers
and a uniform price for the rest. This is a standard model of behavior-based pricediscrimination analyzed in Choe et al. (2018). There are two pure-strategy equilibria,
one being a mirror image of the other. Denote firm i’s uniform price in period 1 by p1i ,and its total profit over two periods by Πi , i = 1, 2. By substituting δc = δf = t = 1
into Proposition 1 in Choe et al. (2018), we obtain the following in the equilibrium
where firm 1’s total profit over two periods is larger than firm 2’s.1
p11 = 8/13, p12 = 10/13, z = 15/26, Π1 = 109/169, Π2 = 387/676.

(A2)

Thus, firm 2 enters the market if E < 387/676 ≈ 0.572.
(ii) Case where firm 1 is fully informed but data sharing is not possible
Suppose now firm 1 has data on all consumers. First, suppose that firm 1 does
not share data with firm 2. Let z N be the marginal consumer in t = 1 upon firm 2’s
entry. Denote firm i’s personalized price for consumer x in period t by pti (x), wheneverrelevant, and firm 2’s uniform price in period t by pt2 . In t = 1, firm 1 serves [0, z N ]with p11 (x) = 2z N − 2x for x ∈ [0, z N ] and firm 2 serves [z N , 1] with p12 = 2z N − 1. In
t = 2, firm 2 serves [z N , 1] with p22 (x) = 2x − 1, and serves additional consumers on
1

In the other equilibrium, we have p11 = 10/13, p12 = 8/13, z = 11/26, Π1 = 387/676, Π2 =109/169.

47

[z ′ , z N ] with p22 such that z ′ = (1 + p22 )/2. Maximizing firm 2’s t = 2 profit leads to
p22 = z N − 1/2, hence z ′ = z N /2 + 1/4 < z N where the inequality is due to z N > 1/2.
In t = 2, firm 1 serves [0, z ′ ] with p21 (x) = 2z ′ − 2x = z N + 1/2 − 2x. Thus, each firm’s
total profit over two periods can be written as follows:

ΠN
1 =

Z zN

Z zN /2+1/4N(2z − 2x)dx +
0

N
N
ΠN
2 = (2z − 1)(1 − z ) +

=

Z0 1

20(z N )2 + 4z N + 1(z + 1/2 − 2x)dx =,16N(2x − 1)dx + (z N − 1/2)(z N /2 − 1/4)

zN

−20(z N )2 + 28z N − 7.
8NMaximizing ΠNand the two firms’ equilibrium profits given by
2 , we find zNz N = 7/10, ΠN
1 = 17/20, Π2 = 7/20.

(A3)Thus, if data sharing is not possible, then firm 2 does not enter if E > 7/20.
Compared to the case where firm 1 does not have any customer data, we can say thatdata plays a role as an entry barrier when 0.35 < E < 0.572.
(iii) Case where firm 1 is fully informed and data sharing is possible
Suppose firm 1 shares customer data on [a, b] with firm 2. Denote firm i’s personalized price for consumer x in period t by pti (x), and firm 2’s uniform price in periodt by pt2 . Consider t = 1. As in our baseline model, firm 2 can use uniform price p12 toserve either [b, 1] only or additional consumers to the left of a, say, [z ′ , a]. We discusseach case below.
First, if firm 2 chooses p12 = 2b − 1 to serve only [b, 1] in t = 1, then its target
segment in t = 2 becomes [a, 1]. In t = 2, as shown previously, firm 2 chooses
p22 = a − 1/2 to serve [a/2 + 1/4, a] and charges p22 (x) = 2x − 1 to serve [a, 1]. Then48

each firm’s total profit over two periods can be written as follows:
Z a/2+1/4

Z a

1 − 12a2 + 4a + 32ab
(a + 1/2 − 2x)dx =
,
16
0
0
Z 1
Z b
(2x − 1)dx + (2b − 1)(1 − b) +(2x − 1)dx + (a − 1/2)[a − (a/2 + 1/4)]
Π2 =(2b − 2x)dx +

Π1 =

a

a

−12a2 − 8b2 + 12a + 16b − 7=.
8Second, if firm 2 chooses p12 = 2z ′ − 1 to serve [z ′ , a] ∪ [b, 1] in t = 1, then its target
segment in t = 1 becomes [z ′ , 1]. Analogous to the previous case, firm 2 chooses
p22 = z ′ − 1/2 to serve [z ′ /2 + 1/4, z ′ ] and charges p22 (x) = 2x − 1 to serve [z ′ , 1]. Firm
2’s total profit in this case is
Π′2 =

Z b

′

Z 1

′

(2x−1)dx+(2z −1)(1−b+a−z )+(2x−1)dx+(z ′ −1/2)[z ′ −(z ′ /2+1/4)].

z′aMaximizing the above profit, we obtain z ′ = (7 − 4(b − a))/10, with the resulting
equilibrium profit given by
−12a
Π′2 =

2

+ 7(2b − 1)2 − 16ab + 28a.
20

As in our baseline model, we can show that firm 1’s profitable data sharing must
induce firm 2 to choose its uniform price in t = 1 to serve only [b, 1]. To see this,
suppose data sharing on [a, b] induces firm 2 to choose its uniform price to serve[z ′ , a] ∪ [b, 1] in t = 1 for some z ′ < a. As shown in the proof of Lemma 2, we havez ′ < z N in this case. Then, it is straightforward to check that firm 1’s t = 1 profit
under data sharing is smaller than that without data sharing. In t = 2, firm 1’s profit
R z′ /2+1/4 ′R zN /2+1/4 N
is 0(z + 1/2 − 2x)dx under data sharing, and 0(z + 1/2 − 2x)dx.
′
N
The latter is larger because z < z .
Then, firm 1’s problem is to choose [a, b] to maximize Π1 subject to Π2 ≥ Π′2 and∗ ∗Π2 ≥ ΠN
2 = 7/20. Solving the problem gives us [a , b ] = [0.710, 0.943], with eachfirm’s total profit over two periods given byΠ∗1 = 1.201, Π∗2 = 0.431.

(A4)Given firm 1’s optimal data sharing, firm 2 enters if E < 0.431. Note that, without49

firm 1’s data sharing, firm 2 enters if E < 0.572. Thus, the possibility of data sharing
mitigates the role of customer data as an entry barrier when 0.35 < E < 0.431.
Analysis for Section 4.8 when firm 1 cannot price discriminate
We show when data sharing can be mutually beneficial even when firm 1 cannot
price discriminate and chooses a uniform price. Other than that, everything else is thesame as in the baseline model. That is, firm 1 is fully informed, firm 2 is uninformed,
and shared data can be used by firm 2 for personalized pricing.
In the benchmark without data sharing, we have the symmetric Hotelling equilibrium where each firm chooses a uniform price equal to 1, firm 1 serves [0, 1/2], andeach firm earns profit 1/2. Consider now the case where firm 1 shares data on [a, b]
for b < 1. We want to construct an equilibrium such that (i) firm 1 charges the local
monopoly’s price p1 for consumers on [0, a], (ii) firm 2 charges the local monopoly’s
price p2 for consumers on [b, 1], and (iii) firm 2 charges personalized prices for consumers on [a, b]. In what follows, we identify necessary and sufficient conditions forsuch an equilibrium.
Step 1. Let us first derive firm i’s local monopoly price. Firm 1’s profit is π1m =p1 min{v − p1 , a}. When v − p1 ≥ a, firm 1’s profit is π1m = p1 a. The optimal priceis p1 = v − a and firm 1 earns (v − a)a. When v − p1 ≤ a (i.e., p1 ≥ v − a), firm 1’s
profit is π1m = p1 (v − p1 ). In this case, firm 1’s optimal price is p1 = v − a iff a ≤ v/2
and is p1 = v/2 iff a ≥ v/2. In sum, firm 1’s optimal local monopoly price isp∗1 =v − a,if 2a ≤ v

v/2,

if 2a ≥ v.

(A5)

Firm 1’s profit is(v − a)a,π1∗ =v 2 /4,

if 2a ≤ v
if 2a ≥ v.

Similarly, we can derive firm 2’s optimal local monopoly price asp∗2 =v − (1 − b),if 2(1 − b) ≤ v

v/2,

if 2(1 − b) ≥ v.

50

(A6)

Firm 2’s profit from the uniform price isπ2∗ =(v − (1 − b))(1 − b),if 2(1 − b) ≤ v

v 2 /4,

if 2(1 − b) ≥ v.

From the above, there are four different cases for a given tuple (v, a, b), based on
the relationship among v, 2a, and 2(1 − b): (i) v ≥ max{2a, 2(1 − b)}, (ii) 2a ≤ v ≤2(1 − b), (iii) 2(1 − b) ≤ v ≤ 2a, (iv) v ≤ min{2a, 2(1 − b)}. In all but case (i), themarket is not fully covered. In what follows, we focus on case (i). Then, we havep∗1 = v − a and p∗2 = v − (1 − b).
Step 2. We check consumers’ purchase decisions are optimal. It is easy to checkthat consumers on [0, a] purchase from firm 1 as their utility from firm 2 is negative;
consumers on [b, 1] purchase from firm 2 as their utility from firm 1 is negative.
Consumer x ∈ [a, b] faces personalized pricep2 (x) = min{v − (1 − x), p∗1 + (2x − 1)} = v − (1 − x).

(A7)

All consumers on [a, b] choose firm 2 because their utility from firm 1 is negative.
Step 3. We now establish the conditions under which no firm has incentive to lower
its price to poach the rival’s customers. Let us start with firm 1. Under deviationp∗ −pdprice pd1 , the marginal consumer is x̂d = 12 + 2 2 1 . A profitable deviation requiresx̂d ≥ b, implying pd1 ≤ v − b. Firm 1 maximizes deviation profit π1d = pd1 (x̂d − b + a) =1 d
p (v+2a−b−pd1 ) subject to pd1 ≤ v−b. First, if (v+2a−b)/2 ≥ v−b, then the optimal
2 1
pd1 = v − b, implying x̂d = b. But such a deviation is never profitable because firm 1
lowers its price without gaining any extra demand. Next, if (v+2a−b)/2 ≤ v−b, thenthe optimal pd1 = (v + 2a − b)/2 and firm 1’s deviation profit is π1d = (v + 2a − b)2 /8.
Therefore, firm 1 has no incentive to poach firm 2’s customers if and only if(v +2a−b)/2 ≥ v −b or (v +2a−b)/2 ≤ v −b and (v +2a−b)2 /8 ≤ (v −a)a. (A8)
Similarly, we can show that firm 2 has no incentive to poach firm 1’s customers if
and only if
v + a + 2b ≤ 3 or v + a + 2b ≥ 3 and (v + a − 2b + 1)2 /8 ≤ (v − (1 − b))(1 − b). (A9)51

Step 4: Each firm should earn higher profit under data sharing than in the benchmark without data sharing. These conditions are
π1∗ = (v − a)a ≥ 1/2 = π1N ,
π1∗ = (v − (1 − b))(1 − b) +

(A10)
Z b

(v − (1 − x))dx ≥ 1/2 = π2N .

(A11)

a

Step 5. If we can find a tuple (v, a, b) such that 2a ≤ v, 2(1 − b) ≤ v, (A8), (A9),(A10), and (A11) hold, then we can say (A5), (A6), and (A7) give us equilibrium
prices and the data sharing [a, b] is mutually beneficial. From the above, we can identify a set of (v, a, b) satisfying the following: (i) 1.45 ≤ v ≤ 2.28 (large v incentivizes afirm to poach the rival’s customers, as shown by (A8) and (A9)); (ii) 0.26 ≤ a ≤ 0.58;(iii) 0.46 ≤ b ≤ 0.84; (iv) b < 1/2 is possible when v ∈ [1.52, 1.86]; (v) b cannot be
too close to 1 because, otherwise, firm 2 has incentives to poach firm 1’s customers,
as shown in (A9).
For example, when v = 1.8, data sharing on (a, b) = (0.5, 0.6) is mutually beneficial
with equilibrium uniform prices given by p∗1 = 1.3 and p∗2 = 1.4. In Figure 3, weplot the set of mutually beneficial data sharing (a, b) when v = 2. An example is(a, b) = (0.45, 0.6), which leads to p∗1 = 1.55 and p∗2 = 1.6. Since π1∗ = (v − a)a which
increases in a for all a ≤ v/2 = 1 when v = 2, firm 1’s optimal data sharing is theone with the largest value of a in the shaded region in Figure 3.

Figure 3: Mutually beneficial data sharing when v = 252

Data sharing on a Salop circle
Consider a unit-perimeter Salop circle where three firms are located equidistantfrom one another and F is a uniform distribution. We label firm 1’s location 0 (aswell as 1), firm 2’s location 1/3, and firm 3’s location 2/3. Firm 1 is fully informed
and firms 2 and 3 are uninformed. The rest is the same as in our baseline model.
1. Benchmark without data sharing
Let z1 denote the marginal consumer’s location on [0, 1/3] where relevant competition is between firms 1 and 2. Then we havep1 (z1 ) + z1 = p2 + (1/3 − z1 ) .
Since p1 (z1 ) = 0, we have z1 = p2 /2+1/6. Let 1/3+z2 denote the marginal consumer’s
location on [1/3, 2/3] where relevant competition is between firms 2 and 3, and firm
1’s personalized price is zero for all consumers on this segment. Then we have
p2 + z2 = p3 + (1/3 − z2 ) .
This gives us z2 = (p3 − p2 )/2 + 1/6. Let 2/3 + z3 denote the marginal consumer’s
location on [2/3, 1] where relevant competition is between firms 1 and 3. Then wehave
p3 + z3 = p1 (z3 ) + (1/3 − z3 ) .
Since p1 (z3 ) = 0, we have z3 = 1/6 − p3 /2.
We now solve for the uniform prices chosen by firms 2 and 3. Firm 2’s profit isπ2 (p2 , p3 ) = p2 (1/3 + z2 − z1 ) = p2 (1/3 + p3 /2 − p2 ) .
Firm 3’s profit isπ3 (p2 , p3 ) = p3 (2/3 + z3 − (1/3 + z2 )) = p3 (1/3 + p2 /2 − p3 ) .
Solving the two firms’ profit maximization problems simultaneously, we obtain p2 =p3 := pN = 2/9, and z1 = 5/18, z2 = 1/6, and z3 = 1/18. Thus the marginal
consumer’s location on each segment is given by 5/18, 1/2, and 13/18. That is, firm 1
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serves [0, 5/18] ∪ [13/18, 1] with personalized price p1 (x) = p2 + (1/3 − 2x) = 5/9 − 2x
on [0, 5/18]. Firm 1’s profit from [13/18, 1] is the same as that from [0, 5/18]. Thus
its equilibrium profit is
π1N = 2

Z 5/18(5/9 − 2x) dx = 25/162.
0Firm 2 serves [5/18, 1/2] with pN = 2/9, and firm 3 serves the rest of the market with
pN = 2/9. Thus they have the same profit given by
π2N = π3N = (2/9) (1/2 − 5/18) = (2/9) (13/18 − 1/2) = 4/81.
2. Beneficial data sharing
The above analysis shows that firm 2’s customer base is [5/18, 1/2], of which[5/18, 1/3] is the segment that lies between firms 1 and 2, and firm 3’s customer
base is [1/2, 13/18] with [2/3, 13/18] being the segment lying between firms 1 and 3.
Unlike the case with linear city, firms 2 and 3 now have their customer base on eachside of their location. For example, the subset [1/3, 1/2] of firm 2’s customer baseis not contestable by firm 1. This implies that firm 1’s withholding of data on thesubset of [5/18, 1/3] and/or [2/3, 13/18] does not have the same effect as in the casewith linear city, mainly because firms 2 and 3 compete for the segment [1/3, 2/3].
On the other hand, firm 1’s problem is qualitatively the same as before. The main
purpose of data sharing is to induce firms 2 and 3 to increase their uniform prices
above pN = 2/9 while the subsequent change in firm 1’s market share, if any, is ofsecondary importance.
We show below that firm 1’s sharing of data on entire [5/18, 1/3] with firm 2 andentire [2/3, 13/18] with firm 3 can benefit all three firms relative to the benchmark
without data sharing. Given data sharing on [5/18, 1/3], firm 2 chooses p2 targeting
only the segment [1/3, 2/3]. Likewise, given data sharing on [2/3, 13/18], firm 3chooses p3 targeting only the segment [1/3, 2/3]. Let 1/3+z be the marginal consumeron [1/3, 2/3]. Then we have
p2 + z = p3 + (1/3 − z) ,which gives us z = (p3 − p2 )/2 + 1/6. Firm 2 chooses p2 to maximize p2 z and firm 354chooses p3 to maximize p3 (1/3 − z). This leads to the symmetric equilibrium uniform
price p∗2 = p∗3 := p∗ = 1/3 and z = 1/6. Note that firm 1 chooses personalized
price equal to zero for all consumers on [1/3, 1/2]. Thus, given p∗2 = p∗3 := p∗ = 1/3,
consumers on [1/3, 1/2] are indifferent between choosing firms 1 and 2, and consumers
on [1/2, 2/3] are indifferent between choosing between firms 1 and 3. From the above,it’s clear that all three firms benefit from data sharing. First, firm 1 does not lose
any market share but can charge higher personalized prices to consumers it servesbecause p∗ > pN . Specifically, firm 1’s personalized price increases from 5/9 − 2x to
2/3 − 2x, hence its profit isπ1∗ = 2

Z 5/18(2/3 − 2x) dx = 35/162 > 25/162 = π1N .

0Firm 2’s market share also stays the same as before but firm 2 charges higher
personalized prices to consumers on [5/18, 1/3] and higher uniform price to consumers on [1/3, 1/2]. Specifically, for consumer x ∈ [5/18, 1/3], firm 2’s personalized price is given by p2 (x) + (1/3 − x) = p1 (x) + x where p1 (x) = 0, hencep2 (x) = 2x − 1/3 ≥ 2/9 = pN for all x ∈ [5/18, 1/3]. Thus firm 2’s profit canbe calculated as
π2∗ =

Z 1/3

(2x − 1/3) dx + p∗ (1/2 − 1/3) = 23/324 > 4/81 = π2N .

5/18

By symmetry, firm 3 has the same profit as firm 2. Thus, all three firms benefit whenfirm 1 shares data on [5/18, 1/3] with firm 2 and data on [2/3, 13/18] with firm 3.
The above is one simple example of beneficial data sharing, and there could beother possibilities. But one clear observation is that sharing additional data on[1/3, 2/3] would not increase uniform prices chosen by firms 2 and 3 above the uniform price p∗ = 1/3. It is because firm 1 chooses personalized price equal to zero for
all consumers on [1/3, 1/2], which sets an upper bound on the uniform price that canbe chosen by firms 2 and 3. For example, consumer x ∈ [1/3, 2/3] chooses firm 2 if
and only if p2 + (x − 1/3) ≤ min{p1 (x) + x, p3 + (2/3 − x)} ≤ p1 (x) + x = x given
p1 (x) = 0. Thus p2 ≤ 1/3.

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