#  Selling Data to a Competitor
(Extended Abstract)

Ronen Gradwohl Department of Economics and Business Administration
Ariel University<EMAIL_ADDRESS>Faculty of Data and Decision Sciences
The Technion – Israel Institute of Technology Moshe Tennenholtz Faculty of
Data and Decision Sciences
The Technion – Israel Institute of Technology<EMAIL_ADDRESS>
###### Abstract

We study the costs and benefits of selling data to a competitor. Although
selling all consumers’ data may decrease total firm profits, there exist other
selling mechanisms—in which only some consumers’ data is sold—that render both
firms better off. We identify the profit-maximizing mechanism, and show that
the benefit to firms comes at a cost to consumers. We then construct Pareto-
improving mechanisms, in which each consumers’ welfare, as well as both firms’
profits, increase. Finally, we show that consumer opt-in can serve as an
instrument to induce firms to choose a Pareto-improving mechanism over a
profit-maximizing one.

##  1 Introduction

In recent years, it has become common wisdom that data is a dominant source of
power. This power is perhaps most clearly illustrated in markets where an
incumbent with access to consumer data competes with an entrant who does not
have such data. As stated in a crisp manner by [26], common wisdom holds that
the incumbent’s key advantage is data superiority: “If you run a market-
leading company, you should never be blindsided by an invader. Locked within
your own records is a huge, largely untapped asset that no attacker can hope
to match: what we call the incumbent’s advantage.” The situation is not
uncommon: In our data-driven economy, competing firms often find themselves in
asymmetric situations where one of them has superior or even exclusive access
to relevant data.

Such data asymmetry has become a major issue for debate. For example, in a
June 2021 press release, the EU declared that it has opened an antitrust
investigation that will “examine whether Google is distorting competition by
restricting access by third parties to user data for advertising purposes on
websites and apps, while reserving such data for its own use,” [14]. One of
the issues in such debates is the question of data sharing: Should the
incumbent share its data in order to increase market competition and consumer
welfare? And can the incumbent profitably sell its data to the entrant?

Strategic decisions about the sale of data to competitors appear in the online
economy frequently, although they are not always stated explicitly. For
example, when an advertiser buys a sponsored-search campaign through an ad
exchange, the advertiser obtains useful information about a segment of
consumers as part of the ad exchange reports. The advertiser might then use
this information when bidding directly for display ads on other platforms,
including platforms on which the ad exchange is also a competitor. The data-
holder (e.g., ad exchange) thus faces a strategic decision about which
consumer segments to sell to a competitor, and at what prices. The data-buyer
(e.g., advertiser), in turn, must decide whether to pay the price and obtain
data about these consumer segments, or whether to enter into competition
without the data on offer.

These strategic considerations raise numerous questions: What are the data-
holder’s costs and benefits from selling data to a data-buyer? What are the
effects of data sale on consumer welfare? And, in a regulated market, can data
sale be regulated in a way that leads to Pareto-improving transactions,
benefitting consumers as well as firms?

In this paper we study these questions in a paradigmatic model of imperfect
competition between two firms who have asymmetric access to data. We consider
the classic Hotelling model of imperfect competition: There are two firms,
each located at a different endpoint of a unit interval, with a unit mass of
consumers distributed across this interval. We model data about a consumer as
information about the consumer’s location on the interval. In the classic
model, neither firm has data about any consumer, and so firms engage in
competition via uniform prices that each offers to all consumers. In our
variation of this model, in contrast, one firm is a data-holder who knows the
locations of all consumers, whereas the other firm is a data-buyer who has no
such data. The data-holder can use its data advantage in order to personalize
prices to consumers, and can thus sometimes undercut the data-buyer’s uniform
price.

In order to study the costs and benefits of data sale to a competitor, we
suppose the firms engage in a data-sharing mechanism. Such a mechanism
consists of a segment of consumers whose data the data-holder shares with the
data-buyer, as well as a price the data-buyer pays the data-holder. After
engaging in such a mechanism, the data-buyer will hold location data about all
consumers in the shared segment, allowing that firm to also personalize prices
to them.

Within this model, we first show that full data-sharing, in which the data-
holder shares all its data with the data-buyer, is harmful to the firms. We
then show that there exist other data-sharing mechanisms—in which only some
consumers’ data is shared—that increase both firms’ profits. In fact, we
identify the mechanism that maximizes total firm profits. This last mechanism,
however, increases firm profits at the expense of consumers. We thus proceed
to show that there exist Pareto-improving mechanisms, in which each consumers’
welfare, as well as both firms’ profits, increase. Finally, we consider the
question of how a regulator can induce firms to utilize a Pareto-improving
mechanism rather than a profit-maximizing one that may harm consumers. We show
that consumer opt-in may serve as such an instrument: If consumers are given
the opportunity to opt-in to having their data sold, and if the data-holder is
only permitted to share data about consumers who have opted-in, then in
equilibrium the firms will choose a Pareto-improving mechanism.

Our results are driven by two forces, which we identify as the direct effect
and the indirect effect of data sharing. The direct effect is the following:
if the data-holder shares data about a particular consumer, then the data-
buyer can now offer that consumer a personalized rather than the uniform
price. This affects both firms’ equilibrium personalized prices to that
consumer, and may thus impact profits and welfare. The indirect effect of data
sharing, on the other hand, is the following: by sharing data about a segment
of consumers, the data-holder changes the set of consumers to whom the data-
buyer’s uniform price applies (since additional consumers will now be offered
personalized prices). And since the uniform price is determined in equilibrium
in part by the locations of consumers to whom that price will apply, a change
in the set of consumers may effect a change in the equilibrium uniform price,
thereby affecting profits and welfare. Our results highlight how the interplay
between the direct and indirect effects of data sharing lead to changes in
firms’ profits and consumers’ welfare.

In addition to identifying the two effects of data sharing, our analysis
generates several general insights. First, and perhaps surprisingly, selling
data to a competitor can be strictly beneficial to both firms.111We note that
this holds even if data is sold at no cost—see Proposition 2. Second, data can
be sold in a way that is Pareto improving. And finally, such Pareto-improving
data-sale can be induced by consumer opt-in regulation.

We note that the idea of selling data to a competitor has been advocated in
financial markets (see, for example, [3]). In that context, the possibility of
data sale allows a decision maker to choose between taking investment risks or
obtaining direct monetary rewards. The incredibly fast-growing data-economy,
in which some firms hold massive amounts of data about consumers, raises calls
to consider such data sale in a broader context: Can it lead to increased
profits to both data-holders and data-buyers? And can it benefit all of
society, including consumers whose data is exchanged?

Related literature This paper is part of a large and growing literature on
data markets (see, e.g., the survey of [6]). Work in this area focuses on
related but orthogonal questions, such as the effects of data-sale by a third-
party data-provider and of information sharing between competitors. Our paper
bridges these strands by considering data sale to a competitor. To the best of
our knowledge, the work of [3] is the only other paper to study such a
scenario, and ours is the first to focus on the effects of such data sale on
firm profits as well as consumer welfare.

The literature on the sale of data by a data provider (e.g., [2, 7, 27, 30,
35] and others) studies how a third-party data-provider can maximize profits
by selling data to a monopolist or to competing firms who use this data to
price discriminate. Within this literature, one paper that is closely related
to ours is that of [13]. [13] consider an information designer who provides
consumer information to oligopolists, and characterize the different market
outcomes that can be achieved by the designer. Our paper differs from this
research in that we suppose data is not held by a third-party, but rather by
one of the competing firms. This firm may sell data to its direct competitor,
affecting both firms’ respective market positions.

A different but related setting is that of [4], where the consumers are
holders of information who may share it with one or both firms so as to
intensify competition. The model and results of [4] bear some similarity to
ours. For example, they also consider a Hotelling model, and show that
consumers are better off whenever those sufficiently closer to one firm than
another share their location with that firm, and those closer to the middle
share their location with both firms. Despite the similarities, our paper
studies an orthogonal question, as we assume one of the firms already has data
about consumers, and focus on whether that firm will sell data to its
competitor. In contrast to the model of [4], in which each consumer chooses
which firm has access to that consumer’s location, in our model the informed
firm strategically chooses whether or not to share this information. Under
consumer opt-in the role of consumers is in determining whether such sharing
could potentially take place, but not in whether it actually takes place.
Finally, while [4] show that, consumers are always better off when they share
some of their information, we show that when firms choose what information to
share this may no longer be the case.

Because our work considers the sale of data from one firm to another, it is
related to the literature on information sharing. Although information sharing
between firms has been studied in a variety of settings,222These include
oligopolistic competition [12, 29], financial intermediation [28, 24, 16],
supply chain management [22, 31], competition between data brokers [21, 23],
and advertising [19]. our paper is most-closely related to that of competitive
price discrimination—see, for example, the surveys of [32] and [15]. One of
the main insights from this literature is that when firms have more data about
consumers, competition between them is more intense, leading to lower prices
and profits. In our paper, in contrast, data is sold by one firm to another in
such a way as to increase profits.

Two papers that specifically analyze the effects of data sharing within a
Hotelling model are [25] and [8]. [25] study a model in which each of two
firms may have data both about consumers’ locations and about their
transportation costs, and consider the eight permutations in which each firm
may have either a dataset about locations, a dataset about transportation
costs, both datasets, or neither datasets. They then analyze the market
effects of firms sharing one or both of their (full) datasets with each other,
and provide conditions under which sharing is beneficial to the firms. [8]
studies a Hotelling model in which locations are two-dimensional, and firms
hold all data about one dimension, both dimensions, or neither dimension. He
analyzes the various scenarios in terms of firm profits and consumer welfare,
with a particular emphasis on the comparison to the regimes of full privacy
(neither firm has any data) and no privacy (both firms have full data).
Interestingly, [8] shows that total firm profits are hump-shaped in the amount
of information they hold; for example, the scenario in which each firm holds
data about a different dimension yields higher profits than both full privacy
and no privacy. Our work differs from both of these papers in that we study
the sale of partial data from one firm to another, with an emphasis on
mutually increasing profits.

In terms of modeling, our paper is most closely related to [27] and [18]. [27]
consider a one-dimensional Hotelling model in which consumers’ locations may
be known to one, both, or neither firm. Their concern is not the sale of data
from one firm to another, but rather the optimal strategy of a data broker who
sells the data to the firms. They also consider the effects of a consumer-side
technology that allows consumers the ability to protect their privacy. [18]
also study a Hotelling model, but suppose that both firms have some data about
consumers. Their main focus is on various forms of mutual data sharing between
the firms.

##  2 The Model

We focus on a standard Hotelling model, in which a unit mass of consumers is
spread over the unit interval according to an atomless distribution $F$ with
continuous, strictly positive density $f$ that has full support. There are two
firms: firm $A$ is located at $\theta_{A}=0$, and firm $B$ is located at
$\theta_{B}=1$. Each consumer chooses at most one firm from which to purchase
a good. Consumers derive value $v$ from the good, but pay two costs: the
price, and a linear transportation cost that scales with the distance between
the consumer and the firm providing the good. Thus, a consumer located at
$\theta$ who buys from firm $i$ at price $p_{i}$ obtains utility
$v-p_{i}-t\left|{\theta-\theta_{i}}\right|$, where $t$ is the marginal
transportation cost. We assume throughout that the market is covered—namely,
that $v>2t$—so that all consumers purchase a good even when there is a
monopolist firm. Finally, we also assume for simplicity that firms’ marginal
costs are 0, and so their profit from the sale of a good is equal to the
price. These are all standard assumptions in Hotelling games.

The standard setup consists of a two-stage game: First, firms simultaneously
set prices; second, consumers choose a firm and make a purchase. In the simple
case where the distribution $F$ of consumers is uniform the game has a unique
subgame perfect equilibrium: firms’ prices are $p_{A}=p_{B}=t$, consumers in
$[0,0.5)$ buy from $A$, and consumers in $(0.5,1]$ buy from $B$ (see, e.g.,
[5]).333The equilibrium is unique up to the choice of the indifferent consumer
located at $\theta=0.5$.

In this paper we will consider a variant of the standard model by supposing
that firms may have additional information about some of the consumers. In
particular, we will suppose that, for each consumer, one or both firms know
the location of that consumer on the unit interval. For such consumers, firms
will be able to offer a personalized price—a special offer specifically
tailored to that consumer. If a firm does not know a consumer’s location,
however, then it cannot distinguish between that consumer and all other
consumers whose location it does not know. All such consumers are offered the
same uniform price.

In our model, firm $B$ is the data-holder and firm $A$ is the data-buyer.
Thus, initially, we assume that firm $B$ knows the locations of all consumers,
whereas firm $A$ does not know any consumer’s location. Given this
informational environment, a data-sharing mechanism $M=(M_{B},r)$ between
firms specifies a subset $M_{B}\subseteq[0,1]$ and a number
$r\in{\mathbb{R}}$, with the interpretation that firm $B$ shares with firm $A$
the locations of consumers in $M_{B}$, and firm $A$ transfers to firm $B$ a
payment $r$. Two simple examples of data-sharing mechanisms are one that
involves no sharing, $M=(\emptyset,r)$, and one that involves full sharing,
$M=([0,1],r)$. Alternatively, firm $B$ may share data about a subset of
consumers. For example, under mechanism $([x,y],r)$, if consumer
$\theta\in[x,y]$ arrives, both firms will know that consumer’s location. On
the other hand, if consumer $\theta\in[0,1]\setminus[x,y]$ arrives, firm $B$
will know that consumer’s location, and firm $A$ will only be able to deduce
that the consumer is not located within $[x,y]$.

In our analysis, we consider the following order of events:

  1. 1.

Firms engage in a data-sharing mechanism $M=(M_{B},r)$.

  2. 2.

Firm $A$ announces uniform price $p_{A}$.444Note that firm $B$ knows all
consumers’ locations, and so personalizes prices to each. It therefore need
not post a uniform price.

  3. 3.

A consumer arrives, and all firms who know the consumer’s location $\theta$
simultaneously offer that consumer a personalized price, $p_{A}(\theta)$ and
$p_{B}(\theta)$.

  4. 4.

The consumer chooses a firm from which to buy, and payoffs are realized.

Note that firms share data, and firm $A$ announces its uniform prices, before
consumers arrive. After a consumer arrives to the market, the firms who know
the consumer’s specific location simultaneously offer personalized prices. If
firm $A$ offers a consumer a personalized price, this offer subsumes the
firm’s original uniform price. Thus, the uniform price $p_{A}$ will apply only
to those consumers who will not subsequently be offered a personalized price
by firm $A$.

Importantly, when firms set personalized prices, they know the uniform price
set by firm $A$ in the previous stage. This is the standard timing considered
in the literature (see, e.g., [34, 11, 10, 27, 9]).555An alternative model
that we do not analyze is one in which firms set uniform and personalized
prices simultaneously, for each consumer. [27] show that, in this case, a
(pure) equilibrium may fail to exist.

For any fixed mechanism $M$, we will consider the pure subgame perfect
equilibria of the game that starts with data-sharing mechanism $M$. Such
equilibria always exists, and consist of a uniform price for firm $A$ followed
by personalized prices for both firms. Once the uniform price is fixed, the
equilibrium personalized prices for each consumer $\theta$ are uniquely fixed.
We will be interested in designing mechanisms $M$ that lead to equilibria with
high firm-profits and high consumer-welfare.

One important desideratum of data-sharing mechanisms (with corresponding
equilibria) is that they be individually rational (IR): That the expected
utility of each firm with data sharing be at least as high as without data
sharing. A data-sharing mechanism should be IR if we expect firms to
participate.

Our main focus will be on mechanisms that are not only IR, but also Pareto-
improving: when sharing takes place, (i) the expected utility of each firm and
every consumer be at least as high as without data sharing, and that (ii)
either firm $A$’s profits, firm $B$’s profits, or total consumer welfare be
strictly higher.

We note that many of our results make no assumptions about the distribution of
consumers. In such a general setting there may be multiple equilibria, even
with no data-sharing, each with different uniform prices. Hence, we will often
describe mechanisms as being IR or Pareto-improving relative to a particular
no-sharing equilibrium.

##  3 No Data-Sharing

We begin by analyzing equilibria under no data-sharing. To this end, define
$\mu(p_{A})=\frac{1}{2}-\frac{p_{A}}{2t}$. If firm $A$ charges uniform price
$p_{A}$, then the consumer located at $\mu(p_{A})$ is indifferent between
purchasing from firm $A$ at that price and purchasing from firm $B$ at price
0. All consumers located to the left of $\mu(p_{A})$ will thus strictly prefer
purchasing from firm $A$ at price $p_{A}$ than from firm $B$ at any
nonnegative price. In contrast, for every consumer located to the right of
$\mu(p_{A})$ there exists a nonnegative price of firm $B$ such that that
consumer will prefer to purchase from $B$ than from $A$. This is formalized in
the following proposition:

###### Proposition 1

Let $P_{A}=\arg\max_{p}p\cdot F\left(\mu(p_{A})\right)$. Without data sharing,
the set of equilibria consist of any uniform price $p_{A}\in P_{A}$ for firm
$A$ and corresponding personalized prices
$p_{B}(\theta)=\max\\{0,p_{A}+t(2\theta-1)\\}$ for firm $B$. In the
equilibrium with uniform price $p_{A}\in P_{A}$, consumers in $[0,\mu(p_{A}))$
purchase from firm $A$, whereas consumers in $[\mu(p_{A}),1]$ purchase from
$B$. The equilibrium with $p_{A}=\max\\{P_{A}\\}$ is strictly dominant for the
firms.

The proof of Proposition 1, and all other propositions, appear in the full
version of this paper [20].

Throughout the paper we will illustrate our results with the simple case in
which consumers are uniformly distributed on $[0,1]$. We note that this is the
standard setup in Hotelling games.

###### Example 1

When consumers are uniformly distributed on $[0,1]$, the set
$P_{A}=\\{t/2\\}$. In the unique equilibrium, then, consumers between 0 and
$\mu(t/2)=1/4$ purchase from $A$ at uniform price $p_{A}=t/2$, whereas the
rest purchase from $B$ at personalized prices
$p_{B}(\theta)=\max\\{t(2\theta-1/2),0\\}$. Total firm profits are
$\pi_{A}=t/8$ and

$\pi_{B}=\int_{1/4}^{1}t(2\theta-1/2)d\theta=\frac{9t}{16},$

whereas consumer welfare is

$CW=\int_{0}^{1}\max\\{v-\theta
t-p_{A},v-t(1-\theta)-p_{B}(\theta)\\}d\theta=\int_{0}^{1}(v-t/2-\theta
t)d\theta=v-t.$

##  4 The Direct Effect and Full Data-Sharing

In this section we begin our analysis of how data-sharing impacts profits and
welfare. Data sharing has a direct effect and an indirect effect. The direct
effect is that if firm $A$ obtains information about a consumer’s locations
via the sharing mechanism, it can now offer that consumer a personalized
price. This affects firm $B$’s equilibrium personalized price to that
consumer, and hence also profits and welfare. The indirect effect of data
sharing is that it may change the set of consumers to whom firm $A$’s uniform
price applies, since additional consumers will now be offered personalized
prices. And since the uniform price is determined in equilibrium in part by
the locations of consumers to whom that price will apply, a change in the set
of consumers may effect a change in the equilibrium uniform price. In this
section we explore the direct effect, and then in Section 5 we explore the
indirect effect.

Suppose that, absent data-sharing, firm $A$’s uniform price is $p_{A}$. If
firm $B$ shares the location $\theta$ of some consumer with $A$, then the
firms compete in personalized prices over that consumer, yielding equilibrium
prices $p_{A}(\theta)=\max\\{t(1-2\theta),0\\}$ and
$p_{B}(\theta)=\max\\{t(2\theta-1),0\\}$. The direct effect of firm $B$
sharing the location of a consumer is summarized in Lemma 1:

###### Lemma 1

Consider mechanism $M=(\\{\theta\\},0)$ relative to no sharing, and suppose
that consumer $\theta$ shows up.

  1. 1.

If $\theta\in(1/2,1]$, consumer $\theta$ still buys from $B$, but now at price
$t(2\theta-1)$. This is a net loss of $p_{A}$ to firm $B$ and a net gain of
$p_{A}$ to the consumer.

  2. 2.

If $\theta\in[\mu(p_{A}),1/2)$, consumer $\theta$ switches to purchasing from
$A$, at price $t(1-2\theta)$. This is a loss of $p_{A}+t(2\theta-1)$ to firm
$B$, a gain of $t(1-2\theta)$ to firm $A$, and a gain of
$p_{A}-t(1-2\theta)\geq 0$ to the consumer. Also, the gain to $A$ is greater
than the loss to $B$ if and only if

$\theta<\frac{1}{2}\left(\mu(p_{A})+\frac{1}{2}\right),$

the midpoint of the interval of $\theta$-s in the case under consideration.

  3. 3.

If $\theta\in[0,\mu(p_{A}))$, consumer $\theta$ still buys from $A$, now at
personalized price $p_{A}(\theta)=t(1-2\theta)>p_{A}$.

Given these direct effects, we now consider full data-sharing, namely,
$M=([0,1],r)$ for some $r$. Under this mechanism, both firms know the location
of every consumer, and so both engage in personalized pricing. Firm $A$’s
uniform price thus applies to no consumer, and so only the direct effect has
any bite. By Lemma 1, relative to the no-sharing mechanism with price $p_{A}$,
consumers $\theta\in[\mu(p_{A}),1]$ are better off, whereas consumers
$\theta\in[0,\mu(p_{A}))$ are worse off, under full data-sharing. For the
firms, naturally firm $B$ is better off with no sharing and firm $A$ with full
sharing. The effect on total profits, however, depends on the distribution
$F$. For the case of uniformly distributed consumers, full data-sharing harms
firms:

###### Example 2

When consumers are uniformly distributed, [33] show that profits are
$\pi_{A}=\pi_{B}=t/4$ (see also [34]). Note that total profits
$\pi_{A}+\pi_{B}$ are higher under no sharing ($t/8+9t/16=11t/16$, by Example
1) than under full sharing ($t/4+t/4=t/2$). This implies that no mechanism
$([0,1],r)$ is IR, regardless of $r$.

Although full data-sharing decreases total firm profits when consumers are
uniformly distributed, there exist distributions of consumers under which full
data-sharing increases profits—for example, this is the case when a
$(1-\varepsilon)$-fraction of consumers are uniformly distributed on the sub-
interval $[0,1/4]$, and the remaining $\varepsilon$ on $(1/4,1]$, for some
small enough $\varepsilon>0$.666Such a consumer distribution does not satisfy
our continuity assumption. However, the same result holds also when the kink
at $1/4$ is smoothed out. However, even then full sharing does not lead to
maximal profits. We now turn to mechanisms that do.

##  5 The Indirect Effect and Firm-Optimal Data-Sharing

In this section we describe firm-optimal mechanisms, which exploit the
indirect effect of data sharing. By Lemma 1, firm $B$’s profit from a consumer
$\theta\in(1/2,1]$ is $p_{A}+t(2\theta-1)$. If $A$’s uniform price were to
increase, this would likewise increase $B$’s profit from consumer $\theta$.
Now, recall that, when there is no sharing, firm $A$ sets its uniform price by
choosing $p_{A}\in P_{A}=\arg\max_{p}p\cdot F\left(\mu(p)\right)$. If $B$ were
to share data about consumers in some interval
$[\underline{\theta},\mu(p_{A})]$, however, then $A$ would offer consumers on
this interval a personalized price. The uniform price would no longer apply to
them, but would instead apply only to consumers
$[0,\underline{\theta})\cup(\mu(p_{A}),1]$. Firm $A$ may then benefit from
increasing (decreasing) the uniform price above (below) $p_{A}$, at the same
time increasing (decreasing) the profits of firm $B$ from consumers
$\theta\in(1/2,1]$. This is the indirect effect of data sharing.

Firm $B$ can exploit both the indirect and direct effects of data sharing by
sharing data both about consumers in $[\underline{\theta},\mu(p_{A})]$ and
about consumers in $(\mu(p_{A}),1/2]$. Note, however, that sharing data about
consumers in $(1/2,1]$ is never beneficial, since it only results in a net
loss to firms and has no indirect effect (by Lemma 1, above).

In general, the firm-optimal mechanism may depend on the distribution of
consumers and other primitives of the model. In Proposition 2, however, we
show that when $v$ (the consumers’ value for the good) is sufficiently high,
then there is an essentially unique mechanism, with a corresponding
equilibrium, that yield the firms maximal joint profits. The mechanism makes
extreme use of the indirect effect of data sharing: Firm $B$ shares data about
consumers $[0,1/2]$, implying that firm $A$’s uniform price no longer applies
to these consumers, and hence that this price can be almost arbitrarily high.
$A$’s uniform price does apply to consumers in $(1/2,1]$, for whom it serves
as an outside option. However, because these consumers will always purchase
from $B$ in equilibrium (by Lemma 1, above), the high outside option allows
that firm to extract these consumers’ entire surplus.

###### Proposition 2

Fix $v>\frac{5t}{2(1-F(1/2))}$. Mechanism $M=([0,1/2],0)$ with equilibrium
uniform price $p_{A}=v-t/2$ maximizes joint firm profits and is IR relative to
any no-sharing equilibrium. Every other firm-optimal mechanism is of the form
$M^{\prime}=([0,1/2],r)$.

###### Example 3

When consumers are uniformly distributed, the mechanism described in
Proposition 2 is actually firm-optimal for all $v>2t$, as we now show. This
mechanism leads to profits $\pi_{A}=t/4$ and

$\pi_{B}=\int_{1/2}^{1}(v-t(1-\theta))d\theta=\frac{v}{2}-\frac{t}{8}>\frac{7t}{8},$

where the inequality follows since $v>2t$. Thus, total profits are at least
$9t/8$. In contrast, consider any mechanism $M^{\prime}=(M_{B},r)$ in which
firm $A$’s uniform price applies to a consumer in $[0,1/2]$, and fix some
uniform price $p_{A}^{\prime}\leq t$. Consumers $\theta\in(1/2,1]$ buy from
$B$, leading to total profits at most $3t/4$ from these consumers. Consumers
$\theta\in[0,1/2]$ either buy from $A$ at uniform price $p_{A}^{\prime}$ or at
personalized price $t(1-2\theta)$, or from $B$ at personalized price
$t(2\theta-1)$ or $p_{A}^{\prime}+t(2\theta-1)$ (depending on whether
$\theta\in M_{B}$). Total profits are maximized when $p_{A}^{\prime}=t$,
consumers $\theta\in[0,1/4]$ buy at $A$’s personalized price, and the rest buy
from $B$ at price $t+t(2\theta-1)$. Profits to $A$ from $[0,1/4]$ and to $B$
from $(1/4,1]$ are each equal to $3t/16$. Total profits from $M^{\prime}$ are
thus bounded above by $3t/4+2(3t/16)=9t/8$, which is equal to the lower bound
on profits from $([0,1/2],0)$.

######  Remark 5.1.

Proposition 2 provides a sufficient condition under which mechanism
$M=([0,1/2],0)$ is firm-optimal for some equilibrium (namely, the one with
uniform price $p_{A}=v-t/2$). However, under this mechanism there are other
equilibria, which involve lower uniform prices, and that yield lower firm
profits. In Proposition 6 in the full version of this paper [20] we describe a
different mechanism with $r=0$ that, while not firm-optimal, yields both firms
strictly higher profits than under no sharing in every equilibrium.

##  6 Pareto-Improving Data-Sharing

In Section 5 above we show that firm $B$ can sell data in a way that maximizes
joint firm profits, and hence allows that firm to charge a high price for the
data. Such sharing, however, comes at the expense of consumers. In particular,
under the equilibrium of Proposition 2, firms extract the entire surplus of
consumers located in $[1/2,1]$. In this section we show that there exist other
data-sharing mechanisms that increase firm profits relative to no sharing,
while at the same time also increasing consumers’ utilities.

Recall that $P_{A}=\arg\max_{p}p\cdot F\left(\mu(p_{A})\right)$, and that, by
Proposition 1, the set of equilibria under no data-sharing consist of uniform
prices $p_{A}\in P_{A}$ by firm $A$ and respective personalized prices
$p_{B}(\theta)=\max\\{0,p_{A}+t(2\theta-1)\\}$ by firm $B$. Denote by
$E(p_{A})$ the no-sharing equilibrium with uniform price $p_{A}$. In the
following proposition we show that for each such no-sharing equilibrium there
exists a Pareto-improving mechanism.

######  Proposition 6.2.

For every $p_{A}\in P_{A}$ there exists $r$ such that mechanism
$M=\left(\left[\mu(p_{A}),\frac{1}{4}+\frac{\mu(p_{A})}{2}\right],r\right)$
with uniform price $p_{A}$ is IR and weakly beneficial to every consumer,
relative to $E(p_{A})$. Furthermore, $M$ yields higher total firm profits than
any other mechanism that is weakly beneficial to every consumer relative to
$E(p_{A})$.

The mechanism $M$ described in Proposition 6.2 does not decrease the utility
of any consumer. Moreover, by bullet 2 of Lemma 1, that mechanism strictly
increases the utilities of a subset of consumers—namely, those located in
$\left(\mu(p_{A}),\frac{1}{4}+\frac{\mu(p_{A})}{2}\right]$.

The main idea underlying the construction for Proposition 6.2 is that firm $B$
shares data about every consumer $\theta$ that satisfies two conditions: (i)
with no sharing, consumer $\theta$ prefers to pay $B$’s personalized price
than $A$’s uniform price; (ii) sharing consumer $\theta$’s location leads to a
net increase in firm profits. Note that these consumers are all closer to $A$
than to $B$, so that the welfare and profit gain is obtained due to an
increase in efficiency. Finally, the construction is such that $A$’s uniform
price under $M$ remains the same as with no sharing, which guarantees that
consumers close to $A$ do not pay a higher price than under no sharing, but
also that firms maximize their joint profits subject to this constraint.

##  7 Consumer Opt-In

Proposition 6.2 above shows that there exist mechanisms that are strictly
Pareto-improving, increasing firm profits as well as consumer welfare.
However, these mechanisms are not optimal for firms—Proposition 2 identifies a
different mechanism as maximizing firm profits, a mechanism that does so at
the expense of consumers. How can a policymaker induce firms to share data in
a Pareto-improving manner, rather than in a profit-maximizing manner? In this
section we identify one way in which a policymaker can do this: by asking each
consumer whether or not they agree to have their data shared, and then
permitting firms to share data only about consumers who have agreed.

In order to analyze such consumer opt-in regulation, we first extend the model
to include a preliminary opt-in stage. After setting up the model, we present
two results. The first, Proposition 7.4 in Section 7.2, states that, under
consumer opt-in, there is an equilibrium of the extended model wherein firms
choose the Pareto-improving mechanism of Proposition 6.2. Now, although
consumer opt-in can lead to the choice of the Pareto-improving mechanism,
there are other equilibria that do not. However, in our second result
here—Proposition 7.5 in Section 7.3—we show that the equilibrium of Section
7.2, where the Pareto-improving mechanism is chosen, is, in a sense, optimal
for the consumers.

###  7.1 The Extended Model

We begin by extending the model of Section 2 with a preliminary stage, in
which each consumer simultaneously chooses whether or not to opt in to having
location data shared. Denote the set of consumers who opted in as $C$. Only
then do firms engage in a data-sharing mechanism $M=(M_{B},r)$; however, firms
are restricted to choosing a mechanism for which $M_{B}\subseteq C$. Such
mechanisms are feasible for $C$.

We assume that firms bargain over the choice of mechanism efficiently—that is,
they choose a mechanism $M$ that maximizes total firm profits, subject to the
opt-in constraint. One way to implement such efficient bargaining is when one
of the firms makes the other a take-it-or-leave-it offer by suggesting a
mechanism $(M_{B},r)$ that is feasible for $C$. Depending on which firm makes
the offer, the chosen price transfer $r$ will vary to favor the offering firm.
Either way, however, firms will choose to offer a mechanism that maximizes
joint firm-profits. This assumption is stated formally in Definition 7.3 below
as part of the solution concept. In addition, as in the previous sections, we
assume that, absent data-sharing, firms play the no-sharing equilibrium
$E(p_{A})$ for some $p_{A}\in P_{A}$.

In this extended model there is an additional, technical complication. We are
assuming that firms choose a mechanism that is feasible for some $C$. However,
since $C$ is generated by the set of consumers who choose to opt in, it may
not be a measurable set. Thus, the firms’ optimization problem may not be
well-defined at every $C$. One way to get around this problem is to consider
the Nash equilibria of this extensive-form game (rather than the subgame
perfect equilibria). However, this is somewhat unsatisfying, as such
equilibria may be sustained by strange off-equilibrium behavior—namely, the
presence of empty threats. Instead, we will use an equilibrium notion that is
weaker than subgame perfect equilibrium but nonetheless suffices to eliminate
empty threats. The general definition is due to [17]; here we give a
specialized version that applies to our specific game.

For the definition, let $L$ denote the set of subsets of $[0,1]$, and let
$\mathcal{M}(C)$ denote the set of all mechanisms feasible for $C$.

######  Definition 7.3.

A set $C^{*}\in L$ and functions $m:L\rightarrow\mathcal{M}(L)$ and
$p:L\rightarrow\hbox{\rm\hbox{I}\kern-1.62498ptR}_{+}$ form a threat-free Nash
equilibrium (TFNE) if

  1. 1.

For every $C$, mechanism $m(C)$ is feasible for $C$, and $p(C)$ is an
equilibrium uniform price for firm $A$ under $m(C)$.

  2. 2.

For every $\theta\in C^{*}$, consumer $\theta$ is weakly better off under
$m(C^{*})$ than under $m(C^{*}\setminus\\{\theta\\})$ (with respective uniform
prices $p(C^{*})$ and $p(C^{*}\setminus\\{\theta\\})$).

  3. 3.

For every $\theta\in[0,1]\setminus C^{*}$, consumer $\theta$ is weakly better
under $m(C^{*})$ than under $m(C^{*}\cup\\{\theta\\})$ (with respective
uniform prices $p(C^{*})$ and $p(C^{*}\cup\\{\theta\\})$).

  4. 4.

For every $\theta\in[0,1]$ and
$C\in\left\\{C^{*}\setminus\\{\theta\\},C^{*}\cup\\{\theta\\}\right\\}$,
mechanism $m(C)$ with uniform price $p(C)$ is IR and jointly firm-optimal
relative to all mechanisms that are feasible for $C$ (with corresponding
uniform prices).

For comparison, in a Nash equilibrium bullet 4 would be replaced by requiring
IR and joint firm-optimality only for $C^{*}$. In a subgame perfect
equilibrium, in contrast, bullet 4 would require these for all sets $C$. A
TFNE is a compromise between the two, requiring IR and joint firm-optimality
for $C^{*}$ and for all sets $C$ that differ from $C^{*}$ by a single
consumer’s unilateral deviation.

###  7.2 Pareto-Improving Equilibrium

Given the extended model above, we can now state our proposition on the
benefit of consumer opt-in.

######  Proposition 7.4.

For every $p_{A}\in P_{A}$ there exists a TFNE $(C^{*},m,p)$ of the extended
model in which $m(C^{*})$ is the Pareto-improving mechanism
$M=\left(\left[\mu(p_{A}),\frac{1}{4}+\frac{\mu(p_{A})}{2}\right],r\right)$,
for some $r$.

Proposition 7.4 shows that, when consumers can choose whether or not to opt in
to having their data shared, and firms are allowed to only share the data of
consumers who have opted in, then the equilibrium mechanism is Pareto
improving. There are other equilibria that lead to a Pareto-improving
mechanism. In fact, as long as consumers $\theta\in[0,\mu(p_{A}))$ do not opt
in to having their data shared, the mechanism that maximizes firms’ profits
will be Pareto improving.

However, there are also other equilibria in which the chosen mechanism is not
Pareto improving. Consider the following strategies: Consumers $[0,1/2]$ opt
in to having their data shared, and firms choose the firm-optimal mechanism
$M=([0,1/2],0)$ from Proposition 2. If some consumer $\theta\in[0,1/2]$ does
not opt in, then firms use the mechanism
$M_{\theta}=([0,1/2]\setminus\\{\theta\\},0)$. This mechanism is identical to
$M$, except that consumer $\theta$ faces firm $A$’s uniform price
$p_{A}=v-t/2$ rather than the personalized price $p_{A}(\theta)=t(1-2\theta)$.
This is no better for consumer $\theta$, and so these strategies form an
equilibrium. Why, then, would consumers choose to collectively opt in as in
Proposition 7.4?

###  7.3 Consumer-Optimal Equilibrium

We now show that the equilibrium of Proposition 7.4 is focal for the
consumers. In particular, we show that it maximizes consumer welfare, relative
to all other equilibria that leave no consumer worse off.

Fix some $p_{A}\in P_{A}$, and observe that there always exists a TFNE of the
extended game in which no consumer opts in, and that this leads to consumer
utilities as derived from equilibrium $E(p_{A})$ in mechanism
$M_{\emptyset}=(\emptyset,0)$. Next, let us consider other opt-in choices for
consumers. For a set $C$ and mechanism $M$ feasible for $C$, say that $M$ is
Pareto-improving for the consumers if the resulting utility of every consumer
is weakly higher than under $M_{\emptyset}$. We now show that consumers’
utilities in the equilibrium of Proposition 7.4 are optimal:

######  Proposition 7.5.

Fix $C\subseteq[0,1]$ and a mechanism $M$ with uniform price $q_{A}$ that is
feasible for $C$ and that is Pareto-improving for the consumers. If $M$ yields
strictly higher total utility to the consumers than $M^{*}$, then $M$ will not
be chosen by the firms in any TFNE in which consumers $C$ opt in.

That is, if we assume consumers make their opt-in decisions in a way that
leads to a weak improvement for each, then they can do no better than the opt-
in strategy of Proposition 7.4.

##  8 Conclusion

In this paper we analyzed the benefits to a data-holder of selling consumer
data to a data-buyer in a Hotelling model of imperfect competition. We
identified the two effects of data sharing, and showed that the interplay of
these effects can lead to Pareto-improving mechanisms that benefit consumers
as well as firms. Finally, we showed that consumer opt-in can induce firms to
choose such a Pareto-improving mechanism.

## Acknowledgements

The work of Tennenholtz was supported by funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant number 740435).

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