# Data Trading with a Monopoly Social Network
_Outcomes are Mostly Privacy Welfare Damaging_

Ranjan Pal, Junhui Li, Yixuan Wang, Mingyan Liu, Swades De, and Jon Crowcroft
R. Pal. M. Liu, J. Li, and Y. Wang are with the Department of Electrical
Engineering and Computer Science, University of Michigan, Ann Arbor, USA.
E-mail: {palr, mingyan, opheelia, joywyx}@umich.edu,S. De is with the
Department of Electrical Engineering, Indian Institute of Technology Delhi,
India. E-mail<EMAIL_ADDRESS>Crowcroft is with the Computer
Laboratory, University of Cambridge, UK, and the Alan Turing Institute, UK,
<EMAIL_ADDRESS>
##  1 abstract

This paper argues that data of strategic individuals with heterogeneous
privacy valuations in a distributed online social network (e.g., Facebook)
will be under-priced, if traded in a monopoly buyer setting, and will lead to
diminishing utilitarian welfare. This result, for a certain family of online
community data trading problems, is in stark contrast to a popular information
economics intuition that increased amounts of end-user data signals in a data
market improves its efficiency. Our proposed theory paves the way for a future
(counter-intuitive) analysis of data trading oligopoly markets for online
social networks (OSNs).

## keywords

distributed community, monopoly, social welfare

##  2 Introduction

Data of billions of online individuals are currently gathered, processed, and
analyzed for personalized advertising or other online service111Facebook
itself has approximately 2.5 billion monthly active individual users.. This
trend is on the high rise with a perennial increase in online apps, IoT
technologies, and advanced AI/ML methodologies. It is a common and age-old
notion in economics (see [1, 2, 3, 4, 5, 6, 7, 8, 9]) that the benefits and
use of sharing individual information with the demand side of an information
market is beneficial to targeted customization, demand side profit, and the
growth of data-‘hungry’ AI/ML controlled businesses. It has also been argued
by economists [10, 11] that because of the above-mentioned benefits that
individual data brings to a market setting, a competitive market mechanism
might generate too little data sharing from the supply side.

In this letter, we rigorously argue, through a counter-example of a simple
application type, that the popularly known economic intuition _does not hold_
in general, atleast for a monopoly information market setting. More
specifically, we show that for some community settings (e.g., Facebook)
trading end-user data/information signals in a monopoly market leads to
diminishing economic utilitarian social welfare. _The intuition behind this
result primarily lies behind the negative externalities created via trading
statistically correlated end-user signals when these heterogeneous users have
varying privacy valuations of their data signals._ The result is contrary to
recent results that intuit/prove that privacy can be detrimental to
information market efficiency [5, 9, 12, 13, 14, 15, 16] if _ideally, one’s
value of privacy is not high, or one’s data is mildly correlated with others._
Specifically, we prove that information markets will be inefficient in _non-
ideal_ community settings - formally hinted earlier in [14]. Our analysis is
complete for a monopoly structure, with a major takeaway being that in
practical social community settings, sub-population privacy will jeopardized
at a monopoly data trading market equilibrium. To the best of our knowledge,
we are the first to mathematically dispel traditional information economics
intuition, albeit for social network settings only. Moreover, it paves the way
for a future (counter-intuitive) analysis of general community data trading
oligopoly markets.

The rest of the paper is organized as follows. In Section II, we provide an
intuition, via an example, towards proving our claim. We then follow this up
in Section III with the description of a formal monopoly market model. In
Section IV, we analyze this market model and formally prove our claim. We
provide illustrative examples of our theory in Section V. We conclude the
letter in Section VI.

##  3 Intuition

We provide an example-driven intuition that leads us to formally investigate
the validity of the hypothesis that OSN user information promotes efficient
data trading markets.

We focus on the widely popular _Cambridge Analytica_ scandal. The company
acquired private information of millions of individuals from data shared by
270,000 Facebook users who voluntarily downloaded an app for mapping
personality traits, called _This is your digital life_. The app accessed
users’ news feed, timeline, posts, and messages, and revealed information
about other Facebook users. The company was finally able to infer valuable
information about more than 50 million Facebook users, which it deployed for
designing personalized political messages and advertising in the Brexit
referendum and the 2016 US presidential election. This scandal highlighted two
important facets: (i) private information (e.g., behavior, habits,
preferences) of users part of an online social community such as Facebook are
correlated and results in knowing such information about other users222Habits
and preferences of a highly educated gay from a particular locality is
informative about others with the same profile and residing in the same area.
whose data is not leaked, and (ii) once it is openly publicized that valuable
user information has been breached to satisfy external objectives, users are
often miffed resulting in a huge social uproar as did happen in the case of
the Cambridge Analytica scandal. These observations motivated us to develop a
skepticism regarding the popular economic notion that more data implies
increased information market efficiency. It could also be that trading in
return for incentives for such community settings might not go down
well333This is a high chance in scenarios of social uproar post publicly known
data breaches, if not in cases where data breaches go un-noticed. with the
user privacy preferences, consequently hampering societal welfare (see Section
III for a definition).

To state our intuition in a relatively more formal manner (courtesy of [15]),
consider a community platform with two users, $i$ = 1, 2. Each user owns its
own personal data, which we represent with a random variable $X_{i}$ (from the
viewpoint of the platform). The relevant data of the two users are related,
which we capture by assuming that their random variables are jointly normally
distributed with mean zero and correlation coefficient $\rho$. The community
platform can acquire or buy the data of a user in order to better estimate her
preferences or actions. Its objective is to minimize the mean square error of
its estimates of user types, or maximize the amount of leaked information
about them. Suppose that the valuation (in monetary terms) of the platform for
the users’ leaked information is one, while the value that the first user
attaches to her privacy, again in terms of leaked information about her, is
$\frac{1}{2}$ and for the second user it is $v>0$. We also assume that the
platform makes take-it-or-leave-it offers to the users to purchase their data.
In the absence of any restrictions on data markets or transaction costs, the
first user will always sell her data (because her valuation of privacy,
$\frac{1}{2}$, is less than the value of information to the platform, 1). But
given the correlation between the types of the two users, this implies that
the platform will already have a fairly good estimate of the second user’s
information. Suppose, for illustration, that $\rho\simeq 1$. In this case, the
platform will know almost everything relevant about user 2 from user 1’s data,
and this undermines the willingness of user 2 to protect her data. In fact,
since user 1 is revealing almost everything about her, she would be willing to
sell her own data for a very low price (approximately 0 given $\rho\simeq 1$).
But once the second user is selling her data, this also reveals the first
user’s data, so the first user can only charge a very low price for her data.
Therefore in this simple example, the community platform will be able to
acquire both users’ data at approximately zero price. Critically, however,
this price does not reflect the users’ valuation of privacy. When $v\leq 1$,
the equilibrium is efficient because data are socially beneficial in this case
(even if data externalities change the distribution of economic surplus
between the platform and users). However, it can be arbitrarily inefficient
when $v$ is sufficiently high. This is because the first user, by selling her
data, is creating a negative externality on the second user.

##  4 System Model

A simple example, such as the one aforementioned, clearly provides an
intuition regarding the inefficiency of information trading in community
settings with heterogeneous privacy valuations. In this section. en route to
generalizing the validity (or invalidity) of our intuition, we propose a
monopoly information trading market model (reproduced from [15])444We use the
same notation for consistency purposes. consisting of $n$ platform users and a
profit-maximizing community platform (e.g., Facebook).

We consider $n$ community users represented by the set
$\mathcal{V}=\\{1,\ldots,n\\}.$ Each user $i\in\mathcal{V}$ has a type denoted
by $x_{i}$ which is a realization of a random variable $X_{i}.$ We assume that
the vector of random variables $\mathbf{X}=\left(X_{1},\ldots,X_{n}\right)$
has a joint normal distribution $\mathcal{N}(0,\Sigma),$ where
$\Sigma\in\mathbf{R}^{n\times n}$ is the publicly known covariance matrix of
$\mathbf{X}.$ Let $\Sigma_{ij}$ designate the $(i,j)$ -th entry of $\Sigma$
and $\Sigma_{ii}=\sigma_{i}^{2}>0$ denote the variance of individual
$i^{\prime}$s type. Each user has some personal data, $S_{i}$, which is
informative about its type, i.e., the ‘DNA’ that drives the user’s tastes (for
example, based on her past behavior, preferences, or contacts). We suppose
that $S_{i}=X_{i}+Z_{i}$ where $Z_{i}$ is an independent random variable with
standard normal distribution555This has taken various forms in the information
privacy literature [17]., i.e., $Z_{i}\sim\mathcal{N}(0,1)$. For any user
joining the community platform, the platform can derive additional revenue
(e.g., due to benefits of targeted advertising) if it can predict the user’s
type. We simply assume that the community platform’s revenue from each user is
a deceasing function of the mean square error of its forecast of the user’s
type, minus what the platform pays to users to acquire their information. More
specifically, the objective of the platform is to minimize

$\sum_{i\in\mathcal{V}}\left(\mathbb{E}\left[\left(\hat{x}_{i}(\mathbf{S})-X_{i}\right)^{2}\right]-\sigma_{i}^{2}+p_{i}\right)$
(1)

where $\mathbf{S}$ is the vector of data the platform acquires,
$\hat{x}_{i}(\mathbf{S})$ is the platform’s estimate of the user’s type given
this information, $-\sigma_{i}^{2}$ is included as a convenient normalization,
and $p_{i}$ denotes payments (be it explicit or implicit) to user $i$ from the
platform for their data (we ignore for simplicity any other transaction costs
incurred by the platform).

Users value their privacy, which we also model in a reduced-form manner
(reflecting both pecuniary and non-pecuniary666As example, the fact that a
user may receive a greater consumer surplus when the platform knows less about
her or she may have a genuine demand for keeping her preferences, behavior,
and information private. There may also be political and social reasons for
privacy, for example, for concealing dissident activities or behaviors
disapproved by some groups. motives) as a function of the same mean square
error. We assume, specifically, that user $i^{\prime}$ s value of privacy is
$v_{i}\geq 0,$ and her payoff is
$v_{i}\left(\mathbb{E}\left[\left(\hat{x}_{i}(\mathbf{S})-X_{i}\right)^{2}\right]-\sigma_{i}^{2}\right)+p_{i}$
This expression and its comparison with objective (1) clarifies that the
platform and users have potentially- opposing preferences over information
about user type. We have again subtracted $\sigma_{i}^{2}$ as a normalization,
which ensures that if the platform acquires no additional information about
the user and makes no payment to it, the payoff is zero. Critically, users
with $v_{i}<1$ value their privacy less than the valuation that the platform
attaches to information about them, and thus reducing the mean square error of
the estimates of their types is socially beneficial. In contrast, users with
$v_{i}>1$ value their privacy more, and reducing their mean square error is
socially costly. In settings without data externalities (where data about one
user have no relevance to the information about other users - an example being
collection agencies not gathering addresses locations), the first group of
users should allow the platform to acquire (buy) their data, while the second
group should not. A simple market mechanism based on prices for data can
implement this efficient outcome, in accordance to the traditional economic
notion that more information implies better market efficiency. However, the
situation could be very different in the presence of data externalities (e.g.,
online community settings such as Facebook).

A key notion for our analysis is breached information, which captures the
reduction in the mean square error of the platform’s estimate of the type of a
user. When the platform has no information about user $i$, its estimate
satisfies
$\mathbb{E}\left[\left(\hat{x}_{i}-X_{i}\right)^{2}\right]=\sigma_{i}^{2}$. As
the platform receives data from this and other users, its estimate improves
and the mean square error declines. The notion of breached information
captures this reduction in mean square error (MSE). Specifically, let
$a_{i}\in\\{0,1\\}$ denote the data sharing action of user $i\in\mathcal{V}$
with $a_{i}=1$ corresponding to sharing. Denote the the profile of sharing
decisions by $\mathbf{a}=\left(a_{1},\ldots,a_{n}\right)$ and the decisions of
agents other than $i$ by $\mathbf{a}_{-i}.$ We also use the notation
$\mathbf{S}_{\mathbf{a}}$ to denote the data of all individuals for whom
$a_{j}=1,$ i.e.,
$\mathbf{S}_{\mathbf{a}}=\left(S_{j}:j\in\mathcal{V}\,\mathrm{s.t.}\,a_{j}=1\right).$
Given a profile of actions a, the breached information of (or about) user
$i\in\mathcal{V}$ is the reduction in the MSE of the best estimator of the
type of user $i$ :
$\mathcal{I}_{i}(\mathbf{a})=\sigma_{i}^{2}-\min_{\hat{x}_{i}(\cdot)}\mathbb{E}\left[\left(X_{i}-\hat{x}_{i}\left(\mathbf{S}_{\mathbf{a}}\right)\right)^{2}\right].$

Notably, because of data externalities, breached information about user $i$
depends not just on her decisions but also on the sharing actions taken by all
users. With this notion at hand, we can write the payoff of user $i$ given the
price vector $p=(p_{1},...,p_{n})$ as

$u_{i}\left(a_{i},\mathbf{a}_{-i},\mathbf{p}\right)=\left\\{\begin{array}[]{ll}p_{i}-v_{i}\mathcal{I}_{i}\left(a_{i}=1,\mathbf{a}_{-i}\right),&a_{i}=1\\\
-v_{i}\mathcal{I}_{i}\left(a_{i}=0,\mathbf{a}_{-i}\right),&a_{i}=0\end{array}\right.$

where recall that $v_{i}\geq 0$ is user’s value of privacy. We also express
the monopoly platform’s payoff more compactly as

$U(\mathbf{a},\mathbf{p})=\sum_{i\in\mathcal{V}}\mathcal{I}_{i}(\mathbf{a})-\sum_{i\in\mathcal{V}:a_{i}=1}p_{i}$
(2)

An action profile $a=(a_{1},...,a_{n})$ of the strategic users and a price
vector $p=(p_{1},...,p_{n})$ for the users constitute a pure strategy
equilibrium of the user-platform game if both users and the community platform
maximize their payoffs given other players’ strategies. More formally, we
define an equilibrium of this game as a _Stackelberg equilibrium_ in which the
monopoly platform chooses the price vector recognizing the user equilibrium
that will result following this choice.

###### Definition 1

_Given the price vector $\mathbf{p}=\left(p_{1},\ldots,p_{n}\right),$ an
action profile $\mathbf{a}=\left(a_{1},\ldots,a_{n}\right)$ is user
equilibrium if for all $i\in\mathcal{V}$,_

$a_{i}\in\operatorname{argmax}_{a\in\\{0,1\\}}u_{i}\left(a_{i}=a,\mathbf{a}_{-i},\mathbf{p}\right).$

We denote the set of user equilibria at a given price vector $\mathbf{p}$ by
$\mathcal{A}(\mathbf{p})$. A pair
$\left(\mathbf{p}^{\mathrm{E}},\mathbf{a}^{\mathrm{E}}\right)$ of price and
action vectors is a pure strategy Stackelberg equilibrium if
$\mathbf{a}^{\mathrm{E}}\in\mathcal{A}\left(\mathbf{p}^{\mathrm{E}}\right)$
and there is no profitable deviation for the platform, i.e.,

$U\left(\mathbf{a}^{\mathrm{E}},\mathbf{p}^{\mathrm{E}}\right)\geq
U(\mathbf{a},\mathbf{p}),\;{\rm
for\;all\;}\mathbf{p}{\rm\;and\;for\;all\;}\mathbf{a}\in\mathcal{A}(\mathbf{p})$

In what follows, we refer to a pure strategy Stackelberg equilibrium simply as
an equilibrium.

We now characterize two important properties of the breached information
function $\mathcal{I}_{i}$ : $\\{0,1\\}^{n}\rightarrow\mathbf{R}$.
1\. _Monotonicity:_ for two action profiles a and $\textbf{a}^{\prime}$ with a
$\geq$ $\textbf{a}^{\prime}$

$\mathcal{I}_{i}(\mathbf{a})\geq\mathcal{I}_{i}\left(\mathbf{a}^{\prime}\right),\quad\forall
i\in\\{1,\ldots,n\\}$

2\. _Submodularity:_ for two profiles a and $\textbf{a}^{\prime}$ with
$\mathbf{a^{\prime}}_{-i}\geq\mathbf{a}_{-i}$,

$\mathcal{I}_{i}\left(a_{i}=1,\mathbf{a}_{-i}\right)-\mathcal{I}_{i}\left(a_{i}=0,\mathbf{a}_{-i}\right)\geq\mathcal{I}_{i}\left(a_{i}=1,\mathbf{a}_{-i}^{\prime}\right)-\mathcal{I}_{i}\left(a_{i}=0,\mathbf{a}_{-i}^{\prime}\right)$

The monotonicity property states that as the set of community users who share
their information expands, the breached information about each user (weakly)
increases. This is an intuitive consequence of the fact that more information
always facilitates the estimation problem of the platform and reduces the mean
square error of its estimates. More important for the rest of our analysis is
the submodularity property, which implies that the marginal increase in the
breached information from individual $i$’s sharing decision is decreasing in
the information shared by others. This too is intuitive and follows from the
fact that when others’ actions reveal more information, there is less to be
revealed by the sharing decision of any given individual. Thus, from the
celebrated result due to Topkis [18], for any $\mathbf{p}$, the set
$\mathcal{A}(\mathbf{p})$ is a complete lattice, and thus has a least and a
greatest element. This implies that the set of user equilibria is always non-
empty.

##  5 Monopoly Market Analysis

Enroute to analyzing the market welfare generated via the aforementioned game
setting, we first define the benchmark _first best_ welfare outcome as the
data sharing decisions that maximize utilitarian social welfare or social
surplus given by the sum of the payoffs of the platform and users. Social
surplus (SoS) from an action profile a is

$SoS(\textbf{a})=U(\mathbf{a},\mathbf{p})+\sum_{i\in\mathcal{V}}u_{i}(\mathbf{a},\mathbf{p})=\sum_{i\in\mathcal{V}}\left(1-v_{i}\right)\mathcal{I}_{i}(\mathbf{a})$

Note that prices do not appear in this expression because they are transfers
from the community platform to users. The first-best action profile,
$\textbf{a}^{W}$, maximizes this expression. The following theorem (built on
[15]) characterizes the first-best action profile.

###### Theorem 1

(due to [15]) _The first best involves $a_{i}^{\mathrm{W}}=1$ if_

$\sum_{j\in\mathcal{V}}\left(1-v_{j}\right)\frac{\left({\rm
Cov}\left(X_{i},X_{j}|a_{i}=0,\mathbf{a}_{-i}^{\mathrm{W}}\right)\right)^{2}}{1+\sigma_{j}^{2}-\mathcal{I}_{j}\left(a_{i}=0,\mathbf{a}_{-i}^{\mathrm{W}}\right)}\geq
0$ (3)

_and $a_{i}^{\mathrm{W}}=0$ if (3) is negative._

Implication \- To understand this result, consider first the case in which
there are no data externalities so that the covariance terms in (3) are zero,
except ${\rm
Cov}\left(X_{i},X_{i}|a_{i}=0,\mathbf{a}_{-i}^{\mathrm{W}}\right)=\sigma_{i}^{2}$,
so that the left-hand side is simply
$\sigma_{i}^{4}/\left(1+\sigma_{i}^{2}\right)$ times $1-v_{i}$. This yields
$a^{W}_{i}=1$ if $v_{i}\leq 1$ (thus a no externality setting becomes
mathematically equivalent to case when all users do not value their privacy
enough) . The situation is different in the presence of data externalities,
because now the covariance terms are non-zero. In this case, an individual
should optimally share her data only if it does not reveal too much about
users with $v_{j}>1$. Note here that the covariance matrix can be robustly
estimated from publicly observed $S_{i}$ values as dependencies are usually
preserved in the addition of noise within a threshold.

In this section, we adopt the more realistic assumption that, to start with,
the monopoly platform does not know the exact valuations of users in a
community (in contrast to assumptions made in existing works such as [19])
that are only private to them, but are informed that the valuations $v_{i}$
come from the cumulative distribution function $F_{i}$ and density function
$f_{i}$ (with upper support denoted by $v_{\max}$). We allow the platform to
design a pricing mechanism that elicits the true privacy valuations from the
users _(as Step 1 of the economy)_ , somewhat similar to the seminal Vickrey-
Clarkes-Groves (VCG) mechanism with a minor variation. More specifically, for
any user $i\in\mathcal{V}$ the price offered to user $i$ _(as Step 2 of the
economy)_ is equal to the surplus of all other users on the platform when user
$i$ is present minus by the surplus when user $i$ is absent. We consequently
have the following result.

###### Theorem 2

(due to [15]) _Let $\mathbf{v}$ be the reported vector of values of privacy.
Then the non-negative pricing scheme_ __

$p_{i}(\mathbf{v})=\left(\mathcal{I}_{i}(\mathbf{a}(\mathbf{v}))+\sum_{j\neq
i}\left(1-v_{j}\right)\mathcal{I}_{j}(\mathbf{a}(\mathbf{v}))\right)-\min_{\mathbf{a}\in\\{0,1\\}^{n}}\left(\mathcal{I}_{i}(\mathbf{a})+\sum_{j\neq
i}\left(1-v_{j}\right)\mathcal{I}_{j}(\mathbf{a})\right)$

_where
$\mathbf{a}(\mathbf{v})=\arg\max_{\mathbf{a}\in\\{0,1\\}^{n}}\sum_{i\in\mathcal{V}}\left(1-v_{i}\right)\mathcal{I}_{i}(\mathbf{a})$
incentivizes users to report their value of privacy truthfully._

###### Definition 2

_An equilibrium is a pair
$\left(\mathbf{a}^{\mathrm{E}},\mathbf{p}^{\mathrm{E}}\right)$ of functions of
the reported valuations $\mathbf{v}=$ $\left(v_{1},\ldots,v_{n}\right)$ such
that each user reports its true value and the expected payoff of the platform
is maximized. That is_,

$\begin{split}&\left(\mathbf{a}^{\mathrm{E}},\mathbf{p}^{\mathrm{E}}\right)=\max_{\mathbf{a}:\mathbb{R}^{n}\rightarrow\\{0,1\\}^{n},\mathbf{p}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}}\mathbb{E}_{\mathbf{v}}\left[\sum_{i=1}^{n}\mathcal{I}_{i}(\mathbf{a}(\mathbf{v}))-\sum_{i:a_{i}(\mathbf{v})=1}p_{i}(\mathbf{v})\right]\\\
&p_{i}(\mathbf{v})-v_{i}\mathcal{I}_{i}(\mathbf{a}(\mathbf{v}))\geq
p_{i}\left(\mathbf{v}_{-i},v_{i}^{\prime}\right)-v_{i}\mathcal{I}_{i}\left(\mathbf{a}\left(\mathbf{v}_{-i},v_{i}^{\prime}\right)\right),\forall
v_{i}^{\prime},\mathbf{v}:i\in\mathcal{V}\end{split}$

We now have the following theorem characterizing the equilibrium of the
monopoly market setting.

###### Theorem 3

(due to [15]) _For any reported vector of values v, the market equilibrium is
given by_ __

$\mathbf{a}^{\mathrm{E}}(\mathbf{v})=\operatorname{argmax}_{\mathbf{a}\in\\{0,1\\}^{n}}\sum_{i=1}^{n}\left(1-\Phi_{i}\left(v_{i}\right)\right)\mathcal{I}_{i}(\mathbf{a})+\Phi_{i}\left(v_{i}\right)\mathcal{I}_{i}\left(\mathbf{a}_{-i},a_{i}=0\right)$

_and_

$p_{i}^{\mathrm{E}}\left(v_{i}\right)=\int_{v}^{v_{\max}}\left(\mathcal{I}_{i}\left(\mathbf{a}^{\mathrm{E}}\left(x,\mathbf{v}_{-i})\right)-\mathcal{I}_{i}\left(\mathbf{a}_{-i}^{\mathrm{E}}\left(x,\mathbf{v}_{-i}\right),a_{i}=0\right)\right)\right)dx+v_{i}\left(\mathcal{I}_{i}\left(\mathbf{a}^{\mathrm{E}}\left(v_{i},\mathbf{v}_{-i}\right)\right)-\mathcal{I}_{i}\left(\mathbf{a}_{-i}^{\mathrm{E}}\left(v_{i},\mathbf{v}_{-i}\right),a_{i}=0\right)\right)$

_Moreover, all users report truthfully and thus the expected payoff of the
platform is_

$\mathbb{E}_{\mathbf{v}}\left[\max_{\mathbf{a}\in\\{0,1\\}^{n}}\sum_{i=1}^{n}\left(1-\Phi_{i}\left(v_{i}\right)\right)\mathcal{I}_{i}(\mathbf{a})+\Phi_{i}\left(v_{i}\right)\mathcal{I}_{i}\left(\mathbf{a}_{-i},a_{i}=0\right)\right]$

_where, $\Phi_{i}(v)=v+\frac{F_{i}(v)}{f_{i}(v)}$ is a non-decreasing function
representing the additional rent that a user will capture in incentive
compatible mechanisms._

Implication \- The theorem guarantees the existence of a unique monopoly
market equilibrium where community platform users elicit their true
valuations. A sufficient condition for $\Phi_{i}(v)$ to be non-decreasing is
for $\frac{f_{i}(v)}{F_{i}(v)}$ to be non-increasing. This requirement is
satisfied for a variety of distributions such as uniform and exponential [20].

We now investigate whether the reachable market equilibrium is efficient. We
have the following result, via [15], in this regard.

###### Theorem 4

(due to [15]) _1\. Suppose high-value users are uncorrelated with all other
users and $\mathcal{V}^{(l)}=\mathcal{V}_{\Phi}^{(l)}$, where
$\mathcal{V}_{\Phi}^{(l)}=\left\\{i\in\mathcal{V}:\Phi_{i}\left(v_{i}\right)\leq
1\right\\}$ denotes the set of users with $\Phi_{i}(v_{i})\leq 1$. Then the
market equilibrium is efficient.
2\. Suppose some high-value users (those in $\mathcal{V}^{(h)}$) are
correlated with users in $v_{\phi}^{(l)}$. Then there exists
$\overline{\mathfrak{v}}\in\mathbb{R}^{|V(n)|}$ such that for
$\mathbf{v}^{(h)}\geq\overline{\mathbf{v}}$ the market equilibrium is
inefficient.
3\. Suppose every high-value user is uncorrelated with all users in
$\mathcal{V}_{\phi}^{(l)},$ but users in a nonempty subset
$\hat{\mathcal{V}}^{(l)}$ of
$\mathcal{V}^{(l)}\backslash\mathcal{V}_{\phi}^{(l)}$ are correlated with at
least one high-value user. Then there exist $\overline{\mathbf{v}}$ and
$\tilde{v}$ such that if $\mathbf{v}^{(h)}\geq\overline{\mathbf{v}}$ and
$v_{i}<\tilde{v}$ for some $i\in\hat{\mathcal{V}}^{(l)}$, the market
equilibrium is inefficient.
4\. Suppose every high-value user is uncorrelated with all low-value users and
at least one high-value user is correlated with another high-value user. Let
$\tilde{\mathcal{V}}^{(h)}\subseteq\mathcal{V}^{(h)}$ be the subset of high-
value users correlated with at least one other high-value user. Then for each
$i\in\tilde{\mathcal{V}}^{(h)}$ there exists $\bar{v}_{i}>0$ such that if for
any $i\in\tilde{\mathcal{V}}^{(h)}v_{i}<\bar{v}_{i},$ the market equilibrium
is inefficient.
5.The social surplus at market equilibrium, $\mathbf{a}^{\mathrm{E}}$, for any
v (either known truthfully or otherwise)_

$SoS\left(\mathbf{a}^{\mathrm{E}}\right)\leq\sum_{i:v_{i}\in\mathcal{V}^{(l)}}\left(1-v_{i}\right)\mathcal{I}_{i}(\mathcal{V})-\sum_{i:v_{i}\in\mathcal{V}^{(h)}}\left(v_{i}-1\right)\mathcal{I}_{i}\left(\mathcal{V}_{\phi}^{(l)}\right)$

Implication \- The theorem provides the conditions under which market
equilibrium in a monopoly community information trading setting is utilitarian
welfare (in)efficient. Note that the market efficiency results in the theorem
are conservative in the sense we assume user privacy valuations are unknown in
the worst case, and the platform does its best to elicit true valuation
responses. Inefficiency in this setting would imply inefficiency in the case
where community user valuations are untruthful (see point #5 in the theorem).
According to the points in the theorem, it is evident that the information
trading market is efficient is when high-value users (those with both
$v_{i},\Phi_{i}(v_{i})\geq 1$ are uncorrelated with all other users -
something practically rare to achieve, and low-value users have
$\Phi_{i}(v_{i})$ values less than one (note here that low-value users always
have $v_{i}<1$ but could have $\Phi_{i}(v_{i})$ values greater than 1). Not
satisfying either will lead to incentive compatibility conditions preventing
efficient allocation. In all other cases, the information trading market is
inefficient (SoS at equilibrium is not optimal), and the extent of
inefficiency depends on whether high-value users are correlated with low-value
users with $\Phi_{i}(v_{i})$ values greater or less than one.

##  6 Examples

In this section, we provide numerical examples (as in [15]) to lucidly
illustrate (a) the existence of a data trading market equilibrium, and (b) the
social surplus (SoS) zone at market equilibrium.

Example 1. Suppose there are two users 1 and 2 with covariance matrix
$\Sigma=\left(\begin{array}[]{lr}1&\rho\\\ \rho&1\end{array}\right)$ and
$v_{1}=v_{2}=v.$ When
$p_{1},p_{2}\in\left[\frac{\left(2-\rho^{2}\right)^{2}}{2\left(4-\rho^{2}\right)},\frac{1}{2}\right],$
both action profiles $a_{1}=a_{2}=0$ and $a_{1}=a_{2}=1$ are user equilibria.
This is a consequence of the submodularity of the leaked information function:
when user 1 shares her data, she is also revealing a lot about user 2 , and
making it less costly for her to share her data. Conversely, when user 1 does
not share, this encourages user 2 not to share. Despite this multiplicity of
user equilibria, there exists a unique (Stackelberg) equilibrium for this game
given by $a_{1}^{\mathrm{E}}=a_{2}^{\mathrm{E}}=1$ and
$p_{1}^{\mathrm{E}}=p_{2}^{\mathrm{E}}=\frac{\left(2-\rho^{2}\right)^{2}}{2\left(4-\rho^{2}\right)}.$
This uniqueness follows because the platform can choose the price vector to
encourage both users to share. _The next example suggests that though there
may be multiple equilibria in the Stackelberg game, all of them yield the same
payoff for the community platform._

Example 2. Suppose there are three users with the same value of privacy and
variance: $v_{i}=1.18$ and $\sigma_{i}^{2}=1$ for $i=1,2,3.$ We let all off-
diagonal entries of $\Sigma$ to be $0.3.$ Any action profile where two out of
three users share their information is an equilibrium, and thus there are
three distinct equilibria. But it is straightforward to verify that they all
yield the same payoff to the platform.

The following example illustrates the social welfare zone at market
equilibrium with variations in the correlation coefficient $\rho$ and privacy
valuation for high-value community users.

Example 3. We consider a setting with two communities, each of size 10.
Suppose that all users in community 1 are low-value and have a value of
privacy equal to 0.9, while all users in community 2 are high-value (with
$v_{h}>1$ ). We also take the variances of all user data to be 1, the
correlation between any two users who belong to the same community to be 1/20,
and the correlation be- tween any two users who belong to different
communities to be $\rho$. depicts equilibrium surplus as a function of $v_{h}$
and $\rho$.

_Two points are worth noting from this example. First, relatively small values
of the correlation coefficient $\rho$ are sufficient for social surplus to be
negative. Second, when $v_{h}$ is very close to 1, the social surplus is
always positive because the negative surplus from high-value users is
compensated by the social benefits their data sharing creates for low-value
users._

##  7 Conclusion

In this paper, we mathematically argued, using recent developments in [15],
that social community data trading is not economically welfare efficient in a
monopoly market setting, going against the popular economic
philosophy/intuition that increased amounts of end-user data signals in a
market improves utilitarian social welfare. The primary reason behind our
result is the significant negative externality (via user signal correlations)
generated by privacy breaches in the information market that cannot be
cancelled out via market equilibrium prices handed over to the users for their
information.

##  8 Acknowledgement

This work NSF-supported under grants CNS-1616575, CNS-1939006, CNS-2012001,
and ARO W911NF1810208.

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