A cooperative game-theoretic approach to quantify the
value of personal information in networks
Michela Chessa
EURECOM
[email protected]
ABSTRACT
The explosion of online business models based on the exploitation of users personal data created a surge of interest
on how to quantify the value of personal information. Recent studies tackling this question, however, have focused
only on the private value of personal information, i.e., the
private cost incurred by users when releasing their information. When users are connected through a social network,
personal information also has a public component: information revealed by a user can benefit his neighbors (e.g., if they
have similar tastes, by improving the recommendations he
receives).
In this paper, we propose a cooperative game-theoretic approach to quantify the value of personal information in networks. We consider a model where users are embedded in a
graph and select the level of personal information to disclose.
Users utility contains the benefits they derive from information modeled as a local public good (each user benefits from
information disclosed by himself and his neighbors), and the
private cost of disclosing information. We propose a natural
extension of this model to a cooperative game and we propose to use classical allocation solutions (the core and the
Shapley value) to quantify the value of personal information.
We show that the game has good properties (in particular
superadditivity) and that the resulting allocations are wellbehaved. Finally, we analyze the impact of the graph on the
values and the resulting incentives for users to creates new
links.
Categories and Subject Descriptors
J.4 [Social and behavioral sciences]: Economics; K.4.1
[Computers and society]: Public Policy Issues—Privacy;
H.4 [Information systems applications]: Miscellaneous;
G.2.2 [Discrete Mathematics]: Graph Theory—Network
problems
General Terms
Patrick Loiseau
EURECOM
[email protected]
Theory
Keywords
Personal Data, Social Network, Cooperative Game Theory,
Core, Shapley Value
1. INTRODUCTION
1.1 Motivations
Inarguably, the Internet has become an essential part of the
citizens’ life and of the economy in many countries. In this
new ecosystem, the overwhelming success of social networking platforms allows the collection of huge amounts of personal data. This data has intrinsic economic value for a
number of social networking applications, such as recommendations and targeted advertising, and this value is extensively exploited by online services. At the same time, also
users benefit from this ecosystem by being granted access to
many services in exchange of their personal data. However,
as such applications are exploding, users are starting to demand appropriate compensation for the release of their valuable data, for instance through data monetization. In order
to guarantee the durability of such an ecosystem and to enable exploring its full potential through a transparent and
competitive Internet in the interest of both the users and the
online service providers, it is essential to understand how to
quantify the value of personal data.
To address the key question of how to quantify the value of
personal data, we propose a game-theoretic approach relying on a model of users and their interactions through the
system. In the choice of how to model and to approach this
task, we identified three crucial points. Firstly, the intrinsic
economic value of personal data is not on the data itself, but
on the information which is possible to derive from it, and
giving a value to information is a difficult task which has
always been problematic in economics [3]. In our model, we
refer to some of the economic literature (e.g., [33]), which
proposes to approach the problem treating information (and
consequently, personal data) as a public good, mainly with
the arguments that it has infinite reproducibility at zero
cost and it is non-excludable among users. Secondly, in a
public good model, cooperation may improve the aggregate
utility of the users upon the inefficiency of the outcome of
a strategic equilibrium. Because of that, economies with
many producers of a public good often converge to the cooperation of the users if this is in their own interest, i.e., if
the final utility share will be beneficial to everyone. Lastly,
online services allow collecting data not only about individual users, but also about the social interaction among them.
In this setting, the graph representing the social relations
between the users plays a crucial role, as some information
about some users may be extracted from the personal data of
the users they are connected with, even though they did not
originally disclose the information. In our analysis, we take
into account these local externalities including in the model
the social networks in which the users are embedded. We
consider the local public good aspect of information, and we
model a situation in which the users can decide to voluntary
cooperate in disclosing their information.
1.2
Summary of our Contributions
In this work we propose a model which takes into account the
three points we have listed above which are crucial to model
personal information. We then propose solutions based on
this model to quantify the value of personal information. In
particular, we focus on the role played by the network and
on the way it influences these solutions. More specifically,
our contributions can be summarized as follows.
First, we propose a model of the interactions between users
who choose strategically the amount of personal information
that they wish to reveal. Our model is a local public good
strategic game inspired from the model of Bramoullé and
Kranton [7], in which the users are embedded in a social
network. Each user’s utility is the difference of a benefits
from information disclosed by herself and her neighbors (the
local public good) and a cost of disclosing information (due
for instance to privacy concerns).
Then, we extend the strategic model to a cooperative game.
When the game is initially presented in strategic form, as
in our case, there are different ways of defining its cooperative extension differing on how the interaction between a
subset of users and its complementary set is modeled. Our
extension assumes that, given a subset of users and the network which represents their relations, they maximize their
aggregate utility, while the other users do not take part in
the game and they do not receive any external utility. This
extension follows in spirit some other extensions previously
adopted in studies of economies with public goods (see Section 2), but extends those to also include the network. We
then prove that the resulting cooperative game is equivalent
to the minimax extension proposed by von Neumann and
Morgenstern [37], from which we can deduce that our game
has the nice properties of monotonicity and superadditivity.
Third, we propose solutions of the cooperative game as ways
to quantify the value of personal information released by the
users. The superadditivity guarantees that it is beneficial for
all the users to cooperate to jointly disclose personal information when the utility is shared according to the solutions
proposed. Specifically, as solutions, we propose the core
and the Shapley value. We prove that the core is nonempty.
This is important because the core has a very nice property
of stability: choosing an allocation in the core, no coalition will have incentives to stop cooperating with the other
players. Alternatively, we also propose to use the Shapley
value. Even if this solution may not be stable, it has other
potentially very desirable properties such as symmetry.
Finally, we analyze how the proposed solutions to quantify
the value of personal information depends on the graph. We
use network games, an extension of cooperative games proposed by Jackson and Wolinsky [22] which permit to model
the dependence of the value function on the graph, to carry
the analysis. In particular, we show that it is beneficial for
the players to create new links and that the only stable network is given by the complete graph, in which each player
is connected to every other player.
1.3
Outline
The remainder of the paper is organized as follows. We review related works in Section 2. In Section 3 we present the
local public good game which models the strategic interactions of the players. Section 4 describes the cooperative
extension, i.e., the model when the players are allowed to
cooperate, and analyzes some properties of the game. In
Section 5, we propose some solutions to quantify the value
of personal information. We study how the graph influence
the model and the solutions in Section 6. Finally, we conclude in Section 7.
2.
RELATED WORKS
Economics of personal data. Academic concerns about
the economic value of personal data date back to the 90’s.
In 1996, Laudon [25] proposed to create a regulated market
of personal data (that he calls “national information market”) and discussed informally some examples and implications. At the same time, Varian [36] also discusses informally ideas about the value of private data and how property
rights should be assigned. More recently, some experimental
studies have been conducted that aim to quantify the value
that users assign to their personal data in different scenarios: Huberman et al. [19], Acquisti et al. [2] and Acquisti
and Grossklags [1] (among others).
The game-theoretic analysis of the private data monetization was pioneered by Kleinberg et al. [24], who proposed
fair compensation mechanisms for personal data based on
cooperative game theory. Their solutions are based on the
core and on the Shapley value like ours, but their simple
game model does not take into account the public good nature nature of personal information which is the main focus of our paper. Recently, a significant thread of research
started on selling personal data using auctions. This was initiated by Ghosh and Roth [17], who studied models where
an analyst runs an auction to buy bits of data from users
with the goal of computing the average of the population’s
bits. They derive near-optimal auctions in the case where
the data valuation is independent from the data itself. A
similar study is conducted by Dandekar et al. [12] in the
case where the analyst’s goal is to estimate the parameters
of a linear model. Ghosh and Roth [17] also emphasize the
inherent bias in auctions when independence between the
data and its valuation does not hold. Elements of solutions
to this issue are proposed by Roth and Schoenebeck [32]
and Ligett and Roth [27]. In all those works, the loss of
privacy by releasing data is quantified using differential privacy [13]. Riederer et al. [31] propose a mechanism called
“transactional privacy” where users can sell access to their
data through an unlimited supply auction; but they do not
analyze this mechanism.
All the aforementioned works only consider the private cost
of revealing information, i.e., the cost incurred by an individual on account of his loss of privacy independently of
the data revealed by others. In a recent work, Ioannidis
and Loiseau [20] propose a different model which takes into
account the public good nature of information: each user
selects the precision of information he reveals to minimize
a cost that comprises a privacy cost and an estimation cost
which decreases when other users increase the precision of
information they reveal. They analyze the game as a noncooperative game and provide results on Nash equilibrium
and on the price of anarchy. Their model, however, does not
take into account the local interactions that happen within
a social network and the fact that information revealed by
a user may not benefit all other users equally.
Network games. In this paper, we consider users embedded in a social network and use a graph to model the
local public good nature of personal information. Games
on graphs have been studied in different contexts and with
different utility functions of the players, see e.g., the books
of Jackson [21] and Easley and Kleinberg [14] and the recent survey by Jackson and Zenou [23]. In particular games
on graphs have recently been studied in two situations: to
model pricing problems with network externalities [8, 5, 15,
16, 11] and to model strategic propagation representing for
instance product adoption [18, 26]. But the literature on the
strategic interaction of the agents in networks is even larger
[35, 4] and, among many others, we mention Bramoullé and
Kranton [7], who described the model we adopt in this paper. These papers differ in the utility model and interactions that they consider. All of these works, however, consider a non-cooperative setting at the exception of Jackson
and Wolinsky [22] who partly use a cooperative approach
to model the strategic interaction of the agents on the link
formation. Moreover, to the best of our knowledge, no prior
work has used games on graphs to model questions related
to personal information and quantification of its value.
Cooperative games for public goods. In this paper, we
use a cooperative game-theory approach: we transform our
local public good model in a cooperative game and use classical allocation solutions to quantify the value of personal information. The use of cooperative game theory methods to
analyze public good problems is standard in the economics
literature. We mention in particular the works of Champsaur [9], Moulin [29] and Chander [10]. To the best of our
knowledge, however, no prior work has analyzed the impact
of possible variations of the network in a the local public
good problems using cooperative game-theory methods.
3.
THE MODEL
In this section, we describe our basic model, in which users
are embedded in a social network and choose strategically
the level of personal data to reveal.
3.1
The Social Network as a Graph
Let N = {1, . . . , n} be a set of players, representing the
users. In our model the players are embedded in a social
network (e.g., the Facebook network), that we represent by
an undirected graph, where the nodes identify the players
and the links their pairwise relations. Formally, an undirected graph g on N is a set of links, i.e., a set of unordered
pairs of players {i, j}, that for simplicity we denote by ij.
For any pair of players i and j, ij ∈ g indicates that players i and j are linked in the graph g (in our example, the
users are “Facebook friends”). Let g N be the complete graph,
i.e., the set of all the unordered pairs of players in N and
G(N ) = {g ⊆ g N } be the set of all the graphs on N . Given
g ∈ G(N ), let Ni (g) = {j ∈ N \ {i} : ij ∈ g} be the set of
neighbors of i in g, i.e., the set of players which are linked to
player i by the graph g, and ni (g) = |Ni (g)| its cardinality.
Player i’s neighborhood in g is defined as herself and her set
of neighbors, i.e., as {i} ∪ Ni (g).
3.2
The Local Public Good Game
In this section, we propose a strategic model of information
disclosure, where information is treated as a local public
good. In particular, we adopt the model in [7] for public
goods in networks to describe the strategic interaction between the users, where the set N of players represents the
users and the undirected graph g represents the social network in which the players are embedded, as seen in Section
3.1. Throughout the paper, at the exception of Section 6,
the graph g is supposed to be a fixed parameter (we refer
to Section 6 for an analysis of our results when the graph
is assumed to be a variable of the model). Each player in
N chooses an effort level ei ∈ [0, +∞) in disclosing personal
information. Depending on the situation at stake and its
interpretation, this effort level can represent the precision in
revealing a single piece of information or the total amount
of information revealed. Let e = (e1 , . . . , en ) denote an effort profile of all players. Our goal is to model situations
in which information disclosed by a user benefits its neighbors. This situation typically occurs with personalization
applications: since neighbors generally have similar tastes,
information revealed by a user leads to better personalization for his neighbors. To model this situation, we consider
a model with positive externalities between neighbors. More
specifically, we assume for simplicity that a neighbor’s effort
is a perfect substitute with one’s own, but a player does not
derive benefits from the effort of players whom she does not
share a link with. A player i’s utility from profile e in graph
g is then given by


X
Ui (e, g) = f ei +
ej  − kei ,
(1)
j∈Ni (g)
where f is a twice differentiable strictly concave function,
with f (0) = 0, f 0 > 0 and f 00 < 0 and k is the cost to
player i for one unit of effort level. The first component f
captures the utility that a player receives when the analyst
gets some information from her and her neighbors; in particular, it is an increasing function of the total level of effort of
the neighborhood of player i. The second component is the
privacy cost; it represents the cost that player i incurs (for
instance due to loss of privacy) by making a given effort and
then it depends only on ei . Note that our model includes
only positive externalities, i.e., benefits obtained by a user
when her neighbors disclose information. We neglect potential negative externalities that could arise; for instance extra
privacy cost that a user could incur if her neighbors disclose
information, due to inference of information about herself.
This assumption is well justified in many situations we intend to model. Assume, for example, that the information
revealed is used by an online service for targeted political
advertising. A user can benefit from a good advertisement,
getting to know a candidate she would like to vote for, but
her loss of privacy for the inferred information about her is
negligible, unless someone gets to know, for sure, who she
is going to vote for (and this happens only if she chooses to
reveal her political position).
We define a n-player strategic-form game
Γ = hN, [0, +∞)n , (Ui )i∈N i ,
where N is the set of players, each player has strategy space
given by [0, +∞) and her utility function is defined in (1).
We define the aggregate utility of the players in N , given an
effort profile e and the graph g, as the sum of their utilities,
i.e., as
X
W (e, g) =
Ui (e, g).
(2)
some cooperative solutions will suggest how to quantify the
value of the personal information of the players. Some cooperative concepts, such as the core, have already been applied
to problems of costs or benefits sharing in economic models
with externalities (see Section 2). The common idea is to
define a cooperative game that, starting from the definition
of the n-player strategic-form game, may describe the utility
generated by each possible coalition of players and then to
analyze the resulting cooperative game. We follow a similar approach, but we include in the analysis the assumption
that the players are embedded in a social network and the
externalities have only a local effect.
We first recall some definitions of cooperative game theory
which are useful in the following analysis.
4.1
i∈N
We say that a profile e is efficient for a given graph g if and
only if there exists no other profile e0 such that W (e0 , g) >
W (e, g). The authors in [7] show the existence of a Nash
equilibrium of Γ, which may not be unique. Moreover,
through a welfare analysis, they show that a Nash equilibrium of this game is in general non efficient. In this setting,
the only way for the players to choose a strategy profile
which maximizes the social welfare is by agreeing to act cooperatively. In the following section, we propose a cooperative extension of this model in which the players can decide
to make binding agreement and to coordinate to choose an
efficient profile.
In our model, both the function f and the cost k are independent of the player i. When collecting real data on a
social network, an online service may easily obtain information about the graph structure of its users, but not as easily
about their privacy concerns. Many works have studied the
problem of estimating how much individuals value their personal data, for example implementing some auction models
(see Section 2), but implementing these mechanisms can be
costly and time consuming for the online service. In this paper, we take an orthogonal approach and choose to propose
solutions to quantify the value personal data when the users
differ only due to their position in the social network, but
are otherwise totally symmetric.
4.
A COOPERATIVE EXTENSION OF THE
MODEL
Public good models often yield non-efficient equilibria. This
kind of models often arises in normative situations, where
the players may be obliged to follow some rules which lead
them to efficiency. This is not the case of our model, where
the users voluntary decide to reveal their personal data.
However, the purpose of achieving efficiency may lead them
to create binding agreement with some other players to commit to play an efficient strategy. In the case of an online
service, this cooperation may be implemented through the
provider which would only give users the possibility of opting
in or not and then impose the level of effort of each player
to social optimum given the subscribed users. The modeling
of such a situation may be done using a cooperative gametheoretic approach; a cooperative analysis will allow us to
predict which coalition will form and the implementation of
Background on Cooperative Game Theory
In a cooperative setting, given a set N of players, we refer to
a subset S ⊆ N as a coalition, with s = |S| its cardinality,
and to N as the grand coalition. Given a coalition S, we refer
to T ⊆ S as a subcoalition of S. A characteristic function is
a function v : 2N → R that associates a real number v(S) to
every coalition S. We impose the normalization condition
v(∅) = 0. For every coalition S, the number v(S) is the total
transferable utility (TU) that is available for division among
the players in S and is called the worth of the coalition. We
denote by V the set of all possible characteristic functions.
A cooperative (TU) game is a couple (N, v) where N is a set
of players and v ∈ V a characteristic function among them.
As we assumed that the players in N are embedded in a
social network represented by a graph g ∈ G(N ), given a
coalition S ⊆ N , we may define the subgraph of g restricted
to S as g|S = {ij|ij ∈ g, i, j ∈ S} ∈ G(S). This is the
subgraph on S obtained by deleting all the links except those
that are between players in S and it represents the social
network restricted to the coalition S of players (observe that
g|S ∈ G(S) means that the remaining players in N \ S are
not considered as singletons, but they are removed).
4.2
The Cooperative Extension of the Local
Public Good Game
In this section, we extend the model proposed in Section 3.2
assuming that the players can choose to cooperate. In particular, given the set of players N , for each coalition S ⊆ N ,
we will define its worth v(S) as the maximal aggregate utility the players in S can get.
Given a coalition S ⊆ N , and, given the graph g, we consider the subgraph of g reduced to S, g|S . Each player in S
can reveal some private information and we denote by eS a
strategy profile for the players in S. The utility of a player
in S is still defined as in (1), where the set of neighbors is
now restricted to the set of neighbors who are in S. Formally, when restricting the analysis to the coalition S, for
each player i ∈ S,


X
Ui (eS , g|S ) = f ei +
ej  − kei .
(3)
j∈Ni (g|S )
We define the aggregate utility of the coalition S given an
effort profile eS as the sum of the utilities of the players in
S on g|S , and we denote it by W (eS , g|S ). Observe that the
aggregate utility of coalition S depends only on the pairwise
relations between its player. Then, the model describes well
also situations in which the graph structure is not know a
priori, but in which the players who are willing to participate
reveal their internal links (and they are not authorized, for
example, to reveal links with other players who are not part
of the coalition). As we are assuming that the players inside
coalition S are cooperating, it is also natural to assume that
these players wish to coordinate in order to maximize their
aggregate utility and they choose jointly an efficient profile
that we denote by e∗S .
We define the cooperative game (N, v) where the worth of a
coalition S is the maximal aggregate utility the coalition can
reach by cooperating, i.e., for each S ⊆ N , the characteristic
function v is defined as
v(S) = W (e∗S , g|S ) = max W (eS , g|S ),
Basic Properties of the Cooperative Game
eS eN \S
=
≤
min
σS (eS )σN \S (eN \S ) f ei +

X

ej  − kei  .
X
max
σN \S ∈∆(EN \S ) σS ∈∆(ES )
max
σS ∈∆(ES )
Ui (σS , σN \S )
i∈S

X X

σS (eS )Ui (eS , eN \S )|e

N \S =0

eS
i∈S

=
max
σS ∈∆(ES )
 


X X
X

σS (eS ) f ei +
ej  − kei 
i∈S
eS
j∈Ni (g)

=
max
X
σS ∈∆(ES )
σS (eS ) 
eS
 
X
f ei +
i∈S
= max
eS ∈ES
X

X
f ei +
i∈S

X

ej  − kei 
j∈Ni (g)
 

ej  − kei 
j∈Ni (g)
= v(S),
where the inequality is because we are minimizing on a
smaller set. Moreover,
v 0 (S) =
=
≥
min
X
max
σN \S ∈∆(EN \S ) σS ∈∆(ES )
max
=
max
σS ∈∆(ES )
σS (eS ) 
eS
 
X
f ei +
i∈S
 
eS ∈ES
X
i∈S
Ui (σS , σN \S )
i∈S

X
Ui (σS , σN \S )
i∈S
X
min
σS ∈∆(ES ) σN \S ∈∆(EN \S )
= max
 
eS eN \S
Proof. At first, we can observe that
i∈S
where ∆(ES ) is the set of correlated strategies available to
coalition S. We let Ui (σS , σN \S ) denote player i’s expected
payoff when the correlated strategies σS and σN \S are implemented, that is
X X
Ui (σS , σN \S ) =
σS (eS )σN \S (eN \S )Ui (eN , g)
X X
for each S ⊆ N .
v (S) =
Deriving a characteristic function from a n-person strategicform game is a standard issue in game theory [30]. When
analyzing cooperative games associated to economies with
public goods and externalities, defining a worth for the coalition S requires to also take into account the strategic choices
made by players who are not members of S. In particular,
the characteristic function should specify explicitly the actions of both the players who are and who are not members
of the coalition. A standard way to get around this problem
has been in the literature to assume that the players outside
the coalition choose the strategies who are the less favorable
to the coalition. We refer in particular to the way of deriving a characteristic function from a n-person strategic-game
form proposed by Von Neumann and Morgenstern [37]. Following their model, the characteristic function starting from
our game Γ is defined ∀S ⊆ N as
X
v 0 (S) =
min
max
Ui (σS , σN \S , g), (5)
=
v(S) = v 0 (S)
0
where ES = [0, +∞)s . Note that even if there exist multiple
efficient cooperation strategies, as they all provide the same
joint cost, the cost function is univocally defined.
σN \S ∈∆(EN \S ) σS ∈∆(ES )
Proposition 1. The cooperative game (N, v) is equivalent to the minimax representation of the game Γ, i.e.,
(4)
eS ∈ES
4.3
Games with characteristic function of a type similar to v are
common in studies of economies with public good (see Section 2). Our characteristic function takes into account the
structure of the social network and the local nature of our
public good. Before studying the properties of the cooperative game (N, v), in the following proposition we show that
the definition of our characteristic function v in (4) and of
the minimax representation v 0 in (5) coincide for our game.
f ei +

ej  − kei 
j∈Ni (g)

X

X

ej  − kei 
j∈Ni (g)
= v(S).
j∈Ni (g)
The idea behind the definition of von Neumann and Morgenstern is to assert that the value of coalition S, v(S), is
the maximum sum of utility payoffs that the players of coalition S can guarantee themselves against the best offensive
threat by the complementary coalition N \ S. The game
(N, v 0 ) is called the minimax representation in cooperative
form of the strategic-form game Γ with transferable utility.
This equivalence is very useful to establish properties of the
game (N, v). We list two important properties. A cooperative game (N, v) is said to be monotonic if
S ⊆ T ⊆ N ⇒ v(S) ≤ v(T ),
meaning that a larger coalition has always a larger worth.
A cooperative game (N, v) is said to be superadditive if
S ∩ T = ∅ ⇒ v(S) + v(T ) ≤ v(S ∪ T ).
Given the game (N, v), the set of imputation is defined as
(
)
X
n
I(v) = x ∈ R |
xi = v(N ), xi ≥ v({i}) ∀i ∈ N .
i∈N
Superadditivity implies that the worth of the coalition N of
all players is at least as large as the sum of the worths of
the members of any partition of N . This property ensures
that the players have incentives to form the grand coalition
in order to maximize their worth. The following theorem
states that our game satisfies these two properties.
Theorem 1. The game (N, v) is monotonic and superadditive.
Proof. Superadditivity follows from Proposition 1, because the minimax representation of a game in strategic form
is always superadditive. Moreover, superadditivity implies
monotonicity.
The superadditivity of our game ensures that, if the players
wish to create binding agreement in order to maximize their
aggregate utility, then they will agree to form the grand
coalition. With the aim of proposing some cooperative solutions as methods to quantify the value of personal information, in the next section we assume that the grand coalition
forms and that the total utility to be shared between the
players is equal to v(N ).
5.
COOPERATIVE SOLUTIONS TO QUANTIFY THE VALUE OF PERSONAL INFORMATION
In the previous section we have defined the cooperative game
(N, v), which is the extension of the n-player strategic-form
game Γ, when the players decide to coordinate to earn a
higher aggregate utility. In particular, we have seen that
the superadditivity of the game (N, v) allows us to predict
that, when the players wish to create binding agreement in
order to maximize their aggregate utility, the grand coalition
N will form. In this section, we tackle the problem of how to
share the worth of the grand coalition between the players
in N . In fact, to take part in the game, the players will
demand an adequate reward for their contribution. In our
model, sharing the utility represents the way to quantify the
personal information of each player, based on how much she
contributes to the aggregate utility of any possible coalition
of players. Cooperative game theory provides many different
solution concepts to solve the problem. In this paper, we
propose two of them, which are the most classical ones: the
core and the Shapley value.
A payoff allocation is a vector x = (x1 , . . . , xn ) ∈ Rn ,
where each component xi is interpreted as the utility payoff
to player i. A solution for the game (N, v) is a function
which associates to every characteristic function v a (possibly empty) set of payoff allocations. In a cooperative situation, before taking part in the game, the players in N
must agree to a solution. Then, for the players to be able to
agree, the solution must propose a “reasonable” allocation.
In particular, a payoff allocation is often required to belong
to the set of imputations.
P
where the first property ( i∈N xi = v(N )) is referred to
as efficiency and the second one (xi ≥ v({i}) ∀i ∈ N ) as
individual rationality. Efficiency guarantees that the whole
amount v(N ) is distributed and moreover that the allocation
is feasible, i.e., it does not propose to share an amount which
is bigger than the total available utility. Individual rationality means that the solution has to assign to a player at least
what she is able to do by herself; otherwise the player will
deviate and will not participate to the game.
The first solution we propose, the core, provides as possible payoff allocations a subset of the set of imputation. In
particular, the core is given by all the imputations which
have also the property of coalitional rationality. Formally,
the core of the game (N, v) is defined as
(
)
X
C(v) = x ∈ I(v)|
xi ≥ v(S), ∀S ⊂ N .
i∈S
The fundamental idea of the core is that an agreement among
the players in N can only be binding if every coalition S ⊂ N
receives collectively at least its worth, i.e., the value that
the players in S could obtain without collaborating with the
players in N \ S. In this way, not only single players, but
also every coalition in N does not have incentives to deviate.
The core is a really appealing solution due to this stability
property. Unfortunately, the core may be empty. The emptyness of the core provides a very important property of the
game, which is that no matter what payoff allocation may
occur, there is always a coalition that could gain by deviating. The following result guarantees that in our game this
is not going to happen and it is possible to provide a payoff
allocation which is stable, in the sense of the core.
Theorem 2. The game (N, v) has a nonempty core.
To prove the nonemptyness of the core, we introduce the
notion of balancedness. A family B of coalitions is called
a balanced collection provided that for each
P S ∈ B, there
exists λS > 0 such that, for all i ∈ N ,
S∈B, B3i λS =
1. A very important result, known as Bondareva-Shapley
Theorem ([6] and [34]), states that the cooperative game
(N, v) has a nonempty core if and only if for every balanced
collection B,
X
λS v(S) ≤ v(N ).
(6)
S∈B
We use this result in our proof of the non-emptiness of the
core below.
Proof of Theorem 2. To prove the non-emptyness of
the core, we show that (N, v) verifies (6). At first, we write
the characteristic function v as
(
X
v(S) = max
Ūi (f i )|f = (f 1 , . . . , f n ) ∈ [0, +∞)2n ,
f
i∈S
fij = 0 ∀ij ∈
/ g|S , fij = fji ∀i, j ∈ S} ,
P
where Ūi (f i ) = f ( j∈N fij )−kfii . This definition is equivalent to the definition in (4). Then let B be any balanced
collection, and for each S ⊆ N , let f S = (f S
i )i∈S be an
arg max allocation which provides v(S). Define the allocation f , where for each i ∈ N , f i is the following convex
combination of the f S
i ,
X
fi =
δS f S
i .
S∈B, B3i
The allocation f is feasible for v(N ), since for each i, j ∈ N
X
X
fij =
δS f S
δS f S
ij =
ij
S∈B, B3i
X
=
S∈B, B3i,j
δS f S
ji = fji .
S∈B, B3i
Then, by the concavity of the Ui , we have that
X
δS v(S)
S∈B
=
X
δS
S∈B
X
Ui (f S
i ) =
i∈S
X
X
δS Ui (f S
i )
i∈N S∈B, S3i
!
≤
X
i∈N
Ui
X
S∈B, S3i
δS f S
i
=
X
Ui (f i ) ≤ v(N ).
i∈N
This proves (6) and, consequently, the nonemptyness of the
core.
The second solution we propose is the Shapley value. Differently from the core, this solution proposes a unique payoff
allocation and it is always defined. The Shapley value, that
we denote by φ, is defined by the following equation: for
every i ∈ N and every v ∈ V
X s!(n − s − 1)!
(v(S ∪ {i}) − v(S)). (7)
φi (v) =
n!
value function (4) involving the max function for each coalition, the convexity of our game does not follow from any
classical result. In future work, we plan to investigate further the convexity of the game, as proving convexity (if it
holds) would allowing adopting the Shapley value as stable
solution.
6.
THE IMPACT OF THE NETWORK ON
THE MODEL AND ON THE ALLOCATION OF UTILITIES
Untill now, we have assumed that the graph g was a fixed
parameter and we have designed a model and proposed some
solutions which strongly depend on this network structure.
In this section, we analyze the importance of this network,
by considering the graph as a variable of the model. In particular, we want to understand how the two solutions we
proposed in the previous section, the core and the Shapley
value, change when we modify the graph and if they can
induce the players to create or to remove some links strategically, in order to get a higher payoff. To this end, we
use the formalism introduced by Jackson and Wolinsky [22],
who propose to define the notions of network games and
allocation rules for network games as natural extensions of
cooperative games and of allocation rules for cooperative
games, respectively. These notions allow keeping track of
the overall value generated by a particular network as well
as how it is allocated across players, and of the way they
vary depending on the graph. In this last part of the paper,
we write our cooperative game (N, v) as a network game, in
order to focus on the graph structure in which the players
are embedded.
First, we introduce some additional notations about graphs,
which are complementary to the ones given in Section 3.1
and which are necessary for the last part of our analysis.
S⊆N \{i}
Equivalently, the Shapley value can also be defined axiomatically as the unique solution satisfying the following four
properties. The first one is (a) efficiency: it is the same
property we have already discussed defining the set of imputations. Second, (b) symmetry: if two players i and j
contribute to any coalition in exactly the same way, they
must be granted the same payoff. The third one is (c) null
player : when a player contributes nothing to any coalition,
should have zero. Last, (d ) additivity: the Shaplye value
of the sum of two games has to be equal to the sum of the
Shapley value of the two games.
Due to these four properties, also the Shapley value represents a very appealing solution for cooperative games. In
particular, the superadditivity of our game ensures that the
Shapley value is an element of the set of imputations, meaning that it is stable with respect to possible deviations of
single players. Unfortuantely, differently from the core, the
Shapley value may not have the property of coalitional rationality. On the other hand, allocations in the core (different
from the Shapley value, if it is one of them) do not have all
the properties satisfied by the Shapley value. The convexity of the game (N, v) would ensure that the Shapley value
is an allocation in the core, and then it is also coalitional
rational. However, due to the particular structure of our
6.1
Background on Graphs
In this section, we assume that a graph g ∈ G(N ) may contain loops, where a loop is a link ii which connects a player
with herself.1 Allowing the presence of loops will be necessary in order to be able to extend our model to the model of
Jackson and Wolinsky [22], as we explain in the next section.
Given the set of players N and a graph g ∈ G(N ), we denote
by g + ij = g ∪ {ij} the graph obtained by adding the link ij
to the existing graph g and by g − ij = g \ {ij} the graph obtained by removing the link. Let N (g) = {i|∃j s.t. ij ∈ g}
be the set of players who have at least one link in the graph
g. A path in g connecting i1 and it is a set of distinct nodes
{i1 , . . . , it } ⊆ N (g) such that {i1 i2 , . . . , it−1 it } ⊆ g. The
subgraph g 0 ⊆ g is a component of g if either (a) N (g 0 ) = {i}
and for all j ∈ N (g), ij ∈ g implies that i = j (i.e., g 0 contains a single node which is connected only to itself in g);
or (b) for all i ∈ N (g 0 ) and j ∈ N (g 0 ), i 6= j, there exists
a path in g 0 connecting i and j, and for any i ∈ N (g 0 ) and
j ∈ N (g), ij ∈ g implies that ij ∈ g 0 (i.e., any two nodes in
N (g 0 ) are connected by a path and there is no path from a
node in N (g 0 ) to any other node in N (g) \ N (g 0 )). The set
of components of g is denoted by C(g).
1
The definition of graph in Section 3.1 does not allow the
presence of loops. However, the presence of loops does not
influence the model and the analysis of the previous sections.
6.2
Background on Network Games
A value function for graphs on N is a function w : G(N ) →
R. For consistency with the cooperative game model, we impose the normalization condition w(∅) = 0 and we refer to
w(g) as the worth of graph g. We denote by W the set of all
possible value functions. A network game is a pair (N, w),
where N is the set of the players and w ∈ W is a value
function on graphs among those players. Differently from a
characteristic function, a value function allows the value to
depend in arbitrary ways on the structure of the network.
The special case in which the value function depends only
on the coalition of players, corresponds to the cooperation
structures of a cooperative game. A network game is an extension of such a model in which the particular structure of
the network, and not only the subsets of players, matters.
Differently from the original model of Jackson and Wolinsky
[22], we allow the graph to contain loops. This is necessary
in our case to represent our cooperative game as a network
game because we need to be able to distinguish between two
different cases. In the first one, when an isolated player
does not belong to the coalition S, she does not contribute
to the aggregate utility; in the network game, we model the
situation by assuming that she does not have links. In the
second one, the isolated player belongs to the coalition, but
she does not share links with any other player and then she
maximizes her utility independently from the others; then,
we assume that this player is connected only to herself by a
loop. It the original model, it was not necessary to distinguish between these two cases, as the utility of an isolated
player was assumed to be zero.
Given the network game (N, w),
Pwe say that the value function is component additive if
h∈C(g) w(h) = w(g). This
condition rules out externalities across components, but still
allows them within components. Given a permutation of
players π (a bijection from N to N ), and any g ∈ G(N ),
let g π = {π(i)π(j)|ij ∈ g}. Thus, g π is a network that
shares the same architecture as g but with the players relabeled according to π. A value function is anonymous is
for any permutation of the set of players π, v(g π ) = v(g).
Anonymity says that the value of a network is derived from
the structure of the network and not the labels of the players who occupy various positions. A graph g ∈ G(N ) is said
strongly efficient if w(g) ≥ w(g 0 ) for all g 0 ∈ G(N ), i.e., if it
guarantees maximal total value.
An allocation rule for network games is a function Y : G(N )×
W → Rn that associates, to a couple of a graph and a value
function, a payoff allocation (x1 , . . . , xn ). As such a function depends on both g and w, it takes full account of the
role of player i in the network and keeps track of how the
valuePis allocated among the players in the society. If Y is
s.t.
i∈N Yi (g, w) = w(g) for all g and w, then we say that
the allocation rule
P is efficient. An allocation rule is component efficient if i∈S Yi (g, w) = w(g|S ) for each g ∈ G and
S ∈ C(g) and each component-additive value function w.2
An allocation rule Y is anonymous if, for any permutation
π, Yπ(i) (g π , wπ ) = Yi (g, w), wher wπ (g π ) = w(g). An allo2
Jackson and Wolinsky [22] refer to balanced and component
balanced allocation rules. We prefer the term “efficient” for
consistency with the terminology for cooperative games and
to avoid possible confusion with the notion of “balanced”
which has been used in Section 5.
cation rule Y satisfies equal bargaining power if for all w, g
and ij ∈ g, with i 6= j,
Yi (g, w) − Yi (g − ij, w) = Yj (g, w) − Yj (g − ij, w).
Under such a rule, every i and j gain equally from the existence of their link relative to their respective “threats” of
severing the link. Now, we provide a notion of stability for
the graph. A network g is pairwise stable with respect to
allocation rule Y and value function w if
(i) for all ij ∈ g, Yi (g, w) ≥ Yi (g − ij, w) and Yj (g, w) ≥
Yj (g − ij, w), and
(ii) for all ij ∈
/ g, if Yi (g + ij, w) > Yi (g, w) then Yj (g +
ij, w) < Yj (g, w).
In this definition, we assumed that the formation of a link requires the consent of both players involved, while severance
can be done by one player unilaterally. Then, a network is
pairwise stable if (i) no player wishes to severe a link that
she is involved in; and (ii) if one link is not in the graph
and one of the involved players would benefit from adding
it, then it must be that the other player would suffer from
the addition of the link.
6.3
A Graph Analysis of the Model
The cooperative extension of the public good game Γ, can
be naturally embedded into the definition of network games
(where, we recall, this is possible thanks to the fact that we
allow the presence of loops). In particular, given the cooperative game (N, vg ) with characteristic function given by (4)
(we need now to explicitly show the dependence on the graph
g to avoid confusion), this is equivalent to a corresponding
network game (N, w) where, for each g ∈ G(N ),
w(g) = vg (N (g)).
(8)
In the same way, given the corresponding graph game, it is
possible to go back to the original cooperative game in (4),
as for each g ∈ G(N ) and for each S ⊆ N
vg (S) = w(g|S ).
(9)
Notice that these two representations are equivalent, in the
sense that they describe the same situation. However, as
they focus on two different aspects, the graph in one case
and the coalitions when the graph is fixed in the other one,
they permit to analyze different aspects. In particular, the
network game representation allows us to analyze how the
social network structure influences the model and the payoff
the users can get.
First, the following proposition provides some properties of
the value function defined in (8).
Proposition 2. The value function w define in (8) is
component additive and anonymous.
Proof. Both the properties follows from the definition of
w. In fact, for each g ∈ G(N ),
w(g) = vg (N (g)) =
max
eN (g) ∈EN (g)
W (eN (g) , g),
where
 
W (eN (g) , g) =
X
f ei +
i∈N (g)

X

ej  − kei  .
j∈Ni (g)
The aggregate utility W as a function of g is component additive, as for each i ∈ N (g), her utility depends only on her
neighborhood, which is a subset of the component to which
player i belongs to. Then, also the maximum value of W as
a function of g is component additive, as the players of each
component maximize independently their aggregate utility,
and this gives the component additivity of w. Moreover,
as f and k do not depend on i, the utility of each player
depends only on her position on the graph and not on her
label, and this gives the anonymity property for w.
The property of component additivity is a direct consequence of the fact that our model allows externalities only
between connected players, in particular only between neighbors. Anonymity guarantees that our model really focuses
on the network structure and that it captures the utility
that a set of players can generate because of their pairwise
relations, independently from their labels.
The following proposition provides another important property of the model: in order to maximize the aggregate utility,
players should create links with any other possible player.
Proposition 3. The complete graph g N is the unique
strongly efficient graph.
Proof. At first, observe that, given a graph g ∈ G(N ),
the monotonicity of the game (N, vg ) in Theorem 1 implies
that w(g + ii) ≥ w(g) for each i ∈ N . From this it immediately follows that, given g, the graph g 0 which contains all
the links of g and all the loops, will provide a higher aggregate utility than g. Such a graph is s.t. every player in N
has at least one link, i.e., N (g 0 ) = N . Moreover, for each
i ∈ N (g 0 ), we observe that




X
X
f ei +
ej  ≤ f ei +
(10)
ej  ,
j∈Ni (g 0 )
j∈Ni (g N )
because f is an increasing function and because g 0 ⊆ g N
implies that Ni (g 0 ) ⊆ Ni (g N ). Given the graph g 0 with
N (g 0 ) = N (g N ) = N , (10) implies that W (eN , g 0 ) ≤ W (eN , g N )
for each eN ∈ EN , from which it follows
w(g 0 ) = vg0 (N ) = max W (eN , g 0 )
eN
≤ max W (eN , g N ) = vgN (N ) = w(g N )
eN
and this proves that g N is strongly efficient. To prove the
uniqueness, let g 6= g N ∈ G(N ) be s.t. N (g) = N and e∗N
a vector which maximizes the aggregate utility W (eN , g).
We want to show that two players who are not directly connected can improve the aggregate utility, by forming a link.
Adding a link ij when player i is s.t. e∗i 6= 0, the aggregate utility becomes W (e∗N , g + ij) > W (e∗N , g) (meaning
that the maximum value for the graph g + ij will be also
greater). Finally, if we add a link ij and both the players
are s.t. e∗i = e∗j = 0, then we can observe that the vector
e∗N cannot be a argmax for the new model (as it does not
satisfy the first order conditions) and this implies that the
maximum for g + ij has to be strictly bigger than w(g).
Proposition 3 states that the complete graph produces a
maximal aggregate utility for the grand coalition N . We can
generalize this result noticing that for each possible coalition
S, the aggregate utility of S is maximal when all the links
are present. This observation implies the following result.
Corollary 1. Given a graph g and the corresponding
cooperative game (N, vg ), for each payoff allocation in the
core C(vg ) there exists a payoff allocation in the core C(vgN ),
i.e., the core of the game with complete graph, which assigns
a higher payoff to each player in N .
This translates in the fact that, implementing the core as a
solution, it is beneficial for the players to create the grand
coalition.
Now, observe that, given the network game (N, w) equivalent through (8) to the cooperative game (N, vg ) defined in
(4), the Shapley value computed on (N, v) is an allocation
rule Φ for (N, w), where Φi (g, w) = φi (vg ). This definition
corresponds also to the definition of the Shapley value for
network game in [22], which satisfies the property of anonimity, component efficiency and equal bargaining power. As
a direct consequence of Proposition 3, the following corollary
provides an important stability result for the Shapley value.
In our model, implementing the Shapley value as solution
to quantify the value of personal information, every player
gains by forming a new link. In this setting, the only social
network which is stable is the one in which each player is
connected to every other player.
Corollary 2. The complete graph g N is the only pairwise stable graph with respect to the Shapley value allocation.
In particular, every player has incentives to create a new link
to augment her own payoff.
Proof. To show this result, it is sufficient to show that
φi (g + ij, w) ≥ φi (g, w) for each i ∈ N . By the definition
of Shapley value, in particular the result is true if all the
marginal contributions are s.t., w(g+ij|S∪{i} )−w(g+ij|S ) ≥
w(g|S∪{i} ) − w(g|S ), i.e., if the marginal contributions of
player i are always bigger when the link ij is present. We
notice that w(g + ij|S ) = w(g|S ), and this implies that it
sufficient to show that w(g + ij|S∪{i} ) ≥ w(g|S∪{i} ), and
this is true, because of what we showed in the last part
of the proof of Proposition 3. When the set of players is
assumed to be S ∪ {i}, by adding a link the aggregate utility
increases.
Note that for our game (N, vg ) (or its network equivalent),
we have an anonymous and component efficient allocation
rule (the Shapley value) which satisfies equal bargaining
power and which supports a strongly efficient network (g N )
as pairwise stable. This game, together with this proposed
solution concept, may suggest that, in a situation in which
the players are rewarded for they data, they have incentives
to create artificial links, in order to get a higher final payoff.
More simply, however, our results should be interpreted as
showing that when the players in N have many links, i.e.,
when the graph is dense, this can be beneficial for the users.
To fully understand the creation of artificial links that could
occur, we would need to introduce in the model a cost of link
creation and potential negative externalities that would arise
(for instance, a fake link could bring bad recommendation
and consequently to a decreasing of the utility of the users).
We leave this interesting extension of the model as future
work.
7.
CONCLUDING REMARKS
In this paper, we propose a cooperative game-theoretic approach to quantify the value of personal information in social
networks with positive externalities. We consider a model
where personal information has a local public good component in addition to its private cost and extend it to a cooperative game. We propose to use the core and the Shapley
value, two standard cooperative game solution concepts to
quantify the value of personal information. We show that
the core is non-empty and that the game is super-additive,
which indicates that these allocations are efficient and that
the grand coalition will form; in particular that the grand
coalition is stable.
Our cooperative game extension assumes that players can
form a coalition and maximize the welfare of their coalition
when all links to players outside the coalition are removed
and players outside the coalition do not take part in the
game. This is a sensible assumption since it is reasonable
that players who decide to join a platform within which information is shared will cooperative towards a higher welfare
(as long as the benefits are appropriately shared). Then our
results on quantifying the value of personal information can
have multiple applications. For instance, they can be used
for monetization, i.e., to price the information. They could
be used by an online service to determine the users with
most valuable information, e.g., to offer discounts, or to target them as participants of an online survey, when involving
all the players can be costly.
Our results in Section 6 show how the relations between
users may increase the value of personal information, due
to the externalities. In particular, users belonging to dense
networks have very high incentives to reveal their data. This
suggest that the individuals may have incentives to create
artificial links in order to increase their own payoff when a
solution such as the Shapley value is adopted to quantify
her data. In fact, fake links may not be associated with the
same positive externality as true links and may have negative
externality that could decrease the aggregate utility. A very
interesting extension of our model would be to analyze if,
regardless of that, a couple of users could still gain by adding
a link (and which couples of users could gain). This will
permit to discuss our results in terms of whether or not a
user has incentive to reveal truthfully her links in our system.
Our model proposes a new approach to the quantification of
personal data which is focused on the network structure of
the users. Differently from the existing literature, this model
allows to analyze how the users can strategically manipulate
the graph (and not only the information revealed). The
same analysis could be applied to other public good models
in networks.
In our study, we have made a number of simplifying assumptions. In particular, in order to focus on the effect of the local public good aspect of personal information derived from
the graph on its value, we have assumed that the private
cost is the same for all players. As all previous researches
have focused only on the private cost of personal information, we believe that it was necessary to clearly separate the
effect of the position of a user in the graph from its private
cost and that our paper provides useful results in this direction. As a next step of our future work, we plan, however,
to extend our analysis to the case of heterogeneous private
costs in order to quantify the value of personal information
in more realistic scenarios that combine heterogeneous private costs and local public good components. We have also
considered only positive externalities that result from information shared by neighbors and neglected negative externalities that could arise (for instance by imposing an additional
privacy cost due to possible inference of information from a
user to its neighbor). Finally, we have considered that contribution from neighbors are additive which is a good first
approximation but may be refined. We plan to relax these
two assumptions in future work.
A typical problem suffered by cooperative game-theory solutions is their computational complexity. In this paper, we
provide results about the core, which can be computed with
linear programming. The Shapley value, on the other hand
cannot be computed efficiently. There exist, however, various approximations of the Shapley value that can be computed efficiently (such as some variations of Monte Carlo
sampling methods [28]). In the future, we plan to extend our
work in two directions: (i) analyze how the use of approximations affect our results and (ii) study the limit regime
where the number of players is very large and how this can
provide more computationally efficient or more intuitive results.
8.
ACKNOWLEDGMENTS
This work was funded by the French Government (National
Research Agency, ANR) through the “Investments for the
Future” Program reference # ANR-11-LABX-0031-01.
9.
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