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IncentIvIzIng effIcIency In LocaL PubLIc good games and aPPLIcatIons to the QuantIfIcatIon of PersonaL data In networksDocuments de travail GREDEG GREDEG Working Papers Series
Michela ChessaPatrick Loiseau
GREDEG WP No. 2018-02https://ideas.repec.org/s/gre/wpaper.html
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Incentivizing e�ciency in local public good games and
applications to the quanti�cation of personal data in networks∗
Michela ChessaUniversité Côte d'Azur, GREDEG, CNRS, France
Patrick LoiseauUniv. Grenoble Alpes, CNRS, Inria, Grenoble INP, LIG, Grenoble, France
Max-Planck Institute for Software Systems (MPI-SWS), Saarbrücken, [email protected]
GREDEG Working Paper No. 2018�02
Abstract
A well established principle a�rms that the privacy of individuals is respected whenever they
are entitled to control the dissemination of their personal data and they are fairly compensated.
From this perspective, quantifying the value of personal data is a crucial task in the Internet
economics. This task is di�cult, however, as the privacy attitude of the users is often charac-
terized by a contradictory behavior, known as the privacy paradox, in which they declare to be
sensitive to privacy losses but also often release large amounts of data to enjoy free services.
In this paper, we model this trade-o� as a local public good game and propose some quanti�-
cations of the users' personal data depending on their position in the social network, based on
enhancing an e�cient solution of the local public good model. In a �rst part, we present some
non-cooperative approaches, based on an internalization of the local externalities. In a second
part, we extend the model to a cooperative game and we apply some well-known solutions from
cooperative game theory to suggest fair ways to compensate the users and to perform a network
analysis.
Keywords: Personal Data, Social Network, Local Public Good Game, Cooperative Game Theory,Core, Shapley value
JEL Code: C71, C72, D62, D85, H41
∗The authors would like to thank the participants of the SING11-GTM2015 European meeting on game theorywhich was held in July 2015 at the St Petersburg University, St. Petersburg, Russia and of the NetEcon'17 workshopwhich was held in June 2017 at the Massachusetts Institute of Technology, Cambridge, MA, U.S.A. for their usefulcomments and suggestions.
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1 Introduction
1.1 Motivations
The Internet has become an essential part of the citizens' life and of the economy in many countries.In this online ecosystem, service providers collect large amounts of personal data about individuals(e.g., their shopping behaviors collected through tracking or login mechanisms) and use it to o�erservices (e.g., product recommendations or targeted advertising) from which they derive high pro�ts.Personal data therefore has clear intrinsic economic value, which is extensively exploited by onlineservices. At the same time, users also bene�t from this ecosystem by being granted free accessto services, but it is unclear whether this appropriately compensates them for the release of theirpersonal data and for the intrusion into their private sphere. As the amount of data collected byonline services is exploding, users and organizations increasingly ask for more fair, transparent, andpersonalized mechanisms for managing this new connected world.
Recent works in the area have mostly focused on the central challenge of de�ning in a rigorousway the concept of privacy and identifying ways to preserve it (see a few examples in [4, 6, 24, 25,32, 41, 43, 58]). This term, however, is often used to denote a largely misguided and contradictoryprinciple that users want to see applied but are often not capable of explaining or quantifyingthemselves [59]. This leads to an evident discrepancy between their declared privacy concerns, andtheir actual behavior, which is well known as privacy paradox [9]. In our formalization, we adopt thewell known principle underlying the privacy de�nition proposed by Kleinberg et al. [38], according towhich the privacy of individuals is respected whenever they are entitled to control the disseminationof their personal data and they are fairly compensated. This naturally raises the question of howmuch is personal data worth?, which recently gained very high interest [48] and is recognized as oneof the crucial points around privacy issues.
Quantifying the value of personal data is a very di�cult task. In particular, we identify threedi�culties which we believe are key to the problem. Firstly, most research studies have focusedon quantifying the private cost of releasing personal data, that is the amount at which users arewilling to give it away [3]. An appropriate quanti�cation, however, should also take into accountthe pro�t that can be extracted from the data by online service providers. Secondly, pro�t is notextracted directly from the personal data itself but from the information the provider can derivefrom it (e.g., users movie interests or sports activities), which has a very di�erent economic nature.Indeed, while personal data is a private good that users can voluntarily decide to disclose at agiven private cost, the information derived from it has a strong public component: informationderived from personal data of a user may bene�t other users (e.g., by improving the quality ofthe recommendation algorithm). Taking into account such externalities is key for an appropriatequanti�cation. Lastly, as many online services collect data not only about individual users butalso about the social interactions among them, the strongest externalities are local externalities:information can be derived about a user from the personal data released by her neighbors in thesocial graph. The position of the users in the social graph therefore plays a crucial role in thequanti�cation of the value of their personal data.
1.2 Contribution
In this paper, we propose a game-theoretic approach to quantify the value of personal data thataddresses the key di�culties mentioned above. Such an approach may be used by a service provider
2
as a starting point to e�ectively reward the users for their contribution, or, more generally, canbe seen as a theoretical tool to investigate the importance of users related to their position in thenetwork. While presenting the results, we will often refer to the service provider seen as an externalmechanism designer aiming to optimize the interaction between the users, but we will keep in mindthe more general �avor of the proposed analysis. Following some economic literature [5, 54], wemodel the public component of the provider's information by treating it as a public good.1 Morespeci�cally, to take into account the local externalities, we propose to model users interactions asa local public good game à la Bramoullé et al. [13, 14]: users contribute by releasing personal data(at a private cost) and bene�t from the local public good (the information derived about them thatallows the provider to give them good service). The information about the users depends on thedata released by themselves and by their neighbors in the social graph. In the �rst part of thepaper, we provide an analysis of the well known generical model in our speci�c scenario, showinghow a service provider can force the users toward e�ciency, by introducing some payments to rewardthem for their contribution (technically, internalizing the externalities of the game). We do thatin three di�erent ways, putting our attention mostly on the last one, which allows us to performa more theoretical analysis about the role of the network in the game. In the last modi�cationof the game, in fact, we show how we can induce e�ciency by modifying the original game into anew game, which is still a public good game à la Bramoullé et al., but with heterogeneous autarkicplay of the users, where the heterogeneity comes from the network structure. Following a similaranalysis than in [14], we show that the Bonacich centrality still plays a role, but it has now twoopposite e�ects that make the analysis even more challenging. In the second part of the paper,we propose a sensible cooperative extension of the game. Our approach follows in spirit otherextensions previously adopted in studies of economies with public goods (see Section 1.3), but italso includes the role of the network. We prove that the resulting cooperative game is equivalent tothe minimax extension proposed by von Neumann and Morgenstern [60]. Then, we propose to usecooperative solutions, such as the core and the Shapley value, to quantify the value of personal data.These cooperative solutions can provide fair quanti�cation of the value of personal data taking intoaccount both the private cost of users and the pro�t extracted by the provider. We show that thecore is non-empty and that it is therefore possible to �nd a stable allocation of welfare, which is anallocation such that no coalition will have incentives to stop cooperating with the other players. TheShapley value, on the other hand, is not guaranteed to be stable, but has other desirable propertiessuch as symmetry/fairness. We claim that using such solutions to reward users for releasing theirpersonal data can allow sustaining cooperation between the users that will improve the aggregateutility in their own interest and in the interest of the provider. Finally, in the third and last part
of the paper, we analyze how these allocations depend on the graph. We use network games, anextension of cooperative games proposed by Jackson and Wolinsky [36] which permits to modelthe dependence of the value function on the graph, to carry the analysis. In particular, we showthat it is bene�cial for the players to create new links and that the only stable network is given bythe complete graph, in which each player is connected to every other player. Any other networkcon�guration could induce the users to strategically create new links.
While games on graphs and local public good games more speci�cally are well-known and widelyused in economics and computer science, our work is, to the best of our knowledge, the �rst study oncooperative extensions of such games in the �eld of the information economics and, more speci�cally,
1A public good is a good that is non-rivalrous (i.e., such that consumption of the good by a user does not reduceavailability to others) and non-excludable (i.e., it is not possible to exclude users from bene�ting from the good).
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of the Internet economics. Besides the main motivation of quanti�cation of the value of personaldata, which is today a very sensible topic that attracted the attention of many researchers in the �eldof the Internet economics, we expect that our model will �nd applications in other areas that arewell modeled by local public goods, in which a cooperative analysis represents a suitable alternativeapproach to enhance e�ciency.
The remainder of the paper is organized as follows. We review related works in Section 1.3. InSection 2 we present the local public good game. In Section 3 we present how to induce an e�cientoutcome in a strategic setting, introducing some payments. Section 4 describes the cooperativeextension to induce e�ciency, the solution concepts and analyzes some properties of the cooperativegame and solutions. We study the graph in�uence in Section 5. We conclude in Section 6. Omittedproofs can be found in appendix.
1.3 Related Works
Economics of personal data. Academic concerns about the economic value of personal data dateback to the 90's with the informal discussions of Laudon [39] and Varian [59]. More recently, someexperimental studies have been conducted that aim to quantify the value that users assign to theirpersonal data in di�erent scenarios, see e.g., Huberman et al. [33], Acquisti et al. [2] or Acquistiand Grossklags [1]. Pu and Grossklags also conducted a recent experimental study to measure howmuch users value their friend's personal data [50].
The game-theoretic analysis of the private data monetization was pioneered by Kleinberg etal. [38], who proposed fair compensation mechanisms for personal data based on cooperative gametheory. Their solutions are based on the core and on the Shapley value like ours, but their simplemodel does not take into account the public good nature of information which is the main focus ofour paper. Recently, a signi�cant thread of research started on selling personal data using auctions:Ghosh and Roth [29], Dandekar et al. [22], Roth and Schoenebeck [53] and Ligett and Roth [42].In all those works, the loss of privacy by releasing data is quanti�ed using di�erential privacy [24].Riederer et al. [52] propose a mechanism called �transactional privacy� where users can sell accessto their data through an unlimited supply auction.
All the aforementioned works only consider the private cost of revealing personal data, i.e., thecost incurred by an individual on account of his loss of privacy independently of the data revealedby others. In a recent work, Ioannidis and Loiseau [34] (see also extensions of the initial modelby Chessa, Grossklags and Loiseau [18,19]) propose a di�erent model which takes into account thepublic good nature of information. They analyze the game as a non-cooperative game and provideresults on Nash equilibrium and price of anarchy, but they do not investigate cooperative solutionsto reach e�ciency. Their model also does not take into account the local interactions that happenwithin a social network. An extension to a graph setting was proposed in [51], but also only in anon-cooperative setting. A very preliminary version of this work has been presented in [20].
Network games. In this paper, we consider users embedded in a social network and use agraph to model the local public good nature of personal information. Games on graphs have beenstudied in di�erent contexts and with di�erent utility functions of the players, see e.g., the booksof Jackson [35] and Easley and Kleinberg [26] and the recent survey by Jackson and Zenou [37]. Inparticular games on graphs have recently been studied in two situations: to model pricing problemswith network externalities [10, 15, 21, 27, 28] and to model strategic propagation representing forinstance product adoption [31, 40]. But the literature on the strategic interaction of the agentsin networks is even larger [7, 57] and, among many others, we mention Bramoullé et al. [13, 14],
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who described the model we adopt in this paper. These papers di�er in the utility model andinteractions that they consider. All of these works, however, consider a non-cooperative setting atthe exception of Jackson and Wolinsky [36] who partly use a cooperative approach to model thestrategic 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 andquanti�cation of its value.
Cooperative games for public goods. In this paper, we use a cooperative game-theory ap-proach: we transform our local public good model in a cooperative game and use classical allocationsolutions to quantify the value of personal information. The use of cooperative game theory methodsto analyze public good problems is standard in the economics literature. We mention in particularthe works of Champsaur [16], Moulin [45] and Chander [17]. To the best of our knowledge, however,no prior work has analyzed the impact of possible variations of the network in local public goodproblems using cooperative game-theory methods.
2 The Model
In this section, we describe our basic model, in which users are embedded in a social network andchoose strategically the level of personal data to reveal.
2.1 The Social Network as a Graph
Let N = {1, . . . , n} be a set of players, representing the users. In our model, the players areembedded in a social network (e.g., the Facebook network), that we represent by an undirectedgraph, where the nodes identify the players and the links their pairwise relations. Formally, anundirected graph g on N is a set of links, i.e., a set of unordered pairs of players {i, j}, that forsimplicity we denote by ij. For any pair of players i and j, i 6= j, ij ∈ g indicates that players iand j are linked in the graph g (in our example, the users are �Facebook friends�). In some partsof the paper, we will also represent the graph through an adjacency matrix. The adjacency matrixG corresponding to a graph g is de�ned as the n × n matrix such that Gij = 1 if ij ∈ g and 0otherwise. Let gN be the complete graph, i.e., the set of all the unordered pairs of players in N andG(N) = {g ⊆ gN} 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 thegraph g, and ni(g) = |Ni(g)| its cardinality, called degree of player i. Player i's neighborhood in gis de�ned as herself and her set of neighbors, i.e., as Ni(g) = {i} ∪ Ni(g). For the measure of aplayer's centrality, given the adjacency matrix G and a scalar q s.t. I − qG is invertible, the vectorof Bonacich centralities [11] c(q,G) is de�ned by c(q,G) = (I − qG)−1G1. The measure ci(q,G)provides a weighted sum of the paths in G starting with i.2
2.2 The utility functions
Given a set of players N and an undirected graph g, we adopt the model of a strategic substitutesgame with linear best replies for public goods in networks [13, 14] to describe their strategic in-teraction. Throughout the paper, at the exception of Section 5, the graph g is assumed to be a
2In this paper, we use the original Bonacich's measure of centrality, to adopt a notation as close as possible toBramoullé et al. [14]. Most of the literature, however, adopts an analogous a�ne trasformation of this measure (e.g.in [8]).
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�xed parameter (in Section 5, we analyze the case where the graph is assumed to be a variable ofthe model). Each player in N chooses a disclosure level di ∈ [0,+∞) of her personal data, whichcorresponds to her contribution to the public good, the information. Depending on the situationat stake and its interpretation, this disclosure level can represent the precision in revealing a sin-gle piece of data or the total amount of data revealed.3 Let d = (d1, . . . , dn) denote a disclosurepro�le of all players. In our model, data disclosed by a user bene�ts her neighbors. This situationtypically occurs with personalized applications: assuming that neighbors have similar tastes, datarevealed by a user leads to better personalization for her neighbors. We consider a model withpositive externalities between neighbors, in which a neighbor's disclosure is a substitute with one'sown proportionally to a factor δ ∈ [0, 1], but a player does not derive bene�ts from the disclosure ofplayers with whom she does not share a link. Intuitively, δ is the measure of how much i's neighborsdirectly a�ect i. Player i's utility from pro�le d in graph g is then given by
Ui(d, g) = f
di + δ∑
j∈Ni(g)
dj
− kdi, (1)
where f is a twice di�erentiable function, with f(0) = 0, f ′ > 0 and f ′′ < 0 and k ∈ R+ is the costto player i for one unit of disclosure level. The �rst component f captures the bene�t for player
i, i.e., the bene�t that player i receives when the provider infers some information from her dataand her neighbors' (in terms, for example, of good ad hoc recommendation); in particular, it is anincreasing function of the level of disclosure of player i and of the total level of disclosure of herneighbors. The second component is the privacy cost ; it represents the cost that player i incurs (forinstance due to loss of privacy) by choosing a given disclosure level and then it depends only on di.
The utility de�ned in (1) has a positive component, the bene�t, which is a function of the inferredinformation (the public good), and a negative component, the privacy cost, which is proportionalto the amount of personal data a user decides to disclose (the private good). The assumption ofincluding only positive externalities is well justi�ed in many contexts, in which, for example, thepresence of possible negative externalities due to inference of information about player i from hisneighbors data, would be a�ecting only the public good component. In this case, these externalitiescould be captured by our model in the bene�t component f through a smaller δ. By assumingδ ≥ 0, we implicitly model situations in which the bene�ts from neighbors' disclosure (positiveexternalities) is higher than the negative externalities. Possible situations in which the negativeexternalities play a higher role than the positive ones, or in which the externalities also a�ect theprivacy cost, could be the object of a further extension of this work, but are not the objective ofthis present work.
In our model, both the function f and the cost k are independent of the player i. Whencollecting real data on a social network, an online service may easily obtain information aboutthe graph structure of its users, but not as easily about their privacy concerns. Many works havestudied the problem of estimating how much individuals value their personal data (see Section 1.3),but implementing these mechanisms can be costly and time consuming for the online service. Inthis paper, we take an orthogonal approach and choose to propose solutions to quantify the valueof personal data when the users di�er only due to their position in the social network, but areotherwise totally symmetric.
3 Note that we assume for simplicity that there is no bound on the disclosure level (di ∈ [0,+∞)). This assumptionis very mild since, if the disclosure level is bounded by dmax su�ciently large so that the disclosed levels at equilibriumor at the optimum are interior, all our results are unchanged.
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2.3 The Local Public Good Game
We describe the strategic interaction of the players through the following local public good game
Γ = 〈N, [0,+∞)n, (Ui)i∈N 〉 , (2)
where N is the set of players, each player has strategy space given by [0,+∞) and her utilityfunction is de�ned by (1). A Nash equilibrium (in pure strategy) of Γ is de�ned as a disclosurepro�le d∗ ∈ [0,+∞)n s.t.
d∗i ∈ arg maxdi∈[0,+∞)
Ui(di,d∗−i), ∀i ∈ N, (3)
where we use the standard notation d−i to denote the collection of actions of every player but i.A complete analysis of the local public good game Γ is provided by Bramoullé et al. [14]. In
particular, we recall the result that Γ always has a Nash equilibrium, which may not be unique,in general. Uniqueness appears under the condition δ < −1/λmin(G), where λmin(G) denotes thelowest eigenvalue of the adjacency matrix G, which turns out to be a crucial parameter of the gamein the analysis of Bramoullé et al. The authors also show that under the same assumption, i.e., whenthe externalities have a su�cient low in�uence on the public good utility, the unique equilibriumd∗ is interior (meaning that no user is free-riding), it satis�es (in matrix form) the following �rstorder conditions (FOCs):
f ′(d + δG · d) = k, (4)
and it is such thatd∗ = (I + δG)−1 · d = d (1− δc(−δ,G)) ,
where d =(d, . . . , d
)with d s.t. f ′
(d)
= k and c(−δ,G) is the vector of individual Bonacichcentralities. In practice, at equilibrium each player and her neighbors provide an aggregate e�ortequal to d and each player contribution is the highest, the smallest her centrality in the graph. Asa consequence of the graph in�uence to the individual equilibrium play, the authors observe thatcomparing two players whose neighbors are nested sets, the player with the larger set of neighborsplays a lower action. The intuitive idea is that a player linked to more outside sources can do lessherself.
We de�ne the aggregate utility of the players in N , given a disclosure pro�le d and the graph g,as the sum of their utilities:
W (d, g) =∑i∈N
Ui(d, g). (5)
The aggregate utility represents the social welfare of the population. We say that a pro�le d ise�cient for a given graph g if it maximizes the social welfare, i.e., if and only if there exists noother pro�le d′ such that W (d′, g) > W (d, g); and we denote by de an e�cient pro�le. A Nashequilibrium pro�le of Γ is in general not e�cient. This outcome is typical of public good models,which often yield non-e�cient equilibria. To quantify the loss of e�ciency at equilibrium, we de�nethe price of anarchy as the ratio between the social welfare at optimum and the social welfare atthe �worst� Nash equilibrium d∗∗ (the equilibrium which provides the lowest welfare), i.e.,
PoA =W (de, g)
W (d∗∗, g). (6)
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Before moving to the next section, we introduce a last notation. Given a disclosure pro�le dand the graph g, we de�ne the total service value
V (d, g) =∑i∈N
f
di + δ∑
j∈Ni(g)
dj
, (7)
which corresponds to the sum of the value of the service for all users, and can also be interpretedas the turnover of the provider (a fraction of which will become its net revenue).
3 Enhancing e�ciency in the stretegic setting
In this section, we present three alternative ways for the provider to induce an e�cient outcome, in astrategic setting, all based on the idea of introducing some payments: a Pigovian tax-type payment,a VCG-type payment, and what we will call a Bonacich centrality-type payment.
3.1 A Pigovian tax-type payment
The �rst solution establishes a per unit payment p and a �xed payment Pi, as detailed in thefollowing theorem.
Theorem 3.1 Given the game Γ de�ned in (2), the provider can induce the players to strategically
select an e�cient disclosure pro�le de by guaranteeing to each player i ∈ N for a given disclosure
pro�le d a payment p = n−1n k per unit of disclosure level, and a �xed payment
Pi =
∑h∈N f
(dh + δ
∑j∈Nh(g) dj
)n
− f
di + δ∑
j∈Ni(g)
dj
.
This entails a cost to the provider of∑i∈N
(pdei + Pi) =n− 1
n
∑i∈N
kdei .
In Theorem 3.1, the payment p is given per unit of disclosure. This means that players con-tributing more receive a higher payment. The �xed payment Pi is the di�erence between the averagebene�t and the bene�t of player i. As such, it may be negative: a player enjoying a bene�t higherthan average will be creditor to the provider. On the contrary, players receiving a bene�t lowerthan average need to be rewarded by the provider with a positive Pi. It is worth noticing that theprice in Theorem 3.1 (pdi + Pi) is an extension of the classical Pigovian tax [49]: in the case of acomplete graph with δ = 1 (corresponding to a standard global public good), Pi = 0 for all playersi and the price reduces to the constant marginal price p which is the Pigovian tax. When Pi 6= 0,this additional term introduces a correction for the local public good game in case of dissymmetriccontribution.
Introducing the payments of Theorem 3.1 in order to reach e�ciency presents a number ofdisadvantages. In particular, it has a cost to the provider; for big values of n, the analyst almostentirely refunds the privacy cost of the players. This can be convenient for the provider only if theincrease of the social welfare (which is an upper bound on the increase of the provider's revenue)is bigger than the payment to the players. To characterize when this happens, we analyze in thefollowing corollary the price of anarchy.
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Corollary 3.2 It is advantageous for the provider to induce the players to strategically select an
e�cient disclosure pro�le using the payments of Theorem 3.1 only if the price of anarchy PoA is
s.t.
PoA > 1 +n−1n
∑i∈N kd
ei
W (d∗∗, g),
where we recall that d∗∗ denotes the �worst� Nash equilibrium d∗∗.
Another drawback of the payments of Theorem 3.1 is that it is not guaranteed that pdi + Pi isnon-negative for all i.
3.2 A VCG-type payment
The problem of negative payments may be avoided by adopting our second solution for inducingan e�cient outcome in a strategic setting, based on a VCG-type payment, as shown in the nexttheorem.
Theorem 3.3 Given the game Γ de�ned in (2), the provider can induce the players to strategically
select an e�cient disclosure pro�le de by guaranteeing to each player i ∈ N for a given disclosure
pro�le d a payment
Qi =∑
h∈Ni(g)
fdh + δ
∑j∈Nh(g)
dj
− fdh + δ
∑j∈Nh(g)\{i}
dj
.This entails a cost to the provider of
∑i∈N
∑h∈Ni(g)
fdeh + δ
∑j∈Nh(g)
dej
− fdeh + δ
∑j∈Nh(g)\{i}
dej
.With this second approach, the players receiving higher payments are the ones providing a
higher marginal contribution to the public good utility of their neighbors. Compared to the previoussolution, the payments of Theorem 3.3 are guaranteed to be non-negative (i.e., Qi ≥ 0 for all i).However, as a drawback, we observe that the cost to the provider can be even bigger than the costto the provider with the payments of Theorem 3.1 (see Appendix D for an example).
3.3 A Bonacich centrality-type payment
We �nally suggest a third solution on which we focus our attention more particularly as it appearsappealing. As for the �rst solution (Pigovian tax-type payments), this solution introduces a paymentper unit of disclosure level; and as for the second solution (VCG-type payments), the payment toa player is guaranteed to be positive. The appeal of this third type of payment, however, lies inits own nature: following well-known literature on pricing in networks with externalities (e.g., thework of Candogan et al. [15]), this solution suggests that the service provider may induce e�ciencyby providing higher payments to the most central players, where centrality is computed using theclassical notion of Bonacich centrality. In our analysis, this measure of centrality naturally arisesas a measure of the importance of the players in our local public good model and as a tool for thequanti�cation of the value of personal data depending on the social network position.
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Theorem 3.4 Given the game Γ de�ned in (2), the provider can induce the players to strategically
select an e�cient disclosure pro�le de by guaranteeing to each player i ∈ N for a given disclosure
pro�le d a payment per unit of disclosure level proportional to her Bonacich centrality, i.e., a
payment bi = δci(−δ,G)k. This entails a cost to the provider of∑i∈N
bidei =
∑i∈N
δci(−δ,G)kdei .
As shown in detail in Appendix E, this result is obtained by writing the �rst order conditionsat e�ciency as
f ′(d + δG · d) = (I + δG)−1k, (8)
and by observing their similarities with the FOCs in (4). In particular, compared to the originalgame Γ de�ned in in (2), the FOCs in (8) correspond to the best response functions of a similargame, but with heterogeneous autarkic play of the users. The heterogeneity is given by the positionof the players in the network, which make them have di�erent thresholds to reach equal to di s.t.f ′(di)
= (I+ δG)−1i ·k = k (1− δci(−δ,G)). Compared to Γ, we observe that, in this new modi�ed
setting, the centrality of a player plays two contrasting roles: on one side, as we have alrady observedin the previous section, a player connected to more outside sources can do less, because she canexploit the contribution of her neighbors. On the other side, she has a higher threshold di to reach,due to her central position, and this can lead her to contribute more, but also to receive higherpayments for her contribution.
4 The Cooperative Extension
In this section, we focus on cooperation. This represents an alternative way to the payments ap-proach seen in the previous section to induce an e�cient outcome. Agreeing to act cooperatively,the players may increase the aggregate utility to be shared, choosing a strategy pro�le which max-imizes the social welfare. A cooperative e�cient outcome can arise when regulatory mechanismsforce the players to reach an e�cient solution (e.g., by establishing the amount of taxes citizens haveto pay by law, instead of letting them freely contribute). Alternatively, in some di�erent real worldsituations, the purpose of achieving e�ciency may lead the users to spontaneously create bindingagreements between them and to commit to play an e�cient strategy, without having to waste apart of the common social welfare to pay an external mechanism designer. More generally and inthe scope of our analysis, a cooperative model is an alternative and e�ective way to establish the�importance of the players�. It allows us to underline the role of the players in the network and toestimate their importance in terms of value of the personal data they disclose.
4.1 The Cooperative Extension of the Local Public Good Game
We extend the model proposed in Section 2 assuming that the players can choose to centralizetheir choices. Situations in which players can cooperate by establishing binding agreements amongthemselves (possibly through an intermediary) and forming coalitions in order to coordinate theirstrategies and share their joint payo�, are studied by cooperative game theory. We refer to [46] for
10
a complete account on the subject. Throughout the paper, in particular, we model our situation asa TU (transferable utility) cooperative game.4
Given a coalition of players S ⊆ N who decided to sign a binding agreement between them, andgiven the graph g, we consider the subgraph of g reduced to S, g|S = {ij|ij ∈ g, i, j ∈ S} ∈ G(S).The idea is that the players in S, while making an agreement to cooperate, accept to reveal theirconnections, while the group is not informed about the rest of the graph. In particular, informationabout the links between two players in N \S, or between one player in S and one player in N \S isnot available. For each player in S, a disclosure level of personal data is selected. We denote by dSa disclosure pro�le for the players in S. Given a disclosure pro�le dS , the utility of a player in S isstill de�ned as in (1), where the set of neighbors is now restricted to the set of neighbors who arein S. Formally, when restricting to the coalition S, for each player i ∈ S her utility is de�ned as
Ui(dS , g|S) = f
di + δ∑
j∈Ni(g|S)
dj
− kdi. (9)
We de�ne the aggregate utility of the coalition S given a disclosure pro�le dS as the sum of theutilities of the players in S on g|S , i.e., as
W (dS , g|S) =∑i∈S
Ui(dS , g|S). (10)
We denote an e�cient pro�le by deS . Note again that the aggregate utility of coalition S dependsonly on the pairwise relations between its players, and not on the other links outside S.
A cooperative game is de�ned by a couple (N, v), where N is the set of players and v : 2N → R,with v(∅) = 0, is the characteristic function, which assigns to every coalition S a worth v(S). In ourmodel, we de�ne the worth of a coalition as the maximal aggregate utility the coalition can reach,i.e., for each S ⊆ N , S 6= ∅,
v(S) = W (deS , g|S) = maxdS∈[0,+∞)s
W (dS , g|S). (11)
Intuitively, the worth of a coalition is given by the maximum social welfare that its players canreach by cooperating. Note that it is possible that there exist multiple e�cient disclosure pro�les.However, as they all provide the same aggregate utility, function v is univocally de�ned.
If a coalition S ⊆ N of players decide to make an agreement for the disclosure of their personaldata, an e�cient disclosure pro�le deS is selected. The aggregate utility of the players in S will begiven by W (deS , g|S), while the total service value for players in S will be given by V (deS , g|S).
4.2 The Allocation of Utilities
Given a set of players, we quantify the value of their personal data by a sharing of the worth of thecoalition between these players, as a reward for their contribution.
Suppose for simplicity that all the players in N decide to cooperate, and that the worth v(N) =W (deN , g) has to be shared between them (everything still holds for a coalition S ⊆ N). A utilityshare is given by an e�cient payo� allocation, i.e., by a vector x = (x1, . . . , xn) ∈ Rn s.t.,
∑i∈N xi =
4In such a game, we assume that players have the option to freely transfer among themselves a commodity, usuallymoney, to increase or decrease their payo�s.
11
v(N) (e�ciency property). Each component xi is interpreted as the utility payo� to player i. Asolution for the game (N, v) is a function which associates to every characteristic function v a(possibly empty) set of payo� allocations. In the following, we propose to use the two most commonconcepts of solution for cooperative game.
The �rst solution we propose to use is the core [30], which provides the following set of e�cientpayo� allocations
C(v) =
{x|∑i∈N
xi = v(N),∑i∈S
xi ≥ v(S), ∀S ⊂ N
}.
The fundamental idea of the core is that an agreement among the players in N can only be bindingif every coalition S ⊂ N receives collectively at least its worth, i.e., the value that the players inS could obtain without collaborating with the players in N \ S. In this way, single players andcoalitions have no incentive to deviate. The core is a very appealing solution due to this stabilityproperty, but it may be empty.
The second solution we propose to use is the Shapley value [55]. Di�erently from the core, thissolution proposes a unique e�cient payo� allocation and it is always de�ned. The Shapley value,that we denote by φ, is de�ned by the following equation: given the characteristic function v, forevery i ∈ N
φi(v) =∑
S⊆N\{i}
s!(n− s− 1)!
n!(v(S ∪ {i})− v(S)).
It is a weighted sum of the contributions that player i brings while joining all possible coalitionsS ⊆ N \ {i}.
The Shapley value can also be de�ned axiomatically as the unique solution satisfying e�ciency,and the following three properties. (a) Symmetry : if two players i and j contribute to any coalitionin exactly the same way, they must be granted the same payo�. (b) Null player : if a playercontributes nothing to every coalition, he should receive zero. (c) Additivity : the Shapley value ofthe sum of two games must be equal to the sum of the Shapley values of the two games.
In the remaining of this section, we present the properties of our cooperative model, by analyzingthe game and the two solutions we proposed to share the aggregated utility between the playersand to quantify the value of their personal data.
4.3 The Properties of the Cooperative Game
Deriving a characteristic function from an n-person strategic-form game is a standard issue in gametheory [47]. When analyzing cooperative games associated to economies with public goods andexternalities, de�ning a worth for the coalition S requires to also take into account the strategicchoices made by players who are not members of S. In particular, the characteristic function shouldspecify 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 is to assume that the players outside the coalitionchoose the strategies which are the least favorable to the coalition, as in the well-known way ofderiving a characteristic function from an n-person strategic-game form proposed by Von Neumannand Morgenstern [60]. Following their model, the characteristic function starting from our game Γ
12
is de�ned for all S ⊆ N as
v′(S) = minσN\S∈∆(DN\S)
maxσS∈∆(DS)
∑i∈S
Ui(σS , σN\S , g),
where ∆(DS) (resp. ∆(DN\S)) is the set of correlated strategies available to coalition S (resp.
N \ S). Here, Ui(σS , σN\S , g) denotes player i's expected payo� when the correlated strategies σSand σN\S are implemented:
Ui(σS , σN\S , g) =∑dS
∑dN\S
σS(dS)σN\S(dN\S)Ui(dN , g).
The idea behind the de�nition of von Neumann and Morgenstern is to assert that the value ofcoalition S, v′(S), is the maximum sum of utilities that the players of coalition S can guaranteethemselves against the best o�ensive threat by the complementary coalition N \ S. The game(N, v′) is called the minimax representation in cooperative form of the strategic-form game Γ withtransferable utility.
The minimax representation is a very convenient way to derive a characteristic function fromΓ thanks to its properties. In particular, if we plan to apply the core as solution concept, theinequality
∑i∈S xi < v′(S) is necessary and su�cient for the players in S to be able to guarantee
themselves payo�s that are strictly better than in x, no matter what the other players in N \ S do.However, the de�nition of the minimax representation is not trivial, both from an analytic and acomputational point of view, as it is de�ned over all the correlated strategies. Also, in our setting,the minimax representation does not have a clear interpretation as the characteristic function wede�ned in (11) has. Yet, our �rst result shows that the de�nition of our characteristic function v in(11) and of the minimax representation v′ coincide for our game.
Proposition 4.1 The cooperative game (N, v) is equivalent to the minimax representation of the
game Γ, i.e.,
v(S) = v′(S), ∀S ⊆ N.
Proof: To prove the �rst inequality ≤, we can observe that
v′(S) = minσN\S∈∆(DN\S)
maxσS∈∆(DS)
∑i∈S
Ui(σS , σN\S , g)
≤ maxσS∈∆(DS)
∑i∈S
∑dS
σS(dS)
fdi + δ
∑j∈Ni(g|S)
ej
− kdi
= maxσS∈∆(DS)
∑dS
σS(dS)
∑i∈S
fdi + δ
∑j∈Ni(g|S)
dj
− kdi
= maxdS∈DS
∑i∈S
fdi + δ
∑j∈Ni(g|S)
dj
− kdi = v(S),
where the inequality is because we are minimizing on a smaller set (in particular, we are �xing thespace to one point), and then the minimum value is necessarily higher. Moreover, to prove the
13
opposite inequality ≥, we can observe that
v′(S) = minσN\S∈∆(DN\S)
maxσS∈∆(DS)
∑i∈S
Ui(σS , σN\S , g)
≥ maxσS∈∆(DS)
minσN\S∈∆(DN\S)
∑i∈S
Ui(σS , σN\S , g)
= maxσS∈∆(DS)
∑dS
σS(eS)
∑i∈S
fdi + δ
∑j∈Ni(g|S)
ej
− kdi
= maxdS∈DS
∑i∈S
fdi + δ
∑j∈Ni(g|S)
dj
− kdi = v(S),
where the inequality follows from the standard inequality min max ≥ max min, often referred to asthe max-min inequality. �
In particular, this result implies that our game (N, v) enjoys all the nice properties of (N, v′),but through a simpler de�nition of the characteristic function, both from the analytical and fromthe computational point of view, with a clearer interpretation.
We now de�ne two important properties of cooperative games. A cooperative game (N, v) issaid to be monotonic if
S ⊆ T ⊆ N ⇒ v(S) ≤ v(T ),
meaning that a larger coalition always has a larger worth. A cooperative game (N, v) is said to besuperadditive if
S ∩ T = ∅ ⇒ v(S) + v(T ) ≤ v(S ∪ T ).
Superadditivity implies that the worth of the coalition N of all players is at least as large as thesum of the worths of the members of any partition of N . The following corollary states that, as aconsequence of Proposition 4.1, our game satis�es these two properties.
Corollary 4.2 The game (N, v) is monotonic and superadditive, i.e., every player has incentives
to cooperate with the others and the grand coalition will form.
Proof: Superadditivity follows from Proposition 4.1, because the minimax representation of agame in strategic form is always superadditive. Moreover, superadditivity implies monotonicity.� The monotonicity ensures that, under cooperation, the aggregate utility increases when a newplayer decides to make an agreement with the others. Most importantly, superadditivity ensuresthat it is optimal to form the grand coalition. In particular, whenever the worth is shared in anadvantageous way for everyone, it is not convenient for a player to abandon the coalition and it isnot convenient to decentralize the cooperation between two di�erent groups of players. Because ofthese important properties, in the following we can assume that the grand coalition forms and thatthe total utility to be shared is equal to v(N).
Now, we observe that also the total service value increases, when the players decided to cooperateto adopt an e�cient disclosure level.
Theorem 4.3 For any e�cient disclosure pro�le de and any Nash equilibrium pro�le d∗, we
have V (de, g) ≥ V (d∗, g), i.e., the total service value is higher under cooperation. In particular,
V (de, g) > V (d∗, g) if g is not totally disconnected.
14
Proof: For each i ∈ N , the equilibrium condition is given by
f ′(d∗i + δ∑
j∈Ni(g|S)
d∗j ) = k,
and the e�ciency condition is instead given by
f ′(dei + δ∑
j∈Ni(g|S)
dej) + δ∑
j∈Ni(g|S)
f ′(dej + δ∑
h∈Nj(g|S)
deh) = k.
It follows that for each i ∈ N , f ′(dei + δ∑
j∈Ni(g|S) dej) ≤ f ′(d∗i + δ
∑j∈Ni(g|S) d
∗j ) and, as f is
concave, that for each i ∈ N , dei + δ∑
j∈Ni(g|S) dej ≥ d∗i + δ
∑j∈Ni(g|S) d
∗j . These two inequalities are
strict for at least one h ∈ N if the graph is not totally disconnected. As f is increasing, then foreach i ∈ N , f(dei + δ
∑j∈Ni(g|S) d
ej) ≥ f(d∗i + δ
∑j∈Ni(g|S) d
∗j ), and the inequality is strict for h. It
follows that their sum is s.t. V (de, g) ≥ V (d∗, g) for each graph g and the inequality is strict if g isnot totally disconnected. �
Corollary 4.2 and Theorem 4.3 ensure that, as the cooperation can improve the utility of boththe players (the users) and the provider, it is likely to occur and to be sustained by all the actors.In the following, we assume that the players act cooperatively and we analyze the properties of thepayo� allocations we have proposed in the previous section.
4.4 The Properties of the Allocations
The �rst solution we proposed, the core, is very appealing, in view of the assumption that a payo�allocation in the core ensures that any coalition can negotiate e�ectively and that such a solutionis stable against deviation by sel�sh coalitions of players. Unfortunately, for some games, the coremay be empty, and if this happens, no matter what allocation may occur, there is always a coalitionthat would gain by deviating. The following result ensures that this is not the case for our game(N, v), and that there always exists at least a stable payo� allocation.
Theorem 4.4 The game (N, v) has a nonempty core, i.e., there exists an e�cient and stable allo-
cation such that no subset of players has incentives to deviate.
Proof: To prove the nonemptyness of the core, we introduce the notion of balancedness. A familyB of coalitions is called a balanced collection provided that for each S ∈ B, there exists λS > 0such that, for all i ∈ N ,
∑S∈B, B3i λS = 1. A very important result, known as Bondareva-Shapley
theorem [12, 56], states that the cooperative game (N, v) has a nonempty core if and only if forevery balanced collection B, ∑
S∈BλSv(S) ≤ v(N). (12)
To prove the non-emptyness of the core, we show that (N, v) veri�es (12). At �rst, we write thecharacteristic function v as
v(S) = maxf
{∑i∈S
Ui(f i)|f = (f1, . . . ,fn) ∈ [0,+∞)2n,
fij = 0 ∀ij /∈ g|S , fij = fji ∀i, j ∈ S} ,
15
where Ui(f i) = f(fii + δ∑
j∈N\{i} fij)− kfii. This de�nition is equivalent to the de�nition in (11).The equivalence can be shown by writing the �rst order conditions and by seeing they are the samethan the �rst order conditions of our original maximization problem de�ning (11). Then let Bbe any balanced collection, and for each S ⊆ N , let fS = (fSi )i∈S be an arg max allocation whichprovides v(S). De�ne the allocation f , where for each i ∈ N , f i is the following convex combinationof the fSi ,
fi =∑
S∈B, B3iλSf
Si .
The allocation f is feasible for v(N), since for each i, j ∈ N
fij =∑
S∈B, B3iλSf
Sij =
∑S∈B, B3i,j
λSfSij
=∑
S∈B, B3iλSf
Sji = fji.
Then, by the concavity of the Ui, we have that∑S∈B
λSv(S)
=∑S∈B
λS∑i∈S
Ui(fSi ) =
∑i∈N
∑S∈B, S3i
λSUi(fSi )
≤∑i∈N
Ui
∑S∈B, S3i
λSfSi
=∑i∈N
Ui(f i) ≤ v(N).
This proves (12) and, consequently, the nonemptyness of the core. �Theorem 4.4 shows that the core of our game (N, v) is non-empty, but it does not provide a
concrete construction to �nd a vector in the core. Regardless of its importance, when the coreis large, this solution can be di�cult to apply as a predictive theory, due to the high number ofstable allocations. To provide a speci�c vector in the core, it is possible to compute the nucleolus, apowerful pointwise solution that has been shown to be stable whenever the core is nonempty [23,46].The nucleolus is de�ned in terms of objections and counterobjections, or, equivalently, as the uniquevector minimizing the dissatisfaction of the members of any possible coalition, in a lexicographicorder. Unfortunately, computing the nucleolus is, in general, very costly.
The second solution we proposed, the Shapley value, is a very powerful tool because of itsfairness properties, which represents in our opinion a very appealing solution concept, in particularfor our purpose of quantifying the value of personal data of the users. However, it may be unstable.A su�cient condition for the Shapley value to belong to the core is that the game satis�es theconvexity property.5 Unfortunately, our game is not guaranteed to be convex.6 Below, we presenttwo examples in which we can show that the Shapley value belongs to the core. Finding su�cientconditions such that this holds in general remains an open question whose investigation is left asfuture work.
5A cooperative game (N, v) is convex if v(S ∪ {i})− v(S) ≤ v(T ∪ {i})− v(T ) ∀S ⊂ T ⊆ N \ {i}.6For a counterexample, assume N = {1, 2, 3}, g12 = g13 = 1 and g23 = 0 Ui(d, g) =
(di + 0.2
∑j∈Ni(g)
dj)0.1−
0.2di. It can be seen that v(1, 3)− v(1) > v(1, 2, 3)− v(1, 2), meaning that the marginal contribution of player 3 tothe smaller coalition {1} is strictly bigger than the marginal contribution to the larger coalition {1, 2}
16
(a) (b)
Figure 1: Symmetric and star graphs.
Example 1 Let g be a symmetric graph. Then each player i ∈ N has the same Shapley value equal
to
φi(v) =1
nv(N).
This Shapley value allocation belongs to the core.
In particular, we observe that circles and complete graphs belong to the class of symmetric graphs.
Example 2 Let g be a star graph. For simplicity, we denote by Sh a star of cardinality h. Then
the center player has the highest Shapley value, given by
φC(v) =1
nv(Sn) +
1
n
n−2∑k=2
[v(Sk)− kv(S1)],
while each player at the periphery has a Shapley value given by
φP (v) =1
nv(Sn)− 1
n(n− 1)
n−2∑k=2
[v(Sk)− kv(S1)].
This Shapley value allocation belongs to the core.
5 The Impact of the Network
So far, we have assumed that the network structure was a �xed parameter. In this section, we analyzethe impact of this network, by considering the graph as a variable of the model. In particular, wewant to understand how the two solutions we proposed in the previous section change when wemodify the graph and if they can induce the players to create or to remove some links strategicallyin order to get a higher payo�. To this end, we use the formalism introduced by Jackson andWolinsky [36], who propose to de�ne 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 networkas well as how it is allocated across players, and of the way they vary depending on the graph. We�rst introduce some additional notations about graphs complementing the ones in Section 2.1, whichare necessary for the analysis of this section.
17
5.1 Background on Graphs
In this section, we assume that a graph g ∈ G(N) may contain loops, where a loop is a link iiwhich connects a player with herself.7 Given the set of players N and a graph g ∈ G(N), we denoteby g + ij = g ∪ {ij} the graph obtained by adding the link ij to the existing graph g and byg − ij = g \ {ij} the graph obtained by removing the link. Let N(g) = {i|∃j s.t. ij ∈ g} be theset of players who have at least one link in the graph g. A path in g connecting i1 and it is a setof distinct nodes {i1, . . . , it} ⊆ N(g) such that {i1i2, . . . , it−1it} ⊆ g. The subgraph g′ ⊆ g is acomponent of g if either (a) N(g′) = {i} and for all j ∈ N(g), ij ∈ g implies that i = j (i.e., g′
contains a single node which is connected only to itself in g); or (b) for all i ∈ N(g′) and j ∈ N(g′),i 6= j, there exists a path in g′ connecting i and j, and for any i ∈ N(g′) and j ∈ N(g), ij ∈ gimplies that ij ∈ g′ (i.e., any two nodes in N(g′) are connected by a path and there is no path froma node in N(g′) to any other node in N(g)\N(g′)). The set of components of g is denoted by C(g).
5.2 The Network Game
The cooperative extension of the public good game Γ can be naturally embedded into the de�nitionof network games. Given the cooperative game (N, v) with characteristic function de�ned by (11),we now underline the dependence on the graph structure g, denoting it by (N, vg). This game isequivalent to a network game (N,w), where N is the set of players, and w : G(N) → R, withw(∅) = 0, is the value function, which assigns to every graph g the worth w(g). Formally, ourcooperative game is equivalent to the network game de�ned for each g ∈ G(N) by
w(g) = vg(N(g)). (13)
Given the corresponding network game, it is possible to go back to the original cooperative gamein (11), as for each g ∈ G(N) and for each S ⊆ N , vg(S) = w(g|S).
A network game is an extension of a cooperative game, in which the particular structure of thenetwork, and not only the subsets of the players, matters. These two representations, focusing ondi�erent aspects, the graph and the coalitions, permit to analyze di�erent properties of the model.In particular, the network game representation allows us to analyze how the social network structurein�uences the e�cient disclosure pro�le selected under cooperation and the value of the personaldata of the users depending on their position.
Di�erently from the original model in [36], we allow the graph to contain loops. This is necessaryto represent our cooperative game as a network game because we need to be able to distinguishbetween two di�erent cases. In the �rst case, when an isolated player does not belong to the coalitionS, she does not contribute to the aggregate utility; in the network game, we model the situationby assuming that she does not have links. In the second case, the isolated player belongs to thecoalition, but she does not share links with any other player and then she maximizes her utilityindependently from the others; then, we assume that this player is connected only to herself by aloop. In the original model [36], it was not necessary to distinguish between these two cases, as theutility of an isolated player was assumed to be zero.
7Adding the presence of loops does not in�uence the model and the analysis of the previous sections.
18
5.3 The Properties of the Network Game
We now de�ne some important properties for network games and we analyze if they are satis�ed byour model. Given the network game (N,w), we say that the value function is component additiveif∑
h∈C(g)w(h) = w(g). This condition rules out externalities across components, but still allowsthem within components. Given a permutation of players π (a bijection from N to N), and anyg ∈ G(N), let gπ = {π(i)π(j)|ij ∈ g}. Thus, gπ is a network that shares the same architecture as gbut with the players relabeled according to π. A value function is anonymous if for any permutationof the set of players π, w(gπ) = w(g). Anonymity says that the value of a network is derived fromthe structure of the network and not the labels of the players who occupy various positions. Thefollowing proposition states that our game, represented as a network game with value function (13),satis�es both properties.
Proposition 5.1 The value function w de�ned in (13) is component additive and anonymous.
Proof: Both the properties follows from the de�nition of w. In fact, for each g ∈ G(N),
w(g) = vg(N(g)) = maxdN(g)∈DN(g)
W (dN(g), g),
where
W (dN(g), g) =∑
i∈N(g)
fdi + δ
∑j∈Ni(g)
dj
− kdi .
The aggregate utility W as a function of g is component additive, as for each i ∈ N(g), her utilitydepends only on her neighborhood, which is a subset of the component to which player i belongsto. Then, also the maximum value of W as a function of g is component additive, as the playersof each component maximize independently their aggregate utility, and this gives the componentadditivity of w. Moreover, as f and k do not depend on i, the utility of each player depends onlyon 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 consequence of the fact that our model allows ex-ternalities only between connected players, in particular only between neighbors. The property ofanonymity is a consequence of the fact that f and k are independent from i. It guarantees that ourmodel really focuses on the network structure and that it captures the utility that a set of playerscan generate because of their pairwise relations, independently from their labels.
A graph g ∈ G(N) is said to be strongly e�cient if w(g) ≥ w(g′) for all g′ ∈ G(N), i.e., if itguarantees maximal total value. The following theorem shows that the complete graph satis�es thisimportant property for our model.
Theorem 5.2 The complete graph gN is strongly e�cient.
Proof: At �rst, observe that, given a graph g ∈ G(N), the monotonicity of the game (N, vg) inTheorem 4.2 implies that w(g + ii) ≥ w(g) for each i ∈ N . From this it immediately follows that,given g, the graph g′ which contains all the links of g and in addition all the loops, will provide a
19
higher aggregate utility than g. Such a graph is s.t. every player in N has at least one link, i.e.,N(g′) = N . Moreover, for each i ∈ N(g′), we observe that
f
di + δ∑
j∈Ni(g′)
dj
≤ fdi + δ
∑j∈Ni(gN )
dj
, (14)
because f is an increasing function and because g′ ⊆ gN implies that Ni(g′) ⊆ Ni(g
N ). Given thegraph g′ with N(g′) = N(gN ) = N , (14) implies that W (dN , g
′) ≤ W (dN , gN ) for each dN ∈ DN ,
from which it follows
w(g′) = vg′(N) = maxdN
W (dN , g′)
≤ maxdN
W (dN , gN ) = vgN (N) = w(gN )
and this proves that gN is strongly e�cient. �Theorem 5.2 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 aggregateutility of S is maximal when all the links are present. In the following result, we show that theremay exist a strongly e�cient graph which is not complete, but that in this case, missing links areonly between non contributing players.
Corollary 5.3 Any strongly e�cient graph g ⊂ gN is s.t. N(g) = N and if ij /∈ g, then there
exists an e�cient disclosure pro�le de such that dei = dej = 0.
Proof: Let g ⊂ gN be a strongly e�cient graph. Then, N(g) = N follows from the proof ofTheorem 5.2. Now, let de be an e�cient disclosure pro�le such that W (de, g) = w(g). Adding alink ij when player i is s.t. dei 6= 0, the aggregate utility becomesW (de, g+ij) > W (de, g) (meaningthat the maximum value for the graph g + ij will be also greater). As a consequence, in order forg to be e�cient we need to have that if ij /∈ g, then dei = dej = 0. �
To complete our analysis of the network game, we �rst notice that a player that discloses zeroat an e�cient pro�le cannot be isolated. Then, that in order to be strongly e�cient, a graph hasto be fully connected. Moreover we observe that, typically, players contributing zero at the e�cientpro�les are players such that their neighborhood is included in the neighborhood of another player,hence, intuitively, players whose links are not necessary for reaching any other player.
5.4 The Properties of the Allocations for Network Games
After analyzing the property of the game as a function of the graph, we now �nally analyze theproperties of the proposed solutions as functions of the graph. We start by extending them tonetwork games. In particular, the de�nition of Shapley value for our game (N, v) corresponds tothe de�nition of the Shapley value as allocation rule for network games in [36], while also any vectorin the core of the cooperative game (N, v) can be interpreted as an allocation for the correspondingnetwork game (N,w). Formally, 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 payo� allocation (x1, . . . , xn).
A notion of stability for the graph with respect to a given allocation is given by the followingde�nition. A network g is pairwise stable with respect to allocation rule Y and value function w if
20
(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 de�nition, we assume that the formation of a link requires the consent of both playersinvolved, while severance can be done by one player unilaterally. Then, a network is pairwise stableif (i) no player wishes to severe a link that she is involved in; and (ii) if one link is not in the graphand one of the involved players would bene�t from adding it, then it must be that the other playerwould su�er from the addition of the link.
Our last two results provide stability results for the core and the Shapley value as allocationrules of the network game (N,w).
Proposition 5.4 Given a non-complete graph g, for each payo� allocation in the core C(vg) thereexists a payo� allocation in the core C(vg+ij) which assigns at least the same payo� to players i andj.
Proof: At �rst, we observe that w(g|S) ≤ w(g + ij|S) for each S ⊆ N , i.e., vg(S) ≤ vg+ij(S),and , in particular vg(S) = vg+ij(S) for each S ⊆ N \ {i, j}. By writing the core conditions, weobserve that they become more restrictive for each player di�erent from i and j, but eventually lessrestrictive for i and j. �
In particular, Proposition 5.4 implies that there exists a vector in the core such that, if thegraph is complete, no player has an incentive to break a link. Such a payo� vector is then a stableallocation. Finally, the last corollary provides an important stability result for the Shapley value aswell.
Proposition 5.5 The complete graph gN 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
payo�.
Proof: To show this result, it is su�cient to show that φi(g + ij, w) > φi(g, w) for at least onei ∈ N . By the de�nition of Shapley value, in particular the result is true if all the marginalcontributions are s.t., w(g + ij|S∪{i}) − w(g + ij|S) ≥ w(g|S∪{i}) − w(g|S), i.e., if the marginalcontributions of player i are always bigger when the link ij is present, and at least one inequalityis strict. We notice that w(g + ij|S) = w(g|S), and this implies that it su�cient to show thatw(g + ij|S∪{i}) ≥ w(g|S∪{i}) for each S ⊆ N \ {i} and the inequality is strict for at least onecoalition S. We have that the weak inequality is true, because of what we showed in the last partof the proof of Theorem 5.2. �
Proposition 5.5 shows that in our model, if the Shapley value is implemented as solution to sharethe aggregate utility between the players, 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 otherplayer. When the complete graph is the only strongly e�cient graph, this result is intuitive becausethe players may share a higher aggregate utility when creating links. Interestingly though, it stillholds when there exists a strongly e�cient graph g ⊂ gN as the following example illustrates.
Example 3 Suppose that g is a graph as in Figure 2-(a), with parameter δ = 1 (i.e., in a situation
of perfect substitutability). Such a graph is strongly e�cient. However, while implementing the
Shapley value to share their payo�, the two extreme agents have incentives to create a new link as
in Figure 2-(b), at the expense of the central player.
21
(a) (b)
Figure 2: Strategic creation of links.
At �rst sight, this last result may suggest that, in a situation in which players from a socialnetwork are rewarded for their data, they have incentives to create arti�cial links, in order to geta higher �nal payo�. This, however, would be an over interpretation. Indeed, our results supposethat all created links have the same positive externality and a�rm that players do have incentivesto have a higher number of e�ective real relations. On the contrary, to fully understand the creationof arti�cial links that could occur, we would need to introduce in the model a cost of link creationand potential negative externalities that would arise (a fake link could bring bad recommendationand consequently to a decreasing of the utility of the users). We leave this extension of the modelas future work. More simply, in the context of personal data, our results should be interpreted asshowing that when the graph is dense, this is bene�cial for the users and it augments the value oftheir personal contribution.
6 Concluding remarks
Our work represents a contribution to the analysis of private data monetization and, more generally,to the Internet economics, with the �nal scope of helping online services to determine the users withmost valuable information, e.g., to o�er discounts, or to target them as participants of an onlinesurvey, when involving all the players can be costly.
We face the problem of quantifying the value of personal data via a game-theoretic approach.In our model, we take into account both the private and the public component of personal data,adopting a local public good model with positive externalities, which embeds the users in a socialnetwork. We propose at �rst to introduce some payments to reward the users for the disclosureof their information, and then, in a second part, we analyze a cooperative extension of the game,to adopt some cooperative solutions as tools to underline the importance of the players dependingon their network position. Both the approaches (via some payments or some coordination of theagents) are based on the idea of enhancing the e�ciency of the public good model.
The second part of our analysis, in particular, based on a cooperative extension of our initiallocal public good game, is presented as a new and original tool in the Internet economics, oppositeto the majority of the existing works whose theoretical analysis is mainly focused on a strategicinteraction of the players. Cooperation could naturally arise, whenever the players manage tocoordinate to implement a more e�cient solution for everyone, or it could be implemented throughan intermediary between the provider and the users who would then use our solutions to computethe monetary transfers on each side. In both cases, such an ecosystem would be likely sustained byall actors since cooperation yields a strictly higher welfare which can be shared in the interest of allparties. For this reasons, we believe that such an approach could �nd applications and extensionsoutside the area of personal data and outside the few domains in which cooperative game theory isalready a well a�rmed tool of analysis.
22
In the �nal part of the paper, this cooperative model allowed us to conduct a preliminary analysison how the users can strategically manipulate the graph (and not only the information revealed).Our results show how the links between users may increase the value of personal data, due to theexternalities. In particular, users belonging to dense networks have very high incentives to revealtheir data; and users always have incentives to reveal existing links. These results do not suggest,however, that users have incentives to create arti�cial links in order to increase their own payo�.Indeed, our model is too simple to discuss creation of fake links, which may not be associatedwith the same positive externality as true links and may instead have negative externality. A �rstinteresting extension of our work will be to include this e�ect and analyze which pairs of users (ifany) have an incentive to create fake links. This will guide us towards a fully incentive compatiblemodel where users reveal links truthfully.
In our study, we have made a number of further simplifying assumptions. In particular, inorder to focus on the e�ect of the local public good aspect of personal information derived fromthe graph on its value, we have assumed that the private cost is the same for all players. As allprevious researches have focused only on the private cost of personal information, we believe thatit was necessary to clearly separate the e�ect of the position of a user in the graph from its privatecost and that our paper provides useful results in this direction. As future work, we plan, however,to extend our analysis to the case of heterogeneous private costs in order to quantify the value ofpersonal data in more realistic scenarios that combine heterogeneous private costs and local publicgood components.
A typical problem su�ered by cooperative game-theory solutions is their computational com-plexity. In this paper, we provide results about the core. In the case of a convex game, its extremepoints can be easily computed with a greedy algorithm, but the computation becomes less e�cientfor a generic game. Also the Shapley value cannot be computed e�ciently. There exist, however,various approximations of the Shapley value that can be computed e�ciently (such as some vari-ations of Monte Carlo sampling methods [44]). We plan to extend our work in two directions: (i)analyze how the use of approximations a�ect our results and (ii) study the limit regime where thenumber of players is very large and how this can provide more computationally e�cient or moreintuitive results.
A Proof of Theorem 1
Introducing the payments p and Pi, the game Γ becomes now a game Γ′ = 〈N, [0,+∞)n, (U ′i)i∈N 〉where the utility function for each i ∈ N is de�ned as
U ′i(d, g) =
∑h∈N f
(dh + δ
∑j∈Nh(g) dj
)n
− k
ndi.
Di�erently from Γ, Γ′ is a potential game with potential function Φ : [0,+∞)n → R s.t., for eachd ∈ [0,+∞)n
φ(d, g) =
∑h∈N f
(dh + δ
∑j∈Nh(g) dh
)n
−∑i∈N
k
ndi. (15)
By de�nition of potential game, the set of Nash equilibria of the game Γ′ is contained in the set ofthe local maxima of its potential function Φ. In particular, an absolute maximum of Φ is a Nash
23
equilibrium of Γ′. In fact, any deviation of a player i ∈ N decreases the potential function, andconsequently it also decreases the utility of player i, by de�nition of potential. Now, we observethat an absolute maximum of Φ is also an absolute maximum of n ·Φ, and the latter coincide withthe function of the aggregate utilities of Γ. Consequently, a Nash equilibrium of Γ′ which maximizesthe potential function Φ is an e�cient disclosure pro�le for the original game Γ and we denote itby de.
The aggregate utility of the game Γ′ at equilibrium de is equal to
∑i∈N
∑h∈N f
(dh + δ
∑j∈Nh(g) dh
)n
−∑i∈N
k
ndi
=∑h∈N
f
dh + δ∑
j∈Nh(g)
dh
−∑i∈N
k
ndi
= W (de, g) +n− 1
n
∑i∈N
kdei .
The term n−1n
∑i∈N kd
ei represents the total utility added to the game by the provider, formally,
the cost for the provider to obtain an e�cient disclosure pro�le by the strategic players.
B Proof of Corollary 1
The increase of social welfare is bigger than the payments to the players only if
W (de, g)−W (d∗∗, g) ≥ n− 1
n
∑i∈N
kdei .
De�ning the Price of Anarchy as in (6), the result immediately follows.
C Proof of Theorem 2
Introducing the payments Qi, the game Γ becomes now a game Γ′′ = 〈N, [0,+∞)n, (U ′′i )i∈N 〉 wherethe utility function for each i ∈ N is de�ned as
U ′′i (d, g) = f
di + δ∑
j∈Ni(g)
dj
− kdi+
∑h∈Ni(g)
fdh + δ
∑j∈Nh(g)
dj
− fdh + δ
∑j∈Nh(g)\{i}
dj
.Let d∗ be a strategy pro�le. If d∗ is a Nash equilibrium of Γ′′, then it satis�es the �rst order
conditions (FOCs):∂U ′′i∂di
(d∗, g)
{= 0 if d∗i > 0≤ 0 if d∗i = 0
, (16)
24
for all i ∈ N , where
∂U ′′i∂di
(d, g) = (17)
f ′
di + δ∑
j∈Ni(g)
dj
+ δ∑
h∈Ni(g)
f ′
dh + δ∑
j∈Nh(g)
dj
− k.But these are precisely the KKT conditions of the maximization of social welfare W which arenecessary and su�cient since W is concave. Therefore d∗ maximizes W . The cost of the providertrivially follows from the de�nition of the payments.
D Comparison of the total costs of payments of Theorem 1 and 2
In this appendix, we show a simple example in which the total cost of payments is higher forTheorem 3.3 than for Theorem 3.1.
Let n = 2 and let the two players be connected. An e�cient pro�le de is such that
2f ′(de1 + de2) = k.
The payments of Theorem 3.1 are p = k/2 and Pi = 0, so that the total payment is
k
2(de1 + de2) = f ′(de1 + de2) · (de1 + de2). (18)
The payments of Theorem 3.3 are Q1 = f(d1 + d2) − f(d2) and Q2 = f(d1 + d2) − f(d1), so thatthe total payment is
2f(de1 + de2)− f(de1)− f(de2). (19)
By concavity of f , (19) is larger than (18).
E Proof of Theorem 3
We suppose that the players aim at maximizing the social welfare function given in (5), with FOCs
f ′(di + δ∑j∈N
gijdj) + δ∑j∈N
gijf′(dj + δ
∑h∈N
gjhdh) = k
that we can rewrite in matrix form as
f ′(d + δG · d) + δG · f ′(d + δG · d) = k
or equivalently as
f ′(d + δG · d) = (I + δG)−1k. (20)
These FOCs are similar to the FOCs in (4), but with heterogeneous autarkic play di s.t. f′ (di) =
(I + δG)−1i · k = k (1− δci(−δ,G)). We observe that this new game can be induced by Γ by
introducing a unitary cost equal to bi = δci(−δ,G)k.
25
We complete the analysis by observing that the equilibrium disclosure pro�le is now s.t.
de = (I + δG)−1 ·(f ′)−1 (
(I + δG)−1 · k)
=(f ′)−1
(k(1− δc(−δ,G))) ◦ k (1− δc(−δ,G)) ,
where ◦ denotes the Hadamard product of the two vectors.
F Example 1
Given a symmetric graph g, the corresponding game (N, v) is such that each player is symmetricand then she will have Shapley value equal to v(N)/n. Moreover, given the symmetry of the game,the core is a symmetric convex set, which contains the symmetric vector given by the Shapley value.
G Example 2
Because of the symmetry property of the Shapley value, every player at the periphery has the samevalue. Such a value is given by
φP (v) =
n−2∑t=0
(n− 2
t
)t!(n− t− 1)!
n!v(S1)
+
n−2∑t=0
(n− 2
t
)(t+ 1)!(n− t− 2)!
n![v(St+2)− v(St+1)]
=
n−2∑t=0
(n− 2)!
t!(n− t− 2)!
t!(n− t− 1)!
n!v(S1)
+
n−2∑t=0
(n− 2)!
t!(n− t− 2)!
(t+ 1)!(n− t− 2)!
n![v(St+2)− v(St+1)]
=n−2∑t=0
n− t− 1
n(n− 1)v(S1) +
n−2∑k=0
k + 1
n(n− 1)[v(St+2 − v(St+1)]
=
(n(n− 1)
2− 1
)1
n(n− 1)v(S1)
+
n−1∑t=2
(− 1
n(n− 1)
)v(St) +
1
nv(Sn)
=1
nv(Sn)− 1
n(n− 1)
n−1∑t=2
[v(St)− tv(S1)]
where we observe that the terms v(St)− tv(S1) are all bigger than 0 because of the superadditivityof the game.
26
Because of the e�ciency property, the center node has Shapley value given by
φC(v) = v(Sn)− (n− 1)φP (v)
=1
nv(Sn) +
1
n
n−1∑t=2
[v(St)− tv(S1)].
To prove that the Shapley value belongs to the core, we need to prove that for each S ⊆ N ,∑i∈S φi ≥ v(S). At �rst, we observe that this is true for each coalition of cardinality 1, as a
consequence of the superadditivity of the game8. Second, we consider a coalition R of r ≥ 2peripheric players. We need to prove that rφP (v) ≥ v(R), and this is true as
v(R) = rv(S1) ≤ rφP (v)
and the equality holds whenever we have disconnected players. Finally, we consider a coalition Rof r − 1 peripheric players and 1 central player. We need to prove that
(r − 1)φP (v) + φC(v) ≥ v(Sr),
which is equivalent to proving that
r
nv(Sn) +
n− rn(n− 1)
n−1∑t=2
[v(St)− tv(S1)] ≥ v(Sr).
To show that this last inequality holds, it is su�cient to prove that
v(Sn)
n≥ v(Sr)
r(21)
for each r ≤ n. At �rst we observe that, as the neighborhood of a peripheric player is contained inthe neighborhood of the central player, there always exists an e�cient pro�le such that the centralplayer is the only one to disclose a nonzero amount of personal data. Denote det the disclosure levelof such an e�cient pro�le for a star ST with t players. The inequality holds as
v(Sn)
n≥ v(Sr)
r⇔
nf(den)− kdenn
≥ rf(der)− kderr
⇔
f(den)− k
nden ≥ f(der)−
k
rder
and der is the argmax of a function f(x) − krx which is increasing in the parameter r, and then it
will have a higher value at maximum for a higher parameter.
8E�cient payo� allocations which satisfy this inequality for each coalition of cardinality 1 are called imputations.The Shapley value of a superadditive game is always an imputation [23].
27
H Example 3
We may observe at �rst that an e�cient disclosure pro�le for both the graph is given by the playeron the left giving d s.t., f ′(d) = k/3. Then, both the graph are providing a value v(N) = 3f(d)−kd.The �rst graph in Figure 3-(a) is a star S3, s.t. each peripheric player has a Shapley value equal to
φP =v(N)
3− 1
6[v(S2)− 2v(S1)] <
v(N)
3.
While adding the link as in Figure 3-(b), the players become symmetric and the ex-peripheric playersnow get a utility equal to v(N)/3, then strictly bigger than before. Obviously, as the aggregateutility remains the same, the Shapley value of the ex-central player decreases.
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Documents De travail GreDeG parus en 2018GREDEG Working Papers Released in 2018
2018-01 Lionel Nesta, Elena Verdolini & Francesco Vona Threshold Policy Effects and Directed Technical Change in Energy Innovation 2018-02 Michela Chessa & Patrick Loiseau Incentivizing Efficiency in Local Public Good Games and Applications to the Quantification of Personal Data in Networks