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PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Personal Information on Social Networks:Incentives for Data Release and Fair
Monetization
Michela CHESSA
Networking and Security Department, [email protected]
Joint work with: Prof. Patrick Loiseau, EURECOM, FranceStratis Ioannidis, Yahoo! Labs, Los Altos, CA, USAProf. Jens Grossklags, PennState University, University Park, PA, USA
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 1 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Motivation
Motivation
Personal data has intrinsic economic value ⇒ the “New Oil” ofthe 21st Century.
Many companies derive profit from user data acquired through onlinetracking: Google, Facebook, Amazon,...
Data have also societal importance ⇒ personal data is a public good
Crucial points:
for the companies and the society: collecting good data,performing good estimation
for the users: having control over their data
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 2 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Motivation
Our Approach
Game Theory models situations of interacting agents. It providesthe good instruments to solve both the crucial previous points
Large literature on Game Theory to value personal data: [Ghosh andRoth, 2011], [Kleinberg et al., 2001], [Ligett and Roth, 2012], andmany others.
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 3 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Motivation
In This Talk (1)
In this work, we study the strategic interaction of the users. Wemodel personal data as a public good, to balance
privacy concerns: fear of losing control of their data
will to contribute:for societal importance, and because theycan have benefits (targeted advertising,...)
We propose a solution for themonetization of personal data,which depends on the socialnetwork structure
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 4 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Motivation
In This Talk (2)
Linear Regression Games
Fair Monetization
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 5 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Linear Regression Games
Linear Regression Games
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 6 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Linear Regression Games
The Linear Regression Model
N = {1, . . . , n} set of usersyi ∈ R private data of user i (sensible data)x i ∈ Rd public features of i (age, sex, ...)εi inherent noise, 0 mean and variance σ2
β ∈ Rd model parameters
yi = βTx i + εi
BUT while replying to a surveyzi , additional noise 0 mean and variance σ2
i
yi perturbed private data
yi = βTx i + εi + zi
A user reveal to the analyst yi and the precision λi = 1σ2+σ2
i
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 7 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Linear Regression Games
Analyst’s parameters estimation
The analyst estimates the model parameters using the generalizedleast-square (GLS) estimator
β = (XTΛX )−1XTΛy
where Λ =
λ1
. . .
λn
.
This estimator is unbiased and has covariance V = (XTΛX )−1.
Theorem (Aitken)
V is the mimimum covariance estimator between all linear unbiasedestimators.
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 8 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Linear Regression Games
The Cost Function
Each user chooses λi ∈ [0, 1/σ2] in order to minimize a cost function
Ji (λi ,λ−i )︸ ︷︷ ︸Cost function
=ci (λi )︸ ︷︷ ︸
Privacy cost
+F (V (Λ))︸ ︷︷ ︸
Estimation cost
Assumption
The privacy costs ci : R+ → R+, i ∈ N, are twice continuouslydifferentiable, non-negative, non-decreasing and strictly convex.
Assumption
The scalarization F : Sn++ → R+ is twice continuously differentiable,
non-negative, non-constant, non-decreasing in the positivesemidefinite order, and convex.
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 9 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Linear Regression Games
The Linear Regression Game
Linear regression game Γ =⟨N, [0, 1/σ2]n, (Ji )i∈N
⟩It is a public good game
it has a unique non-trivial equilibrium
Theorem (Aitken Type Theorem)
In the strategic setting, GLS gives optimal covariance among linearunbiased estimators.
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 10 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Linear Regression Games
Questions
What if the user may lie? In particular, if they can misreportstrategically the precision they used⇒ GLS with strategic users
Is there a way to convince the users to provide data with ahigher precision?⇒ GLS with minimum precision level
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 11 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Linear Regression Games
Linear Regression GamesGLS with strategic users
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 12 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Linear Regression Games
GLS with strategic users
A user reveal to the analyst yi and the reported precisionλri ∈ [0, 1/σ2]
The analyst estimates the model parameters using the generalizedleast-square (GLS) estimator
βr = (XTΛrX )−1XTΛr y
where Λr =
λr1
. . .
λrn
.
This estimator is unbiased and has covariance
V r = (XTΛrX )−1XTΛrΛ−1ΛrX (XTΛrX )−1 � V
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 13 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Linear Regression Games
GLS as a Mechanism
Each user chooses λi ∈ [0, 1/σ2] and λri ∈ [0, 1/σ2] in order tominimize a cost function
J ri (λi , λri ,λ−i ,λ
r−i )︸ ︷︷ ︸
Cost function
=ci (λi )︸ ︷︷ ︸
Privacy cost
+F (V r (Λ,Λr ))︸ ︷︷ ︸Estimation cost
Game Γr =⟨N, [0, 1/σ2]2n, (J ri )i∈N
⟩Is GLS incentive compatible? Do the users have incentives totruthfully reporting the precision?
TheoremThe GLS mechanism is incentive compatible.
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 14 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Linear Regression Games
Linear Regression GamesGLS with minimum precision level
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 15 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Linear Regression Games
A Simplified Model (1)
Public feature has dimension 0β ∈ R model parameterThe GLS estimator is simply a weighted average
β =
∑i∈N λi yi∑i∈N λi
with variance
v =1∑
i∈N λi∈ [σ2/n,+∞]
Each user chooses λi ∈ [0, 1/σ2] in order to minimize a cost function
Ji (λi ,λ−i ) = c(λi ) + f (v)
TheoremThe game Γ has a unique non-trivial symmetric Nash equilibrium s.t.λ∗i = λ∗(n). Playing at equilibrium, the users reach an estimationlevel v(λ∗(n)).
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 16 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Linear Regression Games
A Simplified Model (2)
Corollary
The equilibrium precision level λ∗(n) satisfies
(i) λ∗(n) is a non-increasing function of the number of agents
(ii) limn→+∞ λ∗(n) = 0.
Corollary
The equilibrium variance satisfies
(i) v(λ∗(n)) is a non-increasing function of the number of agents n
(ii) limn→+∞ v(λ∗(n)) = 0.
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 17 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Linear Regression Games
Minimum Precision Level
The analyst sets a minimum precision level η ∈ [0, 1/σ2].
A user i can
decide to not participate ⇒ λi = 0
decide to participate, respecting the level ⇒ λi ≥ η
Game Γη =⟨N,[{0} ∪ [η, 1/σ2]
]n, (J ri )i∈N
⟩Theorem
When λ∗(n) 6= 1/σ2, the analyst can optimize the estimation(minimize the variance) by choosing an optimum minimumprecision level η = η∗(n) > λ∗(n).
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 18 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Linear Regression Games
Monomial Privacy Costs andLinear Estimation Cost
The cost function is now given by
Ji (λi ,λ−i ) = cλki + v
with c ∈ (0,+∞) and k ≥ 2.
Then
λ∗(n) =
{ (1
ckn2
) 1k+1 if
(1
ckn2
) 1k+1 ≤ 1/σ2
1/σ2 otherwise
η∗(n) =
(
1cn(n−1)
) 1k+1
if(
1cn(n−1)
) 1k+1 ≤ 1/σ2
1/σ2 otherwise .
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 19 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Linear Regression Games
Minimum Precision Level (2)
Asymptotically, when the number n of agents is large, the achievedamelioration converges towards a constant depending only on k:
v(λ∗(n))
v(λ∗(n, η(n)))∼n→∞ k
1k+1
In the example, we observe that it is bounded superiorly, it goes to 1for large k ’s and it is in the range of 25− 30% improvement forvalues of k around 2− 10.
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 20 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Fair Monetization
Fair Monetization
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 21 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Fair Monetization
The Social Network
The social relations between theusers play a crucial role
Some information about a user may be extracted from the personaldata of the users he is connected with, even though he did notoriginally disclose the information.
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 22 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Fair Monetization
User i ’s Utility
Grand coalition N
Ui (e, g) = f
ei +∑
j∈Ni (g)
ej
− kei
Subcoalition S ⊆ N
Ui (eS , g |S) = f
ei +∑
j∈Ni (g |S )
ej
−keiM. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 23 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Fair Monetization
The Cooperative Game
Public good models often yield non-efficient equilibria.How to reach efficiency? COOPERATING!
To reach efficiency, the users in S can cooperateand play an efficient profile e
∗S because:
they are forced by the provider
they spontaneously decide to create a bindingagreement
Cooperating, a coalition S may reach an aggregate utility (a value)
v(S) =∑i∈S
Ui (e∗S , g |S)
⇒ cooperative game (N, v)
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 24 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Fair Monetization
von Neumann and MorgensternCharacteristic Function
The characteristic function of von Neumann and Morgensternfrom our n-person strategic-game Γ is defined ∀S ⊆ N as
v ′(S) = minσN\S∈∆(EN\S )
maxσS∈∆(ES )
∑i∈S
Ui (σS , σN\S , g),
where ∆(ES) is the set of correlated strategies available to coalitionS . We let Ui (σS , σN\S) denote player i ’s expected payoff when thecorrelated strategies σS and σN\S are implemented, that is
Ui (σS , σN\S) =∑eS
∑eN\S
σS(eS)σN\S(eN\S)Ui (eN , g)
=∑eS
∑eN\S
σS(eS)σN\S(eN\S)
fei +
∑j∈Ni (g)
ej
− kei
.M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 25 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Fair Monetization
Properties
Proposition
The cooperative game (N, v) is equivalent to the minimaxrepresentation of the game Γ, i.e.,
v(S) = v ′(S)
for each S ⊆ N.
Theorem
The game (N, v) is monotonic and superadditive.
Meaning that, it is not convenient toform two teams!
Users maximize the aggregate utility by cooperating all together
Proposition
The complete graph is the only strongly efficient graph.
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 26 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Fair Monetization
Valuation of Personal Data
We propose a cooperative solution to valuate personal data of theusers
Definition (The Shapley Value)
φi (v) =∑
S⊆N\{i}
s!(n − s − 1)!
n!(v(S ∪ {i})− v(S)).
Definition
A network g is pairwise stable with respect to allocation rule Y andvalue function w if
(i) for all ij ∈ g , Yi (g ,w) ≥ Yi (g − ij ,w) andYj(g ,w) ≥ Yj(g − ij ,w), and
(ii) for all ij /∈ g , if Yi (g + ij ,w) > Yi (g ,w) thenYj(g + ij ,w) < Yj(g ,w).
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 27 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Fair Monetization
Theorem
The complete graph gN is the only pairwise stable graph with respectto the Shapley value allocation. In particular, every player hasincentives to create a new link to augment her own payoff.
Meaning that, it is convenient to be all connected! Does thismean, in practice, that they have advantage to create fake links?NO!
Fake links would affect thedefinition of the utility!
How to model this situation... ⇒ Future work
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 28 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Summary/Future work
Summary/Future work
Linear Regression Games
Model: A user-centric linear regression game. Noise added to data,the users choose the variance.
Results: An Aitken-type theorem for Nash equilibria. Incentivecompatibility in revealing the variance. Incentives for improving theestimation.
Fair Monetization
Model: A user-centric local public good model. Users maycooperate.
Results: Users have incentive to cooperate. Fair monetization ofpersonal information as a function of the graph
Future work: Other possible solutions for the fair monetization.Properties related to the network structure. Privacy costinfluenced by neighbors. To combine the two approaches!!!
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 29 / 30
PersonalInformation on
Social Networks
M. Chessa
Motivation
Linear RegressionGames
Fair Monetization
Summary/Futurework
Summary/Future work
Thank you for your attention!
M. Chessa (EURECOM) Personal Information on Social Networks February 25th, 2015 30 / 30