Personal Information on Social Networks: Incentives for ...€¦ · Social Networks M. Chessa Motivation Linear Regression Games Fair Monetization Summary/Future work Personal Information

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


Social Networks

M. Chessa

Motivation


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



Social Networks

M. Chessa

Motivation


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.



Social Networks

M. Chessa

Motivation


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



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework

Motivation

In This Talk (2)

Linear Regression Games

Fair Monetization



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework





Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework


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



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework


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.



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework


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.



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework


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.



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework


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



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework


Linear Regression GamesGLS with strategic users



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework


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



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework


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.



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework


Linear Regression GamesGLS with minimum precision level



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework


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)).



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework


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.



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework


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).



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework


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 .



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework


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.



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework

Fair Monetization

Fair Monetization



Social Networks

M. Chessa

Motivation


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.



Social Networks

M. Chessa

Motivation


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


Social Networks

M. Chessa

Motivation


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)



Social Networks

M. Chessa

Motivation


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


Social Networks

M. Chessa

Motivation


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.



Social Networks

M. Chessa

Motivation


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).



Social Networks

M. Chessa

Motivation


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



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework

Summary/Future work

Summary/Future work


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!!!



Social Networks

M. Chessa

Motivation


Fair Monetization

Summary/Futurework

Summary/Future work

Thank you for your attention!


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Personal Information on Social Networks: Incentives for ...€¦ · Social Networks M. Chessa Motivation Linear Regression Games Fair Monetization Summary/Future work Personal Information