Cheap talk theoretic tools for distributed sensing in the presence of strategic sensors

Transcription

1 1 / 37 Cheap talk theoretic tools for distributed sensing in the presence of strategic sensors Cédric Langbort Department of Aerospace Engineering & Coordinated Science Laboratory UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN langbort@illinois.edu IPAM UCLA includes joint work w/ E. Akyol (UIUC), T. Başar (UIUC), F. Farokhi (U Melbourne), A. Teixeira (KTH)

2 2 / 37 Motivation Strategic sensors and CPS Example #1: False data injection attacks picture from Miao et al.

3 3 / 37 Example #2: Participatory sensing Motivation Indirect: Query users devices (e.g., Mobile Millennium) Direct: Ask them to report (e.g., Waze)

4 4 / 37 Example #2: Participatory sensing Motivation What if I intentionally under-estimate?

5 Example #2: Participatory sensing Motivation What if I intentionally under-estimate? What if I intentionally over-estimate? 5 / 37

6 Motivation Sounds far-fetched?... 6 / 37

7 Motivation Sounds far-fetched?... 7 / 37

8 Motivation Sounds far-fetched?... 8 / 37

9 9 / 37 Questions How to model such strategic scenarios? Do self interest/ strategic behavior impose fundamental limits on the quality of estimation? Does the degree of strategic intent matter? Implications for system design?...

10 10 / 37 Cheap talk: a general framework for strategic information transmission Definition A game in which (a) better informed sender(s) are/is communicating with a receiver, who ultimately takes a decision influencing all utilities In example #2, e.g., : Better informed senders: Participants Receiver: Traffic estimation platform (e.g., Waze) Decision influencing utilities: Routing recommendations, via traffic estimate.

11 11 / 37 These talks Lecture 1: review some classical Cheap Talk models and results, mainly [Crawford & Sobel, Strategic Information Transmission, Econometrica, 82] Lecture 2 (in week 3): present new ones that are arguably better suited for CPS-motivated problems

12 12 / 37 CS 82 Model Two agents, a sender S and receiver R with utilities U S (a, x) and U R (a, x), respectively. x X = [0, 1] is the state of Nature, observed only by sender S. Receiver R does not know x and only has a uniform prior on X. He makes a decision a R. S can issue a message z, based on her observation of x to (mis)inform R about its value. Note that utilities do not depend on z: cheap talk!

13 13 / 37 CS 82 Model Central technical assumptions For i {S, R}, U i is smooth with U11 i < 0 and Ui 12 > 0. (1) U S (., x) and U R (., x) have a well-defined unique maximum for all x X which we call a S (x) and a R (x) respectively. Because of (1), a i (.) is a smooth and increasing function of x. For example, when maximum is interior point, 0 = d dx Ui 1 (ai (x), x) = U11(a i i d (x), x) }{{} dx ai (x) + U12(a i i (x), x) }{{} <0 >0

14 14 / 37 CS 82 Model Central technical assumptions a S (x) a R (x) for all x X (2) or, equivalently (since X is compact) ɛ > 0 such that a S (x) a R (x) > ɛ for all x X. Both assumptions together mean that S and R have similar but different interests (both decisions increase with state but exact values are always different). Example U S (a, x) = (a (x + θ)) 2, U S (a, x) = (a x) 2 where θ (the type of S ) measures the degree of conflict.

15 15 / 37 CS 82 Model Equilibrium Let γ S be a stochastic kernel (i.e., γ S (. x) is a probability distribution on R for every x X) and γ R : R R be a map. They form a (Bayesian Nash cheap talk) equilibrium if 1 γ S (z x) > 0 iff z argẑ max U S (γ R (ẑ), x) 2 γ R (z) maximizes 1 0 U R (a, x)µ(x z)dx with respect to a, where µ(x z) = γs (z x) γ S (z t)dt

16 16 / 37 A zoo of possible equilibria Babbling: γ S (. x) does not depend on x, messages are uninformative. Fully Revealing: γ S (. x) has all its mass at a single point m x, with m x0 m x1 whenever x 0 x 1. Partially Revealing: When not empty, the set {x γ S (z x) > 0} is not a singleton but not full X either... Simple": γ S (. x) is a simple map, maybe linear

17 17 / 37 Main result Theorem [C&S, 82] There exists N such that for every 1 N N, there exists an equilibrium involving N-bin quantization. More precisely, X is partitioned into N bins and message space is also X γ S (. x) is uniformly distributed over the bin to which x belongs γ R (z) maximizes receiver s expected utility subject to x belonging to same bin as z. Every equilibrium is essentially equivalent to one such equilibrium. N is a decreasing function of the degree of conflict between S and R s utilities. Fully revealing equilibria do not exist in this model!

18 18 / 37 Some extensions Multiple senders Particular case Two senders, one receiver with quadratic utilities: U S i (a, x) = (a (x + θ i )) 2, U R (a, x) = (a x) 2 X = [ M, M]. Theorem (Battaglini 02) A fully revealing equilibrium exists iff θ 2 θ 1 M.

19 19 / 37 Some extensions Multidimensional variables Particular case X R 2 U R (a, x) = x a 2, U S (a, x) = x + θ a 2 Theorem (Battaglini 02) There exists a fully revealing equilibrium (with message space X) if θ 1 and θ 2 are not on the same ray from the origin.

20 x S Our cheap talk model 1-sensor case z R y ˆx(y, z) of this Gauss matrix Fig. 1. Communication structure between sender S and the side channel with Sender receiver S R. has a private type θ, observes state of Nature x, and sends message z Receiver R observes z and side channel s signal y and that herding between strategic senders (including malicious computes ˆx(.) = arg min ones) is indeed a virtue (for φ(.) E x φ(y, z) 2 the receiver). Sender The chooses rest of z the so that paper is organized as follows. First, we consider the single sender cheap talk game in Section II. In E (x + θ) ˆx(y, z) 2 Section III, we generalize the setup to multiple sender case. is minimized, We focus i.e., on the so cheap that receiver talk game is misled in which in thinking the senders thatare x is x + herding θ. in Section IV. Numerical results are presented in Section V. Finally, we conclude the paper in Section VI. Now, the re TH ˆx and th are de 20 / 37

21 21 / 37 Differences with CS 1 State of nature is assumed Gaussian with zero mean. 2 The private type θ of the sender(s) is a random variable in our model. 3 We focus on Stackelberg equilibria rather than Nash equilibria, i.e., the receiver commits to a strategy. 4 Other unusual notions of equilibria for the multi-sender and multistep cases...

22 22 / 37 Stackelberg equilibrium Theorem Assume x, y, z jointly Gaussian with zero mean, then (i) There exists an equilibrium in which the sender s strategy is affine z = a T x + b T θ + c T y + v, v N(0, V ) (ii) It is possible to rescale a, b, c so that v = 0 and Σ zz = I but we never simply get flat-out lie" (i.e., z = x + θ) (iii) There does not exist an equilibrium where sender s strategy is not affine.

23 Proof Regardless of sender S s strategy, receiver uses ˆx(.) = E(x.) Assuming S uses an affine strategy, R s best response is usual LMSE. In this case, S s strategy is such that E (x + θ) ˆx LMSE (y, z) 2 is minimized w.r.t. z This can be rewritten solely in terms of signals covariance matrices and yields a QCQP of the form min trace with Q 0. s.t. Σ xz Σ θz Σ yz Σ xz Σ θz Σ yz T T Q R Σ [ x θ y Σ [ x θ y ][ x θ y ][ x θ y ] ] Σ xz Σ θz Σ yz Σ xz Σ θz Σ yz I 23 / 37

24 24 / 37 Under the additional constraint that Σ x θ y z x θ y z Proof c ed 0, any solution to the QCQP can be realized with an affine sender strategy. (iii) is proved using the maximal correlation measure theorem... When z is required to be a scalar message, a, b, c can be computed more explicitly: Constraint in QCQP is tight Optimal vector of covariances is the eigenvector with smallest eigenvalue of (R 1 2 ) T QR

25 Multisensor case θ 1 S 1 x θ 2 S 2. y 1 y 2 y n R ˆx((y i ) N i=1 ) θ N S N State of Nature: x N (0, Σ xx ) 25 / 37

26 Multisensor case θ 1 S 1 x θ 2 S 2. y 1 y 2 y n R ˆx((y i ) N i=1 ) θ N S N At the first step, N strategic sensors receive their information: Sensor S i s cost: E{ (x + θ i ) ˆx 2 } S i has perfect measurements of x, θ i, knows nothing about others θ = (θ i ) N i=1 N (0, Σ θθ) 26 / 37

27 27 / 37 Multisensor case x θ 1 θ 2 θ N S 1 S 2. S N y 1 y 2 y n R ˆx((y i ) N i=1 ) At the second step, sensors transmit scalar signals: y i = γ i (x, θ i ) R where γ i (x, θ i ) = a i x + b i θ i + v i v i N (0, Σ vi v i ) The set of such mappings is Γ i (isomorph to R nx R nx R 0 ).

28 Multisensor case θ 1 S 1 x θ 2 S 2. y 1 y 2 y n R ˆx((y i ) N i=1 ) θ N S N At the third step, the receiver announces its estimate ˆx = arg min E{ x ˆx(y 1,..., y n ) 2 } over Ψ, the set of all Lebesgue-measurable functions from R N to R nx. 28 / 37

29 Multisensor case θ 1 S 1 x θ 2 S 2. y 1 y 2 y n R ˆx((y i ) N i=1 ) θ N S N At the fourth step, the cost functions are realized: Receiver: E{ x ˆx 2 }; Sensor i: E{ (x + θ i ) ˆx 2 }. 29 / 37

30 30 / 37 Nash-Stackelberg Equilibrium Definition A tuple (ˆx, (γ i ) N i=1 ) Ψ Γ 1 Γ N constitutes a (Nash-Stackelberg) equilibrium in affine strategies if ˆx (.) arg min E{ x ˆx((γj (x, θ j )) N j=1 ˆx(.) Ψ ) 2 }, γ i (.) arg min γ i (.) Γ i E{ (x + θ i ) ˆx (γ i (x, θ i ), (γ j (x, θ j )) j i ) 2 }, i.

31 31 / 37 Symmetric strategies In a large homogeneous sensor population (N >> 1, i.i.d. types), we would expect all sensors to use the same reporting strategy. In this case, receiver R s best response is LMSE with respect to y = y y N N. From this and previous results, can show There exists a(n essentially) unique symmetric equilibrium in affine strategies of the form y i = a T x + b T θ i + v i, v i N(0, V ). It is explicitly computable when each y i is scalar.

32 32 / 37 How good is the equilibrium? Assume that Σ xθ = 0, Σ θi θ j = δ ij Σ θθ. At equilibrium Receiver s error covariance matrix =: ˆΣ = Σ xx nx nx where α, β R 0 and U R with particular, lim ˆΣ = Σ xx, N 1 α + βn U, 0 < U (α + β)σ xx. In i.e., receiver is essentially getting no useful information from the senders, in aggregate. Too many (strategic) cooks spoil the broth...

33 33 / 37 Another class of equilibrium Herding Equilibrium A tuple (ˆx, γ ) Ψ Γ constitutes a herding equilibrium in affine strategies if ˆx arg min E{ x ˆx((γ (x, θ j )) N j=1 ˆx Ψ )2 }, γ arg min γ Γ E{ (x + θ i) ˆx(γ(x, θ i ), (γ(x, θ j )) j i ) 2 }, i. This captures some notion of bounded rationality: Each sender is strategic enough to think others are optimizers too, but not refined/informed enough to guess what their actions might be. Assumes everyone will act as and reason as self.

34 34 / 37 Herding equilibrium Theorem Under same assumptions. There exists a unique herding equilibrium in affine strategies where the receiver follows ˆx (y) = E{x (y y N )/N} and sender S i, 1 i N, employs a linear policy γ (x, θ i ) = a x + b θ i where ( ) ( b 1/2 NΣ a = θθ 0 0 Σ 1/2 xx ) ζ, and ζ is the unit-norm eigenvector corresponding to the smallest eigenvalue of the matrix ( 0 1 N V 1/2 θθ Σ 1/2 ) xx 1 N Σ 1/2 xx V 1/2. θθ Σ xx

35 35 / 37 When herding is a virtue Assume that Σ xθ = 0, Σ θi θ j = δ ij V θθ, then at the herding equilibrium, lim N ˆΣ = 0. The rate of convergence is faster than with nonstrategic noisy sensors. Estimation Error Independent Senders Herding Senders Nonstrategic Senders N

36 36 / 37 Extensions and Future Works Main lesson Strategic sensing is a fact of CSPS life Results show the importance of degree of strategic intent and can help design mitigation policies for sensing/crowd sourcing in this context

37 37 / 37 Extensions and Future Works Main lesson Strategic sensing is a fact of CSPS life Results show the importance of degree of strategic intent and can help design mitigation policies for sensing/crowd sourcing in this context Other current extensions Dynamic or repeated cheap talk game Arbitrary communication graphs A richer theory of Strategic Information Transmission using IT tools... Applications to adversarial machine learning