Remote Estimation Games over Shared Networks

Transcription

1 October st, 04 Remote Estimation Games over Shared Networks Marcos Vasconcelos & Nuno Martins Dept. of Electrical and Computer Engineering Institute of Systems Research University of Maryland, College Park 5 nd Allerton Conference on Communication, Control, and Computing nmartins@umd.edu

2 Motivation Context Sensor Estimator Networked control systems Sensor Estimator Decision Makers cooperate or compete to achieve certain goals Sensor 3 Wireless Network Estimator Network models Incomplete graphs Rate-limited point-to-point channels Sensor N Estimator N Additive White Gaussian Noise Analog Erasure channel What about interference?

3 Interference Multiple agents sharing a communication medium Physical layer: Multiple Access Channel, Interference Channel MAC/Network layer: Collision Channel S DM S Y Channel Model DM.. S N Collision Channel. Y Y N DM chooses to transmit or not Collision when two or more DMs transmit DMN Y i {?,S i, C} Simplest model for interference

4 Problem Statement X DM S Y Estimator ˆX X i N (0, i ) Collision Channel X? X X DMN Y Estimator ˆX U i = i (X i ) {0, } S i = Xi, if U i =?, if U i )= 8 >< >: (?,?), if s =?,s =? (x,?), if s = x,s =? (?,x ), if s =?,s = x (C, C), if s = x,s = x Each sensor-estimator pair minimizes its own cost functional J i ( i, j, i )=E[(X i ˆXi ) ]

5 Previous work S X DM S Collision Channel R ( ˆX, ˆX ) Estimator X DM J(,, ) =E[(X ˆX ) +(X ˆX ) ] Vasconcelos & Martins (Allerton '3) Team decision problem - Focus on full cooperation Proved the optimality of threshold policies (asymmetric in general)

6 Our main results Focus on competitive behavior. Obtain the structure of security and Nash equilibrium policies. Establish a connection with optimal quantization theory collision channel without and with capture 3. Policy design using the Lloyd-Max algorithm

7 Part I. The collision channel without capture X DM S Y Estimator ˆX Collision Channel X DMN Y Estimator ˆX Assume DM transmits with prob. - selfish behavior Worst case scenario for DM Estimator only receives Y {?, C}

8 Security policies When the channel is always occupied by the opponent: Best communication policy Best estimation policy sec (x) sec (y) = 8 < : E[X X 0] = E[X X <0] = q q, if y = C, if y =?. J i ( sec i, Cost self j, i sec )= i ˆx 0 ˆx x The security policy is determined by the optimal bit quantizer

9 Security policies Proposition : A security policy for DMi in the game over the collision channel has a single threshold structure of the form sec i (x i )=, xi 0; 0, x i < 0. If both DMs use security policies, their incurred costs are: J sec i = 3 4 i

10 Security policies Example: = = J ( sec, J ( self, self, sec )= sec, self )= J sec = 3 4 J sec = 3 4 =0.75 = A security policy accesses the channel with probability =0.5 Question: What is the structure of the optimal communication policy when the channel is occupied with probability <?

11 Structure of Nash equilibrium policies Analysis from the perspective of a single DM Assume the opponent transmits with probability D X Sensor S Collision Channel Y Remote Estimator ˆX D B( ) U = (X) {0, } U =0) Y =? U =,D =0) Y = X U =,D =) Y = C (X) =X (?) =ˆx 0 (C) =ˆx J(, ) =E[(X ˆx 0 ) U = 0] Pr(U = 0) + E[ (X ˆx ) U = ] Pr(U = ) Binary quantization with asymmetric distortion

12 Structure of Nash equilibrium policies J(A 0, ˆx 0, ˆx )= Z A 0 (x ˆx 0 ) f X (x)dx + Z R\A 0 (x ˆx ) f X (x)dx Necessary optimality condition: x A 0, (x ˆx 0 ) apple (x ˆx ) p(x) def =(x ˆx 0 ) (x ˆx ) A 0 = {x R p(x) apple 0} p 00 (x) 0 ) A 0 is a convex set Theorem : The Nash equilibrium policies for the game over the collision channel without capture have the following threshold structure nash (x) = 0, if apple x apple ;, otherwise.

13 Design via Lloyd-Max Algorithm. From a pair of representation points compute the roots of p(x) ˆx (k) =(ˆx (k) 0, ˆx(k) ) (ˆx (k) )= ˆx(k) 0 + p ˆx (k) + p (ˆx (k) )= ˆx(k) 0 p (k) ˆx p. The new representation points are the centroids of A (k) 0, A(k) = R\A (k) 0 A (k) 0 = h i (ˆx (k) ), (ˆx (k) ) ˆx (k+) = 4 R A (k) 0 R A (k) 0 f X (x)dx f X (x)dx 3 5 Z A (k) 0 xf X (x)dx This algorithm converges globally to a local minimum

14 Design via Lloyd-Max Algorithm Example : =0.5,X N (0, ) 0, if apple x apple ; (x) =, otherwise. 8 < x, if y = x; (y) = , if y =?; : 0.570, if x = C. J =0.488 transmit ˆx idle ˆx 0 transmit x Example : X N (0, ) X N (0, ) nash (x )= nash (x )= 0, apple x apple ;, otherwise 0, apple x apple ;, otherwise J nash =0.786 J nash =0.5573

15 Remarks. The structural result is independent of the densities of X,X. The convergence of the Lloyd-Max algorithm depends on the pdfs 3. The Nash equilibrium policies perform worse than the security ones: J nash =0.786 J nash = J sec =0.75 J sec = There is an incentive to be conservative even in the absence of communication costs

16 Part II. Collision channel with capture power packet packet time Capture mechanism: In case of a collision, the packet transmitted with the highest power captures the channel and the other is lost. Allow DMs to choose among 3 power levels: U i {0,, } Cost functional must take into account the communication cost: J cap i ( i, j, i )=E[(X i ˆXi ) + U i ]

17 Part II. Collision channel with capture D X Sensor S Collision Channel Y Remote Estimator ˆX Pr(D = i) = i, i {0,, } U =0) Y =? U>0,D <U ) Y = X U =,D ) Y = C U =,D =) Y = C (X) =X (?) =ˆx 0 (C )=ˆx (C )=ˆx J cap (, ) =E[(X ˆx 0 ) U = 0] Pr(U = 0)+ E[( + )(X ˆx ) + U = ] Pr(U = )+ E[ (X ˆx ) + U = ] Pr(U = ) Ternary quantization problem with asymmetric distortion

18 Security Policies Worst case scenario: the opponent always transmits with full power = Z Z Z J cap = A 0 (x ˆx 0 ) f X (x)dx + A [(x ˆx ) + ]f X (x)dx + A [(x ˆx ) + ]f X (x)dx Necessary optimality conditions: x A 0, h (x) apple 0,h (x) apple 0 x A, h (x) > 0,h 3 (x) apple 0 x A, h (x) > 0,h 3 (x) > 0 A i, i {0,, } h (x) =x(ˆx ˆx 0 ) (ˆx ˆx 0 + ) h (x) =x(ˆx ˆx 0 ) (ˆx ˆx 0 + ) h 3 (x) =x(ˆx ˆx ) (ˆx ˆx + ) are convex Theorem : The security policy for the game over the collision channel with capture is determined by a regular quantizer (convex quantization regions).

19 Nash equilibrium policies Z J cap = (x ˆx 0 ) f X (x)dx + A 0 Necessary optimality conditions: x A 0, p (x) apple 0, p (x) apple 0 x A, p (x) > 0, p 3 (x) apple 0 Z [( + )(x A Z ˆx ) + ]f X (x)dx+ [ (x A ˆx ) + ]f X (x)dx p (x) =(x ˆx 0 ) ( + )(x ˆx ) p (x) =(x ˆx 0 ) (x ˆx ) p 3 (x) =( + )(x ˆx ) + (x ˆx ) p 00 i (x) 0 ) {x R p i (x) apple 0} is a convex set A 0 is the intersection of two convex sets ) A 0 is convex A is the set di erence of two convex sets ) A is the union of at most two convex sets Theorem 3: The structure of a Nash equilibrium policy for 8 the game over the collision channel with capture is < nash (x) = : 0, if apple x apple, if 3 apple x apple 4 or 5 apple x apple 6, otherwise.

20 Examples Example : Example : X N (0, ), = X N (0, ), =0.5, =0.5 (x) sec (x) ˆx ˆx 0 ˆx x ˆx ˆx 0 ˆx x A A 0 A

21 Conclusion Two new problems in networked estimation/control: - Collision channel without capture - absence of communication costs - Collision channel with capture - presence of communication costs Obtained the structure of security and Nash equilibria policies Results rely on optimal quantization theory with asymmetric distortion Several open problems: - Existence of optimal quantizers - Uniqueness of Nash equilibrium policies - Convergence of the Lloyd-Max algorithm - Dynamic games and many more