Incentive-Based Pricing for Network Games with Complete and Incomplete Information

Transcription

1 Incentive-Based Pricing for Network Games with Complete and Incomplete Information Hongxia Shen and Tamer Başar Coordinated Science Laboratory University of Illinois at Urbana-Champaign hshen1, 11th ISDG Symposium, Tucson, Arizona December 18 21, 2004

2 Incentive-Based Pricing for Network Games Previous Work Outline Outline A Significant New Direction Dynamic Pricing Complete Information Incomplete Information Multiple Users: Incentive-Design Problem Formulation Conclusions and Extensions Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

3 Previous Work Previous Work Başar and Srikant (2002a) Hierarchical Stackelberg Game Başar and Srikant (2002b) Linear Network Shen and Başar (2004a) Differentiated Pricing Shen and Başar (2004b) Incomplete Information T. Başar and R. Srikant (2002a), Revenue-maximizing pricing and capacity expansion in a many-users regime, Proc. IEEE INFOCOM 2002, pp T. Başar and R. Srikant (2002b), A Stackelberg network game with a large number of followers, J. Optimization Theory and Applications, 115(3): H.-X. Shen and T. Başar (2004a), Differentiated Internet pricing using a hierarchical network game model, Proc. IEEE ACC 2004, pp H.-X. Shen and T. Başar (2004b), Network game with a probabilistic Description of User Types, to appear in Proc. IEEE CDC Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

4 Previous Work Two-Level Hierarchical Stackelberg Game ISP max {pi } n i=1 R Leader Stackelberg game User 1 max x1 F 1 User n max xn F n Followers n-player noncooperative game Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

5 Dynamic Pricing A Significant New Direction Dynamic Pricing ISP max {γi (x i )} n i=1 R Leader Reverse Stackelberg game User 1 max x1 F 1 User n max xn F n Followers n-player noncooperative game Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

6 Dynamic Pricing Incentive-Design Problem Incentive Policy usage-based policy Team Solution Pareto-optimal solution (action outcome desired by the leader) Solution of the Incentive-Design Problem incentive policy to achieve the team solution Incentive Controllability existence of the solution of the incentive-design problem Complete Information the ISP knows the user types Incomplete Information the ISP knows only the probability distribution T. Başar and G. J. Olsder (1999), Dynamic Noncooperative Game Theory, pp Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

7 Complete Information Complete Information Incentive-Design Problem Formulation Team Solution Incentive-Design Problem Solution Incentive Controllability Team Solution vs Stackelberg Game Solution Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

8 Complete Information Incentive-Design Problem Formulation User s net utility: F w (x; r) := w log(1 + x) 1 1 x r, 0 < x < 1 Restriction: for x = 0, r 0 and F w (0; r) 1 Team solution: (x t, r t ) = arg max 0 x<1,r 0 r, s. t. F w (x; r) 1 Incentive-Design Problem Solution: γ : [0, 1) R, γ(0) 0, arg max 0 x<1 F w(x; γ(x)) = x t, γ(x t ) = r t Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

9 Complete Information Team Solution Define: Q(x; w) := w log(1 + x) 1 1 x + 1 Solve: (x t, r t ) = arg max 0 x<1,r 0 r, s. t. F w (x; r) = Q(x; w) 1 r w = 100 ( x t, r t ) 40 F w ( x; r ) = 1 ( r = Q ( x; w ) ) r x Team solution: x t = α(w) := 1+2w 1+8w 2w, w > 1; r t = Q(α(w); w) Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

10 Complete Information Incentive-Design Problem Solution γ : [0, 1) R : γ(0) 0; arg max F w(x; γ(x)) = x t, 0 x<1 No linear incentive γ(x t ) = r t Unique quadratic incentive: γ(x) = a 1 x + a 2 x 2, a 1 = 2rt x t, a 2 = rt (x t ) ( x t, r t ) 40 r = γ ( x ) = a 1 x + a 2 x 2 F w ( x; r ) = 1 ( r = Q ( x; w ) ) r x General incentive Incentive controllable? Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

11 Complete Information ǫ-incentive Controllability Make a small dip : (x t, r t ) (x t, r t ǫ) Unique maximizing point for the user: F w (x t ; r t ǫ) = 1 + ǫ ISP s profit: r t ǫ w = 100 ( x t, r t ) 40 F w ( x; r ) = 1 ( r = Q ( x; w ) ) r x T. Başar and G. J. Olsder (1999), Dynamic Noncooperative Game Theory, pp Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

12 Complete Information Team Solution vs Stackelberg Game Solution r t / r s r t r s x t 20 F w ( xs ; r s ) 0.6 x s F w ( x t ; r t ) w w Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

13 Incomplete Information Incomplete Information Incentive-Design Problem Formulation Team Solution Incentive-Design Problem Solution Incentive Controllability Numerical Examples Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

14 Incomplete Information Incentive-Design Problem Formulation User s type: w = w i w.p. q i (0, 1), i = 1,,m; Assumption: w 1 > > w m > 1 Team solution: {(x it, r it )} m i=1 = arg max {E[r] = {0 x i <1,r i 0} m i=1 s. t. F w i(x i ; r i ) 1, 1 i m, m j=1 q j = 1 m q j r j }, j=1 F w i(x i ; r i ) F w i(x j ; r j ), 1 i j m Incentive-Design Problem Solution: γ : [0, 1) R, γ(0) 0, arg max 0 x<1 F w i(x; γ(x)) = xit, 1 i m, γ(x it ) = r it, 1 i m Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

15 Incomplete Information Constraint Reduction Team solution constraints: F w i(x i ; r i ) 1; F w i(x i ; r i ) F w i(x j ; r j ), 1 i j m Lemma: for w i > w j > 1, F w i(x i ; r i ) F w i(x j ; r j ) F w j(x j ; r j ) F w j(x i ; r i ) and x i x j Constraint reduction: ➊ F w m(x m ; r m ) = 1 ➋ F w m 1(x m 1 ; r m 1 ) = F w m 1(x m ; r m ); x m 1 x m ➌ F w k 1(x k 1 ; r k 1 ) = F w k 1(x k ; r k ); x k 1 x k Reduced constraints: x 1 x m 0; F w m(x m ; r m ) = 1 and F w i(x i ; r i ) = F w i(x i+1 ; r i+1 ), 1 i m 1 Equivalently, r m = Q(x m ; w m ); r i = r i+1 + Q(x i ; w i ) Q(x i+1 ; w i ) Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

16 Incomplete Information Team Solution Two Types E[r] = q 1 Q(x 1 ; w 1 ) + q 2 Q(x 2 ; v 2/2 := w2 q 1 w 1 q 2 ) Team solution: x 1t = α(w 1 ) and x 2t = α(v 2/2 ) 25 ( x 1t, r 1t ) 20 w 1 = 70, w 2 = 40 q 1 = q 2 = 1/2 ( x 2t, r 2t ) 15 r x Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

17 Incomplete Information Team Solution Multiple Types E[r] = q 1 Q(x 1 ; w 1 ) + m k=2 q kq(x k ; v k/m ) If w 1 > v 2/m > > v m/m > 1 : x 1t = α(w 1 ); x it = α(v i/m ) for 2 i < m w 1 = 60, w 2 = 40, w 3 = 30 w 4 = 24, w 5 = 20 q 1 = q 2 = q 3 = q 4 = q 5 = 1/5 ( x 2t, r 2t ) ( x 1t, r 1t ) r 8 6 ( x 5t, r 5t ) ( x 4t, r 4t ) ( x 3t, r 3t ) x Otherwise? Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

18 Incomplete Information Team Solution Induction Reduce m types to m 1 or m 2 types: ➊ if v m 1/m v m/m, x (m 1)t = x mt and E[r] = q 1 Q(x 1 ; w 1 )+ m 2 k=2 q kq(x k ; v k/m )+(q m 1 +q m )Q(x m 1 ; v m 1,m/m ) ➋ if 1 v m 1/m > v m/m, x (m 1)t = x mt = 0 and E[r] = q 1 Q(x 1 ; w 1 ) + m 2 k=2 q kq(x k ; v k/m ) ➌ if v m 1/m > 1 v m/m, x mt = 0 and E[r] = q 1 Q(x 1 ; w 1 ) + m 1 k=2 q kq(x k ; v k/m ) ➍ if v m 1/m > v m/m > 1 and v m 2/m v m 1/m, x (m 2)t = x (m 1)t ➎ if v m 2/m > v m 1/m v m/m > 1, at some point v i 1/m v i/m and x (i 1)t = x it Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

19 Incomplete Information Induction Example v 1/5 = 70, v 2/5 = 10, v 3/5 = 19, v 4/5 = 1, v 5/5 = 0 ➊ 1 v 4/5 > v 5/5 : x 4t = x 5t = 0 ➋ v 2/5 v 3/5 : x 2t = x 3t and v 2,3/5 = 14.5 ➌ v 1/5 > v 2,3/5 > 1: x 1t = α(v 1/5 ) and x 2t = x 3t = α(v 2,3/5 ) w 1 = 70, w 2 = 40, w 3 = 33 w 4 = 25, w 5 = 20 q 1 = q 2 = q 3 = q 4 = q 5 = 1/5 ( x 3t, r 3t ) = ( x 2t, r 2t ) ( x 1t, r 1t ) r ( x 5t, r 5t ) = ( x 4t, r 4t ) = ( 0, 0 ) x Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

20 Incomplete Information Incentive-Design Problem Solution γ : [0, 1) R: γ(0) 0; arg max 0 x<1 F w i(x; γ(x)) = xit, γ(x it ) = r it, 1 i m w 1 = 60, w 2 = 40, w 3 = 30 w 4 = 24, w 5 = 20 q 1 = q 2 = q 3 = q 4 = q 5 = 1/5 ( x 2t, r 2t ) ( x 1t, r 1t ) r 8 6 ( x 5t, r 5t ) ( x 4t, r 4t ) ( x 3t, r 3t ) x Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

21 Incomplete Information ǫ-incentive Controllability Make a small dip : (x it, r it ) (x it, r it ǫ i ) ➊ ǫ 1 > ǫ 2 > ǫ 3 > ǫ 4 > ǫ 5 > 0 ➋ ǫ i s are small enough Unique maximizing point for type i: F w (x it ; r it ǫ i ) = F w (x it ; r it ) + ǫ i ISP s expected profit: E[r t ] m j=1 ǫj w 1 = 60, w 2 = 40, w 3 = 30 w 4 = 24, w 5 = 20 q 1 = q 2 = q 3 = q 4 = q 5 = 1/5 ( x 2t, r 2t ) ( x 1t, r 1t ) r 8 6 ( x 5t, r 5t ) ( x 4t, r 4t ) ( x 3t, r 3t ) x Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

22 Incomplete Information γ : [0, 1) R: γ(0) 0; Another Example arg max 0 x<1 F w i(x; γ(x)) = xit, γ(x it ) = r it, 1 i m w 1 = 70, w 2 = 40, w 3 = 33 w 4 = 25, w 5 = 20 q 1 = q 2 = q 3 = q 4 = q 5 = 1/5 ( x 3t, r 3t ) = ( x 2t, r 2t ) ( x 1t, r 1t ) r ( x 5t, r 5t ) = ( x 4t, r 4t ) = ( 0, 0 ) x Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

23 Incomplete Information ǫ-incentive Controllability Make a small dip : (x it, r it ) (x it, r it ǫ i ) ➊ ǫ 1 > ǫ 2 = ǫ 3 > ǫ 4 = ǫ 5 = 0 ➋ ǫ i s are small enough Unique maximizing point for type i: F w (x it ; r it ǫ i ) = F w (x it ; r it ) + ǫ i ISP s expected profit: E[r t ] m j=1 ǫj w 1 = 70, w 2 = 40, w 3 = 33 w 4 = 25, w 5 = 20 q 1 = q 2 = q 3 = q 4 = q 5 = 1/5 ( x 3t, r 3t ) = ( x 2t, r 2t ) ( x 1t, r 1t ) r ( x 5t, r 5t ) = ( x 4t, r 4t ) = ( 0, 0 ) x Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

24 Multiple Users Multiple Users Complete Information User i s net utility: F wi (x i, x i ; r i ) := w i log(1+x i ) 1 n x i x i r i Team solution: {(x t i, r t i)} n i=1 = arg max {x i 0, n j=1 xj <n,r i 0 } n i=1 n r j, j=1 s. t. F wi (x i, x i ; r i ) 1 n x i, 1 i n Incentive-Design Problem Solution: {γ i } n i=1, γ i(0) 0, arg max F wi (x i, x t i; γ i (x i )) = x t i, 0 x i <n x t i γ i (x t i) = r t i Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

25 Multiple Users Multiple Users Incomplete Information Independent Users: w i = w j i i w.p. q j i i ; Team solution: { Jt {(x Jt i, r i )} n i=1 s. t. F w j i i F w j i i } Jl J= J f = arg max J l J := (j1,,j n ) T q j1 1 qj n n J= J f (x J i, x J i ; r J i ) 1 n x J i, 1 i n, (x J i, x J i ; r J i ) F w j i i n J r j, j=1 (x J i, x J i ; r J i ), 1 i n, J J Incentive-Design Problem Solution: {γ i } n i=1, γ i(0) 0, arg Jt max F j 0 x i <n xjt w i (x i, x Jt i; γ i (x i )) = x i, i i γ i (x Jt i ) = r Jt i Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,

26 Conclusions and Extensions Conclusions Dynamic pricing Incentive-design problem formulation Single ISP, single user ǫ-incentive controllability ǫ-optimal incentive policy Extensions Multiple users, multiple ISPs, continuously distributed user types,... End of the Talk Hongxia Shen and Tamer Başar, Tucson, Arizona, December 18 21,