Pricing Network Services by Jun Shu, Pravin Varaiya

Transcription

1 Prcng Network Servces by Jun Shu, Pravn Varaya Presented by Hayden So September 25, 2003

2 Introducton: Two Network Problems Engneerng: A game theoretcal sound congeston control mechansm that s ncentve compatble Unlke TCP, whch s not ncentve compatble Use prcng for congeston control Economc: Derve a prcng scheme for QoS Free ISP from Flat Rate woes

3 SPAC mechansm Smart Payment Admsson Control Mechansm Assume users compete for network bandwdth n noncooperatve manners Fnd an economcally effcent way to provde statstcally guaranteed servce to users, based on ther value on servces SPAC mechansm to be mplemented on the edge of network management doman Treats aggregated packet flows of dfferent QoS as agents n the mechansm SPAC calculates servce prces usng VCG mechansm

4 VCG Mechansm Revst A famly of mechansms Each agent has quas-lnear preference Characterstc of VCG mechansm: Drect revelaton Strategy-proof Allocatvely-Effcent Wth careful desgn, can also be: Indvdual-ratonal (Weak) budget-balance

5 where * VCG: Payoff to each Agent Agent 's payoff s x u ˆ θ = v x ˆ θ θ p * ( ) ( ( ), ) ( ˆ θ ) s the optmal choce of mechansm ( ˆ θ) = arg max (, ˆ θ j) * x vj x x X j ˆ θ s the reported type profle θ p s the true type of agent s the payment from agent to the mechansm

6 VCG: Payment Payment has the form: ˆ * ˆ ˆ = j j j p h( θ ) v ( x ( θ), θ ) h( ˆ θ ) s a value ndependent of agent 's presence For Indvdual Ratonalty, h( ˆ θ ) s usually defned as: Therefore, ˆ * ˆ ˆ = j j j h( θ ) v ( x ( θ ), θ ) ˆ * ˆ * ˆ ˆ * ˆ ˆ = + j j j j j j u ( θ ) v ( x ( θ), θ ) v ( x ( θ), θ ) v ( x ( θ ), θ )

7 Intuton behnd VCG mechansms u ˆ * ˆ * ˆ ˆ * ˆ ˆ = v x + vj x j vj x j j j ( θ ) ( ( θ), θ ) ( ( θ), θ ) ( ( θ ), θ ) My Happness: Hgher = better My own happness mght lead to a socal choce that makes Mss j less happy I better make a sensble choce so that I am happy, mss j s are happy, and thus everyone n the socety s happy!! ~ Truth Tellng ~ Don t care: Nothng I can do

8 Requrements for SPAC mechansm Incentve Compatble Truth-tellng domnant strategy Strategy-proof User tends to cheat f not strategy-proof Indvdual Ratonal User wll not jon the mechansm f t were not IR ISP wll not mplement the mechansm f t were not IR to them Effcent allocaton Fall back to best-effort Everyone admtted, must and s wllng to play Aucton for addtonal servce Charge only f user nduces congeston

9 SPAC Mechansm Defntons (1) n + 1 players (n agents + 1 prncpal) m levels of statstcally guaranteed QoS (delvery rate), denoted d = (d 0,d 1,,d m-1 ), and 0 d 0 < d 1 < < d m-1 < 1 For smplcty, assume a constant number A k of agent admtted wth guaranteed delvery rate of d k Pror to recevng servce, each agent announces ther values, called bds, of the servce: b = (b 1,,b,,b n ) Each agent s true value of the servce s denoted: θ = (θ 1,,θ,,θ n )

10 SPAC Mechansm Defntons (2) Denote the fnal orderng of all n bds be b (1),b (2),,b (n) Based on b, the prncpal computes a soluton vector x = (x 1,,x,,x n ) x s the servce delvery rate agent gets For mathematcal notaton convenence, defne Q k ( x ) 1 f x = dk = 0 otherwse

11 SPAC Mechansm Defntons (3) Based on b, the prncpal computes a congeston prce vector p = (p 0,p 1,,p m-1 ) for each QoS level where p ( b) 0 0 and k = 1,..., m 1, pk( b) = pk 1( b) + ( dk dk 1) b m The utlty for agent s therefore: m 1 k k k= 0 1 ( n A ) u ( b, θ ) θ x p Q ( x ) = l= k l

12 Illustraton of SPAC defntons Xr rate d m-1 d m-2 d q d q-1 d 1 d 0 $ A m-1 A m-2 A q A q-1 A 1 A 0 x n = x n-1 = d m-1 x e = d q x e-1 = d q-1 Best Effort Zone b (n) b (n-1) b (e) b (e-1) b (j) > 0 b (1) =0

13 Soluton Requrements The fnal soluton to the aucton, x*(b), must be effcent and feasble, therefore x*( b) = argmax bx x X = 1 And subject to capacty constrant n = 1 Q ( x ) A And subject to unversal servce coverage m 1 k= 0 n k k A k n

14 Soluton Implementaton Sort all the bds n descendng order The hghest A m-1 bdders are admtted to QoS level m-1 wth delvery rate d m-1 The next hgher A m-2 bdders receve d m-2 Contnue untl all bdders are admtted (at least to level 0) What about strategy-proofness?

15 Is SPAC strategy-proof? SPAC s a VCG mechansm mplementaton It s strategy proof How? It doesn t look lke so Each agent s utlty (at QoS level k) n SPAC s: u ( b, θ ) = θ x p ( b) k In word, t s agent s happness x θ mnus the congeston prce, p k, she has to pay: pk( b) = pk 1( b) + ( dk dk 1) b m 1 ( n A ) l= k l

16 Is SPAC strategy-proof?? Expandng the recursve defnton of p k yelds u ( b, θ ) = θ x ( d d ) b m Now, note that q q 1 q= 1 b k m 1 l = q ( n A ) denotes the hghest bd of agents who are rejected from the q-th or above levels of QoS In other word: the payment term of agent s utlty s to compensate those suffered because agent enters the game WHO suffers? l 1 ( n A ) l= q l

17 Illustraton of Congeston Prce $ d m-1 d m-2 d q d q-1 d 1 d 0 A m-1 A m-2 A q A q-1 A 1 A 0 Xr rate b b b (e-1) (e) (e+1) b (n-1) b (n) After arrval of agent NOT HAPPY!! Utlty decreases from d q b (e) to d q-1 b (e) b (n+1) b (n) b (e+2) b (e+1) b (e-1) b (e) A 0 not affected not affected not affected

18 Relaton to VCG mechansm Therefore, agent s utlty u ( b, θ ) = θ x ( d d ) b q q 1 q= 1 can be rewrtten as: u ( b, θ ) = θ x + b x b x k j j j j j j Now, t looks very much lke the VCG quaslnear utlty functon of an agent, wth v ( x( θ ), θ ) = xθ m 1 ( n A ) where x denotes j's allocaton before jons the game j l= q l

19 SPAC s strategy-proof VCG mechansms are strategy-proof By showng SPAC s an nstance of VCG mechansms, t s nformally proved to be strategyproof The formal proof s provded n the paper In the case of SPAC, strategy-proofness makes the actual mplementaton very smple

20 Concluson and Open Questons SPAC mechansm provdes congeston control through QoS prcng SPAC mechansm s a VCG mechansm Promotes truth-tellng and provdes effcent soluton SPAC mechansm s smple Implementaton not covered n ths presentaton How often s ths game played on the gateway? Per lfetme of aggregated stream? Per lfetme of ISP? Is x θ a good measure of a user s value on servce?