Lecture 8: S-modular Games and Power Control

Transcription

1 CDS270: Otmzaton Game and Layerng n Commncaton Networ Lectre 8: S-modlar Game and Power Control Ln Chen /22/2006

2 Otlne S-modlar game Sermodlar game Sbmodlar game Power control Power control va rcng A general framewor for dtrbted ower control 2

3 Sermodlar game Characterzed by trategc comlementarte Sermodlar game are remarable Pre trategy Nah eqlbrm ext The eqlbrm et ha an order trctre wth extreme element Many olton concet yeld the ame redcton Behave well nder varo learnng or adatve algorthm Encoma many aled model 3

4 Monotone comaratve tatc Def: oe X R and T ome artally ordered et. A fncton f : X T R ha ncreang dfference ermodlar n x t f for all x x and t t f x t f x t f x t f x t. The ncremental gan to chooe a hgher x greater when t hgher. The ncreang dfference ymmetrc.e. f t t then f x t f x t nondecreang n x. 4

5 Lemma: f f twce contnoly dfferentable f ha ncreang dfference ff t t mle f x t x f x t for all x x or alternatvely that f x t 0 for all x xt t. 5

6 A central qeton: when x t arg max f x t x X wll be ncreang n t? Theorem To: Let X R be comact and T a artally ordered et. Soe f : X T R ha ncreang dfference n x t and er emcontno n x. Then For all t xt ext and ha a greatet and leat element xt and xt. xt and xt are ncreang n. t 6

7 Proof: Extence: X comact and f er emcontno Tae a eqence { x } n xt. From comactne there ext a lmt ont x lm x. Then for all x f x t f x t f x t f x t. Th x xt and xt therefore cloed. It follow that xt ha a greatet and leat element. Let x xt and x xt. Then f x t f mn x x t 0 whch mle f max x x t f x t 0. By the ncreang dfference f max x x t f x t 0. Th max x x maxmze f t. Now c x xt and x xt t follow that x x. A mlar argment ale to xt. 7

8 Sermodlar game Def: the game G { N S N N } a ermodlar game f for all a comact bet of er emcontno n ha ncreang dfference n Corollary: oe G { N S N N } a ermodlar game. Defne the bet reone fncton B arg max. Then S B S ha greatet and leat element and B and are ncreang n R B B B 8

9 Examle: Bertrand game Two frm: form and frm 2 wth rce Payoff 2 + It a ermodlar game nce > 0. Solve by terated trct domnance Let S [0] then [ / 4 / 2]. 0 If < / 4 then If > / 2 then S Let S [ ] then /3 /3 the only Nah eqlbrm. 2 > < 4 < > trctly trctly / 4 + / 4 / 4 + /6 + /6 / / 4 + / / 4 + / 4 / 4 + /6 + /6 L / 4 + L + / 4 + / 4 L L [0] domnated. domnated. 9

10 Theorem: let G { N S N N } be a ermodlar game. Then the et of tratege rvvng terated trct domnance ISD ha greatet and leat element and whch are re trategy Nah eqlbra. Corollary: Pre Nah eqlbrm ext. The larget and mallet tratege comatble wth ISD ratonalzablty correlated eqlbrm and Nah eqlbrm are the ame. If a ermodlar game ha a nqe Nah Eqlbrm t domnance olvable. 0

11 Proof: let and be the larget element of. Let and. If.e. then t domnated by. By ncreang dfference Alo note that Iterate and defne and. Now f then. So a decreang eqence and ha a lmt denoted by. Only the tratege are ndomnated. S S N L S 0 B } : { 0 S S S > < B } : { S S 0 B B + { }

12 2 Smlarly tart wth the mallet element of and dentfy. Show and are Nah eqlbra. For all and Tae the lmt a. Smlarly rove a Nah eqlbrm N L S +

13 Illtratve dagram mallet element Bet reone larget element Bet reone Bet reone 3

14 Sbmodlar game Def: oe X R and T ome artally ordered et. A fncton f : X T R ha decreang dfference bmodlar n x t f for all x x and t t A game a bmodlar game f the ayoff fncton are bmodlar. We ll foc on taton where layer mnmze the ayoff fncton. More generalzaton f x t f x t f x t f x t. 4

15 Monotoncty Def: let A and B are two et. We ay A B f for any a A and b B mn a b A and max a b B. For contrant et S S f S S then the et S oe the decendng roerty. The acendng roerty can be defned when the relaton revered. 5

16 Theorem: for a bmodlar game wth decendng S An Nah eqlbrm ext. The bet reone trategy B mn{arg max S monotoncally converge to an eqlbrm. Proof: Follow monotoncty of the bet reone. Smlar to the roof of former theorem. Smlar relt ext for a ermodlar game wth acendng. S } 6

17 Power control An mortant comonent of rado reorce management Meet target BER or SIR whle lmtng nterference Increae caacty by mnmzng nterference Extend battery lfe Uer agned tlte that are fncton of the ower they conme and the gnal-tonterference rato SIR they attan Try to fnd a good balance between hgh SIR or meetng target SIR and low ower conmton 7

18 Power control va rcng Conder a ngle-cell networ wth a et N of er at ln Each er can chooe a ower The SIR for er mn max [ ] γ h h + σ where h the channel gan from MS to BS and the noe varance. BS 2 2 σ h h h 2 N h N MS MS 2 MS N- MS N 8

19 9 Conder ayoff amed to be ncreang When the tlte are ermodlar? Reqre Examle: ome concave fncton f α γ f f f h h c f γ γ γ γ γ γ 0 < + f f γ γ γ

20 Power control algorthm mn At tme t 0 let 0. At each tme t et er ower mn{arg max } The above algorthm converge to a Nah eqlbrm that the mallet eqlbrm. 20

21 A general framewor for dtrbted ower control Conder a et N of er and a et of M bae taton Uer e ower Denote by the gan of er at bae taton h The SIR of er at bae taton µ wth µ h h + σ 2 2

22 Dfferent ower control cheme Fxed agnment: the er agned to BS wth a SIR reqrement γ. The contrant I FA Mnmm ower agnment Macro dverty lmted dverty and mltle receton are have the contrant of the ame form µ a γ I a 22

23 Standard nterference fncton The tandard nterference fncton atfe the followng roerte Potvty: I > 0 Monotoncty: f Scalablty: for > I I a ai I a I 23

24 Defne a bmodlar game Payoff Contrant et wth a feable olton to S { : I 0 } I Theorem: f a feable olton ext then There a fxed ont to eqaton I The bet reone trategy converge to an eqlbrm. 24

25 Hay Thangvng! 25