GAME THEORY AND INTERFERENCE AVOIDANCE IN DECENTRALIZED NETWORKS

Transcription

1 GAME TEORY AD ITERFERECE AVOIDACE I DECETRALIZED ETWORS Rekha Menon, (MPRG Vrgna Tech, Blacksburg, VA, USA, rmenon@vt.edu) Dr. Allen B. Macenze, (Electrcal and Comuter Engneerng Deartment, Vrgna Tech) Dr. R. Mchael Buehrer, (MPRG, Vrgna Tech) Dr. Jeffrey Reed, (MPRG, Vrgna Tech) ABSTRACT In networks wth transmttng users havng searate uncoordnated recevers, waveform adataton by greedy nterference avodance (IA) algorthms ([], [2] and [3]) mght not lead to far network resource allocatons. A game theoretc framework for ths scenaro, based on Potental game theory s resented n ths aer. Ths model rovdes nsght nto develoment of algorthms that are farer than the greedy IA algorthms and are amenable to dstrbuted mlementatons.. ITRODUCTIO Gven the rofuson of offered servce tyes, transmsson rotocols and software rado caabltes avalable today, networks are becomng less structured and ncreasngly nvolve dstrbuted decson makng. odes need to ndeendently and erodcally adat themselves to changes n the nterference envronment that result from changes n network confguraton (nodes enterng or leavng the network), moblty and the nature of the wreless channel. The alcaton of game theory to dstrbuted waveform adataton/selecton technques amed at reducng the nterference n a network s the focus of ths aer. * In general, transmttng nodes have lttle or no nformaton about the nterference seen at the recever. One aroach to avod nterference s to develo dstrbuted waveform-adataton algorthms where a mnmal amount of feedback s requred between recevers and transmtters. A dstrbuted algorthm s roosed n [] that sequentally udates the sgnature sequences assocated wth a artcular recever n a synchronous CDMA system. The sequences are found to decrease the total sum correlaton (TSC) of the set. The mnmzaton of the TSC s shown to be equvalent to maxmzaton of the sum caacty [2], whch forms a convenent nformaton theoretc socal welfare measure. Ths teratve algorthm (wheren users greedly ncrease ther SIR) converges to a set of orthogonal sequences when the number of users s less than or equal to the rocessng gan * Ths materal s based on work sonsored by Offce of aval Research (OR) grant no: and to a set of Welch Bound Equalty sequences otherwse. Reference [2] generalzes ths aroach to the stuaton where nodes can adat ther modulaton/demodulaton methods usng a general sgnal sace aroach. In [3], game theory s used to model and analyze ths nteracton of nodes communcatng wth a common recever. It s shown n the aer that for two-layer games, any combnaton of metrc and recever tyes results n a best resonse otental game that mnmzes the total sum correlaton functon. In the resence of multle uncoordnated recevers and asymmetrc ower constrants for the users, waveform adataton becomes more dffcult as comared to the systems analyzed n [], [2] and [3], where all nodes transmt to a sngle common recever. Greedy IA algorthms based on maxmzaton of sum-caacty mght not lead to a far allocaton of the resources n the network. ence waveform udate decsons that mrove some system welfare functon n addton to reducng the nterference at the recever assocated wth a artcular transmttng node are requred. A game theoretc aroach to analyzng these systems and constructng solutons that can lead to a far utlzaton of network resources s resented n ths aer. Ths framework s based on otental game theory, whch makes t easly amenable to a dstrbuted mlementaton. The system model for the network scenaro under consderaton s descrbed n Secton 2. Secton 3 gves a bref overvew of game-theory and otental games. Secton 4 resents the game-theoretc framework for far waveform adataton n a decentralzed network. Secton 5 resents an examle formulaton of a far waveform adataton game. Secton 6 summarzes the aer and resents drectons for future research. 2. SYSTEM MODEL The network beng analyzed s made u of a cluster of nodes. Each transmttng node has a searate node of nterest,.e., a searate recevng node. Ths leads to the exstence of multle un-coordnated recevers n the network. Fgure shows the system model wth arrows connectng the transmttng node to ts ntended recever node.

2 The transmttng nodes are allowed to have dfferent transmt ower constrants and the ower constrants are assumed to be fxed by a rocess ndeendent of waveform adataton. Interference s caused at the recever nodes by transmssons from user nodes dfferent from the one assocated wth the artcular recever node. The nterference caused s nfluenced by the ower constrants and assocated channel characterstcs of the user nodes. Transmtter ode Recever ode Fgure : etwork wth multle un-coordnated recevers Let be the number of transmttng nodes n the network. Let s be the sgnature sequence assocated wth transmttng node. The number of avalable sgnal dmensons s. The sgnature sequences are allowed to have real values (as oosed to b-olar sequences). Wthout loss of generalty, the sgnature sequences are s =. The ower receved at assumed to have unt norm ( ) the th recever node from the th transmttng node s denoted by. Ths term takes nto consderaton both the transmt ower constrant and the nfluence of the channel between the transmttng and recevng nodes. The data bt to be transmtted from the th transmt node s denoted by b. The receved sgnal at the receve node s gven by z r = bs + z () = The vector,, models addtve whte Gaussan nose wth zero mean and unt varance. The channel s assumed to be constant over the tme requred for waveform adataton. 3. GAME TEORY AD POTETIAL GAMES Consder a normal form game reresented [4] as Γ= M, A, u, where Γ s a game and { } { } M M The waveforms used by transmttng nodes are assumed to be fully secfed by the sgnature sequence n dfferent sgnal dmensons [2]. ence the waveform of a node s analogous to the sgnature sequence of a node. {,2,, } M= M, s the set of layers of the game. The set of actons avalable for layer s denoted by A and the utlty functon assocated wth each layer by u. If the set of all avalable actons for all layers s reresented by A = A, then u : A. The utlty functon for each M layer s thus a functon of the actons n the game. Players select actons that maxmze ther utlty functons. A ash Equlbrum (E) for a game s an acton rofle from whch no layer can ncrease hs utlty by unlateral devatons. An acton rofle a A s a E f and only f u a u b, a M, b A, a A (2) ( ) ( ) ash equlbra form the steady states of the game. Suose that a normal form game s layed reeatedly. At each stage of the game, layers choose actons that mrove ther utlty functons. The crtera for a artcular choce of acton gves rse to the best and better resonse dynamcs defned below. Best Resonse Dynamc: At each stage, a layer s ermtted to devate from a A to some acton b A ff (, ) (, ) (, ) ( ) u b a u c a c A andc b A and u b a u a. > Better Resonse Dynamc: At each stage, a layer s ermtted to devate from a A to some acton b A ff (, ) ( ) u b a u a. > A otental game ([5], [6]) s a normal form game such that any changes n the utlty functon of any layer n the game due to a unlateral devaton by the layer s also corresondngly reflected n a otental functon. A functon P: A, s called an exact otental functon f u ( a, a ) u ( a, a ) = P( a, a ) P( a, a ), and a, a A. A game that has an exact otental functon s called an exact otental game. A functon P: A, s called an ordnal otental functon, f u( a, a ) u( a, a ) P( a, a ) P( a, a), and a, a A. A game that has an ordnal otental functon s called an ordnal otental game. E of otental games are maxmzers of ther otental functons. All otental games followng a best resonse dynamc converge to a E. Potental games wth fnte acton saces have been establshed to follow a better resonse dynamc to converge to a E as well.

3 4. FORMULATIO AS A POTETIAL GAME Total sum caacty, defned as the sum of the rates achevable by users n the network, s used as a metrc to evaluate the erformance of the network n the centralzed recever scenaro. All the IA algorthms ntroduced for the centralzed network scenaro that greedly mrove ther own utltes (rate or SIR), are desgned to maxmze the sum caacty of the network. owever, when multle uncoordnated recevers exst n the network, and when users could have asymmetrcal ower constrants, the total sum caacty mght not be an arorate metrc. Maxmzaton of ths metrc could lead to allocaton of network resources that are based towards stronger user-recever ars, resultng n very oor erformance of the weaker user-recever ars. A game-theoretc model based on otental game theory for the far allocaton of resources n decentralzed networks s suggested here. Potental games are chosen as these are easy to analyze and gve a framework where users can serve the greater good by followng ther own best nterest,.e., can maxmze a global utlty by only tryng to maxmze ther own utltes. ence t can lead to smle game formulatons where maxmzatons of utlty of users can lead to mrovng a global network farness measure as well. To ncororate the noton of farness, the utlty functon of each user s modfed such that n addton to mrovng the beneft assocated wth the user s erformance at ts ntended recever, the functon also contrbutes to the farness n the network. The utlty functon s of the form gven n Equaton 3. (, ) = (, ) 2 ( (, ),, ) u s f s, =, = ( ( ),, ) 3, Functon f quantfes the beneft assocated wth a artcular choce of sgnature sequence and ower. Functons f 2 s the nterference measure for user erceved at ts assocated recever due to the other users resent n the system. Functon I s some functon of two sgnature sequences s and s. Functon f 3 s the nterference caused by a artcular user at the recevers assocated wth other users. Coeffcent s a weghtng factor. The frst two terms of the utlty functon are ntended to reduce the nterference at the recever assocated wth a artcular user. The thrd term attaches some beneft to beng nce to other users (reducng network nterference) and s ntended to contrbute to the farness n the network. (3) A ossble smle formulaton of the otental functon for ths network s gven by Equaton 4. f ( s, ) α f2 ( I( s, s ),, ), = Pot( SP, ) = = (4) β f 3 ( I( s s, ),, ), = Ths functon conssts of the sum of the utltes of all users. Coeffcents α and β are weghtng factors, vector S = [ s, s2,, s ] and vector = [, 2,, ] Searatng all the terms nvolvng the th user, ( ) = ( ) α 2 (, ) ( ( ),, ) ( ( ),, ) ( ( ),, ) P. ( ) Pot s, S,, P f s, f I s s,,, = β α β 3,, =, =, = 2, 3, f ( s, ) k k α f 2 ( I( s, sk ),, k) kk, = k,, = β k f 3 ( I( sk, s ), k, ) k,, = on contrbutngterms Formulaton of the utlty functon and the otental game as an exact otental game requres the followng condton (Equaton 6) to be satsfed. (, ) (, ) = Pot( s, S,, P ) Pot( s, S,, P) u s u s Ths s ossble f α lsted below Case: Case2: = β = 2 (5) (6 ) and under the two scenaros ( ( ),, ) = ( ( ),, ) ( ( ),, ) = ( ( ),, ), 2, 2,, 3, 3, =,, f ( ) = f ( 2 3 ) =,, (7) (8)

4 In the second case, another formulaton of an exact otental functon could be as follows (Equaton 9). (, ) = (, ) 3 ( (, ),, ) Pot SP f s (9) =, = 4.. Ordnal Potental Functon The utlty functon and the otental functon could be formulated as an ordnal otental game f the followng condton (Equaton 0) s satsfed. u s, u s, ( ) ( ) Pot( s, S,, P ) Pot( s, S,, P) (0) Ths s ossble when ( ) = ( ) = ( 2 3 u ) f ( u ) s an ordnal transformaton of ot ( ) ( ), ( ), ( ) 2 3 ot f f f, where f or when f f f are all ordnal functons where the utlty functon assocated wth each user s gven by, (, ) = (, ) 2 ( (, ),, ) u s f s, = ( ( ),, ) 3,, = and the otental functon s gven by, (, ) = (, ) ot ( (, ),, ) =, = () Pot SP f s (2) Ths formulaton allows each user to have a dfferent utlty functon, the only restrcton beng that the utlty functons are ordnal transformatons of each other. 5. EXAMPLE FORMULATIO OF A POTETIAL GAME A ossble formulaton of a far waveform adataton game, based on the game-theoretc framework descrbed n the revous secton, s resented here. Consder the utlty functon for the th user defned n Equaton 3, modeled on the formulaton n Equaton 3. s s s s ssss u ( s, ) = (3) ere, functon f 2 (from Equaton 3) s the nterference caused by other users n the network at the recever assocated wth user weghted by the receve ower of user. Functon f 3 s the nterference caused by user at recevers assocated wth other users, weghted by the receve ower of the other users. As functons f 2 and f 3 are smlar n structure f = f ), Equaton 9 can be used to form the ( ( ) ( ) 2 3 followng exact otental functon for ths utlty. where, (, ) s = s V S P s s = = The utlty functon can be rewrtten as, (, ) = s R s ssss u s = R R s s I s s (4) (5) = = (6) and R s the cross correlaton matrx of the receved sgnal at recever assocated wth user, = = (7) R = E r r = s s + E zz = s s + I The frst term n the utlty functon can be dentfed as s the nverse Sgnal to Interference Rato (SIR) at the recever assocated wth user. The second term s a weghted sum of nterference caused by the user at other recevers n the system and hence contrbutes to network farness. User, thus tres to maxmze ts SIR and n addton, also tres to reduce the nterference t causes at recevers corresondng to other users. A smlar utlty, whch ncreases a socal measure, s resented n ([7], [8]). owever ths s secfc to a system where each user talks to multle recevers and does not take nto account farness n the network. 5.. Convergence Characterstcs Maxmzers of the otental functon, V, form the steady states of the network. Exact otental games exhbt best and better resonse convergence to these steady states. Best resonse can be mlemented by havng each user change ts sgnature sequence such that the user derves the best utlty for the current state of the network. The utlty functon of the th user can be re-wrtten as (Equaton 8) s s s s u( s, ) = s + s (8) Let, s s s s R = + (9) Then the best resonse of the th user s the egenvector corresondng to the mnmum egenvalue of R. To allow the mlementaton of ths best resonse teraton at the user end, the users are assumed to know the sgnature sequences of all the other users n the network and the value for all { },,,.

5 Alternately, better resonse teratons can also be mlemented at the user end. In ths aroach, the user randomly alters ts sgnature sequence. All the recevers n the network that can hear a artcular user send back the change n SIR caused by ths sgnature alteraton. If the sum of the nverses of the changes n SIR s negatve, the user stcks to ts udate, else t reverses ts sgnature alteraton. The convergence characterstc for a best resonse mlementaton of the algorthm s shown n Fgure 2. The smulated network has sx users and three sgnal sace dmensons. The users are numbered accordng to decreasng receved owers at ther corresondng recever nodes. It s seen that the game converges n about ffteen teratons. Best resonse s mlemented by allowng each user to udate sequentally n a round robn fashon. As each user utlty udate leads to an ncrease n the otental functon, the algorthm converges even wth non-sequental udates. L T = qlog q log = L (20) where q s the share of the th grou, and L s the total number of grous. The ndex has a otental range from zero to nfnty (for very large values of L), wth hgher values ndcatng more unequal dstrbuton of resources. Let r be defned as follows. r log2 + s s = and q = r r = + I (2) (22) The roosed algorthm leads to a farness measure of as oosed to for the greedy algorthm. ence the game formulaton for waveform adataton resented n ths aer ndeed results n farer allocaton of resources than the greedy IA algorthms for waveform adataton. In addton to the examle exact otental game formulaton resented n ths aer, several ordnal transformatons ndcated n [3] could also be used to form far games for waveform adataton. 6. SUMMARY Fgure 2: Best-resonse convergence for a network wth multle un-coordnated recevers (Sold Lnes Far IA Algorthm, Dashed Lnes-Greedy IA Algorthm). Fgure 2 also shows the convergence of a greedy nterference algorthm (smlar to the algorthms dscussed n [], [2] and [3] for the centralzed recever) extended to the multle un-coordnated network scenaro. In the greedy nterference avodance algorthm, users are concerned only wth the maxmzaton of ther own SIR. Thel s entroy measure ([9]) an nequalty ndex (defned below), s used to nvestgate and comare the farness of the two aroaches. Thels entroy measure Inequalty Index: Ths s a measure of nequalty roosed by Thel that derves from the noton of entroy n nformaton theory. The entroy measure, T, s gven by: A game theoretc framework based on Potental games, to model waveform adataton n networks wth multle uncoordnated transmt-receve ars s resented. Ths formulaton leads to solutons that are farer than the greedy IA algorthms. An examle waveform adataton algorthm based on ths model that leads to a farer allocaton of resources than a greedy IA algorthm s also resented. owever, the mlementaton of the descrbed far algorthms requres consderable feedback from the recever nodes n the network. ence more effcent feedback mechansms need to be nvestgated. Alternately, an nvestgaton of more effcent better resonse schemes mght be of sgnfcance, as otental games exhbt better resonse convergence. 6. REFERECES [] S. Ulukus, R.D. Yates, Iteratve Constructon of Otmum Sgnature Sequence Sets n Synchronous CDMA systems,

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