On the convergence of the best-response algorithm in routing games

Transcription

1 On the convergence of the best-response algorthm n routng games Olver Brun, Balakrshna Prabhu, Tatana Seregna To cte ths verson: Olver Brun, Balakrshna Prabhu, Tatana Seregna. On the convergence of the best-response algorthm n routng games. 7th Internatonal Conference on Performance Evaluaton Methodologes and Tools ValueTools, Dec 2013, Turn, Italy. 27p., <hal > HAL Id: hal Submtted on 16 Sep 2013 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 On the convergence of the best-response algorthm n routng games O. Brun 1,2, B.J. Prabhu 1,2 and T. Seregna 1,3 1 CNRS, LAAS, 7 avenue du colonel Roche, F Toulouse, France 2 Unv de Toulouse, LAAS, F Toulouse, France 3 Unv de Toulouse, INSA, F Toulouse, France {brun, balakrshna.prabhu, tseregn}@laas.fr Abstract We nvestgate the convergence of sequental best-response dynamcs n a routng game over parallel lnks. Each player controls a nonneglgble porton of the total traffc, and seeks to splt ts flow over the lnks of the network so as to mnmze ts own cost. We prove that best-response operators are lpschtz contnuous, whch mples that a suffcent condton for the convergence of the best-response dynamcs s that the ont spectral radus of Jacoban matrces of best-response operators be strctly less than unty. We establsh the specfc structure of these Jacoban matrces for our game, and show that ths condton s met n two cases: a two-player game for an arbtrary number of lnks and for a wde class of cost functons; and b for arbtrary numbers of players and lnks n the case of lnear latency functons. For latency functons satsfyng reasonable convexty assumptons, we conecture that the proposed suffcent condton s met for arbtrary numbers of players and lnks. 1 Introducton Game theory has emerged as a fundamental tool for the desgn and analyss of decentralzed resource allocaton mechansms n networks. It has found applcatons n as dverse areas as load-balancng n server farms [5, 14, 12, 20], power control and spectrum allocaton n wreless networks [11, 26, 23], or congeston control n the Internet [1, 18, 24, 33]. In recent years, substantal research effort has been devoted to the study of non-cooperatve routng games n whch each orgn/destnaton flow s controlled by an autonomous agent that decdes how ts own traffc s routed through the network cf. [3, 13, 21, 28] and reference theren. Apart from the gan n scalablty wth respect to a centralzed routng, there are wde-rangng advantages to such a decentralzed routng scheme, ncludng ease of deployment and robusteness to falures and envronmental dsturbances. However, several questons arse when seekng to desgn and mplement such a non-cooperatve routng scheme. One of the most studed one pertans to the neffcency of non-cooperatve routng mechansms. Indeed, n general, the Nash equlbrum resultng from the nteractons of many self-nterested agents does not correspond to an optmal routng soluton. Numerous works have therefore focused on obtanng performance guarantees for non-cooperatve routng schemes [32, 30, 31, 35, 4, 10]. Ths s usually done by evaluatng the Prce of Anarchy, a standard measure of the neffcency of decentralzed algorthms ntroduced by Koutsoupas and Papadmtrou [22]. A small value of the Prce of Anarchy ndcates that, n the worst case, the gap between a Nash Equlbrum and the optmal soluton s not sgnfcant, and thus that good performances can be acheved even wthout a centralzed control. In ths work, we address a dfferent queston: do uncoordnated routng agents converge to a Nash equlbrum? Thus, rather than the qualty of the resultng routng strategy, we are concerned wth the convergence of autonomous routng agents to a Nash equlbrum under some natural dynamcs. More precsely, we address ths queston assumng the well-known myopc best-response dynamcs. Best-response dynamcs play a central role n game theory [8]. For nstance, the Nash equlbrum concept s mplctly based on the assumpton that players follow best-response dynamcs untl they reach a state from whch no player can mprove hs utlty. In a game, the best-response of player s defned as ts optmal strategy 1

3 2 condtoned on the strateges of the other players. It s, as the name suggests, the best response that the player can gve for a gven strategy of the others. Best-response dynamcs then conssts of players takng turns n some order to adapt ther strategy based on the most recent known strategy of the others wthout consderng the effect on future play n the game. In ths paper, we wll restrct ourselves to the sequental or round robn best-response dynamcs, where players play n a cyclc manner accordng to a pre-defned order. The focus of ths paper s the convergence of sequental best-response dynamcs n a network of parallel lnks, shared by a fnte number of selfsh users. Each user controls a nonneglgble porton of the total traffc, and seeks to splt hs flow over the lnks of the network so as to mnmze hs own cost. Ths model was ntroduced n the semnal paper of Orda et al. [29], where t shown that there exsts a unque Nash equlbrum under reasonable convexty assumptons on the edge latency functons. The users may have dfferent traffc demands. When all users control the same amount of traffc, the convergence to the Nash equlbrum follows from the fact that the symmetrc game s a potental game, that s, the Nash equlbrum corresponds to the mnmum of a convex optmzaton problem [13]. For the asymmetrc game, convergence results are avalable only n some specal cases. In [29], the convergence to the unque Nash equlbrum of the two-player routng game was proved when there are only two parallel lnks. As ponted out by the authors, the convergence proof s not readly extendble to more general cases. Altman et al. also study the two-lnk case [2]. Assumng lnear latency functons for the lnks, they prove the convergence of the sequental best-response dynamcs for any number of players. More recently, Mertzos has proven that, for the large class of edge latency functons ntroduced n [29], the two-player routng game converges to the unque Nash equlbrum n a logarthmc number of steps [27]. Hs proof of convergence reles on a potentalbased argument. Namely, he shows that the amount of flow that s reallocated n the network at each step s strctly decreasng. Unfortunately, ths argument does not seem to readly extend to more than two players. We also refer to [19, 17, 16] for convergence results on related, but dfferent, problems. Contrbutons: We propose a dfferent approach to study of the convergence of best-response dynamcs. The key dea to prove the convergence s to study the Jacoban matrces of best-response functons, and to analyze how long products of such matrces grow as a functon of the number of best-response updates. One of the most promnent quanttes characterzng the growth rate of matrx products s the so-called ont or generalzed spectral radus. We show that the best-response functon s Lpschtz, and establsh the specfc structure of ther Jacoban matrces for our game. Then, a suffcent condton for the convergence of the best-response dynamcs s that ther ont spectral radus be strctly less than unty. We thus obtan a purely structural suffcent condton that allows to reduce the analyss of the convergence of the sequental best-response dynamcs to the analyss of the ont spectral radus of certan matrces. Ths condton s used to prove the convergence of the two-player game for an arbtrary number of lnks. We also prove the convergence to the Nash equlbrum for arbtrary numbers of players and lnks n the case of lnear latency functons. Furthermore, although we were not able to prove t, we conecture that the proposed suffcent condton s vald for any numbers of players and lnks. The paper s organzed as follows. In Secton 2, we descrbe the non-cooperatve routng game under nvestgaton and ntroduce best-response dynamcs as well as some notatons. In Secton 3, we outlne the non-lnear spectral radus approach to convergence, and present several propertes of the best-response functon, and compute the structure of ts Jacoban matrx. In Secton 4, we state our man result, and prove the convergence of the best-response functon for the two-player game wth general cost functons and of the K-player game wth lnear cost functons. 2 Problem statement 2.1 Notatons In the followng, IR + denotes the set of non-negatve real numbers. Recall that the 1-norm of a vector x IR S s x 1 = S x. For x X, B o x,r wll denote the open ball of radus r centered at pont x,.e., B o x,r = {z X : x z 1 < r}. Let 1 denote the column vector 1,1,...,1 T. We let I and 0 denote the dentty and the zero matrces, respectvely ther szes wll be clear from the context. A matrx A s postve, and we wrte A 0, f and only f a, 0,,, and that t s negatve

4 3 f A s postve. We recall that the 1-norm of a matrx A s A 1 = max a. denote by σa the λ ts spectral radus, spectrum of the matrx A,.e., σa = {λ IR : x 0,Ax = λx}, by ρa = max λ σa and we recall that ρa A 1. If A 1,...,A n s a collecton of matrces, we denote by n A the product A n A n 1...A 1. For any functon f that s dfferentable at pont x, we denote by Dfx ts Jacoban matrx at x. 2.2 Non-cooperatve routng game We nvestgate a non-cooperatve routng game wth K routng agents and S lnks n whch each routng agent can control how ts own traffc s routed over the parallel lnks. Ths routng game s depcted on Fgure 1. λ 1 r 1, c 1 λ s x, r, c t λ K r S, c S Fgure 1: Traffc classes route ther packets over parallel lnks. Denote by S = {1,...,S} the set of lnks. Lnk S has capacty r and a holdng cost c per unt tme s ncurred for each packet sent on ths lnk. We let π = c /r denote the cost per unt capacty for lnk. We let C = {1,...,K} be the set of routng agent and λ be the traffc ntensty of routng agent. We shall also refer to routng agent as traffc class, or user. Each class can control how ts own traffc s spltted over the parallel lnks and seeks to mnmze ts own cost. Let x = x, S denote the routng strategy of class, wth x, beng the amount of traffc t sends over lnk. We let X denote the set of routng strateges for class,.e., the set of vectors x IR S such that 0 x, < r for all S, and S x, = λ. A strategy profle s a choce of a routng strategy for each user such that the stablty condton C x, < r s satsfed for all lnks S. It s thus a vector x = x C belongng to the product strategy space C X such that C x, < r, for all S. It wll be assumed throughout the paper that C λ < S r, so that X =. The optmzaton problem solved by class, whch depends on the routng decsons of the other classes, can be formulated as follows: mnmze T x,x = Sπ x, φρ BR- subect to x X, 1 y = x, + k x k,, S, 2 ρ = y /r, S, 3 ρ < 1, S, 4 In the above formulaton, y represents the total traffc offered to lnk, ρ s the utlzaton rate of ths lnk, and φ s the cost assocated to the lnk when there s a traffc of y flowng through t. In transportaton or communcaton networks, φ models the delay on the road or the lnk. The total cost ncurred by user s then the sum of the cost of ndvdual lnks weghted by the amount of traffc the user sends on each 5

5 4 of the lnks. Thus, gven the strateges of the others, user seeks to mnmze ts total cost subect to flow conservaton and stablty constrants. Assumpton 1 We shall make the followng assumptons on the cost functon φ: A 1 φ : [0,1 [0,, A 2 lm ρ 1 φρ = +, A 3 contnuous, strctly ncreasng, convex functon, and s twce contnuously dfferentable. Remark 1 At frst glance, t appears that the assumptons are not loose enough to nclude polynomal cost functons, whch are wdely used n transportaton networks. However, t wll be shown n Appendx E that any functon satsfyng B 1 φ : [0, [0,, B 2 lm ρ φρ = +, and B 3 A 3, has an equvalent functon whch satsfes assumptons A 1 A 3. Two functons are sad to be equvalent f the soluton of BR- wth one functon s also the soluton of BR- wth the other. Thus, results obtaned for functons satsfyng A 1 A 3 wll be applcable to functons that satsfy B 1 B 3. We note that Problem BR- s well-defned for all ponts x X snce k x k, < r for all lnks. 2.3 Nash equlbrum A Nash equlbrum of the routng game s a strategy profle from whch no class fnds t benefcal to devate unlaterally. Hence, x X s a Nash Equlbrum Pont NEP f x s an optmal soluton of problem BR- for all classes C, that s, f x = arg mn z X T z,x, C, where x s the vector of strateges of all players other than player at the NEP. It follows from our assumptons on the functon φ, that the lnk cost functons are a specal case of type-b functons, as defned n reference [29]. As proved n Theorem 2.1 of ths reference, ths mples the exstence of a unque NEP for our routng game. In the followng, we shall denote by x ths Nash equlbrum pont. 2.4 Best response dynamcs The best-response of player s defned as ts optmal strategy condtoned on the strateges of the other players. It s, as the name suggests, the best response that the player can gve for a gven strategy of the others. Let x u : X X, defned as x u x = arg mn T u z,x u,x u, 6 z X u be the best-response of user u to the strategy x u of the other players. From the defnton of T u, t can be shown that for each x X, there s a unque x u x. Gven a pont x X, the strategy profle x u x descrbes the strateges of all the players after the best response of user u. Best-response dynamcs then conssts of players takng turns n some order to adapt ther strategy based on the most recent known strategy of the others wthout consderng the effect on future play n the game. Defne a round to be a sequence of best-responses n whch each player plays exactly once. Once an order s fxed n the frst round, t s assumed to be the same n each subsequent round. The order n whch the players best-respond n the frst-round can be arbtrary. Let us fx ths order to be 1, 2,..., K. Defne ˆx 1 : X X as ˆx 1 x = x K x K 1... x 1 x, 7

6 5 be the pont reached from x after one round of play. One can recursvely defne ˆx n x = ˆx 1 ˆx n 1 x, 8 whch s the pont reached after n rounds. The best-response dynamcs can then be defned as the sequence {ˆx n x 0 } n 1 coorespondng to the strategy of players after each round of best-response when x 0 s the ntal strategy. A NEP has the property that each player s strategy s a best-response to strateges of the other players. Therefore f x 0 s a NEP then sequence wll reman at x 0. The man queston we seek to answer s: do the best-response dynamcs for the routng game converge from any startng pont? If they converge, then the converge to the Nash equlbrum pont. 3 The Non-lnear Spectral Radus Approach A usual method to prove the convergence of terates of an operator ˆx 1 : X X s to show that ths operator s a contracton. For ths, one needs to fnd a sutable norm, say, for whch there exsts a constant c [0,1 such that ˆx 1 x ˆx 1 y c x y, for every par of ponts x and y n the set X. The contracton condton says that the dstance between terates of the functon startng from two dfferent ponts decreases wth each teraton. The constant c depends on the norm, and for a contnuously dfferentable operator, t can be computed as sup x Dˆx 1 x, whch s the supremum of the Jacoban over all ponts n the doman of the operator. It s then suffcent to fnd a norm n whch the above condton s satsfed. For the best-response functon, t turns out that t s non-trval to fnd such a norm, ndependently of the startng pont, n whch the dstance decreases wth every teraton. Instead, as wll be seen later t wll be suffcent to fnd a norm n whch the dstance decreases asymptotcally and not wth every teraton. Ths weaker condton can be formalzed usng the noton of the non-lnear spectral radus descrbed below. For a functon f : X X, defne the set Jf = {Dfx : f s dfferentable at x}. 9 whch s the set of Jacoban matrces of the functon f evaluated at all ponts at whch f s dfferentable. Defnton 1 The non-lnear spectral radus of a functon f : X X s defned as [25]: n 1/n ρf = lm sup sup A. n A Jf The non-lnear spectral radus of f s related to the noton of ont spectral radus of a set M of matrces whch s defned as: n 1/n ˆρM = lm sup sup M, 10 n M M and s ndependent of the nduced matrx norm. It measures the worst case growth rate of a sequence of lnear transformatons that are taken from the set M. It can been seen that the non-lnear spectral radus of f s n fact the ont spectral radus of the set of Jacoban matrces of f, Jf. When there s only one matrx n M, from Gelfand s formula t follows that the ont spectral radus s equal to the spectral radus of that matrx. For a set wth several matrces, there s an equvalent result n terms of the generalzed spectral radus of M whch s defned as: ρm = lmsup n sup M M ρ n M 1 n, 11 where ρa s the spectral radus of the matrx A. If M s bounded then the generalzed spectral radus and the ont spectral radus of M are equal [7].

7 6 Consder a lnear dynamcal system of the form x n+1 = A n x n, where the matrces A M can be chosen dfferently n each step. Such a system s called a swtched lnear system. When all the matrces are the same, one can determne the stablty of such a system by checkng whether the spectral radus of ths matrx s less than 1 or not. In case of swthced lnear systems, the same condton wth the ont spectral radus n place of the spectral radus can be used to ascertan the stablty of the system, see for example [34]. For non-lnear operators, the followng convergence crteron was stated n [25]. Theorem 1 [25] Theorem 1 If f : X X s Lpschtz-contnuous and has a non-lnear spectral radus smaller than 1, then the terates of f are globally asymptotcally stable. Moreover, the rate of exponental decay, r, satsfes 0 < r log ρf. Thus, nstead of requrng the best-response to be a contracton, one can show the convergence of the bestresponse dynamcs by showng that: 1. ˆx 1 s Lpschtz-contnuous; and 2. ρˆx 1 < 1. In the rest of ths secton, frst we shall show a few propertes of the best-responsefuncton, and then compute the structure of ts Jacoban matrces, before arrvng at our man result. 3.1 Propertes of the best-response functon The purpose of ths secton s to establsh varous propertes of best-response functon, manly related to ts contnuty and dfferentablty. Let us defne S u x = { S : x u u, x > 0} 12 as the set of lnks used by player u n ts best-response to the strateges x u of other players. We have the followng result. Theorem 2 The best-response functon x u of player u s Lpschtz-contnuous on X wth x u z x u w 1 < 2 z w 1, z,w X. 13 Proof. See Appendx A. Corollary 1 Snce the best-response over one round, ˆx 1, s a composton of best-responses of each of the players cf. 6, t then follows that ˆx 1 s Lpschtz contnuous. Remark 2 The contnuty of the best-response functons s a drect consequence of Berge s Theorem on the contnuty of correspondances [6] see also page 64 of [9]. However, Lpschtz contnuty requres some more work than that. Once the Lpschtz contnuty of ˆx 1 has been establshed, t remans to be shown that ts non-lnear spectral radus s smaller than 1. For ths, we shall nvestgate the ponts at whch the ˆx 1 s dfferentable and compute the structure of ts Jacoban. We note that, accordng to Rademacher s theorem [15], a consequence of Theorem 2 s that the bestresponse functon x u s Fréchet-dfferentable almost everywhere n X; that s, the ponts n X at whch x u s not dfferentable form a set of Lebesgue measure zero. To compute the ponts at whch the dervatve s defned, we shall need the followng defntons:

8 7 Let g, x = T x = π φ y + x, φ y, 14 x, r r r where y = k x k,, be the margnal cost of player on lnk under strategy profle x. We say that lnk s margnally used by user u at pont x whenever the flow of user u on that lnk s 0 although the margnal cost of that player on that lnk s mnmum, that s x u, = 0 and g u, x = mn k S g u,kx. 15 we say that the set S u x s locally stable at pont x f t does not change for an nfntesmal varaton on the strateges of the other players, that s ǫ > 0, z B o x,ǫ,s u x = S u z. 16 From our assumptons on the functon φ, the contnuty of the best-response functons mply that of the margnal costs g, defned n 14 under the best-response dynamcs. In the followng, we say that no lnk s margnally used by user u n ts best-response at pont x f there s no lnk that s margnally used by user u at pont x u x. The two notons ntroduced above are related through the followng result. Lemma 1 f there s no lnk that s margnally used by player u n ts best-response at pont x, then the set of lnks S u x s locally stable at pont x. Proof. See Appendx A. Our frst result regardng the dfferentablty of best-response functons s the followng. Proposton 1 The best-response functon x u s dfferentable at every pont x X such that no lnk s margnally used by player u n ts best-response at pont x. Proof. See Appendx A. 3.2 Structure of the Jacoban matrces The Jacoban matrx of ˆx 1 s the product of Jacoban matrces of best-responses of ndvdual players. So, we shall start by computng the Jacoban of the best-response functons of ndvdual players. Consder a player u and a pont x X at whch x u s dfferentable. The Jacoban matrx of ths functon s then the block matrx x u 1 x 1 x... Dx u x =. x u K x 1 x... x u 1 x K x., x u K x K x where the,-block xu x x measures the senstvty of the strategy of player obtaned after the best response of player u wth respect to a change n the strategy of player. The best-response of a player u s senstve only to the strateges of the other players v u, and these senstvtes are reflected by the block matrces xu u x v whch appear n the uth row of the Jacoban matrx. Recallng that x u u u x u, x = x, 17 x v x v, S, S we shall dstngush between lnks / S u x and lnks S u x. We assume n the followng that the set S u x s locally stable cf. Secton 3.1, and thus that t does not change for an nfntesmal varaton on the strategy x v of player v C.

9 8 Lemma 2 For all lnks / S u x, x u u, x v x = 0, v C, 18 Proof. See Appendx B. For lnks S u x, we have: Lemma 3 There exst a vector θ IR S + and a vector γ IR S + satsfyng γ = 0 for all S u x and S γ = 1 such that x u { u, θ γ 1 f k =, = 19 x v,k θ k γ otherwse, for all players v u and all lnks S u x and k S. Proof. See Appendx B. Remark 3 The vectors θ and γ depend upon the strategy profle x and upon the player u that updates ts strategy. We have not made ths dependence explct n order to smplfy the notaton. Further, the vector θ has the followng mportant property whch wll be helpful n establshng the desred nequalty on the non-lnear spectral radus of ˆx 1. Lemma 4 There exsts a constant q < 1 such that 1 2 θ k q, k S, x X, u C. 20 Proof. See Appendx B. The structure of the Jacoban matrces of the best-response functons s summarzed n the followng result. Theorem 3 The Jacoban matrx of the best response functon x u of player u C has the followng form I Dx u x = M u x M u x, I and M u x = ΨΓB IΘ, where B s the S S matrx wth 1 n every entry,.e., B = 1 T 1, Γ = dagγ and Θ = dagθ, the vectors γ and θ beng those defned n Lemma 3, Ψ a postve dagonal matrx such that Ψ, = 1 f S u x, and Ψ, = 0 otherwse. Proof. The proof s broken down n three steps. Frstly, the uth row follows drectly from Lemma 3. Secondly, the strateges of all players except player u do not change followng the best response of player. Therefore, for all u and all v C, we have x u x v x = { I f v =, 0 otherwse. 21

10 9 Ths explans the appearance of the dentty matrx on the dagonal and the 0 matrx n other columns of each row except the row correspondng to the player dong the best-response that s, row u. Fnally, snce the best response of player u at pont x s nsenstve to her strategy at that pont and depends only on the strateges of the other player, we can conclude that, for all u C, Ths explans why the dagonal block n the uth row s 0. x u u x u x = Corollary 2 The Jacoban matrx of ˆx 1 has the form where the ndex u goes down from K to 1. Dˆx 1 x = 1 Dx u x, u=k 4 Convergence of best-response dynamcs In ths secton, we shall frst formulate a conecture on the non-lnear spectral radus of ˆx 1 on whch the man result of ths paper hnges. Then, ths conecture wll be shown to be true for two partcular cases : a two-player routng games; b K player routng games wth lnear lnk cost functon, φ. Conecture 1 For a fxed K and S, let J ˆ be the set of matrces of the form gven n Corollary 2. Then, the ont spectral radus of J ˆ s strctly less than 1. On the extensve numercal experments that we conducted, the above conecture was ndeed true. The man result of ths paper s then: Theorem 4 If Conecture 1 s true, then the best-response dynamcs 8 for the routng game BR- converges to the unque Nash equlbrum of the game. Whle we were unable to prove the conecture, and hence the convergence of best-response dynamcs, n ts generalty, we can show ts valdty for two non-trval cases the two player game, and the K player game wth lnear lnk cost functon, whch we show below. 4.1 Two-player routng game Frst, we shall prove a general result related to the Jont spectral radus of a certan class of matrces. The clamed result on the convergence of the best-response for the two-player game wll then follow drectly from that result. Let D + be the set of postve dagonal matrces, and G be the set of dagonal matrces Γ D + whose dagonal entres satsfy n addton S γ = For any natural number k 0, the above two types of dagonal matrces are used to defne the set M of S S matrces as follows. M s the set of matrces M that can be wrtten as M = ΓB IΘ for some matrces Γ G and Θ D +. We also defne M k for k 0 as the set of matrces that can be wrtten as the product of k matrces belongng to M, where by conventon M 0 contans only the dentty matrx. For q 0,1, we say that a matrx M s n the set M q f M = ΓB IΘ M and n addton Θ 1 q. We smlarly defne M k q as the set of matrces that can be wrtten as the product of k matrces belongng to M q. We note that the set M q s obvously bounded.

11 10 Accordng to Theorem 3 and Lemma 4, the Jacoban matrces of the best-response functons of players 1 and 2 have the followng smple form: Dx 1 0 Ψ1 M x = 1, and Dx 2 I 0 x =, 24 0 I Ψ 2 M 2 0 where M 1,M 2 M q for some q < 1 and where Ψ 1,Ψ 2 are dagonal matrces wth 0-1 entres on the dagonal. Usng Corollary 2, the Jacoban of the best-response functon over one round has the form Dˆx 1 0 Ψ1 M = 1, 0 Ψ 2 M 2 M 1 where M 1,M 2 M q. It then follows that the structure of the product of n Jacoban matrces has the followng form. Lemma 5 If J 1,J 2,...,J n J, then n 0 Ψ J = 1 X 2n Ψ 2 X 2n, 25 2 where Ψ 1,Ψ 2 are postve dagonal matrces wth 0-1 entres on the dagonal, X 2n 1 1 M q 2n 1, and X 2n 2 M q 2n. Proof. See Appendx C. Lemma 5 shows that the behavour of a large product of Jacoban matrces s governed by the asymptotc behavour of the matrces X n 1,X n 2. These matrces are themselves the product of matrces that belong to M q. Ths suggests to frst characterze the asymptotc growth rate of products of matrces n M q. Our key result regardng ths characterzton s stated n theorem 5. Theorem 5 For any k 1 and any matrx M = k Γ B IΘ n M k, t holds that where θ max = max 1 S θ for all = 1,...,k. Proof. See Appendx D. ρm k θmax, 26 The above theorem holds for any product of matrces belongng to M. If we now restrct our attenton to matrces belongng to M q, we obtan the followng mmedate corollary. Corollary 3 For any product M n M n 1...M 1 of matrces belongng to M q, we have ρm n M n 1...M 1 q n, mplyng that ρm q q. Proof. See Appendx C. We are now n poston to prove that sequental best-response dynamcs converges to the unque Nash equlbrum x. Theorem 6 For the two player routng game over parallel lnks, the sequental best-response dynamcs converges to the unque Nash equlbrum for any ntal pont x 0 X. Proof. See Appendx C.

12 K player games wth lnear lnk cost functons Consderφx = x, adelayfunctonwhchsoftenusedncongestongamestomodeldelaysnroadnetworks. From 55, t follows that θ k = 1/2. Thus, the matrx M u n Theorem 3 s of the form 1 2 ΓB I for some Γ G. Theorem 7 For the K player routng game over parallel lnks and lnear delay functon, the sequental best-response dynamcs converges to the unque Nash equlbrum for any ntal pont x 0 X. Proof. See Appendx F. References [1] A. Akella, S. Seshan, R. Karp, S. Shenker, and C. Papadmtrou. Selfsh behavor and stablty of the nternet: a game-theoretc analyss of tcp. In Proceedngs of the 2002 conference on Applcatons, technologes, archtectures, and protocols for computer communcatons, Pttsburgh, Pennsylvana, USA, August [2] E. Altman, T. Basar, T. Jménez, and N. Shmkn. Routng nto two parallel lnks: Game-theoretc dstrbuted algorthms. Journal of Parallel and Dstrbuted Computng, 619: , September [3] E. Altman, T. Boulogne, R. E. Azouz, T. Jmenez, and L. Wynter. A survey on networkng games n telecommuncatons. Computers and Operatons Research, 332: , February [4] J. Anselm and B. Gaual. Optmal routng n parallel, non-observable queues and the prce of anarchy revsted. In 22nd Internatonal Teletraffc Congress ITC, Amsterdam, [5] U. Ayesta, O. Brun, and B. J. Prabhu. Prce of anarchy n non-cooperatve load balancng games. Performance Evaluaton, 68: , [6] C. Berge. Espaces topologques et fonctons multvoques. Dunod, Pars, [translaton: topologcal spaces. new york: macmllan, ] edton, [7] M. A. Berger and Y. Wang. Bounded semgroups of matrces. Lnear Algebra and ts Applcatons, 166:21 27, [8] N. Berger, M. Feldman, O. Neman, and M. Rosenthal. Dynamc neffcency: Anarchy wthout stablty. In 4th Symposum on Algorthmc Game Theory, October [9] K. C. Border. Fxed pont theorems wth applcatons to economcs and game theory. Cambrdge Unversty Press, [10] O. Brun and B. Prabhu. Worst-case analyss of non-cooperatve load balancng. Techncal Report no 12590, LAAS-CNRS, [11] D. E. Charlas and A. D. Panagopoulos. A survey on game theory applcatons n wreless networks. Computer Networks, 5418: , December [12] H. L. Chen, J. Marden, and A. Werman. The effect of local schedulng n load balancng desgns. In Proceedngs of IEEE Infocom, [13] R. Comnett, J. R. Correa, and N. E. Ster-Moses. The mpact of olgopolstc competton n networks. Operatons Research, Publshed onlne n Artcles n Advance, DOI: /opre , June [14] A. Czuma, P. Krysta, and B. Vockng. Selfsh traffc allocaton for server farms. In Proceedngs of STOC, 2002.

13 12 [15] L. C. Evans and R. F. Garepy. Measure theory and fne Propertes of Functons. Studes n Advanced Mathematcs. CRC Press, Boca Raton, Florda, [16] E. Even-Dar, A. Kesselman, and Y. Mansour. Convergence tme to nash equlbra. In Proceedngs of the 30th nternatonal conference on Automata, languages and programmng, ICALP 03, pages , Berln, Hedelberg, Sprnger-Verlag. [17] A. Fabrkant, C. Papadmtrou, and K. Talwar. The complexty of pure nash equlbra. In A. Press, edtor, Proceedngs of the 36th annual ACM Symposum on Theory of Computng STOC 04, pages , [18] R. Garg, A. Kamra, and V. Khurana. A game-theoretc approach towards congeston control n communcaton networks. ACM SIGCOMM Computer Communcaton Revew, 323:47 61, July [19] M. Goemans, V. Mrrokn, and A. Vetta. Snk equlbra and convergence. In Proceedngs of the 46th annual IEEE Symposum on Foundatons of Computer Scence FOCS 05, pages , [20] M. Havv and T. Roughgarden. The prce of anarchy n an exponental mult-server. Operatons Research Letters, 35: , [21] I. Kedar, R. Melamed, and A. Orda. Equcast: Scalable multcast wth selfsh users. Computer Networks, 5313, August [22] E. Koutsoupas and C. H. Papadmtrou. Worst-case equlbra. In STACS 1999, [23] A. Leshem and E. Zehav. Cooperatve game theory and the gaussan nterference channel. IEEE Journal on Selected Areas n Communcatons, 26: , [24] L. López, G. del Rey Almansa, S. Paquelet, and A. Fernández. A mathematcal model for the tcp tragedy of the commons. Theoretcal Computer Scence, :4 26, October [25] K. Mak, J. Peng, Z. Xu, and K. Yu. A new stablty crteron for dscrete-tme neural networks: Nonlnear spectral radus. Chaos, Soltons and Fractals, 312: , [26] R. Menon, A. MacKenze, J. Hcks, R. Buehrer, and J. Reed. A game-theoretc framework for nterference avodance. IEEE Transactons on Communcatons, 574: , [27] G. Mertzos. Fast convergence of routng games wth splttable flows. In Proceedngs of the 2nd Internatonal Conference on Theoretcal and Mathematcal Foundatons of Computer Scence TMFCS, pages pp , Orlando, FL, USA, July [28] N. Nsan, T. Roughgarden, E. Tardos, and V. Vazran. Algorthmc Game Theory. Cambrdge Unversty Press, New York, NY, USA, [29] A. Orda, R. Rom, and N. Shmkn. Compettve routng n mult-user communcaton networks. IEEE/ACM Transactons on Networkng, 1: , October [30] T. Roughgarden. The prce of anarchy s ndependent of the network topology. J. Comput. Syst. Sc., 672: , [31] T. Roughgarden. Intrnsc robustness of the prce of anarchy. In STOC 09, [32] T. Roughgarden and E. Tardos. How bad s selfsh routng? J. ACM, 492, March [33] S. J. Shenker. Makng greed work n networks: a game-theoretc analyss of swtch servce dscplnes. IEEE/ACM Transactons on Networkng TON, 36: , December [34] J. Theys. Jont Spectral Radus: theory and approxmatons. PhD thess, Center for Systems Engneerng and Appled Mechancs, Unversté Catholque de Louvan, May [35] T. Wu and D. Starobnsk. On the prce of anarchy n unbounded delay networks. In GameNets 06: Proceedng from the 2006 workshop on Game theory for communcatons and networks, page 13, New York, NY, USA, ACM.

14 13 A Proofs of results n Secton 3.1 Proof of Theorem 2. Consder two ponts z and w n X. Let the vectors a,b IR S + be such that a = u z, and b = u w, for all S. In other words, a and b are the total traffc sent on lnk by users other than u n confguratons z and w, respectvely. To smplfy notatons, we denote by x z u, and x w u, the traffc sent on lnk by player u after hs best-response at ponts z and w, respectvely, that s x z u, = xu u, z and xw u, = xu u, w. For the purpose of the proof, we also defne f x,y = π φ x+y + x φ x+y, r r r for all lnks S. Then the margnal costs of player u on lnk after the best-response of that player at ponts z and w can be wrtten as g u, x u z = f x z u,,a and g u, x u w = f x w u,,b. From the KKT condtons, there exst µ z and µ w such that f x z u,,a µ z, wth equalty f S u z, and f x w u,,b µ w, wth equalty f S u w. Wthout loss of generalty, we assume that µ z µ w. As a consequence, we have Consder now the sets f x z u,,a f x w u,,b, S u w. 27 and S = { S : x z u, < x w } u,, 28 S + = { S : x z u, x w } u,. 29 Assume frst that S =. Then x z u, xw u, for all S. However, snce Sx z u, = x w u, = λ u, 30 S ths mples that x z u, = xw u, for all S. It yelds x z u, x w u, = 0 31 S Assume now that S. Snce S = S S +, we obtan from 30 that x z u, x w u, = x z u, x w u,, 32 whch leads to r S + S S x z u, x w u, = 2 x z u, x w u,. 33 For S, we have by defnton 0 x z u, < xw u,, and hence S uw. Thus, S S u w. Wth 27, t yelds f x z u,,a f x w u,,b, and thus x z u, +a φ + xz u, x z φ u, +a x w u, +b φ + xw u, x w φ u, +b, r r r for all S. However, snce for S we have x z u, < xw u, and snce φ and φ are strctly ncreasng, ths mples that x z u, +a > x w u, +b, from whch we deduce that S r r It yelds 0 < x w u, x z u, < a b S. 34 x z u, x w u, < a b 35 S S

15 14 Wth 33, we thus obtan x z u, x w u, < 2 a b 2 a b 36 S S S From 31 and 36, we obtan that, whether S be empty or not, we have Snce x u, x z u, x w u, < 2, S S uz w, u < 2 z, w, S u < 2 z, w, 37 C S z = xu, w for all S and all u, we also have x u, z xu, w = 0 38 u S Fnally, from 37 and 38, we conclude that x u, z xu, w < 2 z, w,, 39 that s, as clamed. C S C S x u z x u w 1 < 2 z w 1, 40 Proof of Lemma 1. Let Ω u be the set of ponts x X where S u x s locally stable. Let us defne f x,y = π φ x+y + x φ x+y, r r r for all lnks S. Note that f x,y s contnuous and strctly ncreasng n both x and y. Then the margnal cost of player u on lnk after the best-response of that player can be wrtten as g u, x u x = f x u u, x, k u x k,. From the KKT condtons, the functon µ : X u IR defned by s such that µx u = mn S g u,x u x S u x f 0, k ux k, < µx u. 41 Note that the contnuty of the best-response functon x u on X cf. Theorem 2 mples that of the margnal costs, and therefore the contnuty of µ on X u. Let x be a pont such that no lnk s margnally used by player u n ts best-response at pont x. Let us frst consder S u x. From 41, there exsts δ > 0 such that f 0, k u x k, µx u δ. Snce f x,y s contnuous n y and µx u s contnuous on X u, there exsts ǫ 1 > 0 such that, for all z B o x,ǫ 1, f 0, k u z k, < f 0, k ux k, + δ 2 < µx u δ 2, and µz u > µx u δ 2. It yelds

16 15 f 0, k uz k, < µx u δ 2 < µz u, z B o x,ǫ 1, and thus, accordng to 41, we have S u z for all z B o x,ǫ 1 f S u x. As a consequence, f S u x = S, then S u z = S for all z suffcently close to x, and thus x Ω u. Otherwse we can fnd S \S u x. Snce no lnk s margnally used by player u n ts best-response at pont x, there exsts β > 0 such that f 0, k ux k, µx u +β, S \S u x. 42 Proceedng as above, we can show that there exsts ǫ 2 > 0 such that, for all z B o x,ǫ 2, µz u < µx u + β 2 and f 0, k u z k, > f 0, k ux k, β 2 > µx u+ β 2, from whch we conclude that f 0, k u z k, > µz u, for all z B o x,ǫ 2. Ths mples that f S u x, then S u z for all z B o x,ǫ 2. Choosng ǫ = mnǫ 1,ǫ 2, we thus conclude that for all z B o x,ǫ, S u x S u z and S \ S u x S \S u z, whch s equvalent to S u x = S u z. Ths shows that f no lnk s margnally used by player u n ts best-response at pont x, then S u x s locally stable. Proof of Proposton 1. From Lemma 1, we know that f x s such that no lnk s margnally used by user u n ts best-response at pont x, then the set of lnks S u x s locally stable at x. As shown n Theorem 3, ths condton s suffcent to compute the partal dervatves of x u at pont x. It can be seen from 21 and 22 that the partal dervatves xu x v x, u,v C, and xu u x u x are contuous at x. Accordng to Lemma 2, the contnuty of the partal dervatves xu u, x v x at x for / S u x follows from the local stablty of S u x at x. Fnally, a closed-form formula s gven n 19 for the partal dervatves xu u, x v,k for v u and for S u x,k S. In vew of equatons 50-54, the contnuty of these partal dervatves follows from our assumptons on φ and from the contnuty of x u at x. Thus, all partal dervatves of x u exst and are contnuous at x, and therefore x u s contnuously dfferentable at x. B Proofs of results n Secton 3.2 Proof of Lemma 2. The proof follows from the assumpton that S u x s locally stable at x. We have x u u, x+hy = xu u, x = 0 for any vector y and h > 0 suffcently small. Ths mples that the drectonal dervatves of x u u,, and thus ts partal dervatves, are 0. Proof of Lemma 3. The proof s based on two observatons: at a best-reponse strategy, the change n margnal cost of player u due to a change n the strategy of player v s the same n every lnk that s used at the best-response strategy; and the total flow s conserved for player u rrespectve of the change n the strategy of player 1. Recall that g u, x u x := T u x u, x u x. s the margnal cost of player u at lnk under strategy profle x u x,.e., after the best-response of player u. For S u x, from the KKT condtons, the best-response strategy of player u, x u u, s such that the margnal cost s the same n all the lnks that receve a postve traffc at ths strategy. That s, g u, x u x = µx u S u x, 43

17 16 where the constant µ depends upon the strateges of the players but not on the ndex of the lnk. The local stablty of S u x mples that the set of lnks used by user u does not change for an nfntesmal varaton on the strateges of the other players. Ths leads to our frst observaton whch s that the change n the margnal cost of player u at ts best-response strategy due to the change n the strategy of player v u at lnk k s the same at all lnks that receve a postve flow of player u. Thus, g u, x v,k x u x = µ 2, S u x, 44 where µ 2 depends upon the strateges of the players. We have not made ths dependence explct n order to smplfy the notaton. For a functon of the form hfx,x, ts dervatve wth respect to x s gven by dhfx,x dx = dhf,x df df dx + dhf,x, dx where n the frst term on the RHS, h s treated to as a functon of f only, whereas n the second term t s treated as a functon of x only. Snce x u u, s a functon of x v,k, we can use the above formula to rewrte 44 as u u, dg u, x + dg u, = µ 2, S u x, 45 dx u, x v,k dx v,k where the partal dervates are replaced by full dervates n order to ndcate that the functon s dfferented n one varable whle treatng the other as constant. The partcular form of the cost functon gven n problem BR- permts a smplfcaton of the LHS of the above equaton by notng that the margnal cost n a lnk depends only on the traffc that s routed to that lnk. Thus, u u, dg u, x +δ k dg u, = µ 2, S u x, 46 dx u, x v,k dx v,k where δ k s unty f = k, and s zero otherwse. The value of µ 2 can be computed usng the second observaton that the total flow of player u s conserved rrespectve of the strategy of player v. That s, S ux x u u, x v,k = 0 47 We thus obtan µ 2 = = l S ux dg u,k dx v,k δ k l dg u,l dx v,k dgu,k dx u,k dgu,l dx u,l 1 1 l S ux dgu,l dx u,l l S ux 1 dgu,l dx u,l 1 1 1, 48 and x u u, x v,k = θ k γ δ k, S u x, 49 where θ k = dg u,k dx v,k dgu,k dx u,k 1, 50

18 17 and γ = l S ux dgu,l dx u,l dgu,. 51 dx u, We wll now show that 0 < θ k < 1 and 0 < γ < 1. We have g u,k = π k φρ k + x u,k φ ρ k. 52 r k Thus, snce φ s an ncreasng and convex functon, ndependently of the player v u, and Thus, from 50, θ k > 0 and dg u,k dx v,k = π k r k dg u,k dx u,k = π k r k φ ρ k + x u,k r k φ ρ k > 0, 53 2φ ρ k + x u,k φ ρ k > r k θ k = φ ρ k + x u,k r k φ ρ k 2φ ρ k + x < u,k r k φ ρ k We thus obtan that θ k s ndependant of v and that 0 < θ k < 1. Smlarly, we note that γ s postve and smaller than unty due to the fact that dg 0,l s postve for all l. To conclude the proof, we note that dx 1 0,k S xγ = 1 from the defnton of the vector γ n 51. Thus, lettng γ = 0 for S u x, we obtan u S γ = 1. In order to prove Lemma 4, we need the followng result. Lemma 6 There exsts a strctly postve constant ρ max < 1, ndependant of u and x, such that the utlzaton rate of each and every lnk S u x s upper bounded by ρ max after the best-response of user u at pont x, that s, ρ u x ρ max, S u x, x X, u C, 56 where ρ u x = 1 r C xu, x. Proof of Lemma 6. Observe that x X mples that k u x k, < r for all lnks, and thus that the optmzaton problem for player u s well-defned. Let z = x u x be the pont reached after the best response of player u. To smplfy notatons, we let ρ = ρ u x. From the KKT condtons, there exsts µ u x u such that [ π φρ + z ] u, φ ρ = µ u x u, S u x 57 r Snce 0 z u, /r ρ, S u x, 57 leads to the nequaltes Moreover, ρ and φ ρ are non-negatve. Thus, 58 leads to π φρ µ u x u, / S u x 58 π φρ µ u x u, 59 µ u x u π ρ φ ρ +φρ. 60 µ u x u π ρ φ ρ +φρ, / S u x,

19 18 whch combned wth 60 gves the nequalty µ u x u π ρ φ ρ +φρ, S. 61 Let f : [0,1 [c, be defned by f x := π xφ x + φx. Note that f s ncreasng and nonnegatve, and hence nvertble. The nverse on f s defned on [c,. Let us defne h : [0, [0,1 n the followng way : { f 1 h x = J x f x [c, ; 0 f x [0,c. The functon h s contnuous and non-decreasng. Further, from 61, k h µ u x u ρ = z k,. r Summng over all the lnks, we obtan the followng functonal nequalty on µ u x u : hµ u x u := r h µ u x u z k, = λ, 62 k that s µ u x u s such that the above nequalty s satsfed. A bound on µ u x u tself can now be obtaned by makng use of the followng observatons. Snce h s contnuous and non-decreasng for all S, h s contnuous and non-decreasng. It has [0, as ts doman and [0, r as ts mage. Further, lm x hx = r. From the stablty condton, λ < r. Usng these propertes and 62, we can conclude that µ u x u µ max <. It then follows from 59 that ρ β = φ 1 µmax, S u x, π and the upper bound β depends nether upon u nor upon x. Moreover, φ s such that x < φ 1 x < 1, and hence β < 1. By defnton of S u x, we also have ρ > 0 and thus β > 0. Takng ρ max = max S ux β yelds the proof. Proof of Lemma 4. We note from 55 that, snce x u,k r k φ ρ k 0, we have θ k φ ρ k + x u,k r k φ ρ k /2φ ρ k +2 x u,k r k φ ρ k, mplyng that θ k Snce φ s ncreasng and convex, θ k s an ncreasng functon of x u,k consderng ρ k = ρ u k x as fxed, and snce x u,k /r k ρ k, we also have the followng nequalty: θ k φ ρ k +ρ k φ ρ k 2φ ρ k +ρ k φ ρ k φ ρ k 1 2φ ρ k +ρ k φ ρ k. 64 Snce the numerator and the denomnator of the fracton appearng on the rght-hand sde of 64 are strctly ncreasng n ρ k, Lemma 6 mples that θ k q = 1 φ 0 2φ ρ max +ρ max φ < ρ max A consequence of Theorem 3 and Lemma 4.

20 19 Proposton 2 The set J of Jacoban matrces s bounded. Proof of Proposton 2. Consder a player u C and a pont x where the best-response functon x u s dfferentable. Theorem 3 mples that Dx u x I 1 + M u x max θ m γ n δ m n 66 m S n S ux For m S u x, we have whle for m S u x, we have n S ux θ m γ n δ m n = θ m < 1, 67 n S ux θ m γ n δ m n = θ m n S ux,n m = 2θ m 1 γ m γ n + γ m 1 < Wth 66, 67 and 68, we obtan that Dx u x 1 < 3 for all players u C and all ponts x where x u s dfferentable. From ts defnton n 9, we thus conclude that the set J s bounded. From the submultplcatvty of norms and relaton 7, t follows that Corollary 4 The set J s bounded. C Proofs of results n Secton 4.1 Proof of Lemma 5. The proof s by nducton. The clam s true for n = 1. Gven that the form s true for some n, t wll be shown that the form holds for n+1. By defnton, n+1 J = J n+1 n J 0 Ψ1 M = 1 0 Ψ 3 X 2n Ψ 2 M 2 M 1 0 Ψ 4 X 2n 2 0 Ψ = 1 M 1 Ψ 3 X 2n 2 0 Ψ 2 M 2 M 1 Ψ 4 X 2n 2 Snce M 1 M q and Ψ 4 s a 0-1 dagonal matrx, t follows that M 1 Ψ 4 M q. Usng the prevous fact and the defnton M q 2n and the fact that X 2n 2 M q 2n, one can deduce that M 1 Ψ 3 X 2n 2 M q 2n+1 1, and M 2 M 1 Ψ 4 X 2n 2 M 2n+1 q. Proof of Corollary 3. Consder M 1,M 2...M n M q. Each matrx M can be wrtten as M = Γ B IΘ, where θ max = Θ 1 q. From theorem 5, we thus obtan ρm n M n 1...M 1 1 n q. As a consequence, sup ρ M 1,...,M n M q n M 1 n q

21 20 Snce M q s bounded, ts ont spectral radus and ts generalzed spectral radus concde. From the defnton n 11, we mmedately obtan that ρm q q. Proof of Theorem 6. Snce J s bounded see Corollary 4, and the Generalzed spectral radus s equal to the Jont spectral radus of a bounded set of matrces, t suffces to prove that ρj < 1. From Lemma 5, we have n det J λi = det λi det Ψ 2 X 2n 2 λi = λ S det Ψ 2 X 2n 2 λi, mplyng that λ 0 s an egenvalue of n J f and only f t s an egenvalue of Ψ 2 X 2n 2. Thus, ρ n J = ρ Ψ 2 X 2n 2. Further, ρ Ψ 2 X 2n 2 Ψ 2 X 2n X 2n Ψ 2 1 = X 2n, and thus, snce X 2n 2 M 2n q, ρ n 2 J sup M 1 = ρ 2n M q, M M q 2n where the last equalty s obtaned usng the defnton of the Jont spectral radus 10. Let ǫ = 1 q 2 > 0. Snce ρ n M q 1 n ρm q as n, there exsts N such that for all n N, ρ n 1 n J ρm q +ǫ q + 1 q = 1+q 2 2, where the last nequalty follows from Corollary 3. Snce the rght hand-sde s ndependant of J 1,...,J n, we deduce that sup ρ J 1,...,J n J n and, accordng to 11, t yelds ρj 1+q 2 < 1. J 1 n 2 1+q, n N, 2 2 D Proof of Theorem 5 The man dffculty n provng Theorem 5 s that the matrces M of M k are nether postve nor negatve. To crcumvent ths dffculty, we shall construct a postve or negatve matrx A such that ρm ρa and A 1 k θ max. Before showng how to construct such a matrx, we state two basc propertes of the matrces n M k n the followng lemma. Lemma 7 For any matrx M M k, the followng two assertons hold: a for each and every column, S m, = 0, b f λ 0 s an egenvalue of M and f x s the assocated egenvector, then S x = 0. Proof. Let us frst prove asserton a. Consder M M k and wrte M as M = ΓB IΘY wth Y M k 1. Then,

22 21 m,1,..., m,s = 1 T M = 1 T ΓB IΘY = γ 1,..., = 0 T, γ 1 Y whch proves the result. Let us now prove asserton b. Let M M k be wrtten n the form M = ΓB IΘY and consder λ σm, λ 0 and x 0 such that λx = Mx. Multplyng on both sdes by 1 T, we obtan λ S x = 0 = 1 T x = 1 T M x = 0, where the last equalty follows from asserton a. Snce λ 0, ths mples that S x = 0. We wll now use property b of Lemma 7 to show that, for any matrx M M k, f we choose the matrx A to be of the form A = DB +M, where D s any dagonal matrx, then ρm ρa. Lemma 8 For any matrx M M k and for any dagonal matrx D, ρm ρdb +M. Proof. Let λ 0 be an egenvalue of M and x be the assocated egenvector. We have DB +Mx = DBx+λx = x D1+λx = λx, 69 where the last equalty s obtaned usng property b of Lemma 7. Snce ths can be done for all non-zero egenvalues of M, we conclude that σm {0} σdb +M. Ths clearly mples that.e., ρm ρdb +M. max λ max λ, λ σm λ σdb+m Gven a matrx M M k, we shall now consder two specfc choces of the dagonal matrx D : the frst choce allows to obtan a matrx A 0 such that ρm ρa, whle the second one produces a matrx A 0 wth the same property. Snce the two choces lead to a postve or negatve matrx A, the evaluaton of A 1 s greatly smplfed, allowng to obtan useful upper bounds on ρm. These bounds are proven n the followng proposton. Proposton 3 For any matrx M M k, the two followng nequaltes on ρm are vald: ρm ρm S mn m,k, 70 1 k S S max m,k, 71 1 k S Proof. Let us frst consder the dagonal matrx D defned as D = dag mnm 1,k,mnm 2,k,...,mnm S,k, k k k and consder the matrx A = DB +M. Snce a, = m, mn k m,k,,, we have A 0. We know from Lemma 8 that ρm ρa A 1. Hence