Uppr Bounding th Pric of Anarchy in Atomic Splittabl Slfish Routing Kamyar Khodamoradi 1, Mhrdad Mahdavi, and Mohammad Ghodsi 3 1 Sharif Univrsity of Tchnology, Thran, Iran, khodamoradi@c.sharif.du Sharif Univrsity of Tchnology, Thran, Iran, mahdavi@c.sharif.du 3 Sharif Univrsity of Tchnology, Thran, Iran, ghodsi@sharif.du Abstract. Slfish bhavior of nods of a ntwork such as snsors of a gographically distributd snsor ntwork, ach of which ownd and opratd by a diffrnt stakholdr may lad to a gam thortic stting calld slfish routing. Th fact that vry nod strictly aims at maximizing its own utility can caus dgradations of social wlfar. An issu of concrn would b th quantitativ masur of this infficincy. W study th pric of anarchy in slfish routing gams, a quantitativ masur of infficincy of worst quilibrium of th gam imposd by noncooprativ bhavior of playrs. For th most of this papr, w considr atomic modls of slfish routing in which th ntwork is a multicommodity flow ntwork with multipl sourcs and sinks, and ach playr controls a considrabl amount of th ovrall traffic. Th pric of anarchy for many variants of atomic splittabl instancs is not wll undrstood, and uppr bounding this paramtr in prsnc of affin cost functions is th problms w tackl in this papr. 1 Introduction A vry popular qustion to addrss in ral world communication ntworks is th way diffrnt lmnts of th ntwork rout information through th ntwork to an intndd dstination. Th nd of communicating bits of information in a wid sprad ntwork is trivial, and it is not a surpris that so much ffort has bn put into finding bttr and mor fficint mthods of this communication. Svral algorithms hav bn proposd for maximizing th ovrall prformanc of ad hoc ntworks, assuming that vry nod is willing to contribut to th social wlfar. But this would not b th cas whn ths intrmdiat lmnts act slfishly; thy wish to incur as littl cost of transmission (nrgy, dlay, tc.) as possibl, and to that nd, th can calculat vrything to find th prfct stratgy. This is likly to happn is svral sttings. Considr a ntwork of snsors installd in a spcific sit by diffrnt stakholdrs. Each snsor is programmd in a way that solly tris to minimizd its own cost in th ntwork it sks to bnfit th corrsponding stakholdr. Th lack of coordination btwn ths slf intrstd lmnts usually rsults in infficincy.
Slfish bhavior of ntwork lmnts, such as snsors hav bn mostly dalt with in trms of nrgy consumption in thortical computr scinc litratur. Each slfish snsor wishs to do th last amount of work in th ntwork, hnc rmaining aliv as much as possibl. Enrgy consumption can b critical whn ach snsor is quippd with a small battry. Yt it should b mntiond that satisfactory prformanc of such an lmnt can not b masurd just by th duration it kps working. Th quality of working mattrs too. In our study of slfish bhavior of ntwork lmnts, w us a simplifid masur of quality th dlay ach nod incurs. This is mostly du to th fact that adding mor ralistic, application-orintd paramtrs of quality in a communication would mak our modl so sophisticatd that it would b too hard to prform a mathmatical analysis on th modl. Mor prcisly, w look for a quantitativ masur of infficincy in trms of th ovrall communication dlay tim, causd by slfish bhavior of participating nods. Th suitabl notion that bst modls th outcom of this noncooprativ bhavior is Nash Equilibrium. A vast portion of currnt litratur on th infficincy of quilibria is ddicatd to slfish routing gams in which th playrs ar not allowd to rout thir flow of data or information through or mor diffrnt paths in th ntwork (S nonatomic slfish routing gams in [1], or atomic unsplittabl slfish routing gams in [1] for xmapl). Howvr, this is not th cas in many ral world communication ntworks. Thrfor, w choos to study th pric of anarchy, th ration btwn th worst cas quilibria of a gam to th optimal flow (minimizing th ovrall dlay of th ntwork) in sttings whr playrs ar allowd to frly split thir amount of flow into diffrnt paths btwn thir intndd sourc and dstination in th ntwork. This modl of slfish routing is quit naturally calld atomic splittabl slfish routing [13]. Th modl w us is dscribd in dtails in th nxt subsction. 1.1 Th Modl W us a slightly modifid modl of slfish routing that is wll studid in th litratur [1][4][6][7][8][9][11][13][14][1]. W considr sourc routing for th routing gams discussd in this papr, in which ach playr chooss a complt path from th sourc to th dstination for routing its traffic through. In gnral, thr ar two diffrnt classs of slfish routing gams, namly non-atomic and atomic slfish routing of which, w only discuss th lattr. Th modifir nonatomic rfrs to multi-commodity ntworks in which ach commodity rprsnts a vry larg numbr of playrs, ach controlling a ngligibl amount of th ovrall traffic. Th atomic slfish routing diffrs with th formr on in that in atomic sttings, ach commodity rprsnts a playr who controls a considrabl amount of traffic. Mor formally, w dnot an atomic slfish routing gam with a tripl (G, d, c) whr G rprsnts a multi-commodity flow ntwork (hncforward w rfr to it as just a ntwork for simplicity),and is givn in th form of G = (V, E) with vrtx st V and dirctd dg st E. In addition, commoditis ar givn in th form of th st {(s 1, t 1 ), (s, t ),..., (s k, t k )} of sourc-sink vrtics, whr k is
3 th numbr of playrs. Each playr nds to rout hr traffic through diffrnt paths in th ntwork originating in s i and travling to t i. W dnot th st of all ths s i -t i paths by P i. W do not considr th ntworks in which P i = for som i = 1,,..., k. W also dnot th st of all paths in th ntwork by P = k P i. In gnral, no capacity limit is imposd on th dgs, and th graph G is allowd to contain paralll dgs. Th scond coordinat in th tripl, d = (d 1, d,..., d k ), is a vctor of dmands, a prscribd positiv amount of th traffic that ach playr should rout from th corrsponding sourc to th sink. A flow f is said to b fasibl for a vctor d if for ach i {1,,..., k}, P P i f P = d i, whr f P is usd to dnot th amount of traffic going through th path P. Finally, c is th cost function for th ntwork. Mor prcisly, w dnot th cost of ach dg by c : IR + IR +. W must considr som limitations for th choic of cost functions as w always assum that th cost function is nonngativ, continuous, and nondcrasing. Th rason bhind ths assumptions is that thy ar all rasonabl in applications whr cost rprsnts a quantity that incrass with ntwork congstion (.g. dlay tim). Dfin th ovrall cost of a flow f incurrd by a playr i as C (i) (f) = E c (f ) (1) f (i), th amount of traffic snt through th dg by playr i can b dnotd by f (i) = P () P P i: P in turn. Also lt f b th total amount of traffic travling through th dg. It is immdiat from out discussions that f = k f (i). Rmark. Not that th costs associatd with th dgs ar in fact, cost pr unit of traffic, maning that if a playr routs x amount of traffic on an dg, sh will b chargd x c (x) units of cost. A fasibl flow f in atomic splittabl modl in indxd by both playrs, and paths. In othr words, th flow is a vctor of th form f = (f (1), f (),..., f (k) ), in which ach dnots th flow corrsponding to playr i, and in turn is a vctor of th form = ( P 1, P,..., P ), whr P Pi i dnots th numbr of s i -t i paths. Not that w can also dnot th total cost incurrd by a playr as a function of flow vctors of th individuals: C (i) (f) = C (i) (f (1), f (),..., f (k) ). (3) Basd on ths basic notations, w giv th dfinition of an quilibrium flow: Dfinition 1. (Atomic Splittabl Equilibrium Flow) Lt (G, d, c) b an atomic splittabl instanc, and f a fasibl flow for this instanc. f is an quilibrium flow for th ntwork if, for vry playr i {1,,..., k} and vry fasibl flow ˆf i, C (i) (f (1), f (),...,,..., f (k) ) C (i) (f (1), f (),..., ˆ,..., f (k) ).
4 1. Rlatd Work Th nonatomic modl of slfish routing was first formally dfind by Wardrop [16]. Bckmann, McGuir, and Winstn [3] provd th xistnc and uniqunss of th quilibria in nonatomic modls. Nonatomic slfish routing is immdiatly applicabl for ntworks using sourc routing. In ral world communication ntworks such as th Intrnt, anothr mthod of routing calld distributd shortstpath routing is typically usd instad of sourc routing. In this modl, th usr is not rsponsibl for slcting a full path from th sourc to th rcivr. For a discussion of this modl s [9]. Th pric of anarchy in nonatomic slfish routing gams was first studid by Roughgardn and Tardos [14]. For rsults on uppr and lowr bounds on th pric of anarchy in this modl, rfr to [1], [11] and [14]. Finally, Roughgardn and Tardos suggstd som way to rduc th pric of anarchy in nonatomic sttings [1]. Atomic slfish routing gams wr first considrd by Rosnthal [10]. Rosnthal also proposd th concpt of congstion gams and also showd that quilibrium flows nd not xist in wightd atomic instancs th instancs in which th playrs do not control th sam amount of traffic. Th pric of anarchy of atomic instancs was first studid by Suri, Toth, and Zhou [15] in th contxt of th asymmtric schduling gams. For mor rsults on th pric of anarchy in atomic slfish routing, s Awrbuch t al. work in [1]. Th notion of quilibrium usd by th authors so far was pur-stratgy Nash quilibrium. For dtaild bounds on th pric of anarchy using mixd-stratgy Nash quilibrium, rfr to [1] and [7]. Th authors of [5] considrd th pric of anarchy and stability in diffrnt classs of asymmtric schduling instancs. S [], [6], and [5] for rsults on th pric of stability in atomic slfish routing gams. Finally, svral rsarchrs hav studid slfish routing in th atomic splittabl modl. Equilibrium flows in th atomic splittabl modl can bhav in countrintuitiv ways, and th pric of anarchy in this modl is not wll undrstood. It was initially claimd that th uppr bounds on th pric of anarchy for nonatomic instancs carry ovr to atomic splittabl ons [8][13], but Comintti, Corra, and Stir Moss [4] rcntly gav countrxampls to ths claims in multicommodity ntworks. Obtaining tight bounds on th pric of anarchy in this modl rmains an important opn qustion. 1.3 Our Rsults In this sction, w first giv a proof of xistnc of quilibrium in atomic splittabl slfish routing instancs with affin (i.., linar) cost functions using a powrful mthod call th potntial function mthod. Spaking gnrally, w propos a function on th outcom of th gam (th flows) which rach its minimum if applid to an quilibrium flow. Thn, w giv a proof of th convxity of th function proposd and discuss th importanc of convxity for any function to b a potntial function for out modl. Finally, w procd to our mail rsult and propos an uppr bound on th pric of anarchy of such modls.
5 Existnc of Equilibrium Th proof prsntd in this papr is basd on a powrful tool, namly th Potntial Function. Roughly spaking, a potntial function is dfind on th outcoms of a slfish routing gam in a way that a flow is an quilibrium on if, it is th global minimizr of potntial function. Mor formally, w stat this thorm: Thorm 1. Lt (G, d, c) b an atomic splittabl instanc with k playrs and affin cost functions of th form ax + b. Such an instanc always admits an quilibrium flow. Proof. In th proof of this thorm w assumd that cost functions associatd with ach dg ar all affin. Mor prcisly, w assum c (f) = a f + b. Lt Φ as dnot th potntial function for atomic splittabl instanc, dscribd as Φ as (f) = E a j=i f (i) f (j) + b. (4) First, w show that Φ as is convx. Obsrv that for any ral-valus function dfind on a vctor spac such as Φ as (f) to b convx, a ncssary and sufficint condition is that Φ as ( f + f ) Φ as(f) + Φ as (f ). So, w must prov that a E E 1 j=i a E Thrfor, a E ( j=i ( j=i + f (i) ) ( f (j) + f (j) ) f (i) f (j) + f (j) + b f (i) f (j) + f (j) + b ( + f + f (t). (t) ) f (i) f (j) + f (i) f (j) + f (i) f (j) + f 4 (j) ) 0. So, a E ( j=i f (j) + f (i) f (j) f (i) f ) (j) f (i) f (j) 0. From which w can writ: a E j=i ( f (i) )(f (j) f (j) ) 0. (5)
6 It can also asily b vrifid that: (f (i) f (i) )(f (j) j=i E ( ) f (j) 1 ) = (f (i) f (i) ) + 1 By substituting (6) in (5), w gt ( a (i)) 1 f (i) f + 1 ( f (i) f (i) ) ( ( f (i) ) ). 0, which is always tru sinc a s ar nonngativ. So, Φ as (f) always admits a minimum valu on its domain. W nxt show that vry global minimizr of th potntial function corrsponds to an quilibrium flow of th ntwork. For th sak of contradiction, assum that this dos not happn. Mor formally, w choos a flow f such that it minimizs th potntial function valu, but w assum a playr l can mitigat his cost by dviating to anothr split of th flow among th s l -t l paths. W call this nw split ˆf. W can immdiatly writ C (f (1),..., f,..., f (k) ) C (f (1),..., ˆf,..., f (k) ). As long as th othr playrs do not dviat, for th flow ˆf w hav { ˆ =, for i l. Thrfor: ˆ = 0 > C (f (1),..., ˆf,..., f (k) ) C (f (1),..., f,..., f (k) ) = ( c ( ˆf ) ) ˆf c (f ) f E = [( ) ( ) ] a ˆf (i) + b ˆf a f (i) + b f E = [ ] a ˆf (i) ˆf + b ˆf a f (i) f b f E = ( a ( ˆf f (i) ) ˆf + a ( ˆf ) (f ) ) + b ( ˆf f ). E i {1,...,k},i l (7) W claim that quation (7) is xactly th amount of chang that taks plac in th valu of potntial function in cas of dviation: Th trm a i {1,...,k},i l ( ˆf f (i) ) ˆf, is th chang in first trm of (4) (th potntial function) whn xactly (6)
7 on of th variabls i or j is qual to l. In cas of i, j l, no changs occur in nithr th potntial function and th quation (7). Th scond trm of (7) is qual to th magnitud of chang in first trm of (4) whn both i and j ar qual to l. Finally, Th last xprssion (with b cofficint) also appars in th amount of chang in th potntial function valu if a playr l dviats. Thrfor potntial function valu dcrass as playr l dviatd from th flow f to ˆf, which is in contradiction with th choic of f. Thrfor, th global minimizr of Φ as is in fact an quilibrium flow for th instanc. As w hav alrady showd th xistnc of a global minimizr for Φ as, th proof is complt. 3 Uppr Bounds on th Pric of Anarchy In this sction, w obtain an uppr bound on th pric of anarchy of atomic splittabl modls basd on th potntial function. First, w prov th following two lmmas. Lmma 1. Lt k b a positiv intgr and V b k-lmnt vctor of ral numbrs. Also lt v i dnot th i th lmnt of th vctor. Th following is always tru: k + 1 v i k i v i k (k + 1 i) v i. Proof. dfin th ral variabl α as and β as β = α = i v i, (k + 1 i) v i. W know from basic algbra that th arithmtic man is always gratr than or qual to gomtric man. Th arithmtic man of α and β is [ ] α + β = 1 i v i + (k + 1 i) v i = k + 1 i + i v i = k + 1 v i. Th gomtric man of α and β is α β = i v i k (k + 1 i) v i. W always hav that α+β α β, hnc th claim.
8 Lmma. Lt (G, d, c) b an atomic splittabl instanc and f, a fasibl flow for th ntwork. Also suppos that Φ as (f) dnots th potntial function for this instanc. Thn f c (f ) k k + 1 a j=i Proof. From th dfinition of f w hav f c (f ) = a = a j=1 f (i) f (j) + b j=1 f (i) f (j) + b W dfin Γ and Ψ in th following mannr: Γ = a Ψ = a j=1 j=i f (i) f (j) + b f (i) f (j) + b + b.., (8). (9) Not that with abov notation dfind, th lmma w sk to prov would b of th form of Γ k k + 1 Ψ. To prov so, w first comput Γ Ψ. Γ Ψ = a f ( j) + b j=1 = a = a j=i = Ψ b f k j=i+1 a f (i) f (j) a f (i) f (j) j=i f (i) f (j) + b f (t). (10)
9 On th othr hand, using Cauchy-Schwarz inquality from linar algbra, w gt f (i) f (j) f (i) f (j). (11) It is asily validatd that = (k)f (1) f (j) = (1)f (1) + (k 1)f () + ()f () +... + (1)f (k) +... + (k)f (k) = (k + 1 i) = i f (i). Thrfor from ths two quations togthr with (11) w can writ: f (i) f (j) k (k + 1 i)f (i) i f (i). (1) Also from Lmma 1 it is straightforward to s k (k + 1 i)f (i) i f (i) k + 1. (13) Combining th two last rsults in (1) and (13) w gt a f (i) a k + 1 Th following inquality is also trivial for k 1 From (14) and (15), a k + 1 j=1 b f (i) f (j) + b f (t) b k + 1 From substitution of (16) in (10) w arriv at a, f (i) f (j). (14). (15) f (i) + b. (16) Thrfor Γ Ψ Ψ k + 1 Ψ = k 1 k + 1 Ψ. Γ Ψ + k 1 k + 1 Ψ = k k + 1 Ψ.
10 In what follows, w propos our uppr bound on th pric of anarchy of atomic splittabl slfish routing gams with affin cost functions: Proposition 1. Lt (G, d, c) b an atomic splittabl instanc with k playrs and affin cost function. Also lt th flow f b th quilibrium and th flow f b th optimum flow that minimizs th ovrall cost. Thn, C(f) k k+1 C(f ). Proof. From Lmma, and by summing th costs ovr th dgs, th rsults would b C(f) k k + 1 Φ as(f) k k + 1 Φ as(f ) k k + 1 C(f ). Th first inquality is immdiat from Lmma. Th scond inquality is bcaus th quilibrium flow f is a global minimizr of th potntial function. Th third inquality is dducd from th fact that C(f) always has som additional trms in comparison with Φ as (f). Mor formally E a Φ as (f) = E a = C(f) a E ( i,j {1,...,k},i>j f (j) i,j {1,...,k},i>j + b f (j). ) f (j) is always gratr than or qual to zro sinc for all dgs E and all playrs i {1,,..., k} w hav f (i) choos a s to b nonngativ. This complts th proof. Rfrncs 0, and also w [1] B. Awrbuch, Y. Azar, and L. Epstin, Th Pric of Routing Unsplittabl Flow, in Procdings of th 37th Annual ACM Symposium on Thory of Computing (STOC), 005, pp. 57-66. [] E. Anshlvich, A. Dasgupa, J. Klinbrg,. Tardos, T. Wxlr, and T. Roughgardn, Th Pric of Stability for Ntwork Dsign with Fair Cost Allocation, in Procdings of 45th Annual Symposium on Foundation of Computr Scinc (FOCS), 004, pp. 95-304. [3] M. J. Bckmann, C. B. McGuir, and C. B. Winstn, Studis in Economics of Transportation, Yal Univrsity Prss, 1956. [4] R. Comintti, J. R. Corra, and N. E. Stir Moss, Ntwork Gams with Atomic Playrs, in Procdings of th 33rd Annual Intrnational Colloquium in Automata, Languags, and Programming (ICALP), vol. 4051 of Lctur Nots in Computr Scinc, 006, pp. 55-536. [5] I. Caragiannis, M. Flammini, C. Kaklamanis, P.Kanllopoulos, and L. Moscardlli, Tight Bounds for Slfish and Grdy Load Balancing, in Procdings of th 33rd Annual Intrnational Collquium in Automata, Languags, and Programming (ICALP), vol. 4051 of Lctur Nots in Computr Scinc, 006, pp. 311-3.
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