On the performance of approximate equilibria in congestion games

On the p erformance of appro ximate equilibria in congestio n games George Christo doulou ∗ Elias Koutsoupias † P aul G. Spirakis ‡ Abstract W e study the performa nce of approximate Nash equilibria for linear congestion games. W e consider how m uch the pr ice of anarch y w o rsens and how mu ch the price o f stability improv es as a function of the approximation factor ǫ . W e give (almost) tight upper and low er b ounds for b oth the price o f anarch y and the price of stability for a tomic and non-a tomic congestion ga mes. Our results not o nly enco mpa ss and g eneralize the existing re s ults of exact equilibria to ǫ -Nash equilibria, but they also provide a unified appro ac h which reveals the co mmon threads of the atomic and non- atomic price of anar chy results. By expanding the sp ectrum, we als o ca st the existing r esults in a new light. F or example, the Pigou netw o rk, which gives tight results for exact Nash e q uilibria of selfish routing, remains tight for the price of stability of ǫ -Nash equilibria. 1 In tro duction A cen tral concept in Game Th eo r y is the notion of equ il ib rium and in p a r tic u la r the notion of Nash equilibrium. Algorithmic Game Theory has s tudied extensiv ely and with remark able success the com- putational issues of Nash equilibria. As a result, w e understand almost completely the computational complexit y of exact Nash equilibria (they are PP AD-complete for games describ ed explicitly [11, 5] and PLS-complete for games d escribed su cc in c tly [13]). The results established a long susp ected drawbac k of Nash equilibria, namely that they cannot b e computed effectiv ely , thus up grading the imp ortance of appro ximate Nash equ ilibr ia . W e don’t un derstand completely the compu tational issues of app ro x- imate Nash equ ilibr ia [15, 13, 12, 26], but they p ro vide a more r ea sonable equilibrium concept: It mak es sense to assume th a t an agen t is willing to accept a situation that is almost optimal to him. In another dir ec tion, a large b o dy of researc h in Algorithmic Game Th eo r y concerns the degree of p erformance degradation of systems du e to the selfish b eha vior of its users. Central to th is area is the notion of pr ice of anarch y (PoA) [14, 19] and the price of stabilit y (P oS) [1]. Th e firs t notion compares the so cial cost of th e w orst-case equilibr ium to th e so cial optimum, which could b e obtained if every agen t follo we d ob edien tly a central authorit y . The second n otion is ve r y similar bu t it considers the b est Nash equilibrium instead of the worst one. A n a tu ral question then is how the p erformance of a sys te m is affected when its users are ap- pro ximately s elfis h: What is the appr oximate pric e of anar chy and the appr oximate pric e of stability ? Clearly , b y allo wing the pla y ers to b e almost rational (within an ǫ factor), w e expand the equilibrium concept and w e exp ect the price of anarc h y to get w orse. On the other h and, the p rice of stabilit y should impro ve. The question is how they c hange as functions of the p aramet er ǫ . This is exactly the question that w e address in this w ork. W e study tw o fun damen tal classes of games: the class of congestion games [20, 17] and the class of n o n -a tomic congestion games [16]. The latte r class of games includes the selfish routing games whic h pla yed a central r o le in the dev elopmen t of the pr ic e of anarc hy [21, 22]. The former class pla ye d also an imp ortant role in the d ev elopmen t of the area of th e pr ic e of anarc hy , since it relates to the task allo ca tion p roblem, whic h w as the first p roblem to b e studied within the framewo r k ∗ Max-Planc k -Institut f ¨ ur Informatik, Saarbr ¨ uck en, German y . Email: { gchr isto } @mpi-inf.mpg. de † Department of I nfo rmatics, Un iv ersity of Athens. Email: elias@di.uoa.gr ‡ Computer Engineering an d In fo rmatics Dep artmen t, Patras Un iv ersity , Greece. Email: { spirakis } @cti.gr 1 s t l ( f ) = 1 + ǫ l ( f ) = f Figure 1: The Pigou net work. of the price of anarc h y [14]. Al th o u gh the price of anarch y and stabilit y of these games for exact equilibria w as established long ago [21, 8, 7, 2]—and actually Tim Rough gard en [23, 25] addressed partially the question for the price of anarc h y of appro ximate equ ilibr ia — ou r results add an unexp ected understand ing of the iss ues in v olve d . While these t wo classes of games are conceptually very similar, dissimilar tec hniques w er e emp loy ed to answ er the questions concerning the PoA and P oS . Moreo v er, the qualitativ e asp ects of the an s w ers w ere quite d ifferen t. F or instance, the Pigou net work of tw o parallel links captures the hard est net work situation for the pr ic e of anarc hy for the selfish routing. In fact, Roughgarden [24] p ro v ed that th e Pigou netw ork is the worst case scenario for a v ery br o ad class of dela y functions. On the other hand, the lo wer b ound for congestion games is differen t and somewhat more inv olv ed [2, 8]. F or the selfish routing games, the price of s ta b ili ty is not different than the pr ice of anarch y b ecause these games h a v e a unique Nash (or W ardrop as it is called in these games) equilibr iu m. On the other hand, for the atomic congestion games, the pr oblem prov ed more c hallenging [7, 4 ]. New tec hniques exploiting the p oten tial of these games needed in order to come up with an upp er b ound. The low er b ound is quite complicated and , unlik e the selfish routing case, it has a dep endency on the n u mb er of pla ye r s (it attains the maxim um v alue at the limit). The main differen ce b et w een th e tw o classes is the “in tegralit y” of atomic conge stion games: In congestion games, when a p lay er consid ers switc hing to another strategy , he has to tak e in to acco u n t the extra cost that he will add to the edges (or facilities) of th e new strategy . The num b er of pla ye r s on the new edges increases b y one and this c h ange s the cost. On the other hand, in the selfish r outing games the c hange of strategi es h as no additional cost. A simple—although n ot en tirely rigorous—w a y to think ab out it, is to consider the effects of a tin y amount of flo w that p onders w h ether to change path: it will not really affect the flow on the new ed ges (at least for contin uous cost functions). Is integrali ty the reason w h ic h lies b ehind the differen ce of th ese t wo classes of games? It s e ems so for the exact case. But our w ork could b e interpreted as r e vealing that the uniquen ess of the Nash equilibrium is also an imp ortan t factor. Because when w e m ov e to the w id er class of ǫ -Nash equilibria, the uniqueness is dropp ed and the problems lo ok quite similar qualitativ ely; the integral ity difference is still th er e, but it only manifests itself in different quan titativ e or algebraic differences. 1.1 Our con tribution and related w ork Our work encompasses and generalizes some fundamental results in the area of the p rice of anarc hy [21, 8, 7, 2] (see also th e recen tly published b o ok [18] f o r bac kground in formati on ). Ou r techniques not only provide a u nifying approac h bu t they cast the existing results in a n ew ligh t. F or instance, the Pigou netw ork (Figure 1) is still the tight example for the p rice of stabilit y , but n ot th e pr ic e of anarc hy . Instead for the price of anarc hy , the net work of Figure 2 is tigh t; in fact, th is net work is tigh t only for ǫ ≤ 1; a more complicated n e tw ork is required for larger ǫ . W e consider ǫ appr o ximate Nash equilibria. W e use th e multiplic ative definition of appr oximate e quilibria : In congestion games, a play er do es not switc h to a new strateg y as long as his curren t cost is less than 1 + ǫ times the new cost. In the selfish routing games, w e use exactly the same d efinition: the flo w is at an equilibrium wh en th e cost on its paths is less than 1 + ǫ times the cost of ev ery 2 1 1’ 2 2’ 3 3’ f γ f γ f γ Figure 2: Lo wer b ound for selfish routing. T here are 3 distinct edge latency f unctions: l ( f ) = f , l ( f ) = γ (a constan t whic h d epend s on ǫ ), l ( f ) = 0 (omitte d in the p ic tu re). There are 3 commo dities of rate 1 with source i and d estination i ′ . The tw o p at h s for the first commo dit y are sh o wn in b old lines. alternate path. There h a v e b een other d e fi nitions for approximate Nash equilibr ia in the lite r a tu re. The most-studied is the additiv e case [15, 11]. In [6], they consider approximate equilibria of the m ultiplicativ e case and they stu d y con verge n ce issues for congestion games. O u r definition differs sligh tly and our results can b e naturally adapted to the definition of [6]. There is a large b ody of wo r k on the price of anarc hy in v arious mo dels [18]. More relev ant to our work are the follo wing pub lic ations: In [2, 8], it is p r o v ed that the price of anarc hy of congestion games for pur e equilibr ia is 5 2 . Later in [7], it is sho wed that the ratio 5 / 2 is tigh t ev en for correlated equilibria, and consequen tly for mixed equilibria. F or symmetric games, it is something less: 5 n − 2 2 n +1 [8], where n is the n umb er of pla yers. F or weig hted congestion games, the price of anarch y is 1 + φ ≈ 2 . 618 [2]. Later in [7], it w as pr o v ed that the same ratio holds ev en for correlated equilibria. In [7, 4], it w as sho w n that the price of stabilit y of lin ear congestion games is 1 + √ 3 / 3. F or the s el fi sh routing paradigm, the price of anarc hy (and of stabilit y) for linear latencies is 4 / 3 [21] (see also [9] for a simplified version of this pr oof ), and the results are extended to non-atomic games in [22]. Th e most relev ant w ork is [23, 25] w hic h give s tigh t b ounds for the price of anarc hy of approximate equilibria when ǫ ≤ 1. W e extend th is to ev ery p ositiv e ǫ u sing differen t tec hniqu e s . In this w ork, we giv e (almost) tigh t u pp er and low er b ound s f or the Po A and P oS of atomic and non-atomic linear congestio n games. Our results are summarized in the T able 1 (where atomic r efers to congestions games). A tomic Non-atomic Anarc hy (1 + ǫ ) z 2 +3 z +1 2 z − ǫ where z = ⌊ 1+ ǫ + √ 5+6 ǫ + ǫ 2 2 ⌋ (Section 3) (1 + ǫ ) 2 (esp ecia lly for ǫ ≤ 1: 4(1+ ǫ ) 3 − ǫ ) (Section 4) Stabilit y 1+ √ 3 ǫ + √ 3 (Section 5) 4 (3 − ǫ )(1+ ǫ ) (Section 6) T able 1: The upp er b ounds (with p oin ters to r ele v ant sections). The results in the ab o v e table includ e the upp er b ounds. W e h a v e matc hin g lo we r b ound s except for the atomic Po S (and for n on -integral v alues ǫ > 1 for the PoA of non-atomic case). The price of stabilit y reduces to 1 for ǫ ≥ 1, whic h means that the optimal is a 1-Nash equilibriu m, for b oth the atomic and non-atomic case. Also, the price of anarc hy is app ro ximately (1 + ǫ )(3 + ǫ ) and (1 + ǫ ) 2 for large ǫ , the atomic and non-atomic case resp ectiv ely . The pr ice of anarc hy for ǫ ≤ 1 h as b een established b efore in [23, 25]. 3 The interesting case is p robably when ǫ is small. F or ǫ ≤ 1 / 3 the resu lts are su mmarized in th e T able 2: A tomic Non-atomic Anarc hy 5(1+ ǫ ) 2 − ǫ 4(1+ ǫ ) 3 − ǫ Stabilit y 1+ √ 3 ǫ + √ 3 4 (3 − ǫ )(1+ ǫ ) T able 2: The upp er b ounds for ǫ ≤ 1 / 3. A usefu l to ol, in teresting in its o wn righ t, is a generaliza tion of the notion of p otent ial function for b oth th e atomic and non-atomic case (Theorems 6 an d 10) f o r the case of ǫ -Nash equilibria. Remark ably , the parameter ǫ app ears only in the lin ea r part of the (quadratic) p oten tial fu nctio n . Our app roac h is similar to [8, 7], but it is muc h more inv olv ed tec hn ica lly and requ ir es a deep er understand ing of the p oten tial function iss u es in vo lved. W e wan t also to dra w atten tion to our tec h - niques in b ounding the ap p ro ximate price of anarch y for the selfish r o u ting whic h differ considerab ly from the techniques of [21] and others [18]. The main difference is that w e mov e fr om a domain with unique equilibrium to a domain with a set of solutions. 2 Definitions A congestion game [20], also called an exact p otent ial game [17], is a tuple ( N , E , ( S i ) i ∈ N , ( f e ) e ∈ E ), where N = { 1 , . . . , n } is a set of n p la y ers, E is a set of facilitie s , S i ⊆ 2 E is a set of pure s trat egies for pla ye r i : a pu re strateg y A i ∈ S i is simply a subset of facilities and l e is a cost (or latency) fun ct ion, one for eac h facilit y e ∈ E . Th e cost of play er i f or th e pure strategy profile A = ( A 1 , . . . , A n ) is c i ( A ) = P e ∈ A i l e ( n e ( A )), wh ere n e ( A ) is th e n umb er of pla y ers who use f a cilit y e in the strategy profile A . Definition 1. A pu re s tr a tegy profile A is an ǫ equilibrium iff for ev ery pla y er i ∈ N c i ( A ) ≤ (1 + ǫ ) c i ( A i , A − i ) , ∀ A i ∈ S i (1) W e b eliev e that the m ultiplicativ e d efinition of approximat e equilibria mak es more sense in the framew ork that we consider. T h is is b ec aus e the costs of the p la yers u sually v ary in this setting and a uniform ǫ do es n ot mak e muc h sense. Giv en that the p rice of anarc hy is a r at io, we need a defin itio n that is insensitive to scaling. The so cial cost of a p ure strategy profile A is the sum of th e p la y ers cost S C ( A ) = Sum ( A ) = X i ∈ N c i ( A ) The p ure appro x im ate price of anarc hy , is the so cial cost of the worst case ǫ equ ili b rium o ver th e optimal so cial cost P oA = max A is a ǫ -Nash S C ( A ) opt , while the pure app ro ximate price of stabilit y , is the so cial cost of the wo r s t case ǫ equ ili b rium o ve r the optimal so cial cost P oS = min A is a ǫ -Nash S C ( A ) opt . 4 Instead of defi n ing formally the class of nonatomic congestion games, we prefer to fo cus on the more illustrativ e–more restrictiv e though–class of selfish routing games. The d ifference in the tw o mo dels is that in a non-atomic game, th er e do es not exist any net work and the str a tegies of the pla yers are just sub set s of facilitie s (as in th e case of atomic congestio n games) and they do n o t necessarily form a path in a netw ork. The most desirable results are obtained when the upp er b ound s hold for general non-atomic games and matc hing lo wer b ounds hold f o r the sp ecial case of selfish r outing. Our results almost follo w this pattern, with a few exceptions of lo wer b ounds. This is b ecause w e put emphasis on the simp lic ity and w e didn’t attempt to extend them to th e selfish routing case. Let G = ( V , E ) b e a directed graph, w here V is a set of v ertices and E is a set of edges. In this net work w e consider k commo dities: s ou r ce -n ode pairs ( s i , t i ) with i = 1 . . . k , th at define the sources and d e s ti n at ions. The set of simple paths in ev ery pair ( s i , t i ) is d enot ed by P i , while with P = ∪ k i =1 P i w e den o te their union. A flo w f , is a mappin g from the set of p at h s to the set of nonn egativ e reals f : P → R + . F or a giv en flow f , the flo w on an edge is defined as the sum of the flo ws of all the paths that use this edge f e = P P ∈P ,e ∈ P f P . W e relate with ev ery commo dit y ( s i , t i ) a traffic rate r i , as the total traffic that needs to m ov e fr om s i to t i . A flo w f is feasible, if for ev ery commo dit y { s i , t i } , the traffic r at e equ a ls the flo w of eve r y path in P i , r i = P P ∈P i f P . Every edge in tro duces a d el ay in the net work. Th is dela y dep ends on the load of the edge and is d ete r mined by a dela y function, l e ( · ). An instance of a routing game is d enote d b y the triple ( G, r , l ). The latency of a path P , for a giv en fl o w f , is defined as the sum of all the latencies of the edges that b elong to P , l P ( f ) = P e ∈ P l e ( f e ). T he so cia l cost that ev aluates a giv en flo w f , is the total dela y due to f C ( f ) = X P ∈P l P ( f ) f P . The total delay can also b e expressed via ed ge flo ws C ( f ) = P e ∈ E l e ( f e ) f e . F r om no w on, when w e are talking ab out flo w s, w e mean feasible flo ws. In [3, 10], it is sho w n that ther e exists a (uniqu e) equilibrium flo w , kno wn as W ardrop equilibr ium[27 ]. I n analogy to their definition, we define the ǫ W ardrop equ ili b rium flo ws, as follo ws Definition 2. A feasible flo w f , is an ǫ -Nash (or W ardrop) equilibrium, if an d only if for ev ery commo dit y i ∈ { 1 , . . . , k } and P 1 , P 2 ∈ P i with f P 1 > 0, l P 1 ( f ) ≤ (1 + ǫ ) l P 2 ( f ). In this w ork w e restrict our attent ion to line ar latency functions : l e ( x ) = a e x + b e , wh ere a e and b e are nonnegativ e constants. Our results naturally extend to mixed and correlated equilibria. W e also b eliev e that they can b e also extended to m ore general latency fu nctions suc h as p olynomials. 3 Congestion Games – P oA In this section w e study the d epend ency on the p aramet er ǫ , of the p rice of anarch y for the case of atomic congestion games. F or large ǫ th e pr ice of anarc hy is roughly (1 + ǫ ) 2 . Th e same holds the non-atomic case as w e are going to establish in the next section. W e will need the follo wing arithmetic lemma. Lemma 1. F or every α, β , z ∈ N : β ( α + 1) ≤ 1 2 z + 1 α 2 + z 2 + 3 z + 1 2 z + 1 β 2 Pr o of. Consid er the fu nctio n f ( α, β ) whic h we obtain when w e sub tract the left part of the statemen t’s inequalit y from the right part and multiply the result by 2 z + 1. 5 f ( α, β ) = a 2 + ( z 2 + 3 z + 1) β 2 − (2 z + 1) β ( α + 1) =  α − 2 z + 1 2 β  2 + (8 z + 3) β 2 − (8 z + 4) β 4 . F or β = 0, and for an y β ≥ 2, f ( α, β ) is clearly p ositiv e. F or β = 1 it tak es the form f ( α, 1) = ( α − z )( α − z − 1) ≥ 0, and the lemma follo ws . Our first theorem giv es an up p er b ound for the price of anarc hy for congestio n games; this is tigh t, as we are going so on to establish. This r esult generalizes the b ound in [2, 8] to appr o ximate equilibria. The p roof is for linear latency fu n cti ons of the form l e ( x ) = x , bu t it can b e easily extended to latencies of the form l e ( x ) = a e x + b e , with nonnegativ e a e , b e . Theorem 1 (At omic-Po A-Upp er-Bound) . F or any p ositive r e al ǫ , the appr oximate pric e of anar chy of gener al c ongestion games with line ar latencies is at most (1 + ǫ ) z 2 + 3 z + 1 2 z − ǫ , wher e z ∈ N is the maximum inte ger with z 2 z +1 ≤ 1 + ǫ (or e quivalently for z = ⌊ 1+ ǫ + √ 5+6 ǫ + ǫ 2 2 ⌋ ). Pr o of. Let A = ( A 1 , . . . , A n ) b e an ǫ -appr o ximate pure Nash, and P = ( P 1 , . . . , P n ) b e the op timum allocation. F r o m the defin ition of ǫ -equilibria (Inequalit y (1)) we get X e ∈ A i n e ( A ) ≤ (1 + ǫ ) X e ∈ P i ( n e ( A ) + 1) . If we sum up for ev ery pla yer i and u se L emm a 1, we get Sum ( A ) = X i ∈ N c i ( A ) = X i ∈ N X e ∈ A i n e ( A ) = X e ∈ E n 2 e ( A ) ≤ (1 + ǫ ) X e ∈ E n e ( P ) ( n e ( A ) + 1) ≤ 1 + ǫ 2 z + 1 X e ∈ E n 2 e ( A ) + (1 + ǫ )( z 2 + 3 z + 1) 2 z + 1 X e ∈ E n 2 e ( P ) = 1 + ǫ 2 z + 1 Sum ( A ) + (1 + ǫ )( z 2 + 3 z + 1) 2 z + 1 opt . F r om this w e obtain the theorem Sum ( A ) ≤ (1 + ǫ ) z 2 + 3 z + 1 2 z − ǫ opt . The ab o v e is a t ypical pr oof in this work. All our upp er b ound p roofs ha ve similar form. T h e pro ofs of the price of stabilit y are more c hallenging how ev er, as they require the use of appropriate generalizat ions of the p oten tial function. W e no w sho w th at the ab o v e up per b ound is tigh t. 6 Theorem 2 (A tomic-P oA-Lo wer-Bo u nd) . F or any r e al p ositive ǫ , ther e ar e instanc es of c ongestion games with line ar latencies, for which the appr oximate pric e of anar chy of gener al c ongestion games with line ar latencies, is at le ast (1 + ǫ ) z 2 + 3 z + 1 2 z − ǫ , wher e z ∈ N is the maximum inte ger with z 2 z +1 ≤ 1 + ǫ . Pr o of. Let z ∈ N b e the maxim um in teger with z 2 z +1 ≤ 1 + ǫ . W e will constru ct an instance w ith z + 2 pla ye r s and 2 z + 4 facilities. There are t wo t yp es of facilities: • z + 2 facilities of t yp e α , with latency l e ( x ) = x and • z + 2 facilities of t yp e β with latency l e ( x ) = γ x = ( z +1) 2 − (1+ ǫ )( z +2) (1+ ǫ )( z +1) − z 2 x . Pla y er i has t wo alternativ e pure strategies, S 1 i and S 2 i . • The fi rst strateg y is to pla y the t wo facilit ies α i and β i , i.e. S 1 i = { α i , β i } . • The second strategy is to play eve r y facilit y of type α except for α i and z + 1 facilities of type β starting at facilit y β i +1 . More precisely , the second strategy has th e f a cilities S 2 i = { α 1 , . . . , α i − 1 , α i +1 , . . . , α z +2 , β i +1 , . . . , β i +1+ z } , where the in dices m ay require computations ( mo d z + 2). First we p ro v e that pla ying the second strategy S 2 = ( S 2 1 , . . . , S 2 n ) is a ǫ -Nash equilibrium. The cost of play er i is c i ( S 2 ) = ( z + 1) 2 + γ z 2 , as there are exactly z + 1 pla y ers using facilities of t yp e α and exactly z pla yers usin g facilities of t yp e β . If pla yer i un ila terally switc hes to the other av ailable strategy S 1 i he has cost c i ( S 1 i , S 2 − i ) = ( z + 2) + γ ( z + 1) = c i ( S 2 ) 1 + ǫ , whic h sho ws that S 2 is an ǫ -Nash equ ilibr ium. The optim um allo cation is the strategy profile S 1 , where ev ery pla y er has cost c i ( S 1 ) = 1 + γ and so the pr ice of anarc hy is c i ( S 2 ) c i ( S 1 ) = ( z + 1) 2 + γ z 2 1 + γ = (1 + ǫ ) z 2 + 3 z + 1 2 z − ǫ . Notice that the parameter z is an int eger b ecause it expresses a n umb er of faciliti es. The ab o v e theorems (lo wer and upp er b ound) emplo y , for any p ositiv e real ǫ , an inte ger z ( ǫ ), whic h is th e maximum int eger that satisfies z 2 z +1 ≤ 1 + ǫ . S o f o r ǫ ∈ [0 , 1 / 3], z ( ǫ ) = 1 and the p rice of anarc hy is 5(1+ ǫ ) 2 − ǫ , f or ǫ ∈ [1 / 3 , 5 / 4], z ( ǫ ) = 2 and the price of anarch y is 11(1+ ǫ ) 4 − ǫ and so on. Roughly the price of anarc hy gro ws as (1 + ǫ )(3 + ǫ ). 7 4 Selfish Routing – P oA In this section w e estimate the price of anarc hy f o r non-atomic conge s ti on games and consequen tly for its sp ecial case, the selfish routing. Our results generalize th e resu lts in [21 , 22] to the case of appro ximate equilibr ia. The p roof has the same form with the p r oof of the atomic case in the previous section. Again, w e will need an arithmetic lemma. The m a in change now is that w e deal with contin uous v alues instead of integ r al s . Lemma 2. F or every r e als α, β , λ it holds, β α ≤ 1 4 λ α 2 + λβ 2 , wher e Pr o of. Simp ly b ecause α 2 + 4 λ 2 β 2 − 4 λαβ = ( α − 2 λβ ) 2 ≥ 0. Theorem 3 (Selfish -P oA-Upp er-Bo u nd) . F or any p ositive r e al ǫ , and for every λ ≥ 1 , the appr oximate pric e of anar chy of non-atomic c ongestion games with line ar latencies is at most 4 λ 2 (1 + ǫ ) 4 λ − 1 − ǫ . Pr o of. Let f b e an ǫ -app r o ximate Nash flo w , and f ∗ b e the optimum flow (or any other feasible fl o w). F r om the definition of app ro ximate Nash equilibria (Inequalit y (1)), we get that for eve r y path p with non-zero flow in f and an y other path p ′ : X e ∈ p l e ( f e ) ≤ (1 + ǫ ) X e ∈ p ′ l e ( f ∗ e ) . W e s u m these inequalities for all pairs of p a th s p and p ′ w eight ed with the amoun t of flo w of f and f ∗ on these paths. X p,p ′ f p f ∗ p ′ X e ∈ p l e ( f e ) ≤ (1 + ǫ ) X p,p ′ f p f ∗ p ′ X e ∈ p ′ l e ( f ∗ e ) X p ′ f ∗ p ′ X e ∈ E l e ( f e ) f e ≤ (1 + ǫ ) X p f p X e ∈ E l e ( f ∗ e ) f ∗ e ( X p ′ f ∗ p ′ ) X e ∈ E l e ( f e ) f e ≤ (1 + ǫ )( X p f p ) X e ∈ E l e ( f ∗ e ) f ∗ e But P p f p = P p ′ f ∗ p ′ is equal to the total rate for the feasible flows f and f ∗ . Simplifying, we get X e ∈ E l e ( f e ) f e ≤ (1 + ǫ ) X e ∈ E l e ( f e ) f ∗ e . This is the generalization to app r o ximate equilibr ia of the inequalit y established by Bec kmann , McGuire, and Winston [3] for exact W ardrop equilibr ia. Since we consid er linear functions of th e f o r m l e ( f e ) = a e f e + b e , we get X e ∈ E  a e f 2 e + b e f e  ≤ (1 + ǫ ) X e ∈ E a e f e f ∗ e + (1 + ǫ ) X e ∈ E b e f ∗ e . 8 Applying Lemma 2, w e get X e ∈ E  a e f 2 e + b e f e  ≤ (1 + ǫ ) X e ∈ E a e  1 4 λ f 2 e + λf ∗ e 2  + (1 + ǫ ) X e ∈ E b e f ∗ e . from which w e get X e ∈ E  a e  1 − (1 + ǫ ) 1 4 λ  f 2 e + b e f e  ≤ λ (1 + ǫ ) X e ∈ E a e f ∗ e 2 + (1 + ǫ ) X e ∈ E b e f ∗ e , and for λ ≥ 1 4 λ − 1 − ǫ 4 λ S C ( f ) ≤ (1 + ǫ ) λS C ( f ∗ ) . This giv es price of anarch y at most of 4 λ 2 (1 + ǫ ) 4 λ − 1 − ǫ , for ev ery λ ≥ 1. The expression 4 λ 2 (1+ ǫ ) 4 λ − 1 − ǫ of the theorem is min imize d for λ = (1 + ǫ ) / 2 wh en ǫ ≥ 1 (and λ = 1 w hen ǫ ≤ 1). W e therefore obtain th e follo wing t w o corollaries by substituting λ = 1 and λ = (1 + ǫ ) / 2. The first corollary was pro ved b efore in [23, 25] usin g d iffe r en t tec hniqu es. Corollary 1. F or any nonne gative r e al ǫ ≤ 1 , the app r oximate pric e of anar chy of non-atomic c on- gestion games with line ar latencies is at most 4(1 + ǫ ) 3 − ǫ . Corollary 2. F or any p ositive r e al ǫ ≥ 1 , the appr oximate pric e of anar chy of non-atomic c ongestion games with line ar latencies is at most (1 + ǫ ) 2 . W e now show that the ab o v e up per b ounds are tigh t. T o b e precise, w e sh o w that Corollary 1 is tigh t and that Corollary 2 is partially tight —on ly for integral v alues of ǫ . The follo win g th eo r em for the case of ǫ ≤ 1 w as fir st shown in [23, 25]. W e in clud e a different pro of here for completeness and b ecause it is similar to the generalization f o r ǫ > 1, in Th eorem 5 . Theorem 4 (Selfish-PoA -Low er-Bound for ǫ ≤ 1) . F or any nonne gative r e al ǫ ≤ 1 , ther e ar e instanc es of c ongestion games with line ar latencies, for which the appr oxima te pric e of anar chy of gener al c on- gestion games with line ar latencies, is at le ast 4(1 + ǫ ) 3 − ǫ . Pr o of. W e will construct an in sta n ce with 3 commodities, eac h of them with unit fl o w, an d 6 facilities (a sligh tly more inv olv ed net wo r k case app ears in Figure 2. Th ere are tw o typ e s of facilities: • 3 facilities of t yp e α , with latency l ( x ) = x and • 3 facilities of t yp e β w it h constan t latency l ( x ) = γ = 2(1 − ǫ ) 1+ ǫ . Commo dit y i h as tw o alternative pur e strategies, S 1 i and S 2 i . 9 • The fi rst strateg y is to c ho ose b oth the facilities α i and β i , i.e. S 1 i = { α i , β i } • As a second alternativ e, pla yers of commo dit y i ma y c h oose every facilit y of t yp e α except for α i ; w e denote this set by S 2 i = { α − i } . First we prov e th a t pla y in g the s e cond strategy S 2 = ( S 2 1 , S 2 2 , S 2 3 ) is a ǫ -Nash equilibrium. T he cost of ev ery pla y er in commo dit y i is c i ( S 2 ) = 4, as there are exactly z + 1 pla y ers us ing facilities of type α and exactly z pla yers us in g facilities of t yp e β . If pla yer i un ila terally switc hes to the other av ailable strategy S 1 i he gets c i ( S 1 i , S 2 − i ) = 2 + γ = c i ( S 2 ) 1 + ǫ and so S 2 is an ǫ − ap p ro ximate equilibrium. In the optim u m case, the pla ye r s u se strategy pr ofi le S 1 , where commo dit y i has cost c i ( S 1 ) = 1 + γ and so the p rice of anarc hy is c i ( S 2 ) c i ( S 1 ) = 4 1 + γ = 4(1 + ǫ ) 3 − ǫ . F or larger ǫ ( ǫ > 1), w e ha ve : Theorem 5 (Selfis h -P oA-Lo w er-Bound for ǫ ≥ 1) . F or any r e al p ositive ǫ , ther e ar e instanc es of c ongestion games with line ar latencies, for which the appr oximat e pric e of anar chy of gene r al c ongestion games with line ar latencies, is at le ast (1 + ǫ ) z ( z + 1) 2 z − ǫ = (1 + ǫ ) z 2 + z 2 z − ǫ , wher e z = ⌊ 1 + ǫ ⌋ . Pr o of. Let z = ⌊ 1 + ǫ ⌋ . W e will constru ct an in sta n ce with z + 2 commo dities and 2 z + 4 facilities. There are tw o t yp es of facilitie s : • z + 2 facilities of t yp e α , with latency 1 and • z + 2 facilities of t yp e β with latency γ = ( z +1) 2 − (1+ ǫ )( z +1) (1+ ǫ ) z − z 2 . Commo dit y i h as tw o alternative pur e strategies, S 1 i and S 2 i . • The fi rst strateg y is to c ho ose b oth the facilities α i and β i , i.e. S 1 i = { α i , β i } . • As a second alt ern at ive, commo dit y i ma y choose every facilit y of t yp e α except for α i and z facilities of type β as defin ed in the follo wing: S 2 i = { α 1 , . . . , α i − 1 , α i +1 , . . . , α z +2 , β i +1 , . . . , β z + i +1 } , where the in dices are computed ( mo d z + 2). First w e prov e that pla yin g the second strategy S 2 = ( S 2 1 , . . . , S 2 n ) is a ǫ -Nash equilibriu m. The cost of commo dit y i is c i ( S 2 ) = ( z + 1) 2 + γ z 2 , as th ere are exactl y z + 1 commod itie s us in g facilities of t yp e α and exactly z pla y ers u sing facilitie s of t yp e β . 10 If commo dit y i unilaterally switc hes to the other a v ailable strategy S 1 i , its cost b ecomes c i ( S 1 i , S 2 − i ) = ( z + 1) + γ z = c i ( S 2 ) 1 + ǫ , whic h sho ws that S 2 is an ǫ − ap p ro ximate equilibrium. The optimum is the strategy profi le S 1 , wh ere ev ery commo dit y h as cost c i ( S 1 ) = 1 + γ . It follo ws that the p rice of anarc hy is c i ( S 2 ) c i ( S 1 ) = ( z + 1) 2 + γ z 2 1 + γ = (1 + ǫ ) z ( z + 1) 2 z − ǫ . 5 A tomic Games – P oS An upp er b oun d of th e p rice of stabilit y is p erhaps m ore difficult to obtain b ecause w e hav e to find a w ay to c haracterize the b est ǫ -N ash equilibrium. W e don’t kno w ho w to do this, so w e use an indir ec t approac h: W e id entify a p roperty such th at ev ery profile satisfying the pr o p erty is guarantee d to b e an ǫ -Nash equilibrium. W e then upp er b ound the pr ic e of anarc hy of all profiles satisfying this p r operty . T o this end, we generalize the notion of p oten tial [17]; a c haracteristic p r operty of congestion games is that they p ossess a p ote ntial function. W e defin e the ǫ -p oten tial function of a profile A to b e q Φ ǫ ( A ) = 1 2 X e ∈ E ( a e n e ( A ) + b e ) n e ( A ) + 1 2 1 − ǫ 1 + ǫ X e ∈ E ( a e + b e ) n e ( A ) . F or ǫ = 0, this reduces to the classical p oten tial function for congestio n games. More imp ortantl y , it generalizes the f ollo w ing in teresting prop ert y to ǫ -Nash equilibrium. Theorem 6. Any pr ofile A which is a lo c al minimum of Φ ǫ , is an ǫ -Nash e quilibrium. Pr o of. Consid er a profile A = ( A 1 , . . . , A n ). W e w ant to compu te th e c h ange in the ǫ -p oten tial function when pla yer i c h an ges from strategy A i to a strategy P i ∈ S i . Th e r esulting profile ( P i , A − i ) has n e ( P i , A − i ) =    n e ( A ) + 1 , e ∈ P i − A i n e ( A ) − 1 , e ∈ A i − P i n e ( A ) , otherwise. F r om this w e can compute the d ifference Φ ǫ ( P i , A − i ) − Φ ǫ ( A ) = X e ∈ P i − A i  a e n e ( A ) + 1 1 + ǫ ( a e + b e )  − X e ∈ A i − P i  a e n e ( A ) + 1 1 + ǫ ( − a e ǫ + b e )  . W e can rewrite this as Φ ǫ ( P i , A − i ) − Φ ǫ ( A ) = X e ∈ P i  a e n e ( A ) + 1 1 + ǫ ( a e + b e )  − X e ∈ P i ∩ A i a e − (2) X e ∈ A i  a e n e ( A ) + 1 1 + ǫ ( − a e ǫ + b e )  . 11 Supp ose no w th a t profile A is a lo ca l minimum of Φ ǫ . This translates to Φ ǫ ( P i , A − i ) ≥ Φ ǫ ( A ) for all i . The cost for pla ye r i b efore the c h a n ge is c i ( A ) = P e ∈ A i ( a e n e ( A ) + b e ) and after the c hange is c i ( P i , A − i ) = P e ∈ P i ( a e n e ( P i , A − i ) + b e ). W e w ant to s ho w that A is an ǫ -Nash equ ili b rium: c i ( A ) ≤ (1 + ǫ ) c i ( P i , A − i ). The ǫ -potent ial consists of t wo p a r ts that can b e u sed to b ound the cost of p la yer i at pr ofile A and ( P i , A − i ): c i ( A ) = X e ∈ A i ( a e n e ( A ) + b e ) ≤ X e ∈ A i (1 + ǫ )  a e n e ( A ) + 1 1 + ǫ ( − a e ǫ + b e )  , (whic h h olds b ec aus e n e ( A ) ≥ 1 when e ∈ A i ), and c i ( P i , A − i ) = X e ∈ P i ( a e ( n e ( A ) + 1) + b e ) − X e ∈ P i ∩ A i a e ≥ X e ∈ P i  a e n e ( A ) + 1 1 + ǫ ( a e + b e )  − X e ∈ P i ∩ A i a e (whic h h olds for ǫ ≥ 0). It follo ws immediately th a t c i ( A ) ≤ (1 + ǫ ) c i ( P i , A − i ). Consequently , A is an ǫ -Nash equilibrium. First we presen t an easy upp er b ound, th a t uses only the previous theorem. Prop os it ion 1. F or line ar c ongestion games, the pric e of stability is at most 2 1+ ǫ . Pr o of. Let A b e the allocation that minimizes the ǫ p oten tial Φ ǫ , and let P b e th e optimum allo cat ion. W e hav e Φ ǫ ( A ) ≤ Φ ǫ ( P ) and so Sum ( A ) + 1 − ǫ 1 + ǫ X e ∈ E ( a e + b e ) n e ( A ) ≤ Sum ( P ) + 1 − ǫ 1 + ǫ X e ∈ E ( a e + b e ) n e ( P ) , (3) from which w e get Sum ( A ) ≤ 2 1 + ǫ Sum ( P ) . The previous theorem give s us an easy wa y to b oun d the price of stabilit y . Clearly this is not tigh t: f or ǫ = 0, it do esn’t pro vid e us the r ig ht answe r 1 + √ 3 / 3 [7, 4], although it giv es us a go od estimation. T o get a b etter upp er b ound w e need to wo r k h arder. Lemma 3. F or inte gers α, β and for γ = ( 3+2 √ 3 )( e − 3+2 √ 3 ) 3 e +3+2 √ 3 γ β 2 + 1 − γ ǫ 1 + ǫ β − γ − ǫ 1 + ǫ α + (1 − γ ) β α ≤  2 √ 3 − 3  ( e − 1) 3 e + 3 + 2 √ 3 α 2 + 2 3 + √ 3 3 e + 3 + 2 √ 3 β 2 12 Pr o of. Let f b e the expression that we tak e if we subs ti tu te γ = ( 3+2 √ 3 )( e − 3+2 √ 3 ) 3 e +3+2 √ 3 , and then sub strac t the first p art from the second and d ivid e by ( 2 √ 3 − 3 ) ( e − 1) 3 e +3+2 √ 3 . W e can study f as a fu n cti on of in tegers α and β . f ( α, β ) = 1 / 4  2 √ 3 + 3 − 4 b − 2 b √ 3 + 2 a  2 + 1 / 8  5 + 3 √ 3   8 β − 3 − 3 √ 3  . W e wan t to pr o ve that f ( α, β ) ≥ 0, for eve r y α, β ∈ N . One can easily verify that f ( α, 0) =  3 + a + 2 √ 3  a ≥ 0 , f ( α, 1) = α ( α − 1) ≥ 0 , while for β ≥ 2 it gets only p ositiv e v alues. W e can now pro ve the most imp ortan t result of this section. Theorem 7 (A tomic-P oS-Upp er-Bound ) . F or any p ositive r e al ǫ ≤ 1 , the appr oximate pric e of stability of gener al c ongestion games with line ar latencies is at most √ 3 + 1 √ 3 + ǫ . Pr o of. Let A b e the allocation that minimizes the ǫ p oten tial Φ ǫ , and let P b e th e optimum allo cat ion. Since A is a lo cal minimum of Φ ǫ , if we sum (2) for all pla y ers i , we get X e ∈ E n e ( A )  a e n e ( A ) + 1 1 + ǫ ( − a e ǫ + b e )  ≤ X e ∈ E n e ( P )  a e n e ( A ) + 1 1 + ǫ ( a e + b e )  − X i ∈ N X e ∈ P i ∩ A i a e . F or simplicit y let’s assume a e = 1 , b e = 0, although th e results hold in general. W e get X e ∈ E  n 2 e ( A ) − ǫ 1 + ǫ n e ( A )  ≤ X e ∈ E n e ( P )  n e ( A ) + 1 1 + ǫ  (4) If we m ultiply (3) with γ and (4) with (1 − γ ), for γ = ( 3+2 √ 3 )( e − 3+2 √ 3 ) 3 e +3+2 √ 3 and add them, we get X e ∈ E n 2 e ( A ) ≤ γ β 2 + 1 − γ ǫ 1 + ǫ X e ∈ E n e ( P ) − γ − ǫ 1 + ǫ X e ∈ E n e ( A ) + (1 − γ ) X e ∈ E n e ( P ) n e ( A ) ≤  2 √ 3 − 3  (1 − ǫ ) 3 ǫ + 3 + 2 √ 3 X e ∈ E n 2 e ( A ) + 6 + 2 √ 3 3 ǫ + 3 + 2 √ 3 X e ∈ E n 2 e ( P ) and so X e ∈ E n 2 e ( A ) ≤ √ 3 + 1 √ 3 + ǫ X e ∈ E n 2 e ( P ) . Theorem 6 implies that the so cia lly optimal allocation is 1 − equilibrium. So for ǫ ≥ 1, trivially the price of stabilit y is 1. The follo wing th eo r em sho ws that this is tigh t, in the sense th a t th e so cial cost of the b est (1 − δ )-appro ximate equilibrium, is strictly greater than the so cia l optim um. 13 Theorem 8. Ther e exist instanc es of c ongestion games, (even with two p ar al lel links), wher e a the optimum al lo c ation is not a (1 − δ ) − appr oximate e quilibrium, for any arbitr arily smal l p ositive δ . This me ans that the pric e of stability for (1 − δ ) -appr oximate e quilibria is strictly gr e ater than 1. Pr o of. Consid er a game with t wo facilities ( p ar al lel links) e 1 , e 2 and n pla ye r s. The facilities ha ve latencies l e 1 ( x ) = (2 n − 1) · x − γ , for some arbitrary small p ositiv e γ and l e 2 ( x ) = x . Consider the allo catio n P , th at is pro duced when one p la y er, (say the first), c ho oses the first link and the rest of the pla yers use the second link. This has cost Sum ( P ) = 2 n − 1 − γ + ( n − 1) 2 , whic h is optimal: Any other allocation, in whic h k 6 = 1 p la yers u se the first link and n − k the second, has cost (2 n − 1 − γ ) k 2 + ( n − k ) 2 ≥ (2 n − 1 − γ ) + ( n − 1) 2 . In strateg y profile P , the first pla ye r has cost 2 n − 1 − γ , while the rest of the p la y ers hav e cost n − 1 eac h. If the first pla y er unilaterally deviates to the second link h e will ha ve cost n . This means th at opt is a (1 − 1+ γ n )-appro ximate equ ilibr ium. Therefore, for an y δ , there is an in sta n ce with sufficien tly large n u m b er of pla ye r s n ( δ ), where opt is not a (1 − δ )-appro ximate equilibrium. W e no w giv e an almost matc hing lo we r b ound for the p rice of stabilit y . Th e upp er and lo wer b ounds are not equal b ut they matc h at the extreme v alues of ǫ = 0 and ǫ = 1. F or ǫ = 0, we get the kno wn price of stabilit y [7, 4 ]. Th e pr ic e of stabilit y d ecrea ses as a function of ǫ , an d drops to 1 for ǫ = 1. Theorem 9 (A tomic-P oS-Lo wer-Bound) . Ther e ar e line ar c ongestion games whose appr oximate Nash e quilibria (even their domina nt e quilibria as the pr o of r eve als) have pric e of stability of the S um so cial c ost appr o aching 2 3 + ǫ + θ ǫ 2 + 3 ǫ 3 + 2 ǫ 4 + θ + θ ǫ 6 + 2 ǫ + 5 θ ǫ + 6 ǫ 3 + 4 ǫ 4 − θ ǫ 3 + 2 θ ǫ 2 , wher e θ = √ 3 ǫ 3 + 3 + ǫ + 2 ǫ 4 . Pr o of. W e describ e a game of n + λ p la y ers with parameters α , β , and λ w hic h we w ill fix late r to obtain the d esir ed p roperties. Eac h play er i has t wo strategies A i and P i , where the str ategy profile ( A 1 , . . . , A n ) will b e the equilibr ium and ( P 1 , . . . , P n ) will hav e optimal so cial cost. There are also λ play ers that hav e fixed strategies; they d on’t ha ve an y alternativ e. They pla y a fixed facilit y f λ . There are 3 t yp es of facilities: • n facilities α i , i = 1 , . . . , n , eac h with cost function l ( x ) = αx . F acilit y α i b elongs only to strategy P i . • n ( n − 1) facilities β ij , i, j = 1 , . . . , n and i 6 = j , eac h with cost l ( x ) = β x . F acilit y β ij b elongs only to str ategies A i and P j . • 1 facilit y f λ with unit cost l ( x ) = x . W e will fir s t compute the cost of ev ery p la yer and ev ery strategy profile. By symmetry , we n ee d only to consider the cost cost A ( k ) of pla yer 1 and the cost cost P ( k ) of pla yer N of the strategy profile ( A 1 , . . . , A k , P k +1 , . . . , P n ). Therefore, cost A ( k ) = (2 n − k − 1) β + ( λ + k ) . 14 Similarly , w e compute cost P ( k ) = α + ( n + k − 1) β . W e n o w wan t to select the parameters α and β so that the strategy profile ( A 1 , . . . , A n ) is (1 + ǫ )- dominan t. Equiv alen tly , at ev ery strategy profi le ( A 1 , . . . , A k , P k +1 , . . . , P n ), play er i , i = 1 , . . . , k , has n o reason to switc h to strategy P i , b ecause it’s (1 + ǫ ) times less profitable. W e use d ominan t strategies b ecause it is easier to guaran tee th a t there is n o other equilibrium. This is expressed b y the constrain ts (1 + ǫ ) · cost A ( k ) ≤ cost P ( k − 1) , for ev ery k = 1 , . . . , n . (5) All these constraints are linear in k and they are satisfied by equalit y wh en α = (1 + ǫ ) (2 nǫ − ǫ + ǫλ + n + 2 λ + 1) 2 + ǫ and β = 1 + ǫ 2 + ǫ , as one can verify with straight f o r w ard, alb eit tedious, s ubstitution. In summ a r y , for the ab o ve v alues of the parameters α and β , we obtain the desired p rop er ty that the strategy profile ( A 1 , . . . , A n ) is a (1 + ǫ )-domin ant strategy . If we increase α by any small p osit ive δ , inequalit y (5 ) b ecomes strict and the (1 + ǫ )-dominant strategy is un ique (and therefore unique Nash equilibrium). W e n ow w an t to s elect the v alue of the parameter m so that the p rice of anarc hy 1 of this equilibriu m is as high as p ossible. The p rice of anarc hy is cost A ( N ) + λ ( λ + n ) cost P (0) + λ 2 whic h for the ab o ve v alues of α and β can b e s im p lified to 3 n 2 + 2 n 2 ǫ − n − nǫ + 4 nλ + 2 nλǫ + 2 λ 2 + ǫλ 2 4 n 2 ǫ − nǫ + 3 nλǫ + 2 n 2 + 2 nλ + 2 n 2 ǫ 2 − nǫ 2 + nǫ 2 λ + 2 λ 2 + ǫλ 2 . If we optimize the parameter λ and tak e the limit of n to infinit y w e get the theorem. 6 Selfish Routing – P oS Here w e follo w the ideas of the pr evi ous section to define an appropr iate p ot ential function for ǫ -Nash equilibrium for the selfish rou tin g pr ob lem or more generally n on-a tomic congestion games. It is easier to deal with the m ore general case of non-atomic congestion games rather than the selfis h routing case, since we d on ’t h a v e to concern our selv es with the und erlying net work. In fact, our approac h reve als ho w little w e really need to establish results that encompass many influential results in the literature. Consider a fl o w f for the selfish routing with flow f e through ev ery edge e . W e d efi ne the ǫ -p oten tial function Φ ǫ ( f ) = X e ∈ E  1 2 a e f 2 e + 1 1 + ǫ b e f e  . W e will show that the global m inim um of Φ ǫ ( f ) is an ǫ -Nash equilibr ium: 1 Since this is the unique 1 + ǫ Nash Equilibrium of this game, the terms price of anarc hy and price of stabilit y are equiv alent. 15 Theorem 10. In a non-atomic c ongestion g ame, the flow f which minimizes the ǫ -p otential function is an ǫ -Nash e quilibrium. F urthermor e, for any other flow f ′ the fol lowing ine quality holds: X e ∈ E  a e f 2 e + 1 1 + ǫ b e f e  ≤ X e ∈ E  a e f e f ′ e + 1 1 + ǫ b e f ′ e  . Pr o of. Consid er a flow f and t wo paths p and p ′ of the same commo dit y . Supp ose th at the fl o w f on path p is p ositiv e. W e wan t to compute ho w m uc h Φ ǫ ( f ) c hanges when we shift a small amount δ > 0 of flo w from path p to path p ′ . More pr ecisely , if f ′ denotes the new fl o w, w e compute the follo wing limit lim δ → 0 Φ ǫ ( f ′ ) − Φ ǫ ( f ) δ = X e ∈ p ′  a e f e + 1 1 + ǫ b e  − X e ∈ p  a e f e + 1 1 + ǫ b e  (6) If f minimizes Φ ǫ , then the ab o v e quantit y is nonnegativ e. But w e can b ound the cost of paths p and p ′ with the tw o terms of this quantit y as follo w s: l p ( f ) = X e ∈ p ( a e f e + b e ) ≤ (1 + ǫ ) X e ∈ p ( a e f e + 1 1 + ǫ b e ) and l p ′ ( f ) = X e ∈ p ′ ( a e f e + b e ) ≥ X e ∈ p ′ ( a e f e + 1 1 + ǫ b e ) . It follo ws that l p ( f ) ≤ (1 + ǫ ) l p ′ ( f ), whic h imp lie s that f is an ǫ -Nash equilibr ium. F or th e second part, we observe that the expression (6), which is n onnega tive for f which minimizes Φ ǫ , implies that for ev ery path p on w hic h f is p ositiv e and every other path p ′ w e must ha ve X e ∈ p  a e f e + 1 1 + ǫ b e  ≤ X e ∈ p ′  a e f e + 1 1 + ǫ b e  . Consider now another flow f ′ whic h satisfies the rate constraints for the commod iti es and let us su m the ab o v e inequalities for all paths p and p ′ w eight ed with the amoun t of flow in f and f ′ . More precisely: X p,p ′ f p f ′ p ′ X e ∈ p  a e f e + 1 1 + ǫ b e  ≤ X p,p ′ f p f ′ p ′ X e ∈ p ′  a e f e + 1 1 + ǫ b e  X p ′ f ′ p ′ X e ∈ E  a e f 2 e + 1 1 + ǫ b e f e  ≤ X p f p X e ∈ E  a e f e f ′ e + 1 1 + ǫ b e f ′ e  But P p ′ f ′ p ′ = P p f p is equal to the sum of the rates for all commo dities. If we remo ve from the expression this common factor, the second part of the theorem follo ws. F r om Lemma 2, if we substitute λ with 1 / (1 + ǫ ), w e get that for any reals α, β , and ǫ ∈ [0 , 1] αβ ≤ 1 + ǫ 4 α 2 + 1 1 + ǫ β 2 . (7) Theorem 11 (Selfish-Po S -Upp er -Bound ) . The pric e of stability is at most 4 (3 − ǫ )(1+ ǫ ) . 16 Pr o of. Let f b e the p oten tial minimizer of Φ ǫ and f ∗ b e the optimum flo w . F r om Theorem (10) and (7) w e get that X e ∈ E a e f 2 e + 1 1 + ǫ b e f e ≤ X e ∈ E a e ( 1 + ǫ 4 f e 2 + 1 1 + ǫ f ∗ e 2 ) + 1 1 + ǫ b e f ∗ e or X e ∈ E a e 3 − ǫ 4 f 2 e + 1 1 + ǫ b e f e ≤ 1 1 + ǫ C ( f ∗ ) , and since 1 / (1 + ǫ ) ≥ (3 − ǫ ) / 4, we get 3 − ǫ 4 C ( f ) ≤ 1 1 + ǫ C ( f ∗ ) , whic h giv es the desired result: C ( f ) ≤ 4 (3 − ǫ )(1 + ǫ ) C ( f ∗ ) . W e no w establish that the P igou netw ork (extended to tak e into accoun t the p a r amet er ǫ , Figure 1) giv es tigh t r esu lts. Theorem 12 (Selfish-Po S -Lo w er-Bound) . The pric e of stability is at le ast 4 (3 − ǫ )(1+ ǫ ) . Pr o of. Consid er the Pigou net work of Figure 1. Ther e is a un it of flo w that wa nts to m ov e from s to t . Clearly , the only (1 + ǫ )- W ardrop flo w is to c ho ose the low er edge, for ǫ < 1. Th is giv es a so cial cost of 1. On th e other h and the optimum is to r o u te (1 + ǫ ) / 2 of the traffic fr om the lo w er edge and (1 − ǫ ) / 2 of the traffic from the upp er edge. This giv es a so cial opt of (1+ ǫ )(1 − ǫ ) 2 + (1+ ǫ ) 2 (1+ ǫ ) 2 = (1+ ǫ )(3 − ǫ ) 4 , and so the pr ice of stabilit y is 4 (3 − ǫ )(1+ ǫ ) as needed. Ac kno wle dgements The authors w ould lik e to thank Ioannis Caragiannis for many helpful discu s - sions and Tim Roughgarden for useful p ointe r s to literature. References [1] Elliot Ansh ele vich, Anirban Dasgupta, Jon M. Klein b erg, ´ Ev a T ardos, T om W exler, and T im Roughgarden. Th e p rice of stabilit y for netw ork design with fair cost allocation. 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