Network Design with Weighted Players

Transcription

1 Nwork Dign wih Wighd Playr Ho-Lin Chn Tim Roughgardn Jun 21, 27 Abrac W conidr a modl of gam-horic nwork dign iniially udid by Anhlvich al. [2], whr lfih playr lc pah in a nwork o minimiz hir co, which i prcribd by Shaply co har. If all playr ar idnical, h co har incurrd by a playr for an dg in i pah i h fixd co of h dg dividd by h numbr of playr uing i. In hi pcial ca, Anhlvich al. [2] provd ha pur-ragy Nah quilibria alway xi and ha h pric of abiliy h raio bwn h co of h b Nah quilibrium and ha of an opimal oluion i Θ(log k), whr k i h numbr of playr. Lil wa known abou h xinc of quilibria or h pric of abiliy in h gnral wighd vrion of h gam. Hr, ach playr i ha a wigh w i 1, and i co har of an dg in i pah qual w i im h dg co, dividd by h oal wigh of h playr uing h dg. Thi papr prn h fir gnral rul on wighd Shaply nwork dign gam. Fir, w giv a impl xampl wih no pur-ragy Nah quilibrium. Thi moiva conidring h pric of abiliy wih rpc o α-approxima Nah quilibria oucom from which no playr can dcra i co by mor han an α muliplicaiv facor. Our fir poiiv rul i ha O(log w max )-approxima Nah quilibria xi in all wighd Shaply nwork dign gam, whr w max i h maximum playr wigh. Mor gnrally, w ablih h following rad-off bwn h wo objciv of good abiliy and low co: for vry α = Ω(log w max ), h pric of abiliy wih rpc o O(α)-approxima Nah quilibria i O((log W)/α), whr W i h um of h playr wigh. In paricular, hr i alway an O(log W)- approxima Nah quilibrium wih co wihin a conan facor of opimal. Finally, w how ha hi rad-off curv i narly opimal: w conruc a family of nwork wihou o(log w max / log log w max )-approxima Nah quilibria, and how ha for all α = Ω(log w max /log log w max ), achiving a pric of abiliy of O(log W/α) rquir rlaxing quilibrium conrain by an Ω(α) facor. Dparmn of Compur Scinc, Sanford Univriy, 393 Trman Enginring Building, Sanford CA Rarch uppord in par by NSF Award holin@anford.du. Dparmn of Compur Scinc, Sanford Univriy, 462 Ga Building, Sanford CA Suppord in par by by ONR gran N , DARPA gran W911NF-4-9-1, and an NSF CAREER Award. im@c.anford.du. 1

2 1 Inroducion 1.1 Th Pric of Sabiliy in Nwork Dign Gam Undranding h inracion bwn incniv and opimizaion in nwork i an imporan problm ha ha rcnly bn h focu of much work by h horical compur cinc communiy. Dpi h walh of rul obaind in hi ara ovr h pa fiv yar, nwork dign and formaion rmain a fundamnal opic ha i no wll undrood. Whil conomi and ocial cini hav long udid gam-horic modl for how nwork ar or hould b crad wih lf-inrd agn (.g. [6, 14, 15] and h rfrnc hrin), h mahmaical chniqu for quanifying h prformanc of uch nwork ar currnly limid. Th goal of quanifying prformanc (or lack hrof) in h prnc of lfih bhavior naurally moiva h win concp of h pric of anarchy and h pric of abiliy. To dfin h, fir rcall ha a (pur-ragy) Nah quilibrium i an aignmn of all of h playr of a noncoopraiv gam o ragi o ha h following abiliy propry hold: no playr can wich ragi and bcom br off, auming ha all ohr playr hold hir ragi fixd. A h oucom of lfih, uncoordinad bhavior, Nah quilibria ar ypically infficin and do no opimiz naural objciv funcion [11]. Th pric of anarchy and h pric of abiliy ar wo way o maur h infficincy of Nah quilibria of a gam, wih rpc o a noion of ocial good (uch a h oal co incurrd by all of h playr). Th pric of anarchy of a gam, fir dfind in Kououpia and Papadimiriou [16], i h raio of h objciv funcion valu of h wor Nah quilibrium and ha of an opimal oluion. Th pric of anarchy i naural from h prpciv of wor-ca analyi an uppr bound on h pric of anarchy bound h infficincy of vry poibl abl oucom of a gam. Th pric of abiliy, by conra, i h raio of h objciv funcion valu of h b Nah quilibrium and ha of an opimal oluion. Th pric of abiliy wa fir udid in Schulz and Sir Mo [25] and wa o-calld in Anhlvich al. [2]. Th pric of abiliy ha primarily bn udid in nwork dign gam [2, 3], wih h inrpraion ha h nwork will b dignd by a cnral auhoriy (for u by lfih agn), bu ha hi auhoriy i unabl or unwilling o prvn h nwork ur from acing lfihly afr h nwork i buil. In uch a ing, h b Nah quilibrium h b nwork ha accoun for h incniv facing h nwork ur i an obviou oluion o propo. In hi n, h pric of abiliy maur h ncary dgradaion in oluion qualiy caud by impoing h gam-horic conrain of abiliy. 1.2 Shaply Co Sharing wih Unwighd Playr Th goal of analyzing h co of nwork crad by or dignd for lfih ur wa fir propod by Papadimiriou [21] and iniially xplord indpndnly by Anhlvich al. [3] and Fabrikan al. [12]. Th wo papr udid diffrn yp of nwork dign gam; alo, h fir conidrd h pric of abiliy (whr i wa calld h opimiic pric of 2

3 anarchy ), h cond h pric of anarchy. (S alo [1, 1, 18] for mor rcn work on h and rlad modl.) Clo o h prn work i a variaion on h modl of [3] ha wa propod and udid by Anhlvich a. [2], which hy calld nwork dign wih Shaply co haring and w will abbrvia o Shaply nwork dign gam. Th mo baic modl conidrd in [2] i h following. Th gam occur in a dircd graph G = (V, E), whr ach dg ha a nonngaiv co c, and ach playr i i idnifid wih a ourc-ink pair ( i, i ). Evry playr i pick a pah P i from i ourc o i dinaion, hrby craing h nwork (V, i P i ) a a ocial co of i P i c. Thi ocial co i hard among h playr in h following way. Fir, if dg li in f of h chon pah, hn ach playr chooing uch a pah pay a proporional har π = c /f of h co. Th ovrall co c i (P 1,...,P k ) o playr i i hn h um P i π of h proporional har. Thi proporional haring mhod njoy numrou dirabl propri. I i budg balancd, in ha i pariion h ocial co among h playr; i can b drivd from h Shaply valu, and a a conqunc i h uniqu co-haring mhod aifying crain fairn axiom (.g. [19]); and, a hown in [2], i coax bnign bhavior from h playr. Spcifically, Anhlvich al. [2] howd ha a pur-ragy Nah quilibrium alway xi in conra o h mor gnral co-haring ha wa allowd in h prdcor modl [3] and ha h pric of abiliy undr Shaply co-haring i a mo h kh harmonic numbr H k = O(log k), whr k i h numbr of playr. Anhlvich al. [2] alo providd an xampl howing ha hi uppr bound i h b poibl, and provd numrou xnion. 1.3 Shaply Co Sharing wih Wighd Playr A naural and imporan xnion ha Anhlvich al. [2] idnifid bu provd fw rul for i ha o wighd playr. In mo nwork dign ing, w xpc h amoun of raffic o vary acro ourc-ink pair. Such non-uniformiy ari for many raon. For xampl, playr could rprn populaion of cuomr of Inrn Srvic Providr, which canno b xpcd o po a common iz; playr could rprn individual wih diffrn bandwidh rquirmn; or colluion among vral playr could yild a ingl virual playr wih iz qual o h um of ho of h colluding playr. Th dfiniion of nwork dign wih Shaply co-haring xnd aily o includ wighd playr: if w i dno h wigh of playr i, hn i co har of an dg i c w i /W, whr W i h oal wigh of h playr ha u a pah conaining h dg. Whil ay o dfin, hi wighd nwork dign gam appard challnging o analyz. Indd, prior o h prn work, h primary rul known for hi wighd gam wr nially uggion ha i xhibi mor complx bhavior han i unwighd counrpar. In paricular, Anhlvich al. [2] provd h following: ha h ky ponial funcion proof chniqu for h unwighd ca canno b dircly ud for gam wih wighd playr; and ha h pric of abiliy can b a larg a Ω(k + log W), whr k i h numbr of playr and W = i w i i h um of h playr wigh (auming w i 1 for all i). Th poiiv rul of [2] for wighd gam concrnd only h pcial ca of 2-playr gam and ingl-ourc, ingl-ink gam, whr pur-ragy Nah quilibria 3

4 wr hown o xi. No furhr poiiv or ngaiv rul on ihr h xinc of purragy Nah quilibria or on h pric of abiliy wr known for wighd Shaply nwork dign gam. 1.4 Our Rul In hi papr, w giv h fir gnral rul for wighd Shaply nwork dign gam. W h ag for our work in Scion 3 by xhibiing uch a gam wih no pur-ragy Nah quilibrium. Thi xampl ha only hr playr, mploy a ingl-ink undircd nwork, and h raio bwn h maximum and minimum playr wigh can b mad arbirarily mall. (Pur-ragy Nah quilibria ar known o xi in all wighd Shaply nwork dign gam wih wo playr [2].) Thu hr ar no larg cla of wighd Shaply nwork dign gam ha alway po pur-ragy Nah quilibria byond ho idnifid in [2]. Our xampl moiva conidring a largr cla of quilibria o rcovr a guaran ha quilibria xi. Onc xinc ha bn ablihd, w can hn amp o bound h pric of abiliy wih rpc o hi largr of quilibria. Thr ar vral poibl approach o accomplihing hi goal, and w compar h a lngh in h nx ubcion. In hi papr, w puru h am lin of inquiry a in Anhlvich al. [3] whr for a diffrn bu rlad nwork dign gam, pur-ragy Nah quilibria did no ncarily xi and conidr approxima pur-ragy Nah quilibria. An oucom i an α-approxima Nah quilibrium if no playr can dcra i co by mor han an α muliplicaiv facor by dviaing. Th obviou goal i hn o prov ha α-approxima Nah quilibria alway xi and ha om uch quilibrium ha co wihin a β facor of opimal, whr α and β ar a mall a poibl. Sinc h wo paramr work again ach ohr, w k o undrand mor gnrally h inracion bwn h b-poibl valu of α and β. How much abiliy mu w giv up in ordr o achiv a low-co oluion, and vic vra? I i poibl o ak on or boh of α, β o b an abolu conan? Th prn papr i h fir o udy h rad-off curv for h wo paramr in Shaply nwork dign gam. Our main rul giv a compl oluion o h quion. To dcrib hm, cal playr wigh o ha h minimum playr wigh i 1, and l w max and W dno h maximum wigh and h um of all wigh, rpcivly. On h poiiv id, w how ha vry wighd Shaply nwork dign gam admi an O(log w max )-approxima Nah quilibrium, and ha h pric of abiliy wih rpc o uch quilibria i O(logW). Mor gnrally, w prov h following rad-off bwn h wo objciv: for vry α = Ω(log w max ), h pric of abiliy wih rpc o O(α)-approxima Nah quilibria i O((log W)/α). Thu o implmn a nwork wih co wihin a conan facor of h opimal oluion, i uffic o rlax h quilibrium conrain by a logarihmic (in W) facor. Thi i a nw rul vn for unwighd Shaply nwork dign gam. (Rcall ha in unwighd gam, i i impoibl o approxima h co o wihin an o(log k) facor wihou rlaxing h quilibrium conrain [2].) On h ngaiv id, w dmonra ha hi rad-off curv i vry clo o h b poibl. In our mo involvd conrucion, w xhibi a family of wighd Shap- 4

5 ly nwork dign gam wihou o(log w max / log log w max )-approxima Nah quilibria. Rcovring h xinc of quilibria hrfor rquir rlaxing h quilibrium conrain by a upr-conan (hough only logarihmic) facor. W alo how ha for vry α = Ω(log w max / loglog w max ), a pric of abiliy of O((log W)/α) can only b obaind by rlaxing h quilibrium conrain by an Ω(α) facor. 1.5 Dicuion of Alrnaiv Approach W conclud h Inroducion by juifying our dciion o focu on α-approxima purragy Nah quilibria and by dicuing hr alrnaiv way of rlaxing h problm. Fir, w could ignor h non-xinc of pur-ragy Nah quilibria and prov bound on h pric of abiliy for inanc in which uch quilibria do xi. Thi approach ha bn uccfully applid o bounding h pric of anarchy in wighd unpliabl lfih rouing gam [5, 9], which do no alway po pur-ragy Nah quilibria [23]. Unforunaly, for wighd Shaply nwork dign gam, a conqunc of our conrucion i ha no ublinar bound on h pric of abiliy i poibl in h paramr rang whr pur-ragy Nah quilibria nd no xi. Prcily, w how in Scion 4.4 ha for vry funcion f(x) = o(log x/ log log x), hr i a family of wighd Shaply nwork dign gam in which f(w max )-approxima Nah quilibria xi, bu all uch quilibria hav co an Ω(W) facor im ha of opimal. Scond, w could udy h rcn noion of ink quilibria du o Goman, Mirrokni, and Va [13]. A ink quilibrium of a gam i a rongly conncd componn wih no ougoing arc in h b-rpon graph of h gam (whr nod corrpond o oucom, arc o b-rpon dviaion by playr). No ha onc a qunc of b-rpon dviaion lad o a ink quilibrium, i will nvr again cap i. Sink quilibria alway xi, alhough hy can b xrmly larg (uch a h nir b-rpon graph). Th ocial valu (or co) of a ink quilibrium i dfind in [13] a h xpcd valu of a random a, whr h xpcaion i ovr h aionary diribuion of a random walk in h dircd graph corrponding o h quilibrium. Whil ink quilibria ar a wll-moivad concp and mak analy of h pric of anarchy mor robu and raliic (and hi wa h moivaion in [13]), i i no clar ha hy ar rlvan o pric of abiliy analy, whr w nviion a ingl oluion bing propod o playr a a low-co, abl oucom. No in paricular ha a ink quilibrium offr no guaran o an individual playr xcp for a rivial on: if a nod i rachd via a b-rpon dviaion by ha playr, hn of cour i will no wan o dvia again. Unforunaly, hi i mall conolaion o a playr ha pnd mo of i im in undirabl a whil ohr playr ak hir urn prforming hir own b-rpon dviaion. Third, and prhap mo obviouly, w could udy mixd-ragy Nah quilibria, whr ach playr can randomiz ovr i pah o minimiz i xpcd co. Evry wighd Shaply nwork dign gam admi a la on mixd-ragy Nah quilibrium by Nah Thorm [2]. A wih ink quilibria, howvr, i i no clar how o inrpr mixd-ragy quilibria in h conx of h pric of abiliy of nwork dign ( alo h dicuion in [3]). For xampl, a mixd-ragy Nah quilibrium could randomiz 5

6 only ovr oucom ha ar no α-approxima Nah quilibria for any raonabl valu of α, lading only o ralizaion ha would b xrmly difficul o nforc. On poibl oluion would b o implmn om yp of conrac binding h playr o h ralizaion of a mixd-ragy Nah quilibrium. Onc nforcabl conrac ar aumd, howvr, i i arguably mor raliic o imply build a nar-opimal nwork and approprialy ranfr paymn from playr incurring mall co o ho incurring larg co. Finally, if on ini on making aumpion ha cau mixd-ragy Nah quilibria o b raliically implmnabl, hn w advoca corrlad quilibria [4] a a mor uiabl candida for pric of abiliy analy. Corrlad quilibria ar no hardr o juify han mixd-ragy Nah quilibria for h pric of abiliy of nwork dign. Morovr, inc hy form a convx conaining all mixd-ragy Nah quilibria, hy m likly o b boh mor powrful and mor analyically racabl. W no ha h infficincy of corrlad quilibria in diffrn applicaion ha largly rid analyi o far (hough [8]), and lav hi dircion opn for fuur rarch. 2 Th Modl W now brifly formaliz h modl of nwork dign wih lfih playr ha w oulind in h Inroducion. A wighd Shaply nwork dign gam i a dircd graph G = (V, E) wih k ourc-ink pair ( 1, 1 ),...,( k, k ), whr ach pair ( i, i ) i aociad wih a playr i ha ha a poiiv wigh w i. By caling, w can aum ha min i w i = 1. Finally, ach dg E ha a nonngaiv co c. Th ragi for playr i ar h impl i - i pah P i in G. An oucom of h gam i a vcor (P 1,...,P k ) of pah wih P i P i for ach i. For a givn oucom and a playr i, h co har π i of an dg P i i c w i /W, whr W = j : P j w j i h oal wigh of h playr ha lc a pah conaining. Th co o playr i in an oucom i h um of i co har: c i (P 1,...,P k ) = P i π i. An oucom (P 1,...,P k ) i a (pur-ragy) Nah quilibrium if, for ach i, P i minimiz c i ovr all pah in P i whil kping P j fixd for j i. An oucom (P 1,...,P k ) i an α-approxima Nah quilibrium if for ach i, c i (P 1,...,P i,...,p k ) α c i (P 1,...,P i,...,p k) for all P i P i. Th co C(P 1,..., P k ) of an oucom (P 1,..., P k ) i dfind a i P i c. Th pric of abiliy of a gam ha ha a la on Nah quilibrium i C(N)/C(O), whr N i a Nah quilibrium of minimum-poibl co and O i an oucom of minimum-poibl co. Th pric of abiliy of α-approxima Nah quilibria i dfind analogouly. Finally, w will omim u h xprion (α, β)-approxima Nah quilibrium o man an oucom ha i an α-approxima Nah quilibrium and ha ha co a mo a β facor im ha of opimal. 6

7 3 Non-Exinc of Equilibria wih Wighd Playr In hi cion, w prov ha wighd Shaply nwork dign gam nd no po a pur-ragy Nah quilibrium. Propoiion 3.1 Thr i a 3-playr wighd Shaply nwork dign gam ha admi no pur-ragy Nah quilibrium. Morovr, h undrlying nwork i undircd wih a ingl ink, and h raio bwn h maximum and minimum playr wigh can b mad arbirarily mall. Rcall ha Anhlvich al. [2] provd ha vry wo-playr wighd Shaply nwork dign gam ha a pur-ragy Nah quilibrium. Proof of Propoiion 3.1: W fir prn a dircd nwork wih no pur-ragy Nah quilibrium and hn dcrib how o convr i ino an undircd xampl. Th dircd vrion i hown in Figur 1. L G dno hi graph and w > 1 a paramr. Th playr wih ourc 1, 2, and 3 hav wigh w 2, 1, and w, rpcivly. All hr playr har a common ink. Co for h dg of G ar dfind a in Tabl 1, whr w aum ha ǫ > i much mallr han 1/w 3. S S S T Figur 1: A hr-playr wighd Shaply nwork dign gam wih a ingl-ink nwork and no pur-ragy Nah quilibrium. L c i dno h co of dg i. Our argumn will rly on h following wo chain of inqualii, which follow from our choic of dg co: and c 5 w 2 w c 9 c 6 + c 9 w 2 w > c 7 > c 5 + c 9 w w 2 + w + 1 > c 8 > c 6 w w c 9 w 2 w 2 + w + 1 ; (1) w w + 1. (2) 7

8 Edg Co Edg Co Edg Co 1 2 3ǫ w 3 /(w 2 + w + 1) ǫ 6 w 3 /(w 2 + w + 1) + ǫ 7 [(w 3 + w 2 )/(w 2 + w + 1)] 8 [(w 3 + w)/(w 2 + w + 1)] 9 1 [ǫ(2w 2 + 1)/(2w 2 + 2)] +[ǫ(2w + 1)/(2w + 2)] Tabl 1: Edg co for h graph G in Propoiion 3.1. (For h radr who wih o vrify h, w ugg iniially aking w = 2.) Now uppo for conradicion ha a (pur-ragy) Nah quilibrium xi in G. Suppo furhr ha h cond playr u h pah in hi quilibrium. Th fir half of h inqualiy (2) impli ha h hird playr mu b uing h on-hop pah 8 (i would har dg 6 wih no ohr playr, and in h b ca would har dg 9 wih boh of h ohr playr). Th fir half of inqualiy (1) hn impli ha h fir playr mu u h on-hop pah 7. Bu hn h cond playr would prfr h pah 3 6 9, conradicing our iniial aumpion. Similarly, if h cond playr u h pah in a Nah quilibrium, hn h cond half of inqualiy (2) impli ha h hird playr mu b uing h pah Th cond half of inqualiy (1) hn impli ha h fir playr mu u Sinc hi would cau h pah o b prfrabl o h cond playr, w again arriv a a conradicion. Thr i hu no Nah quilibrium in hi wighd Shaply nwork dign gam. To convr hi dircd xampl ino an undircd on, imply mak all of h dg undircd and add a larg conan M >> w 3 o h co of h dg 1, 2, 3, 4, 7, and 8. Th co of vry pah in h original dircd nwork incra by xacly M; h co of nw pah ar a la 2M. A long a M i ufficinly larg, no playr will u on of h nw undircd pah in an quilibrium, and all of h argumn for h dircd nwork carry ovr wihou chang. 4 Low-Co Approxima Nah Equilibria: Lowr Bound In hi cion w prn ngaiv rul on h xinc and pric of abiliy of α- approxima Nah quilibria in wighd Shaply nwork dign gam. W a our lowr bound on h faibl rad-off bwn co and abiliy in Subcion 4.1. Th chnical har of hi lowr bound i Subcion 4.3, whr w conruc wighd Shaply nwork dign gam wihou o(log w max / log log w max )-approxima Nah quilibria. To illura our main ida, w prn a implr vrion of hi conrucion in Subcion 4.2. Finally, Scion 4.4 prov ha vn whn o(log w max / log log w max )-approxima Nah quilibria xi, uch quilibria can hav arbirarily larg co. W will giv narly maching poiiv rul in Scion 5. 8

9 4.1 Lowr Bound for Trading Sabiliy for Co Th goal of hi cion i o ablih h following lowr bound on h faibl radoff bwn h abiliy and h co of approxima Nah quilibria: for vry α = Ω(log w max / loglog w max ), a pric of abiliy of O((log W)/α) can b achivd only by rlaxing quilibrium conrain by an Ω(α) facor. Prcily, w will prov h following. Thorm 4.1 L f and g b wo bivaria ral-valud funcion, incraing in ach argumn, uch ha vry wighd Shaply nwork dign gam wih maximum playr wigh w max and um of playr wigh W admi an f(w max, W)-approxima Nah quilibrium wih co no mor han a (1 + g(w max, W)) facor im ha of opimal. Thn: (a) for om conan c, for all W w max 1; (b) for om conan c, for all W w max 1. log w max f(w max, W) c log log w max f(w max, W) g(w max, W) c log W A w will in h nx cion, Thorm 4.1 i opimal up o a doubly logarihmic facor in par (a). 4.2 Nwork Wihou Approxima Nah Equilibria Our proof of Thorm 4.1 i fairly chnical. To inroduc h main ida in h proof, w fir brifly dcrib a implr family of nwork. Th nwork can b ud o dfin wighd Shaply nwork dign gam wihou (2 ǫ)-approxima Nah quilibria, whr ǫ > i arbirarily mall. Sinc proving hi fac i no ay, w dicu only h conrucion. In Scion 4.3 w build on hi conrucion o prov Thorm 4.1. W conidr h nwork and ourc-ink pair hown in Figur 2. (Each ourc of h form,j corrpond o h ink.) In h figur, all ourc and ink hav only on incidn arc, xcp for and, which ach hav on incoming and on ougoing arc. Thr ar wo primary pah, which conain all of h dg on h lowr and uppr horizonal pah, rpcivly. Looly paking, ach playr choo bwn wo hor pah (on for ach primary pah), and long pah ha wrap around h nwork and inrc boh primary pah. For uiabl choic of dg co and playr wigh, long pah will no b ud in any approxima Nah quilibrium. Edg no on ihr primary pah hav co. W rfrain from prcily pcifying h co of ohr dg or h playr wigh; roughly, h formr quaniy incra xponnially whil h lar quaniy dcra xponnially from lf o righ in Figur 2. Th plan for proving ha h nwork do no hav approxima Nah quilibria i a follow. Th playr wih ourc-ink pair ( i, i ), which ha h larg wigh, mu choo 9

10 * i * i 1 1,1,2,n To * i i 1 i 2 i 1 3 2,1,2,n 1 * i * i 1 1,1,2,n To * Figur 2: A nwork wih no (2 ǫ)-approxima Nah quilibria. Th wo primary pah ar hown in bold. on of h primary pah. Thi dciion mak h dg on hi pah look chap o h ohr playr. Scond, whichvr primary pah h larg playr choo, i dciion mu cacad hrough h r of h playr. Third, h n playr wih ink hn wrap around o h ohr primary pah, which in urn cau h larg playr o wan o wich o h ohr primary pah, hrby prcluding any abl oucom. W giv a rigorou vrion of hi argumn for a mor complx nwork in h nx cion. 4.3 Nwork Wihou o(log w max / loglog w max )-Approxima Nah Equilibria W nx build on h conrucion in h prviou cion o how a nar-opimal lowr bound on h xinc of approxima Nah quilibria. Thorm 4.2 For vry funcion f(x) = o(log x/ log log x), hr i a family of wighd Shaply nwork dign gam ha do no admi f(w max )-approxima Nah quilibria a w max. Th high-lvl ida bhind h proof of Thorm 4.2 i imilar o ha of h prviou conrucion, wih uppr and lowr primary pah ha wrap around and cro ovr a hir nd. A bfor, only dg on h primary pah hav nonzro co and mo playr choo bwn hor pah on h uppr and lowr primary pah. Th ourc of amplificaion in h nw conrucion i ha, inad of having a qunc of playr wih xponnially dcraing wigh, w will u a group of playr in ach wigh cla. For ach ag of h nwork, hr will b a corrponding qunc of dg on ach of h primary pah inad of ju on. Th dail follow. S h paramr p o h quar of a ufficinly larg ingr. L α dno H p, whr H j = j l=1 1/l lnj i h jh Harmonic numbr; hi will b roughly our lowr bound on h approximaion facor ncary o guaran h xinc of approxima Nah quilibria. S i o 5 log 2 α +2 and n o 2p 2i α. W conidr a nwork ha compri i 1 ag ha ar conncd in ri. All ag bu h fir and la hav h rucur hown in Figur 3(a). Th fir and la ag ar dpicd in Figur 3(b) and (c), rpcivly. 1

11 Primary pah ar dfind a in h prviou cion. A pah i hor if i conain dg from only on of h primary pah, and i long ohrwi. W furhr claify a hor pah a uppr or lowr, dpnding on which of h wo primary pah i inrc. Th co of h dg ar: c( 2i ) = c( 2i ) = p 2i ; c( 2i 2 ) = c( 2i 2 ) = p 2i /α; c( 2i 1 ) = c( 2i 1 ) = 3p 2i α 3 ; c( 2j,l ) = c( 2j,l ) = 2 i j 1 p 2i /lα 2 for j = 1, 2,..., i 2 and l = 1, 2,..., p; c( 2j 1 ) = c( 2j 1 ) = 2 i j 1 p 2i+1 2 for j = 2, 3,..., i 1; c(,j ) = c(,j ) = α, for j = 1, 2,..., n; all ohr dg hav co. Th playr in h nwork gam ar a follow. Evry playr will b claifid a ihr vn, odd, or mall. Playr A 2i, A, and Ā (wih corrponding ourc-ink pair ( i, i ), (, ), and (, )) hav wigh p 2i. Playr A 2i i an vn playr; h ohr wo ar odd. For ach j = 3, 4,..., i 1 and l = 1, 2,..., p, hr i an vn playr A 2j,l wih wigh p 2j and ourc-ink pair ( 2j, 2j,l ). For ach j = 1, 2,..., i 1, hr ar wo odd playr A 2j+1 and A 2j+1 wih ourc-ink pair ( 2j+1, 2j+1 ) and ( 2j+1, 2j+1 ), rpcivly, and wih wigh p 2j+1. Thr ar vn playr A 4 and A 2 wih rpciv wigh p 4 and p 2 and rpciv ourc-ink pair ( 4, 4 ) and ( 2, 2 ). For ach l = 1, 2,..., n, hr i a mall playr A,l wih wigh 1 and ourc-ink pair (,l, ). W now giv h proof. Proof of Thorm 4.2: Conidr h wighd Shaply nwork dign gam dcribd abov, whr h paramr p i ufficinly larg. W bgin wih a fw prliminary obrvaion. Fir, h maximum and minimum playr wigh ar p 2i and 1, rpcivly. Thu w max = p Θ(log log p) whil α = H p = Θ(log w max / log log w max ). Scond, odd playr hav only on availabl (impl) pah. Third, h um W of h playr wigh i 3p 2i + i 1 i 1 p p 2j + 2 p 2j+1 + p 4 + p 2 + 2p 2i α 3p 2i α (3) j=3 for p ufficinly larg. W now ablih h following ix claim in urn. j=1 11

12 2k+3 2k+1 2k+1 2k+3 2k,1 2k,2 2k, p 2k+1 prviou ag 2k 2k+2,1 2k+2,2 2k+2,n nx ag 2k+3 2k,1 2k,2 2k, p 2k+1 2k+3 2k+1 2k+1 * (a) * 2i 1 2i 1 2i 2i 2 2i 1 2i 2i 2 2i 2i 4 nx ag 2i 2i 2 2i 1 * * 2i 1 2i 1 (b) ,1 3,1,2,n To * prviou ag 2 4,1,1,2,n 2 5 2,1 3,1,2,n To * (c) Figur 3: (a) Th rucur of h (i k)-h ag. (b) Th rucur of h fir ag. (c) Th rucur of h la ag. 12

13 (C1) In vry α/6-approxima Nah quilibrium, no mall playr u a pah ha conain boh of h dg 2i and 2i. (C2) In vry α/6-approxima Nah quilibrium, playr A 2i u a hor pah. (C3) In vry α/6-approxima Nah quilibrium, vry vn playr u a hor pah. (C4) In vry α/6-approxima Nah quilibrium in which playr A 2i u i lowr (uppr) hor pah, all of h vn playr alo u lowr (uppr) hor pah. (C5) In vry α/6-approxima Nah quilibrium in which playr A 2i u i lowr (uppr) hor pah, all of h mall playr u pah ha includ dg 2i ( 2i ). (C6) In vry α/6-approxima Nah quilibrium in which ach of h mall playr choo a pah ha conain 2i ( 2i ) bu no 2i ( 2i ), playr A 2i choo i uppr (lowr) hor pah. Sinc α/6 = Θ(log w max / log log w max ) and claim (C1), (C5), and (C6) canno imulanouly hold, claim (C1) (C6) imply h horm. To prov (C1), no ha if a mall playr A,l u a pah ha includ boh 2i and 2i, hn i ravr ihr dg 2i 1 or dg 2i 1. Sinc h co of ach of h dg i 3p 2i α 3, h playr incur co a la 3p 2i α 3 /W, whr W i h um of h playr wigh. (Rcall hi playr ha uni wigh.) By (3), hi i a la α 2. On h ohr hand, if h playr A,l choo a pah conaining only h non-zro co dg,j and 2i or,j and 2i, hn i incur co a mo α + p 2i /(1 + p 2i ) < α + 1 < 2α. (Th playr will har dg 2i or 2i wih h playr A or Ā, rpcivly.) Thu in vry α/6-approxima Nah quilibrium, no mall playr u a pah conaining boh 2i and 2i. Th proof of (C2) i imilar. If playr A 2i do no u a hor pah, hn i u a pah ha conain ihr dg 2i 1 or dg 2i 1 and incur co a la 3p 2i α 3 (p 2i /W) p 2i α 2. If i u a hor pah, hn i incurrd co i a mo p 2i (1 + 1/α) < 2p 2i. For claim (C3), w fir prov h arion for all vn playr of h form A 2j,l, by downward inducion on j. For h ba ca, conidr a playr A 2i 2,l for arbirary l {1, 2,..., p}. Playr A 2i mu u ihr dg 2i 2 or 2i 2. Th odd playr A 2i 1 and A 2i 1 mu occupy h dg 2i 1 and 2i 1. Thu, hr i a hor pah availabl o playr A 2i 2,l wih co a mo p 2i α ( ) ( ) p 2i 2 + 3p 2i α 3 p 2i 2 + p 2i 2 + p 2i p 2i 2 + p 2i 1 l m=1 2p 2i mα p2i 2 2 α + 3p2i 1 α 3 + 2p2i α 3p2i α for p ufficinly larg. On h ohr hand, vry long pah of playr A 2i 2,l conain ihr dg 2i 3 or dg 2i 3. By (C1) and (C2), h oal wigh on hi dg i a mo h wigh of h corrponding odd playr plu h oal wigh of h vn playr ohr han A 2i, which i a mo p 2i 3 + i 1 p p 2j + p 4 + p 2 2p 2i (3/2) (4) j=3 13

14 for p ufficinly larg. Thrfor, if playr A 2i 2,l choo a long pah, i incur co a la ( ) p 2i+(1/2) p 2i 2 > p2i p 2i 2 + 2p 2i (3/2) 2. (5) Inqualii (4) and (5) imply claim (C3) for playr of h form A 2i 2,l. For h induciv p, fix j {3, 4,..., i 2} and aum ha vry playr of h form A 2j,l wih j > j u a hor pah. Conidr a playr A 2j,l for om l {1, 2,..., p}. Arguing a in h ba ca, h playr can choo a hor pah and incur co a mo l 2 i j 1 p 2i + p 2i+(1/2) p 2i 4 mα 2 p 2i 4 + p + 2i j p 2i 2i 3 lα 2 m=1 2i j 1 p 2i α + p 2i (1/2) + 2i j p 2i α 2 < 2i j p 2i α, providd p i ufficinly larg. On h ohr hand, vry long pah conain ihr dg 2j 1 or dg 2j 1. By (C1), (C2), and h induciv hypohi, h oal wigh on hi dg i a mo p 2j 1 + j p p 2m + p 4 + p 2 2p 2j+(1/2) m=3 for p ufficinly larg. Thu, if playr A 2j,l choo a long pah, i incur co a la ( ) 2 i j 1 p 2i+(1/2) p 2j > 2 i j 2 p 2i. p 2j + 2p 2j+(1/2) Thi compl h induciv p. Finally, w ablih (C3) for playr A 4 and A 2. Givn ha (C3) hold for all ohr playr, playr A 4 ha a hor pah on which i would incur co a mo 2 i 3 p 2i + 2 i 2 p 2i (1/2) + 2i 2 p 2i α α 2 < 2i 2 p 2i α, whil vry long pah conain ihr 3 or 3 and cau h playr o incur co a la ( ) 2 i 2 p 2i+(1/2) p 4 > 2 i 3 p 2i+(1/2) p 4 + p 3 + p 2 for larg p. For playr A 2, prviou p imply ha on of h dg 2,1, 2,1 conain playr A 4 whil h ohr i unoccupid by ohr playr. Long pah conain boh of h, o if playr A 2 choo on of hm i incur co a la 2 i 2 p 2i+(1/2). On h ohr hand, hr i a hor pah wih co a mo 2 i 2 p 2i α i 2 p 2i (3/2) + 2p 2i α 2 < 2 i 2 p 2i α 2. Auming p i larg, hi impli ha playr A 2 mu choo a hor pah, compling h proof of (C3). 14

15 For (C4), by ymmry w can aum ha playr A 2i ak i lowr hor pah. W procd by conradicion. Among all vn playr ha choo an uppr hor pah, lc h lfmo on h on wih maximum indx j and, ubjc o hi, wih maximum indx l. Suppo hi playr i A 2j,l wih j 3. If j = i 1, hn h co o hi playr on i minimum-co lowr hor pah i a mo p 2i 2 α + 3p2i 1 α 3 + 2p2i lα < 4p2i 2 lα 2 for p larg. Th ky poin i hi: by (C3) and h dfiniion of l, h only playr ligibl for uing dg 2i 2 ar A 2j,1,...,A 2j,l. Thu, if A 2j,l choo an uppr hor pah, i incur co a la p 2i /lα, providing a conradicion. Similarly, if h playr i A 2j,l wih j {3, 4,..., i 2}, hn our choic of j nur ha h co o h playr on i minimum-co lowr hor pah i a mo 2 i j 1 p 2i 2 + p 2i (1/2) + 2i j p 2i < 2i j+1 p 2i, α lα 2 lα 2 whil our choic of l nur ha h co incurrd on vry uppr hor pah i a la 2 i j 1 p 2i /lα, a conradicion. Finally, uppo ha h lfmo vn playr chooing a hor uppr pah i A 4 or A 2. Th conradicion in h formr ca i nially h am a ha for h prviou ca of a playr A 2j,l wih j {3, 4,..., i 2}. In h lar ca, h co incurrd by playr A 2 on i lowr hor pah i a mo 2 i 2 p 2i i 2 p 2i (1/2) + 2p 2i α 2 < 4p 2i α 2 α 2 for p larg. Sinc dg 2,1 i occupid by no ohr playr, our choic of i = 5 log 2 α + 2 nur ha h co incurrd by playr A 2 on i uppr hor pah i a la 2 i 2 p 2i p 2i α 3, α 2 which compl h proof of (C4). W prov claim (C5) by conradicion. Aum ha playr A 2i choo i lowr hor pah. L l b h minimum indx for which playr A,l choo a pah conaining h dg 2i. By (C1) (C4) and our choic of l, hi playr incur h full α co of dg,l. On h ohr hand, if h playr choo h,l - pah conaining,l and 2i (and no ohr dg wih non-zro co), hn i har h formr dg wih playr A 2 and incur co a mo α 1 + p + p2i < p2i Thi compl h conradicion and h proof of (C5). Finally, aum h hypohi in claim (C6) hold. By (C1) (C3), if playr A 2i choo i lowr hor pah, hn i har dg 2i only wih playr A and incur co a la p 2i /2. On h ohr hand, h co of i uppr hor pah i ( ) ( ) p 2i p 2i + p2i p 2i < 2p2i 2p 2i + 2p 2i α α p 2i + p 2i 1 α, 15

16 which compl h proof of (C6) and of h horm. Wih Thorm 4.2 in hand, w can aily finih h proof of Thorm 4.1. Proof of Thorm 4.1: Par (a) follow immdialy from Thorm 4.2. Par (b) hold vn for h pcial ca of unwighd Shaply nwork dign gam and follow from a minor modificaion of an xampl in [2]. Spcifically, Anhlvich al. [2] prnd an unwighd Shaply nwork dign gam in which h minimum-co oluion ha co 1 + ǫ, whr ǫ > i arbirarily mall, and h uniqu Nah quilibrium ha co H k. Morovr, h wo oucom u dijoin dg. For ach fixd valu of W, w can ak hi xampl wih k = W playr and cal down h co of h dg ud by h Nah quilibrium by an f(1, W) facor. Thi yild an (unwighd) gam in which h only f(1, W)-approxima Nah quilibrium ha co Ω(log W/f(1, W)); hr i ill a oluion wih co 1 + ǫ. Thu f(1, W) g(1, W) = Ω(log W) for all W A Lowr Bound on h Pric of Sabiliy In hi cion, w mploy h nwork of Scion 4.3 o how ha, in addiion o h vaporaion of α-approxima Nah quilibria onc α = o(log w max / log log w max ), in inanc whr uch quilibria do xi, hir co can b xrmly high. Thi and in conra o rcn work on rouing gam [5], whr hr ar good uppr bound on h pric of anarchy vn in cla of nwork whr Nah quilibria ar no guarand o xi. Propoiion 4.3 For vry funcion f(x) = o(log x/ log log x), hr ar wighd Shaply nwork dign gam ha admi pur-ragy Nah quilibria, bu in which all f(w max )- approxima Nah quilibria hav co Ω(W) im ha of opimal. Proof: W adop h noaion ud in Scion 4.3. For vry funcion f(x) = o(log x/ log log x), w can find a ufficinly larg conan p uch ha h corrponding wighd Shaply nwork dign gam G conrucd in ha cion ha no f(w max )-approxima Nah quilibria. W no ha h um of h dg co in G i a mo p 2i+1, providd p i ufficinly larg. W hn conruc a nw nwork gam a follow. W ak m copi G 1, G 2,...,G m of G and rmov h playr A in ach of hm. A hown in Figur 4, w add h xra nod N 1, N 2,...,N m, N T and T. W add h following dg: on dg from N T o T wih co C = mp 4i ; for ach j = 1, 2,..., m, on dg from N j o T wih co C; for ach j = 1, 2,..., m, on zro-co dg from N j o G j copy of h vrx ; for ach j = 1, 2,..., m, on zro-co dg from G j copy of h vrx o N T. W alo add m playr B 1, B 2,...,B m. For ach j, h playr B j ha wigh p 2i and ourcink pair (N j, T). By conrucion, h only playr ha can u dg inid h nwork G j ar h playr inrnal o hi gam and h nw playr B j. 16

17 T C C C C N 1 N N 2 3 N m * * * * G 1 G 2 G 3 * * * * G m N T C To nod T Figur 4: A nwork in which all approxima Nah quilibria hav co far from opimal. Evry rcangl rprn a nwork of h yp dcribd in Scion 4.3. Fir uppo ha a playr B j connc o T via h vric and inrnal o G j. By h argumn in h proof of Thorm 4.2, in vry uch oucom, om playr inrnal o h nwork G j ha a dviaion ha dcra i co by mor han an f(w max ) facor; hr, B j i playing h rol of h dld playr A in h gam G j. Thu no uch oucom i an f(w max )-approxima Nah quilibrium of h gam. On h ohr hand, on uch oucom wih ach B j chooing h pah ha inrc h nwork G j ha co a mo C + mp 2i+1, providd p i ufficinly larg. Now uppo ha ach playr B j avoid G j and connc dircly o T. No playr of h form B j can profiably dvia, inc i would bar h full co of h dg from N T o T. Morovr, conidr h following ragi for h playr inrnal o a gam G j. Each vn playr of G j choo i minimum-co uppr hor pah. Each mall playr of G j choo i minimum-co pah ha includ h dg 2i bginning on h lowr primary pah and wrapping around o h uppr on. W claim ha in hi ca, no playr inrnal o G j ha an incniv o dvia. Thi claim i ay o for h vn playr. (Rcall odd playr only hav on availabl ragy.) No mall playr wan o dvia; inc playr B j i avoiding h gam G j, h dg 2i i unoccupid, and vry dviaion ha includ i would co a mall playr h full p 2i amoun. In concluion, h nwork gam admi a pur-ragy Nah quilibrium, bu vry f(w max )- approxima Nah quilibrium ha co a la mc, which i an Ω(m) facor im largr han h opimal co. Sinc m can b arbirarily larg, h propoiion follow. 17

18 5 Low-Co Approxima Equilibria: Uppr Bound In hi cion w prov our main poiiv rul, ha vry wighd Shaply nwork dign gam admi an approxima Nah quilibrium wih low co. Spcifically, w how ha for all α = Ω(log w max ), vry uch gam admi an O(α)-approxima Nah quilibrium wih co an O((logW)/α) im ha of opimal. (Rcall ha w max and W dno h maximum playr wigh and h um of h playr wigh, rpcivly.) In paricular, vry wighd Shaply nwork dign gam po an O(log W)-approxima Nah quilibrium wih co a mo a conan im ha of opimal. Thi i a nw rul vn for unwighd Shaply nwork dign gam. A a high lvl, our proof i bad on h ponial funcion mhod ha ha bn prviouly ud o bound h pric of anarchy and abiliy in a numbr of diffrn gam ( [24]). A ral-valud funcion Φ dfind on h oucom of a gam i a ponial funcion if, for vry playr i and vry poibl dviaion by ha playr, h chang in h valu of Φ qual h chang in playr i objciv funcion. Thu a ponial funcion rack ucciv dviaion by playr. In paricular, local opima of a ponial funcion ar prcily h pur-ragy Nah quilibria of h gam. Ponial funcion wr originally applid in noncoopraiv gam hory by Bckmann, McGuir, and Winn [7], Ronhal [22], and Mondrr and Shaply [17], in uccivly mor gnral ing, o prov h xinc of Nah quilibria. Ponial funcion can alo b ud o bound h pric of abiliy: if a gam ha a ponial funcion Φ ha i alway clo o h ru ocial co, hn a global opimum of Φ, or any local opimum rachabl from h min-co oucom via b-rpon dviaion, ha co clo o opimal. Indd, Anhlvich al. [2] provd boh h xinc of Nah quilibria and an H k uppr bound on h pric of abiliy in unwighd Shaply nwork dign gam uing a ponial funcion. Propoiion 3.1 impli ha wighd Shaply nwork dign gam do no gnrally admi a ponial funcion ( alo [2]). W nonhl how ha ida from ponial funcion can b ud o driv a narly opimal abiliy v. co rad-off for approxima Nah quilibria of wighd Shaply nwork dign gam. Th iniial ida i impl: w idnify an approxima ponial funcion, which dcra whnvr a playr dvia and dcra i co by a ufficinly larg facor. Thi argumn will imply h xinc of an O(log w max )-approxima Nah quilibrium wih co wihin an O(log W) facor of opimal in vry wighd Shaply nwork dign gam. Exnding hi argumn o obain a abiliy v. co rad-off rquir furhr work. Th raon i ha w will u a common approxima ponial funcion for all poin on h rad-off curv, and hi ponial funcion can ovrima h ru co by a much a a Θ(log W) facor. Thi funcion hrfor m incapabl of proving an o(log W) approximaion facor for h co, vn if w rlax quilibrium conrain by a larg facor. W ovrcom hi problm by mor carfully conidring how xra co i incurrd hroughou b-rpon dynamic aring from a minimum-co oucom. Mor prcily, w how ha a w incra h rlaxaion facor on h quilibrium conrain, h allowabl brpon dviaion lad o mor rapid dcra in h valu of our approxima ponial funcion. Th formal amn i a follow (cf., Thorm 4.1). 18

19 Thorm 5.1 L f and g b wo bivaria ral-valud funcion aifying: (a) (b) f(w max, W) 2 log 2 [(1 + w max )] (6) for all W w max 1; and f(w max, W) g(w max, W) 2 log 2 (1 + W) for all W w max 1. Thn vry wighd Shaply nwork dign gam wih maximum playr wigh w max and um of playr wigh W admi an f(w max, W)-approxima Nah quilibrium wih co a mo (1 + g(w max, W)) im ha of opimal. Bfor proving h horm, w ablih om prliminary rul. Fac 5.2 L x and y b ral numbr, and uppo ha y 1 and ha x = or x 1. Thn: (a) log 2 (1 + x + y) log 2 (1 + x) y x+y ; and (b) log 2 (1 + x + y) log 2 (1 + x) < log 2 [(1 + y)] y x+y. Proof: For boh par, w will u h fac ha (1 + 1 x )x approach monoonically from blow a x. For par (a), fir no ha if x and y 1 + x, hn h inqualiy hold: h righ-hand id i a mo 1 whil h lf-hand id qual log 2 (1 + y ) 1. So 1+x uppo ha y < 1 + x; hn ( 1 + y ) x+y ( y 1 + y )1+x y x 1 + x Raiing boh id of hi inqualiy o h y/(x+y) powr and hn aking logarihm (ba 2) vrifi h claim. For par (b), w hav ( 1 + y ) x+y y 1 + x = < ( 1 + y 1 + x ( 1 + y 1 + x (1 + y). ) 1+x y ) 1+x y ( 1 + y 1 + x ) ( 1 + y 1 + x A bfor, raiing boh id of hi inqualiy o h y/(x + y) powr and hn aking logarihm (ba 2) vrifi h claimd inqualiy. W nx conidr h xinc of approxima Nah quilibria wihou worrying abou hir co. 19 ) y 1 y

20 Lmma 5.3 For vry funcion f(w max, W) aifying f(w max, W) log 2 [(1 + w max )] (7) for all W w max 1, vry wighd Shaply nwork dign gam admi an f(w max, W)- approxima Nah quilibrium. Proof: W dfin an approxima ponial funcion Φ for a wighd Shaply nwork dign gam a follow: for an oucom (P 1,..., P k ) of h gam, dfin Φ(P 1,..., P k ) = E c log 2 (1 + W ), whr W = j : P j w j. Call a dviaion by a playr from on oucom o anohr α- improving if h dviaion dcra h co incurrd by h playr by a la an α muliplicaiv facor. Thu α-approxima Nah quilibria ar ho oucom from which no α-improving dviaion xi. Sinc hr ar a fini numbr of oucom, w can prov h lmma by howing ha f(w max, W)-improving dviaion ricly dcra h approxima ponial funcion Φ. Conidr an α-improving dviaion of playr i from h oucom (P 1,...,P k ), ay o h pah Q i, whr α qual f(w max, W). W aum ha P i and Q i ar dijoin; if hi i no h ca, h following argumn can b applid o P i \ Q i and Q i \ P i inad. By h dfiniion of α-improving, w hav Q i c w i W + w i 1 c wi, (8) f(w max, W) W P i whr W = j : P j w j dno h oal wigh on dg bfor playr i dviaion. W can hn driv h following: Φ = Q i c [log 2 (1 + W + w i ) log 2 (1 + W )] c [log 2 (1 + W ) log 2 (1 + W w i )] (9) P i < [ ] w i c log 2 [(1 + w i )] c wi (1) W Q + w i W i P i log 2 [(1 + w max )] Q i c. w i W + w i P i c wi W c wi f(w max, W) log 2 [(1 + w max )] W P f(w max, W) i (11) In hi drivaion, h qualiy (9) follow from h dfiniion of Φ; h inqualiy (1) follow from Fac 5.2, wih Fac 5.2(b) applid o ach rm in h fir um wih x = W 2

21 and y = w i, and Fac 5.2(a) applid o ach rm in h cond um wih x = W w i and y = w i ; inqualiy (11) follow from (8); and h final inqualiy follow from aumpion (7). W now xnd h argumn in h proof of Lmma 5.3 o accoun for h co of approxima quilibria. Proof of Thorm 5.1: Conidr a maximal qunc of f(w max, W)-improving dviaion ha bgin in a minimum-co oucom wih co C. By Lmma 5.3, hi qunc i fini and rmina a a f(w max, W)-approxima Nah quilibrium. Conidr a dviaion in hi qunc by a playr i from a pah P i o a pah Q i, and l A dno h co of h dg of Q i ha wr prviouly vacan (i.., ud by no playr). W hn hav Φ c wi f(w max, W) log 2 [(1 + w max )] (12) W P f(w max, W) i 1 c wi (13) 2 W P i 1 2 A f(w max, W), (14) whr inqualiy (12) i h am a inqualiy (11) in h proof of Lmma 5.3; inqualiy (13) follow from aumpion (6); and inqualiy (14) follow from h fac ha h co incurrd by playr i bfor i dviaion i a la f(w max, W) im h co i incur afr h dviaion, which i a la h um A of h co of h prviouly vacan dg. Hnc, in h maximal qunc of f(w max, W)-improving dviaion, whnvr h co of h currn oucom incra by an addiiv facor of A, h ponial funcion Φ dcra by a la A f(w max, W)/2. By h dfiniion of Φ, h ponial funcion valu of h opimal oucom i a mo a log 2 (1 + W) muliplicaiv facor largr han i co C. Morovr, h ponial funcion i alway nonngaiv and only dcra hroughou h qunc of dviaion. Thrfor, h co only incra by a 2C log 2 (1+W)/f(w max, W) addiiv facor hroughou h nir qunc of dviaion. Th qunc hu rmina in a (f(w max, W), 1 + (2 log 2 (1 + W)/f(w max, W)))-approxima Nah quilibrium. Rmark 5.4 Our proof of Thorm 5.1 i qui flxibl and carri ovr o xnion known for h unwighd ca [2]. For xampl, Thorm 5.1 and i proof hold for congion gam (whr h ragy of a playr i an arbirary collcion of ub of a ground ) and for concav (inad of conan) dg co. 6 Fuur Dircion Th prn papr giv an nially igh analyi of h faibl rad-off bwn h abiliy and co of approxima Nah quilibria in Shaply nwork dign gam. On h ohr hand, h corrponding rad-off curv in vral naural pcial ca i no 21

22 wll undrood. For xampl, wha ar h faibl rad-off in undircd nwork? In ingl-ink nwork? Or whn hr i only a mall numbr of diinc playr wigh? Acknowldgmn W hank h anonymou rfr for hir commn. Rfrnc [1] S. Albr, S. Eil, E. Evn-Dar, Y. Manour, and L. Rodiy. On Nah quilibria for a nwork craion gam. In Procding of h 17h Annual ACM-SIAM Sympoium on Dicr Algorihm (SODA), pag 89 98, 26. [2] E. Anhlvich, A. Dagupa, J. Klinbrg, É. Tardo, T. Wxlr, and T. Roughgardn. Th pric of abiliy for nwork dign wih fair co allocaion. In Procding of h 45h Annual Sympoium on Foundaion of Compur Scinc (FOCS), pag , 24. [3] E. Anhlvich, A. Dagupa, É. Tardo, and T. Wxlr. Nar-opimal nwork dign wih lfih agn. In Procding of h 35h Annual ACM Sympoium on Thory of Compuing (STOC), pag , 23. [4] R. J. Aumann. Subjciviy and corrlaion in randomizd ragi. Journal of Mahmaical Economic, 1(1):67 96, [5] B. Awrbuch, Y. Azar, and L. Epin. Th pric of rouing unpliabl flow. In Procding of h 37h Annual ACM Sympoium on Thory of Compuing (STOC), pag 57 66, 25. [6] V. Bala and S. Goyal. A non-coopraiv modl of nwork formaion. Economrica, 68(5): , 2. [7] M. J. Bckmann, C. B. McGuir, and C. B. Winn. Sudi in h Economic of Tranporaion. Yal Univriy Pr, [8] G. Chriodoulou and E. Kououpia. On h pric of anarchy and abiliy of corrlad quilibria of linar congion gam. In Procding of h 13h Annual Eurpoan Sympoium on Algorihm (ESA), pag 59 7, 25. [9] G. Chriodoulou and E. Kououpia. Th pric of anarchy of fini congion gam. In Procding of h 37h Annual ACM Sympoium on Thory of Compuing (STOC), pag 67 73, 25. [1] J. Corbo and D. C. Park. Th pric of lfih bhavior in bilaral nwork formaion gam. In Procding of h 24h ACM Sympoium on Principl of Diribud Compuing (PODC), pag 99 17,

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