Nwork Dign wih Wighd Playr SPAA 26 Full Papr Submiion) Ho-Lin Chn Tim Roughgardn March 7, 26 Abrac W conidr a modl of gam-horic nwork dign iniially udid by Anhlvich al. [1], 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. [1] provd ha pur-ragy Nah quilibria alway xi and ha h pric of abiliy h raio in co of a minimum-co Nah quilibrium and an opimal oluion i Θlog 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 α muliplicaiv facor. Our fir poiiv rul i ha Olog 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 radoff bwn h wo objciv of good abiliy and low co: for vry α = Ωlog w max ), h pric of abiliy wih rpc o Oα)-approxima Nah quilibria i Olog W )/α), whr W i h um of h playr wigh. In paricular, hr i alway an Olog 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 olog w max / log log w max )-approxima Nah quilibria, and how ha for all α = Ωlog w max / log log w max ), achiving a pric of abiliy of Olog W/α) rquir rlaxing quilibrium conrain by an Ωα) facor. Dparmn of Compur Scinc, Sanford Univriy, 393 Trman Enginring Building, Sanford CA 9435. Rarch uppord in par by NSF Award 323766. Email: holin@anford.du. Dparmn of Compur Scinc, Sanford Univriy, 462 Ga Building, Sanford CA 9435. Suppord in par by by ONR gran N14-4-1-725, DARPA gran W911NF-4-9-1, and an NSF CAREER Award. Email: im@c.anford.du.
1 Inroducion 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. [5, 12, 13] 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 [9]. 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 [14], 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 [22] and wa o-calld in Anhlvich al. [1]. Th pric of abiliy ha primarily bn udid in nwork dign gam [1, 2], 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 incanly prvn h nwork ur o 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 can b rgardd a h ncary dgradaion in h oluion qualiy caud by impoing h gam-horic conrain of abiliy. 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 [1] and iniially xplord indpndnly by Anhlvich al. [2] and Fabrikan al. [1]. 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 anarchy ), h cond h pric of anarchy. Clo o h prn work i a variaion on h modl of [2] ha wa propod and udid by Anhlvich a. [1], which hy calld nwork dign wih Shaply co haring and w will abbrvia o Shaply nwork dign gam. Th mo baic modl conidrd in [1] 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 aumd o b hard among h playr in h following way. Fir, if dg li in f of h chon pah, hn 1
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. Of all h way o har h ocial co among h playr, hi 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. [16]); and, a hown in [1], i coax bnign bhavior from h playr. Spcifically, Anhlvich al. [1] howd ha a pur-ragy Nah quilibrium alway xi unlik wih h mor gnral co-haring ha wa allowd in h prdcor modl [2] and ha h pric of abiliy undr Shaply co-haring i a mo h kh harmonic numbr H k = Olog k), whr k i h numbr of playr. Anhlvich al. [1] alo providd an xampl howing ha hi uppr bound i h b poibl, and provd numrou xnion. Shaply Co Sharing wih Wighd Playr A naural and imporan xnion ha Anhlvich al. [1] 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 could ari for many raon. For xampl, ach playr could rprn h raffic of a larg populaion, uch a h cuomr of an Inrn Srvic Providr, and all uch populaion 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. Bu 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. [1] 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 Ωk +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 [1] for wighd gam concrnd only h pcial ca of 2-playr gam and of ingl-commodiy gam whr all playr hav boh h am ourc and h am ink). No furhr poiiv or ngaiv rul on ihr h xinc of pur-ragy Nah quilibria or on h pric of abiliy wr known for wighd Shaply nwork dign gam. 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. Purragy Nah quilibria ar known o xi in all wighd Shaply nwork dign gam wih wo playr [1].) Thu hr ar no larg cla of wighd Shaply nwork dign gam ha alway po pur-ragy Nah quilibria byond ho idnifid in [1]. 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 2
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. [2] whr for a diffrn bu rlad nwork dign gam, pur-ragy Nah quilibria did no ncarily xi and conidr approxima pur-ragy Nah quilibria. W ay ha 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 mor gnrally undrand 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? 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 Olog w max )-approxima Nah quilibrium, and ha h pric of abiliy wih rpc o uch quilibria i Olog W ). Mor gnrally, w prov h following rad-off bwn h wo objciv: for vry α = Ωlog w max ), h pric of abiliy wih rpc o Oα)-approxima Nah quilibria i Olog 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 olog k) facor wihou rlaxing h quilibrium conrain [1].) 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 Shaply nwork dign gam wihou olog 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 = Ωlog w max / log log w max ), a pric of abiliy of Olog W )/α) can only b obaind by rlaxing h quilibrium conrain by an Ωα) facor. Dicuion of Alrnaiv Approach W conclud h Inroducion by juifying our dciion o focu on α-approxima pur-ragy 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 rcnly bn uccivly applid o bounding h pric of anarchy in wighd unpliabl lfih rouing gam [4, ], which do no alway po pur-ragy Nah quilibria [11, 2]. 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 will how in h full vrion of h papr ha for vry funcion fx) = olog x/ log log x), hr i a family of wighd Shaply nwork dign gam in which fw max )-approxima Nah quilibria xi, bu all uch quilibria hav co an ΩW ) facor im ha of opimal. Scond, w could udy h rcn noion of ink quilibria du o Goman, Mirrokni, and Va [11]. 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 brpon dviaion by playr). No ha onc a qunc of b-rpon dviaion lad o 3
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 [11] 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 [11]), 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 lowco, 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 [17]. 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 [2]). For xampl, a mixd-ragy Nah quilibrium could ponially randomiz 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 naropimal 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 [3] 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 [7]), 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. 4
Th co of an oucom P 1,..., P k ), dnod by CP 1,..., P k ), i i P i c. Th pric of abiliy of a gam ha ha a la on Nah quilibrium i CN)/CO), 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. 3 Non-Exinc of Nah Equilibria wih Wighd Playr In hi cion, w prov ha wighd Shaply nwork dign gam nd no po a purragy Nah quilibrium. Propoiion 3.1 Thr i a 3-playr wighd Shaply nwork dign gam ha admi no purragy 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. [1] 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 ɛ > i much mallr han 1/w 3. S S S 1 1 2 3 2 3 4 7 5 6 9 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: c 5 w 2 w 2 + 1 + c 9 w 2 w 2 + 1 > c 7 > c 5 + c 9 w 2 w 2 + w + 1 ; 1) 5
Edg Co Edg Co Edg Co 1 2 3ɛ 3 4 5 w 3 /w 2 + w + 1) ɛ 6 w 3 /w 2 + w + 1) + ɛ 7 [w 3 + w 2 )/w 2 + w + 1)] [w 3 + w)/w 2 + w + 1)] 9 1 [ɛ2w 2 + 1)/2w 2 + 2)] +[ɛ2w + 1)/2w + 2)] Tabl 1: Edg co for h graph G in Propoiion 3.1. and c 6 + c 9 w w 2 + w + 1 > c > c 6 w w + 1 + c 9 w w + 1. 2) 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 han h cond playr u h pah 2 5 9 in hi quilibrium. Th fir half of h inqualiy 2) impli ha h hird playr mu b uing h on-hop pah 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 3 6 9 in a Nah quilibrium, hn h cond half of inqualiy 2) impli ha h hird playr mu b uing h pah 4 6 9. Th cond half of inqualiy 1) hn impli ha h fir playr mu u 1 5 9. Sinc hi would cau h pah 2 5 9 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. 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 olog w max / log log w max )-approxima Nah quilibria. W illura a implr vrion of hi conrucion in Subcion 4.2, which i nough o rul ou h xinc of 2 ɛ)-approxima Nah quilibria for arbirarily mall ɛ >. W will giv narly maching poiiv rul in Scion 5. 4.1 Lowr Bound for Trading Sabiliy for Co Th goal of hi cion i o ablih h following lowr bound on h faibl rad-off bwn h abiliy and h co of approxima Nah quilibria: for vry α = Ωlog w max / log log w max ), a pric of abiliy of Olog W )/α) can b achivd only by rlaxing quilibrium conrain by an Ωα) facor. Prcily, w will prov h following. 6
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 fw max, W )-approxima Nah quilibrium wih co a mo 1 + gw max, W )) 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. fw max, W ) c log w max log log w max fw max, W ) gw 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 2 ɛ)-approxima Nah Equilibria W now work oward Thorm 4.1 by giving nwork wihou α-approxima Nah quilibria for α arbirarily clo o 2. W fir dcrib h nwork, hn giv h inuiion bhind h conrucion, and hn giv h dail. W will conidr h nwork hown in Figur 2. In h figur, all ourc and ink hav only on incidn arc, xcp for and, which ach ha on incoming and on ougoing arc. Thr ar wo primary pah, dnod Q and Q, which conain all of h dg on h lowr and uppr horizonal pah, rpcivly. Thr will b vral paramr. W aum ha w ar givn an arbirarily mall poiiv numbr ɛ.1, wih h goal of xhibiing a nwork wih no 2 ɛ )-approxima Nah quilibria. W hn p = /ɛ, ɛ = ɛ /, i = log 1+ɛ 32p) + 2, and n = p i+1. W nx dicu h dg co. To minimiz ubcrip, in hi ubcion and h nx w will u c) o dno h co of an dg. Edg no on ihr primary pah hav co. Th dg co on h primary pah ar a follow: c i ) = cē i ) = pi 2 ; c j ) = cē j ) = pi 1 + ɛ)i j 1, for all j = 1, 2,..., i 1; c,j ) = cē,j ) = 1, for all j = 1, 2,..., n. Th rmaining dg on h primary pah hav co, a in Figur 2. Th playr ar a follow. Playr A i, A, and Ā wih corrponding ourc-ink pair i, i ),, ), and, )) hav wigh p i. For ach j = 1, 2,..., i 1, hr i a playr A j wih wigh p j and ourc-ink pair j, j ). Thr ar n mall playr A,1, A,2,..., A,n wih wigh 1. For vry j, h mall playr A,j ha ourc,j and ink. W can hn prov h following. 7
* i * i 1 1,1,2,n To * i i 1 i 2 i 1 2,1,2,n 3 1 i i 1 1,1,2,n * * To * Figur 2: A nwork wih no 2 ɛ)-approxima Nah quilibria. Thorm 4.2 For vry ɛ,.1), h wighd Shaply nwork dign gam abov ha no 2 ɛ )-approxima Nah quilibrium. In h proof of Thorm 4.2 w will formaliz h following ida. Fir, playr A i mu choo on of h primary pah, which in urn mak h dg on hi pah look chap o h ohr playr. Scond, whichvr primary pah A i choo, i dciion mu cacad hrough h r of h playr. Third, h n mall playr hn wrap around o h ohr primary pah, which in urn cau playr A i o wan o wich o h ohr primary pah, hrby prcluding any abl oucom. W now mak hi proof approach rigorou. Proof kch): W ar wih om rminology. A hor pah of a playr ha i no mall i a pah ha lav h playr ourc wih a hop o on of h primary pah, follow ha primary pah, and hn nd wih a final hop o h playr dinaion. A long pah of a playr ha i no mall i on ha conain dg of boh primary pah. No ha for vry playr ha i no mall, all of i pah ar ihr hor or long; all uch playr hav prcily wo hor pah, xcp for playr A and Ā, who ach hav on. For h proof, w will formaliz h following amn in urn. 1) In vry 2 ɛ )-approxima Nah quilibrium, no mall playr u a pah conaining j or ē j wih j {1, 2,..., i 1}. 2) In vry 2 ɛ )-approxima Nah quilibrium, playr A i u a hor pah. 3) In vry 2 ɛ )-approxima Nah quilibrium in which playr A i u i lowr uppr) hor pah, h playr A 1,..., A i 1 alo u hir lowr uppr) hor pah. 4) In vry 2 ɛ )-approxima Nah quilibrium in which playr A i u i lowr uppr) hor pah, all of h mall playr u pah ha includ h dg ē i i ). 5) In vry 2 ɛ )-approxima Nah quilibrium in which all of h mall playr u pah ha includ h dg ē i i ), playr A i u i uppr lowr) hor pah. Sinc 4) and 5) ar muually xcluiv, proving 1) 5) compl h proof of h horm. Du o pac conrain, w dfr furhr dail o Appndix A.1. 4.3 Nwork Wihou olog w max / log log w max )-Approxima Nah Equilibria W nx build on h conrucion in Thorm 4.2 o how a much rongr and nar-opimal) lowr bound on h xinc of approxima Nah quilibria.
Thorm 4.3 For vry funcion fx) = o log x log log x ), hr i a family of wighd Shaply nwork dign gam ha do no admi fw max )-approxima Nah quilibria a w max. Du o pac conrain, w only dcrib om of h inuiion bhind Thorm 4.3 and dfr all of h dail o h Appndix. Th high-lvl ida i imilar o h prviou conrucion, wih an 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 can choo bwn hor pah on h uppr and lowr primary pah. Th ourc of amplificaion in h nw conrucion i ha, inad of having on playr wih wigh p i for ach i a in h prviou xampl, w will u p playr wih wigh p 2i for ach i. For ach ag, hr will b p dg on ach of h main pah inad of ju on. Th rucur of h argumn ha hr i no α-approxima Nah quilibrium hn coni of vrifying h analogou amn 1) 5). Whil h proof i complicad by h largr valu of α and h incrad numbr of playr and pah, i i concpually vry imilar o h proof of Thorm 4.2. Th dail can b found in h Appndix. Wih Thorm 4.3 in hand, w can aily finih h proof of Thorm 4.1. Proof of Thorm 4.1: Par a) follow immdialy from Thorm 4.3. Par b) hold vn for h pcial ca of unwighd Shaply nwork dign gam and follow from a minor modificaion of an xampl in [1]. Spcifically, Anhlvich al. [1] prnd an unwighd Shaply nwork dign gam in which a minimum-co oluion ha co 1 and h uniqu Nah quilibrium ha co H k. Morovr, h wo oucom u dijoin dg. For ach fixd valu of W, w ak hi xampl wih k = W playr and cal down h co of h dg ud by h Nah quilibrium by a f1, W ) + ɛ facor. Thi yild an unwighd) gam in which h only f1, W )-approxima Nah quilibrium ha co Ωlog W/f1, W )) and h minimum-co oluion ill ha valu 1). Thu f1, W ) g1, W ) = Ωlog W ) for all W 1. 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 = Ωlog w max ), vry uch gam admi an Oα)-approxima Nah quilibrium wih co an Olog W )/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 Olog 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 [21]). 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 [6], Ronhal [19], and Mondrr and Shaply [15], 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. [1] provd boh 9
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 [1]). W nonhl how ha ida from ponial funcion can b ud o driv an nially 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 Olog w max )-approxima Nah quilibrium wih co wihin an Olog 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 radoff curv, and hi ponial funcion can ovrima h ru co by a much a a Θlog W ) facor. On h urfac, hi funcion hrfor m incapabl of proving an olog W ) bound on 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. Spcifically, w how ha a w incra h rlaxaion facor on h quilibrium conrain, h allowabl b-rpon dviaion lad o mor rapid dcra in h valu of our approxima ponial funcion. Roughly, hi allow u o prov ha vry qunc of uch dviaion nd ufficinly quickly, wihou accruing much addiional co. Prcily, w u h ida o prov h following rul cf., Thorm 4.1). Dail ar in h Appndix. Thorm 5.1 L f and g b wo bivaria ral-valud funcion aifying: a) for all W w max 1; and fw max, W ) 2 log 2 [1 + w max )] b) for all W w max 1. fw max, W ) gw max, W ) 2 log 2 1 + W ) Thn vry wighd Shaply nwork dign gam wih maximum playr wigh w max and um of playr wigh W admi an fw max, W )-approxima Nah quilibrium wih co a mo 1 + gw max, W )) im ha of opimal. Rmark 5.2 Our proof of Thorm 5.1 i qui flxibl and carri ovr o many of h xnion known for h unwighd ca [1]. For xampl, Thorm 5.1 coninu o 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. W dfr furhr dail o h full vrion. Rfrnc [1] 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 295 34, 24. 1
[2] 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 511 52, 23. [3] R. J. Aumann. Subjciviy and corrlaion in randomizd ragi. Journal of Mahmaical Economic, 11):67 96, 1974. [4] 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. [5] V. Bala and S. Goyal. A non-coopraiv modl of nwork formaion. Economrica, 6:111 1229, 2. [6] M. J. Bckmann, C. B. McGuir, and C. B. Winn. Sudi in h Economic of Tranporaion. Yal Univriy Pr, 1956. [7] G. Chriodoulou and E. Kououpia. On h pric of anarchy and abiliy of corrlad quilibria of linar congion gam. Manucrip, 25. [] 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. [9] P. Duby. Infficincy of Nah quilibria. Mahmaic of Opraion Rarch, 111):1, 196. [1] A. Fabrikan, A. Luhra, E. Manva, C. H. Papadimiriou, and S. J. Shnkr. On a nwork craion gam. In Procding of h 22nd ACM Sympoium on Principl of Diribud Compuing PODC), pag 347 351, 23. [11] M. X. Goman, V. Mirrokni, and A. Va. Sink quilibria and convrgnc. In Procding of h 46h Annual Sympoium on Foundaion of Compur Scinc FOCS), pag 142 151, 25. [12] M. O. Jackon. A urvy of modl of nwork formaion: Sabiliy and fficincy. In G. Dmang and M. Woodr, dior, Group Formaion in Economic; Nwork, Club, and Coaliion, chapr 1. Cambridg Univriy Pr, 25. [13] K. Jain and V. V. Vazirani. Applicaion of approximaion algorihm o coopraiv gam. In Procding of h 33rd Annual ACM Sympoium on Thory of Compuing STOC), pag 364 372, 21. [14] E. Kououpia and C. H. Papadimiriou. Wor-ca quilibria. In Procding of h 16h Annual Sympoium on Thorical Apc of Compur Scinc STACS), volum 1563 of Lcur No in Compur Scinc, pag 44 413, 1999. [15] D. Mondrr and L. S. Shaply. Ponial gam. Gam and Economic Bhavior, 141):124 143, 1996. [16] H. Moulin and S. Shnkr. Sragyproof haring of ubmodular co: Budg balanc vru fficincy. Economic Thory, 13):511 533, 21. [17] J. F. Nah. Equilibrium poin in N-pron gam. Procding of h Naional Acadmy of Scinc, 361):4 49, 195. 11
[1] C. H. Papadimiriou. Algorihm, gam, and h Inrn. In Procding of h 33rd Annual ACM Sympoium on Thory of Compuing STOC), pag 749 753, 21. [19] R. W. Ronhal. A cla of gam poing pur-ragy Nah quilibria. Inrnaional Journal of Gam Thory, 21):65 67, 1973. [2] R. W. Ronhal. Th nwork quilibrium problm in ingr. Nwork, 31):53 59, 1973. [21] T. Roughgardn. Ponial funcion and h infficincy of quilibria. In Procding of h Inrnaional Congr of Mahmaician ICM), 26. To appar. [22] A. S. Schulz and N. Sir Mo. On h prformanc of ur quilibria in raffic nwork. In Procding of h 14h Annual ACM-SIAM Sympoium on Dicr Algorihm SODA), pag 6 7, 23. A Miing Proof A.1 Miing Proof from Scion 4 A.1.1 Proof of Thorm 4.2 Proof of Thorm 4.2: W prov arion 1) 5) in h proof kch in urn. Bfor bginning, no ha h only impl) pah availabl o ach of A and Ā i i hor pah; hnc h wo playr u h dg i and ē i bu no ohr dg on h primary pah. For 1), fir no ha if a mall playr u an dg j or ē j for om j {1, 2,..., i 1}, hn i alo u ihr 1 or ē 1. W hrfor nd only prov 1) for j = 1. Fix a mall playr A,h. If hi playr u a hor pah a pah ha u only on dg from ach primary pah), i incur a co of a mo 1 + 1 p i 1 + p i 2 3 2, inc whil i migh pay for h nir uni) co of h dg,h or ē,h, i har h dg i or ē i wih a playr of wigh p i A or Ā, rpcivly). Suppo ha playr A,h inad u a pah ha conain 1 or ē 1. Thn h co incurrd by h playr i a la h co of hi dg dividd by h um W of all of h playr wigh rcall ha a mall playr ha uni wigh). Our paramr choic nur ha hi dg co i a la 4p i+1, whil W = p i+1 + 3p i + i 1 j=1 pj. Sinc i 2 4p i+1 4p 16) p i + 3p i 1 + p j, h co incurrd by h playr on hi pah i a la 4 16 p = 4 2ɛ. Thi i ricly grar han 2 ɛ ) 3 2, which ablih 1). For 2), uppo fir ha playr A i u a long pah. Such a pah mu includ dg 1 or ē 1. By par 1), h oal wigh of h playr uing hi dg i a mo p i + p i 1 + + p. Hnc h co incurrd by playr A i on uch a pah i a la 4p i+1 p i p i + p i 1 + + p, which i a la 2p i+1 inc w hav chon p ufficinly larg. On h ohr hand, if A i choo a hor pah, i co i l han 3 4 pi < 2p i+1 /2 ɛ ). j= 12
Arion 3) rquir h mo involvd argumn. Suppo A i u i lowr hor pah h argumn for h ohr ca i ymmric). Fir conidr playr A i 1. If i u i lowr hor pah, hn i har h fir dg i 1 ) wih playr A i and hnc incur co a mo p i 1 p i 1 + p i pi + pi 1 + ɛ) pi ) 1 p + 1 + ɛ = pi 1 + ɛ 4 If playr A i 1 u any ohr pah, i mu in paricular u h dg ē i 1 and ē i 2. Morovr, par 1) and 2) imply ha h only ohr playr ha could b uing h dg ar A i 2,..., A 1. Th co incurrd by A i 1 on uch a pah i hrfor a la p i 1 p i 1 + + p p i Sinc ) + pi 1 + ɛ) 1 ɛ ) p 1 p 2 + ɛ ) pi pi 2 + ɛ) = 1 ɛ > 2 ɛ ) 1 + ɛ 4 ), ). ) 2 + ɛ playr A i 1 will choo i lowr hor pah in vry 2 ɛ )-approxima Nah quilibrium in which playr A i choo i lowr hor pah. Th abov argumn hn appli inducivly o playr A i 2,..., A 2. For a gnric playr j, i co on i lowr hor pah givn ha A j+1 u i lowr hor pah) i a mo p j pi p j + pj+1 1 + ɛ)i j 1 + pi 1 + ɛ)i j < pi 1 + ɛ)i j 1 1 + ɛ 4 whil i co on vry ohr pah givn ha playr A i 1,..., A j+1 u hir lowr hor pah) i a la p j ) p i p j + + p 1 + ɛ)i j 1 + pi 1 + ɛ)i j pi 1 + ɛ)i j 1 1 ɛ ) 2 + ɛ ). A abov, hi impli ha A j will u i lowr hor pah. Finally, for playr A 1, h co of i lowr hor pah givn ha A 2 u i lowr hor pah) i a mo p p + p 2 pi 1 + ɛ)i 2 + n p i+1 + 4p i 1 + ɛ). Evry ohr 1-1 pah conain h dg ē 1 which, a a conqunc of h prviou p, i ohrwi unoccupid. Hnc, vry ohr 1-1 pah ha co a la 4p i+1. Sinc w hav chon p ufficinly larg, hi i ricly grar han 2 ɛ )p i+1 + 4p i 1 + ɛ)), and hnc A 1 will u i lowr hor pah, compling h proof of 3). For 4), fir conidr h mall playr A,1. Sinc par 1) 3) imply ha dg ē,1 i unoccupid xcp poibly for A,1, h playr will incur a co of 1 for uing a pah ha includ hi dg. On h ohr hand, if h playr u i lowr hor pah h pah conaining only,1, ē i, and zro-co dg), i har dg,1 wih playr A 1 and hrfor incur co a mo ) ) 1 1 1 + 1 + p 1 + p i pi 2 1 1 + p + 1 2. Sinc 1 2 + 1 1+/ɛ ) )2 ɛ ) < 1, playr A,1 will no ak a pah ha includ h dg ē,1, and mu hrfor ak a pah ha includ,1 and wrap around o includ h dg ē i. Applying hi am argumn inducivly o h playr A,2,..., A,n hn prov 4). ), ). 13
Finally, o how 5), conidr a 2 ɛ )-approxima Nah quilibrium in which all of h mall playr choo pah ha includ h dg ē i a uual, h ohr ca i ymmric). If playr A i i uing i lowr hor pah, hn par 1) 4) imply ha i har dg i only wih playr A, and hrfor i co har for hi dg i p i /4. On h ohr hand, if A i u i uppr hor pah, i har dg ē i wih all p i+1 of h mall playr, and hrfor i co on hi pah i a mo p i ) p i + p i+1 pi 2 + pi pi 1 + 4 ). p Sinc ɛ = /p, p i 4 > pi 1 + 4 ) 2 ɛ ), p and hnc A i mu u i uppr hor pah. Thi compl h proof of 5) and of h horm. A.1.2 Proof of Thorm 4.3 c log Wmax In hi ubcion, w will prn an xampl wihou log log W max )-approxima Nah quilibrium for om conan c. Similar o h conrucion for Thorm 4.2, hi nwork will coni of i 1 ag conncd rially. All of h ag xcp h fir on and h la on hav h rucur hown in Figur 3a). Th fir ag ha h rucur hown in Figur 3b) and h la ag ha h rucur hown in Figur 3c). Th co of h dg ar dfind a h follow: ce 2i ) = cē2i) = p 2i, ce 2i 2 ) = cē 2i 2 ) = p 2i 1 H p), ce 2i 1 ) = cē2i 1) = 3p 2i H P )) 3, ce 2k, ) = cē2k,) = 2 i k 1 p 2i 1 [H p)] 2 for k = 1, 2,..., i 2, = 1, 2,..., p, ce 2k 1 ) = cē2k 1) = 2 i k 1 p 2i+ 1 2 for k = 2, 3,..., i 2, ce,k ) = cē,k) = H p), for k = 1, 2,..., n Hr, w p o b h quar of a larg ingr, i = 2.5 log 2 H p) + 1, L = n = 2p 2i H p). Th playr in hi nwork gam ar h following: p2i 2H p), Thr ar 3 playr P i, Pi, P i wih wigh p 2i. Thy wan o Connc from S i o T i, from S o T and from S o T rpcivly. Thr i on playr P 4 wih wigh p 4 who wan o connc from S 4 o T 4, and on playr P 2 wih wigh p 2 who wan o connc from S 2 o T 2. Thr ar p playr, P 2j,1, P 2j,1,..., P 2j, p, wih wigh p 2j for ach j = 3, 4,..., i 1. Th playr P 2j,k wan o connc from S 2j o T 2j,k for vry j, k. Thr ar wo playr P 2j+1 and P 2j+1 wih wigh p 2j+1, for ach j = 1, 2,..., i 1. Thy wan o connc from S 2j+1 o T 2j+1 and from S 2j+1 o T 2j+1, rpcivly. 14
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) 5 3 3 5 2,1 3,1,2,n To * prviou ag 2 4,1,1,2,n 2 5 2,1 3,1,2,n To * 5 3 3 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. 15
Thr ar n playr P,1, P,2,..., P,n wih wigh 1. Th k-h playr P,k wan o connc from S,k o T for vry k. Lmma A.1 If any playr bid P 2 u om dg in hr diffrn ag, hn om of h playr can dvia from hi ragy and rduc h co by a facor of H p)/2. Proof: For playr Pi and P i, hr i only on impl pah ha connc from S o T and from S o T. So, hy mu u ha pah and h pah only u dg in h fir ag. Th am proof appli o h playr P 2j+1 and P 2j+1. If h playr P, wih wigh 1 u dg in hr diffrn ag, hn h mu u ihr E 2i 1 or Ē2i 1 o rach h cond ag, and hi paymn i a la ce 2i 1 )/oal wigh) [H p)] 2. Hnc h can dvia o pay for dg E, and Ē2i only, paying a mo H p)+1. Th co i rducd by mor han a facor of H p)/2 by hi wiching. If h playr P 2i u dg in hr diffrn ag, hn h mu u ihr E 2i 1 or Ē2i 1. Hi paymn for any of h dg i a la p 2i [H p)] 2. So, h can dvia o pay for dg E 2i and E 2i 2 only, paying a mo 2p 2i. Th co i rducd by mor han a facor of [H p)] 2 /2. Auming ha if any of h playr wih wigh largr han p 2j u dg in mor han hr ag, hn om of h playr can dvia and rduc h co by a facor of a la H p)/2. Conidr h playr P 2j,1, P 2j,1,..., P 2j, p, if any of hm u dg in hr diffrn ag, hn om of hm mu u ihr E 2j 1 or Ē2j 1. If any playr wih wigh largr han p 2j u ihr E 2j 1 or Ē2j 1, hn h nd o u dg in a la hr diffrn ag, hnc om of h playr may dvia from hi currn ragi and rduc h co by a facor of a la H p)/2. Ohrwi, Any playr P 2j, who u ihr E 2j 1 or Ē2j 1 mu pay a la 2 i j 1 p 2i for ha dg, and h may dvia o u dg E 2j,1, E 2j,2,..., E 2j, p, E 2j+1, and E 2j 2,, paying a mo 2 i j 1 p 2i 1 [H p p)] 2 =1 1 + 2 i j 2 p 2i+ 1 2 1 1+p + 2i j p 2i 1 [H, l han 2 i j p 2i /H p). Th co i rducd by mor p)] 2 han a facor of H p)/2. Inducivly, if any of h playr P 2j,1, P 2j,1,..., P 2j, p, j = 3, 4,,..., i 1 u dg in hr diffrn ag, hn om of h playr can dvia from hi currn ragy and rduc h co by a la a facor of H p)/2. Th am proof alo appli o h playr P 4 Lmma A.2 Thr i no H p)/4)-approxima Nah quilibrium for h xampl w conrucd. Proof: Suppo w ar givn a of ragy S which i a H p)/4)-approxima Nah quilibrium. By lmma A.1, all playr xcp P 2 can only u dg in a mo wo diffrn ag if hy follow h ragy S. Th playr P 2i ha only wo choic: uing dg E 2i, E 2i 2 or dg Ē 2i, Ē 2i 2. Wihou lo of gnraliy, may aum ha h u E 2i, E 2i 2. Now, for h playr P 2i 2,1, P 2i 1,2,..., P 2i 2, p, if om of hm u h dg Ē2i 2, l h P 2i 2, b h on wih h high cond indx among hm. Sinc no ohr playr will u dg Ē2i 2 ohrwi hy mu u dg in a la hr ag), h oal wigh of h playr uing ha dg Ē2i 2 i a mo p 2i 2 p. So, h playr P 2i 2, mu pay a la 2i H p), hnc h can dvia and only pay for dg E 2i 2, E 2i 1 and E 2i 4,. Hi nw paymn i a mo 16
p 2i /1 + p 2 ) + 3p 2i [H p)] 3 /1 + p) + 2p2i [H 4p p)], which i l han 2i 2 [H p)]. So, h can rduc 2 hi paymn by a la a facor of H p)/4 by hi dviaion, which i impoibl inc S i a H p)/4-approxima Nah quilibrium. Thrfor, all h playr P 2i 2, mu u h dg E 2i 2 and hu mu u h dg E 2i 4,. So, h dg E 2i 4, i ud by h playr P 2i 2,, for = 1, 2,..., p and no playr wih wigh p 2i 2 u h dg Ē2i 4,, = 1, 2,..., p. Givn hi, w can apply h am proof o h playr wih wigh p 2i 4 and how ha hy mu u all h dg E 2i 4, and no u dg Ē 2i 4,. Inducivly, w can prov h playr P 2j, mu u h dg E 2j 2, for all j = 3, 4,..., i 1 and = 1, 2,..., p. Alo, playr P 4 mu u h dg E 2 by h am proof. From h dicuion abov and lmma A.1, auming h playr P 2i u E 2i, E 2i 2, hn no playr ohr han P 2 will u h dg Ē2,1 according o h of ragi S. If h playr P 2 u Ē 2,1, hn h mu pay a la cē2,1) = 2i 2 p 2i [H p)], which i mor han p 2i [H p)] 3. H can dvia 2 o u h dg E 2,1, E 3 and dg E,1, E,2,..., E,n, paying a mo ce 2,1 )/1+p 2 )+ce 3 )/1+ p) + nh p), which i l han 3p 2i [H p)] 2. Hi co i rducd by a la a facor of H p)/4, a conradicion. So, h playr P 2 mu u dg E 2,1, E 3 and dg E,1, E,2,..., E,n. If h playr P,1 u h dg Ē,1, h mu pay for ha dg by himlf, hnc having o pay a la H p). Bu hi i impoibl inc h can choo o pay for dg E, and Ē2i only, paying a mo H p)/1 + p 2 ) + 1. Similarly, no playr wih wigh 1 u h dg Ē, and E 2i and all of hm u h dg Ē2i. So, h playr P 2i nd o pay p 2i /2 for h dg E 2i. Bu h only hav o pay a mo p 2i 2+2H p) + p2i H 2p2i p), which i l han H p), if h dvia o u dg Ē2i, Ē 2i 2. Hnc h playr P 2i can dvia and rduc h co by a facor of H p)/2, a conradicion o h original aumpion ha S i a H p)/4)-approxima Nah quilibrium. So, hi of ragy S do no xi, a dird. log 2 1 For h xampl hown in h cion, h maximum wigh i W max = p 2i. Sinc H p) = log W max log log W max, by lmma A.2, hr i no log 2 log W max ) 4 log log W max -approxima Nah quilibrium. A.2 Miing Proof from Scion 5 W now prov Thorm 5.1. W fir ablih om prliminary rul. Fac A.3 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 of Fac A.3: 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+x ) 1. So uppo ha y < 1 + x; hn 1 + y ) x+y y 1 + y ) 1+x y 2. 1 + x 1 + x Raiing boh id of hi inqualiy o h y/x + y) powr and hn aking h logarihm ba 2) of boh id vrifi h claim. 17
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 ) y 1 y A for a), raiing boh id of hi inqualiy o h y/x+y) powr and hn aking h logarihm ba 2) of boh id vrifi h claimd inqualiy. W nx conidr h xinc of approxima Nah quilibria wihou worrying abou hir co. Rcall ha w max and W dno h maximum playr wigh and h um of h playr wigh of a wighd Shaply nwork dign gam, rpcivly, afr wigh hav bn cald o ha h minimum playr wigh i 1. Lmma A.4 For vry funcion fw max, W ) aifying fw max, W ) log 2 [1 + w max )] for all W w max 1, vry wighd Shaply nwork dign gam admi an fw 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 ΦP 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. To prov h lmma, i uffic o how ha fw 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 = fw max, W ). W will 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 w i c 1 c wi, 3) W + w i α W Q i P i whr W = j : P j w j dno h oal wigh on dg bfor playr i dviaion. 1
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 )] 4) P i < ] w i c [log 2 [1 + w i )] w i c 5) W + w i W Q i P i log 2 [1 + w max )] Q i c w i W + w i P i c w i W. w i c fw max, W ) log 2 [1 + w max )] W fw max, W ) P i 6) In hi drivaion, h qualiy 4) follow from h dfiniion of Φ; h inqualiy 5) follow from Fac A.3, wih Fac A.3b) applid o ach rm in h fir um wih x = W and y = w i, and Fac A.3a) applid o ach rm in h cond um wih x = W w i and y = w i ; and h final inqualiy 6) follow from 3) and our choic of α. W now xnd h argumn in h proof of Lmma A.4 o accoun for h co of approxima quilibria, which prov Thorm 5.1. Proof of Thorm 5.1: Conidr a maximal qunc of fw max, W )-improving dviaion ha bgin in a minimum-co oucom wih co C. By Lmma A.4, hi qunc i fini and rmina a a fw 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 w i c fw max, W ) log 2 [1 + w max )] W fw max, W ) P i 1 2 P i c w i W ) 1 2 A fw max, W ), 9) whr inqualiy 7) i h am a inqualiy 6) in h proof of Lmma A.4; inqualiy ) follow from h choic of h funcion f; and inqualiy 9) follow from h fac ha h co incurrd by playr i bfor i dviaion i a la fw 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 fw max, W )-improving dviaion, whnvr h ocial co incra by an addiiv facor of A, h ponial funcion Φ dcra by a la 1 2 fw max, W ) A. Th ponial funcion valu of h ocial opimum i a mo a log 2 1 + W ) muliplicaiv facor largr han i co C, and h ponial funcion only dcra hroughou h qunc of dviaion. Th ocial co can hrfor only incra by a 2C log 2 1 + W )/fw max, W ) addiiv facor hroughou h nir qunc of dviaion. Th qunc mu hrfor rmina in a fwmax, W ), 1 + 2 log 2 1+W ) ) fw max,w ) -approxima Nah quilibrium. 7) 19