Network Design with Weighted Players (SPAA 2006 Full Paper Submission)

Size: px
Start display at page:

Download "Network Design with Weighted Players (SPAA 2006 Full Paper Submission)"

Transcription

1 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 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.

2 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

3 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

4 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

5 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

6 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 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 c 9 w 2 w > c 7 > c 5 + c 9 w 2 w 2 + w + 1 ; 1) 5

7 Edg Co Edg Co Edg Co 1 2 3ɛ 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 c 9 w w ) 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 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 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. 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 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

8 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

9 * 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 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.

10 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

11 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 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 , 24. 1

12 [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 , 23. [3] R. J. Aumann. Subjciviy and corrlaion in randomizd ragi. Journal of Mahmaical Economic, 11):67 96, [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: , 2. [6] M. J. Bckmann, C. B. McGuir, and C. B. Winn. Sudi in h Economic of Tranporaion. Yal Univriy Pr, [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 , 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 , 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 , 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 , [15] D. Mondrr and L. S. Shaply. Ponial gam. Gam and Economic Bhavior, 141): , [16] H. Moulin and S. Shnkr. Sragyproof haring of ubmodular co: Budg balanc vru fficincy. Economic Thory, 13): , 21. [17] J. F. Nah. Equilibrium poin in N-pron gam. Procding of h Naional Acadmy of Scinc, 361):4 49,

13 [1] C. H. Papadimiriou. Algorihm, gam, and h Inrn. In Procding of h 33rd Annual ACM Sympoium on Thory of Compuing STOC), pag , 21. [19] R. W. Ronhal. A cla of gam poing pur-ragy Nah quilibria. Inrnaional Journal of Gam Thory, 21):65 67, [2] R. W. Ronhal. Th nwork quilibrium problm in ingr. Nwork, 31):53 59, [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 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 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 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

14 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 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 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 ɛ)i j 1 + pi 1 + ɛ)i j < pi 1 + ɛ)i j 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 ) ) p 1 + p i pi p Sinc /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

15 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 ). p Sinc ɛ = /p, p i 4 > pi ) 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

16 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. 15

17 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 = i j 2 p 2i 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

18 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 x + y) log x) y x+y ; and b) log x + y) log x) < log 2 [1 + y)] y x+y. Proof of Fac A.3: For boh par, w will u h fac ha 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 y 1+x ) 1. So 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 h logarihm ba 2) of boh id vrifi h claim. 17

19 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 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

20 W can hn driv h following: Φ = Q i c [log W + w i ) log W )] c [log W ) log 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 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 W )/fw max, W ) addiiv facor hroughou h nir qunc of dviaion. Th qunc mu hrfor rmina in a fwmax, W ), log 2 1+W ) ) fw max,w ) -approxima Nah quilibrium. 7) 19

Network Design with Weighted Players

Network Design with Weighted Players 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

More information

Final Exam : Solutions

Final Exam : Solutions Comp : Algorihm and Daa Srucur Final Exam : Soluion. Rcuriv Algorihm. (a) To bgin ind h mdian o {x, x,... x n }. Sinc vry numbr xcp on in h inrval [0, n] appar xacly onc in h li, w hav ha h mdian mu b

More information

Ma/CS 6a Class 15: Flows and Bipartite Graphs

Ma/CS 6a Class 15: Flows and Bipartite Graphs //206 Ma/CS 6a Cla : Flow and Bipari Graph By Adam Shffr Rmindr: Flow Nwork A flow nwork i a digraph G = V, E, oghr wih a ourc vrx V, a ink vrx V, and a capaciy funcion c: E N. Capaciy Sourc 7 a b c d

More information

Poisson process Markov process

Poisson process Markov process E2200 Quuing hory and lraffic 2nd lcur oion proc Markov proc Vikoria Fodor KTH Laboraory for Communicaion nwork, School of Elcrical Enginring 1 Cour oulin Sochaic proc bhind quuing hory L2-L3 oion proc

More information

14.02 Principles of Macroeconomics Fall 2005 Quiz 3 Solutions

14.02 Principles of Macroeconomics Fall 2005 Quiz 3 Solutions 4.0 rincipl of Macroconomic Fall 005 Quiz 3 Soluion Shor Quion (30/00 poin la a whhr h following amn ar TRUE or FALSE wih a hor xplanaion (3 or 4 lin. Each quion coun 5/00 poin.. An incra in ax oday alway

More information

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018 DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Aoc. Prof. Dr. Burak Kllci Spring 08 OUTLINE Th Laplac Tranform Rgion of convrgnc for Laplac ranform Invr Laplac ranform Gomric valuaion

More information

Chapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System

Chapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System EE 422G No: Chapr 5 Inrucor: Chung Chapr 5 Th Laplac Tranform 5- Inroducion () Sym analyi inpu oupu Dynamic Sym Linar Dynamic ym: A procor which proc h inpu ignal o produc h oupu dy ( n) ( n dy ( n) +

More information

Transfer function and the Laplace transformation

Transfer function and the Laplace transformation Lab No PH-35 Porland Sa Univriy A. La Roa Tranfr funcion and h Laplac ranformaion. INTRODUTION. THE LAPLAE TRANSFORMATION L 3. TRANSFER FUNTIONS 4. ELETRIAL SYSTEMS Analyi of h hr baic paiv lmn R, and

More information

Chapter 12 Introduction To The Laplace Transform

Chapter 12 Introduction To The Laplace Transform Chapr Inroducion To Th aplac Tranorm Diniion o h aplac Tranorm - Th Sp & Impul uncion aplac Tranorm o pciic uncion 5 Opraional Tranorm Applying h aplac Tranorm 7 Invr Tranorm o Raional uncion 8 Pol and

More information

Lecture 26: Leapers and Creepers

Lecture 26: Leapers and Creepers Lcur 6: Lapr and Crpr Scrib: Grain Jon (and Marin Z. Bazan) Dparmn of Economic, MIT May, 005 Inroducion Thi lcur conidr h analyi of h non-parabl CTRW in which h diribuion of p iz and im bwn p ar dpndn.

More information

CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 259] B-Trees

CPSC 211 Data Structures & Implementations (c) Texas A&M University [ 259] B-Trees CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 259] B-Trs Th AVL r and rd-black r allowd som variaion in h lnghs of h diffrn roo-o-laf pahs. An alrnaiv ida is o mak sur ha all roo-o-laf pahs

More information

Midterm exam 2, April 7, 2009 (solutions)

Midterm exam 2, April 7, 2009 (solutions) Univrsiy of Pnnsylvania Dparmn of Mahmaics Mah 26 Honors Calculus II Spring Smsr 29 Prof Grassi, TA Ashr Aul Midrm xam 2, April 7, 29 (soluions) 1 Wri a basis for h spac of pairs (u, v) of smooh funcions

More information

Lecture 1: Numerical Integration The Trapezoidal and Simpson s Rule

Lecture 1: Numerical Integration The Trapezoidal and Simpson s Rule Lcur : Numrical ngraion Th Trapzoidal and Simpson s Rul A problm Th probabiliy of a normally disribud (man µ and sandard dviaion σ ) vn occurring bwn h valus a and b is B A P( a x b) d () π whr a µ b -

More information

LaPlace Transform in Circuit Analysis

LaPlace Transform in Circuit Analysis LaPlac Tranform in Circui Analyi Obciv: Calcula h Laplac ranform of common funcion uing h dfiniion and h Laplac ranform abl Laplac-ranform a circui, including componn wih non-zro iniial condiion. Analyz

More information

Elementary Differential Equations and Boundary Value Problems

Elementary Differential Equations and Boundary Value Problems Elmnar Diffrnial Equaions and Boundar Valu Problms Boc. & DiPrima 9 h Ediion Chapr : Firs Ordr Diffrnial Equaions 00600 คณ ตศาสตร ว ศวกรรม สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา /55 ผศ.ดร.อร ญญา ผศ.ดร.สมศ

More information

Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors

Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors Boc/DiPrima 9 h d, Ch.: Linar Equaions; Mhod of Ingraing Facors Elmnar Diffrnial Equaions and Boundar Valu Problms, 9 h diion, b William E. Boc and Richard C. DiPrima, 009 b John Wil & Sons, Inc. A linar

More information

Instructors Solution for Assignment 3 Chapter 3: Time Domain Analysis of LTIC Systems

Instructors Solution for Assignment 3 Chapter 3: Time Domain Analysis of LTIC Systems Inrucor Soluion for Aignmn Chapr : Tim Domain Anali of LTIC Sm Problm i a 8 x x wih x u,, an Zro-inpu rpon of h m: Th characriic quaion of h LTIC m i i 8, which ha roo a ± j Th zro-inpu rpon i givn b zi

More information

1 Finite Automata and Regular Expressions

1 Finite Automata and Regular Expressions 1 Fini Auom nd Rgulr Exprion Moivion: Givn prn (rgulr xprion) for ring rching, w migh wn o convr i ino drminiic fini uomon or nondrminiic fini uomon o mk ring rching mor fficin; drminiic uomon only h o

More information

Discussion 06 Solutions

Discussion 06 Solutions STAT Discussion Soluions Spring 8. Th wigh of fish in La Paradis follows a normal disribuion wih man of 8. lbs and sandard dviaion of. lbs. a) Wha proporion of fish ar bwn 9 lbs and lbs? æ 9-8. - 8. P

More information

UNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS

UNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Answr Ky Nam: Da: UNIT # EXPONENTIAL AND LOGARITHMIC FUNCTIONS Par I Qusions. Th prssion is quivaln o () () 6 6 6. Th ponnial funcion y 6 could rwrin as y () y y 6 () y y (). Th prssion a is quivaln o

More information

CHAPTER CHAPTER14. Expectations: The Basic Tools. Prepared by: Fernando Quijano and Yvonn Quijano

CHAPTER CHAPTER14. Expectations: The Basic Tools. Prepared by: Fernando Quijano and Yvonn Quijano Expcaions: Th Basic Prpard by: Frnando Quijano and Yvonn Quijano CHAPTER CHAPTER14 2006 Prnic Hall Businss Publishing Macroconomics, 4/ Olivir Blanchard 14-1 Today s Lcur Chapr 14:Expcaions: Th Basic Th

More information

On the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument

On the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument Inrnaional Rsarch Journal of Applid Basic Scincs 03 Aailabl onlin a wwwirjabscom ISSN 5-838X / Vol 4 (): 47-433 Scinc Eplorr Publicaions On h Driais of Bssl Modifid Bssl Funcions wih Rspc o h Ordr h Argumn

More information

Let s look again at the first order linear differential equation we are attempting to solve, in its standard form:

Let s look again at the first order linear differential equation we are attempting to solve, in its standard form: Th Ingraing Facor Mhod In h prvious xampls of simpl firs ordr ODEs, w found h soluions by algbraically spara h dpndn variabl- and h indpndn variabl- rms, and wri h wo sids of a givn quaion as drivaivs,

More information

Institute of Actuaries of India

Institute of Actuaries of India Insiu of Acuaris of India ubjc CT3 Probabiliy and Mahmaical aisics Novmbr Examinaions INDICATIVE OLUTION Pag of IAI CT3 Novmbr ol. a sampl man = 35 sampl sandard dviaion = 36.6 b for = uppr bound = 35+*36.6

More information

Spring 2006 Process Dynamics, Operations, and Control Lesson 2: Mathematics Review

Spring 2006 Process Dynamics, Operations, and Control Lesson 2: Mathematics Review Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw. conx and dircion Imagin a sysm ha varis in im; w migh plo is oupu vs. im. A plo migh imply an quaion, and h quaion is usually an

More information

Microscopic Flow Characteristics Time Headway - Distribution

Microscopic Flow Characteristics Time Headway - Distribution CE57: Traffic Flow Thory Spring 20 Wk 2 Modling Hadway Disribuion Microscopic Flow Characrisics Tim Hadway - Disribuion Tim Hadway Dfiniion Tim Hadway vrsus Gap Ahmd Abdl-Rahim Civil Enginring Dparmn,

More information

4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b

4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b 4. Th Uniform Disribuion Df n: A c.r.v. has a coninuous uniform disribuion on [a, b] whn is pdf is f x a x b b a Also, b + a b a µ E and V Ex4. Suppos, h lvl of unblivabiliy a any poin in a Transformrs

More information

PWM-Scheme and Current ripple of Switching Power Amplifiers

PWM-Scheme and Current ripple of Switching Power Amplifiers axon oor PWM-Sch and Currn rippl of Swiching Powr Aplifir Abrac In hi work currn rippl caud by wiching powr aplifir i analyd for h convnional PWM (pulwidh odulaion) ch and hr-lvl PWM-ch. Siplifid odl for

More information

A Condition for Stability in an SIR Age Structured Disease Model with Decreasing Survival Rate

A Condition for Stability in an SIR Age Structured Disease Model with Decreasing Survival Rate A Condiion for abiliy in an I Ag rucurd Disas Modl wih Dcrasing urvival a A.K. upriana, Edy owono Dparmn of Mahmaics, Univrsias Padjadjaran, km Bandung-umng 45363, Indonsia fax: 6--7794696, mail: asupria@yahoo.com.au;

More information

Problem Set If all directed edges in a network have distinct capacities, then there is a unique maximum flow.

Problem Set If all directed edges in a network have distinct capacities, then there is a unique maximum flow. CSE 202: Deign and Analyi of Algorihm Winer 2013 Problem Se 3 Inrucor: Kamalika Chaudhuri Due on: Tue. Feb 26, 2013 Inrucion For your proof, you may ue any lower bound, algorihm or daa rucure from he ex

More information

Network Flows: Introduction & Maximum Flow

Network Flows: Introduction & Maximum Flow CSC 373 - lgorihm Deign, nalyi, and Complexiy Summer 2016 Lalla Mouaadid Nework Flow: Inroducion & Maximum Flow We now urn our aenion o anoher powerful algorihmic echnique: Local Search. In a local earch

More information

2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35

2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35 MATH 5 PS # Summr 00.. Diffrnial Equaions and Soluions PS.# Show ha ()C #, 4, 7, 0, 4, 5 ( / ) is a gnral soluion of h diffrnial quaion. Us a compur or calculaor o skch h soluions for h givn valus of h

More information

Economics 302 (Sec. 001) Intermediate Macroeconomic Theory and Policy (Spring 2011) 3/28/2012. UW Madison

Economics 302 (Sec. 001) Intermediate Macroeconomic Theory and Policy (Spring 2011) 3/28/2012. UW Madison Economics 302 (Sc. 001) Inrmdia Macroconomic Thory and Policy (Spring 2011) 3/28/2012 Insrucor: Prof. Mnzi Chinn Insrucor: Prof. Mnzi Chinn UW Madison 16 1 Consumpion Th Vry Forsighd dconsumr A vry forsighd

More information

Relaxing planarity for topological graphs

Relaxing planarity for topological graphs Rlaing planariy for opological graph Jáno Pach 1, Radoš Radoičić 2, and Géza Tóh 3 1 Ciy Collg, CUNY and Couran Iniu of Mahmaical Scinc, Nw York Univriy, Nw York, NY 10012, USA pach@cim.nyu.du 2 Dparmn

More information

Applied Statistics and Probability for Engineers, 6 th edition October 17, 2016

Applied Statistics and Probability for Engineers, 6 th edition October 17, 2016 Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 CHATER Scion - -. a d. 679.. b. d. 88 c d d d. 987 d. 98 f d.. Thn, = ln. =. g d.. Thn, = ln.9 =.. -7. a., by symmry. b.. d...6. 7.. c...

More information

Chapter 3: Fourier Representation of Signals and LTI Systems. Chih-Wei Liu

Chapter 3: Fourier Representation of Signals and LTI Systems. Chih-Wei Liu Chapr 3: Fourir Rprsnaion of Signals and LTI Sysms Chih-Wi Liu Oulin Inroducion Complx Sinusoids and Frquncy Rspons Fourir Rprsnaions for Four Classs of Signals Discr-im Priodic Signals Fourir Sris Coninuous-im

More information

THE INVOLUTE-EVOLUTE OFFSETS OF RULED SURFACES *

THE INVOLUTE-EVOLUTE OFFSETS OF RULED SURFACES * Iranian Journal of Scinc & Tchnology, Tranacion A, Vol, No A Prind in h Ilamic Rpublic of Iran, 009 Shiraz Univriy THE INVOLUTE-EVOLUTE OFFSETS OF RULED SURFACES E KASAP, S YUCE AND N KURUOGLU Ondokuz

More information

The Science of Monetary Policy

The Science of Monetary Policy Th Scinc of Monary Policy. Inroducion o Topics of Sminar. Rviw: IS-LM, AD-AS wih an applicaion o currn monary policy in Japan 3. Monary Policy Sragy: Inrs Ra Ruls and Inflaion Targing (Svnsson EER) 4.

More information

Mixing Real-Time and Non-Real-Time. CSCE 990: Real-Time Systems. Steve Goddard.

Mixing Real-Time and Non-Real-Time. CSCE 990: Real-Time Systems. Steve Goddard. CSCE 990: Ral-Tim Sym Mixing Ral-Tim and Non-Ral-Tim goddard@c.unl.du hp://www.c.unl.du/~goddard/cour/raltimsym 1 Ral-Tim Sym Mixd Job - 1 Mixing Ral-Tim and Non-Ral-Tim in Prioriy-Drivn Sym (Chapr 7 of

More information

Shortest Path With Negative Weights

Shortest Path With Negative Weights Shor Pah Wih Ngaiv Wigh 1 9 18-1 19 11 1-8 1 Conn Conn. Dird hor pah wih ngaiv wigh. Ngaiv yl dion. appliaion: urrny xhang arbirag Tramp amr problm. appliaion: opimal piplining of VLSI hip Shor Pah wih

More information

Network Congestion Games

Network Congestion Games Ntwork Congstion Gams Assistant Profssor Tas A&M Univrsity Collg Station, TX TX Dallas Collg Station Austin Houston Bst rout dpnds on othrs Ntwork Congstion Gams Travl tim incrass with congstion Highway

More information

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 11

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 11 8 Jun ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER SECTION : INCENTIVE COMPATABILITY Exrcis - Educaional Signaling A yp consulan has a marginal produc of m( ) = whr Θ = {,, 3} Typs ar uniformly disribud

More information

An Indian Journal FULL PAPER. Trade Science Inc. A stage-structured model of a single-species with density-dependent and birth pulses ABSTRACT

An Indian Journal FULL PAPER. Trade Science Inc. A stage-structured model of a single-species with density-dependent and birth pulses ABSTRACT [Typ x] [Typ x] [Typ x] ISSN : 974-7435 Volum 1 Issu 24 BioTchnology 214 An Indian Journal FULL PAPE BTAIJ, 1(24), 214 [15197-1521] A sag-srucurd modl of a singl-spcis wih dnsiy-dpndn and birh pulss LI

More information

Selfish Routing. Tim Roughgarden Cornell University. Includes joint work with Éva Tardos

Selfish Routing. Tim Roughgarden Cornell University. Includes joint work with Éva Tardos Selfih Rouing Tim Roughgarden Cornell Univeriy Include join work wih Éva Tardo 1 Which roue would you chooe? Example: one uni of raffic (e.g., car) wan o go from o delay = 1 hour (no congeion effec) long

More information

Circuit Transients time

Circuit Transients time Circui Tranin A Solp 3/29/0, 9/29/04. Inroducion Tranin: A ranin i a raniion from on a o anohr. If h volag and currn in a circui do no chang wih im, w call ha a "ady a". In fac, a long a h volag and currn

More information

Graphs III - Network Flow

Graphs III - Network Flow Graph III - Nework Flow Flow nework eup graph G=(V,E) edge capaciy w(u,v) 0 - if edge doe no exi, hen w(u,v)=0 pecial verice: ource verex ; ink verex - no edge ino and no edge ou of Aume every verex v

More information

CSE 245: Computer Aided Circuit Simulation and Verification

CSE 245: Computer Aided Circuit Simulation and Verification CSE 45: Compur Aidd Circui Simulaion and Vrificaion Fall 4, Sp 8 Lcur : Dynamic Linar Sysm Oulin Tim Domain Analysis Sa Equaions RLC Nwork Analysis by Taylor Expansion Impuls Rspons in im domain Frquncy

More information

Network Flow. Data Structures and Algorithms Andrei Bulatov

Network Flow. Data Structures and Algorithms Andrei Bulatov Nework Flow Daa Srucure and Algorihm Andrei Bulao Algorihm Nework Flow 24-2 Flow Nework Think of a graph a yem of pipe We ue hi yem o pump waer from he ource o ink Eery pipe/edge ha limied capaciy Flow

More information

14.02 Principles of Macroeconomics Problem Set 5 Fall 2005

14.02 Principles of Macroeconomics Problem Set 5 Fall 2005 40 Principls of Macroconomics Problm S 5 Fall 005 Posd: Wdnsday, Novmbr 6, 005 Du: Wdnsday, Novmbr 3, 005 Plas wri your nam AND your TA s nam on your problm s Thanks! Exrcis I Tru/Fals? Explain Dpnding

More information

The Residual Graph. 11 Augmenting Path Algorithms. Augmenting Path Algorithm. Augmenting Path Algorithm

The Residual Graph. 11 Augmenting Path Algorithms. Augmenting Path Algorithm. Augmenting Path Algorithm Augmening Pah Algorihm Greedy-algorihm: ar wih f (e) = everywhere find an - pah wih f (e) < c(e) on every edge augmen flow along he pah repea a long a poible The Reidual Graph From he graph G = (V, E,

More information

The Residual Graph. 12 Augmenting Path Algorithms. Augmenting Path Algorithm. Augmenting Path Algorithm

The Residual Graph. 12 Augmenting Path Algorithms. Augmenting Path Algorithm. Augmenting Path Algorithm Augmening Pah Algorihm Greedy-algorihm: ar wih f (e) = everywhere find an - pah wih f (e) < c(e) on every edge augmen flow along he pah repea a long a poible The Reidual Graph From he graph G = (V, E,

More information

Midterm Examination (100 pts)

Midterm Examination (100 pts) Econ 509 Spring 2012 S.L. Parn Midrm Examinaion (100 ps) Par I. 30 poins 1. Dfin h Law of Diminishing Rurns (5 ps.) Incrasing on inpu, call i inpu x, holding all ohr inpus fixd, on vnuall runs ino h siuaion

More information

Optimal time-consistent investment in the dual risk model with diffusion

Optimal time-consistent investment in the dual risk model with diffusion Opimal im-conin invmn in h dual rik modl wih diffuion LIDONG ZHANG Tianjin Univriy of Scinc and Tchnology Collg of Scinc TEDA, Sr 3, Tianjin CHINA zhanglidong999@.com XIMIN RONG Tianjin Univriy School

More information

[ ] 1+ lim G( s) 1+ s + s G s s G s Kacc SYSTEM PERFORMANCE. Since. Lecture 10: Steady-state Errors. Steady-state Errors. Then

[ ] 1+ lim G( s) 1+ s + s G s s G s Kacc SYSTEM PERFORMANCE. Since. Lecture 10: Steady-state Errors. Steady-state Errors. Then SYSTEM PERFORMANCE Lctur 0: Stady-tat Error Stady-tat Error Lctur 0: Stady-tat Error Dr.alyana Vluvolu Stady-tat rror can b found by applying th final valu thorm and i givn by lim ( t) lim E ( ) t 0 providd

More information

1 Motivation and Basic Definitions

1 Motivation and Basic Definitions CSCE : Deign and Analyi of Algorihm Noe on Max Flow Fall 20 (Baed on he preenaion in Chaper 26 of Inroducion o Algorihm, 3rd Ed. by Cormen, Leieron, Rive and Sein.) Moivaion and Baic Definiion Conider

More information

7.4 QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS *

7.4 QUANTUM MECHANICAL TREATMENT OF FLUCTUATIONS * Andri Tokmakoff, MIT Dparmn of Chmisry, 5/19/5 7-11 7.4 QUANTUM MECANICAL TREATMENT OF FLUCTUATIONS * Inroducion and Prviw Now h origin of frquncy flucuaions is inracions of our molcul (or mor approprialy

More information

Math 266, Practice Midterm Exam 2

Math 266, Practice Midterm Exam 2 Mh 66, Prcic Midrm Exm Nm: Ground Rul. Clculor i NOT llowd.. Show your work for vry problm unl ohrwi d (pril crdi r vilbl). 3. You my u on 4-by-6 indx crd, boh id. 4. Th bl of Lplc rnform i vilbl h l pg.

More information

Mundell-Fleming I: Setup

Mundell-Fleming I: Setup Mundll-Flming I: Sup In ISLM, w had: E ( ) T I( i π G T C Y ) To his, w now add n xpors, which is a funcion of h xchang ra: ε E P* P ( T ) I( i π ) G T NX ( ) C Y Whr NX is assumd (Marshall Lrnr condiion)

More information

where: u: input y: output x: state vector A, B, C, D are const matrices

where: u: input y: output x: state vector A, B, C, D are const matrices Sa pac modl: linar: y or in om : Sa q : f, u Oupu q : y h, u u Du F Gu y H Ju whr: u: inpu y: oupu : a vcor,,, D ar con maric Eampl " $ & ' " $ & 'u y " & * * * * [ ],, D H D I " $ " & $ ' " & $ ' " &

More information

2. The Laplace Transform

2. The Laplace Transform Th aac Tranorm Inroucion Th aac ranorm i a unamna an vry uu oo or uying many nginring robm To in h aac ranorm w conir a comx variab σ, whr σ i h ra ar an i h imaginary ar or ix vau o σ an w viw a a oin

More information

INTRODUCTION TO AUTOMATIC CONTROLS INDEX LAPLACE TRANSFORMS

INTRODUCTION TO AUTOMATIC CONTROLS INDEX LAPLACE TRANSFORMS adjoint...6 block diagram...4 clod loop ytm... 5, 0 E()...6 (t)...6 rror tady tat tracking...6 tracking...6...6 gloary... 0 impul function...3 input...5 invr Laplac tranform, INTRODUCTION TO AUTOMATIC

More information

(1) Then we could wave our hands over this and it would become:

(1) Then we could wave our hands over this and it would become: MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and

More information

Chap.3 Laplace Transform

Chap.3 Laplace Transform Chap. aplac Tranorm Tranorm: An opraion ha ranorm a uncion ino anohr uncion i Dirniaion ranorm: ii x: d dx x x Ingraion ranorm: x: x dx x c Now, conidr a dind ingral k, d,ha ranorm ino a uncion o variabl

More information

Phys463.nb Conductivity. Another equivalent definition of the Fermi velocity is

Phys463.nb Conductivity. Another equivalent definition of the Fermi velocity is 39 Anohr quival dfiniion of h Fri vlociy is pf vf (6.4) If h rgy is a quadraic funcion of k H k L, hs wo dfiniions ar idical. If is NOT a quadraic funcion of k (which could happ as will b discussd in h

More information

Admin MAX FLOW APPLICATIONS. Flow graph/networks. Flow constraints 4/30/13. CS lunch today Grading. in-flow = out-flow for every vertex (except s, t)

Admin MAX FLOW APPLICATIONS. Flow graph/networks. Flow constraints 4/30/13. CS lunch today Grading. in-flow = out-flow for every vertex (except s, t) /0/ dmin lunch oday rading MX LOW PPLIION 0, pring avid Kauchak low graph/nework low nework direced, weighed graph (V, ) poiive edge weigh indicaing he capaciy (generally, aume ineger) conain a ingle ource

More information

I) Title: Rational Expectations and Adaptive Learning. II) Contents: Introduction to Adaptive Learning

I) Title: Rational Expectations and Adaptive Learning. II) Contents: Introduction to Adaptive Learning I) Til: Raional Expcaions and Adapiv Larning II) Conns: Inroducion o Adapiv Larning III) Documnaion: - Basdvan, Olivir. (2003). Larning procss and raional xpcaions: an analysis using a small macroconomic

More information

REPETITION before the exam PART 2, Transform Methods. Laplace transforms: τ dτ. L1. Derive the formulas : L2. Find the Laplace transform F(s) if.

REPETITION before the exam PART 2, Transform Methods. Laplace transforms: τ dτ. L1. Derive the formulas : L2. Find the Laplace transform F(s) if. Tranform Mhod and Calculu of Svral Variabl H7, p Lcurr: Armin Halilovic KTH, Campu Haning E-mail: armin@dkh, wwwdkh/armin REPETITION bfor h am PART, Tranform Mhod Laplac ranform: L Driv h formula : a L[

More information

H is equal to the surface current J S

H is equal to the surface current J S Chapr 6 Rflcion and Transmission of Wavs 6.1 Boundary Condiions A h boundary of wo diffrn mdium, lcromagnic fild hav o saisfy physical condiion, which is drmind by Maxwll s quaion. This is h boundary condiion

More information

Economics 302 (Sec. 001) Intermediate Macroeconomic Theory and Policy (Spring 2011) 4/25/2011. UW Madison

Economics 302 (Sec. 001) Intermediate Macroeconomic Theory and Policy (Spring 2011) 4/25/2011. UW Madison conomics 302 (Sc. 001) Inrmdia Macroconomic Thory and Policy (Spring 2011) 4/25/2011 Insrucor: Prof. Mnzi Chinn Insrucor: Prof. Mnzi Chinn UW Madison 21 1 Th Mdium Run ε = P * P Thr ar wo ways in which

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!

More information

Randomized Perfect Bipartite Matching

Randomized Perfect Bipartite Matching Inenive Algorihm Lecure 24 Randomized Perfec Biparie Maching Lecurer: Daniel A. Spielman April 9, 208 24. Inroducion We explain a randomized algorihm by Ahih Goel, Michael Kapralov and Sanjeev Khanna for

More information

Estimation of Mean Time between Failures in Two Unit Parallel Repairable System

Estimation of Mean Time between Failures in Two Unit Parallel Repairable System Inrnaional Journal on Rcn Innovaion rnd in Comuing Communicaion ISSN: -869 Volum: Iu: 6 Eimaion of Man im bwn Failur in wo Uni Paralll Rairabl Sym Sma Sahu V.K. Paha Kamal Mha hih Namdo 4 ian Profor D.

More information

CIVL 8/ D Boundary Value Problems - Quadrilateral Elements (Q4) 1/8

CIVL 8/ D Boundary Value Problems - Quadrilateral Elements (Q4) 1/8 CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) 1/8 ISOPARAMERIC ELEMENS h inar rianguar mn and h iinar rcanguar mn hav vra imporan diadvanag. 1. Boh mn ar una o accura rprn curvd oundari, and. h provid

More information

SOLUTIONS. 1. Consider two continuous random variables X and Y with joint p.d.f. f ( x, y ) = = = 15. Stepanov Dalpiaz

SOLUTIONS. 1. Consider two continuous random variables X and Y with joint p.d.f. f ( x, y ) = = = 15. Stepanov Dalpiaz STAT UIUC Pracic Problms #7 SOLUTIONS Spanov Dalpiaz Th following ar a numbr of pracic problms ha ma b hlpful for compling h homwor, and will lil b vr usful for suding for ams.. Considr wo coninuous random

More information

ON FINITE MORSE INDEX SOLUTIONS OF HIGHER ORDER FRACTIONAL LANE-EMDEN EQUATIONS

ON FINITE MORSE INDEX SOLUTIONS OF HIGHER ORDER FRACTIONAL LANE-EMDEN EQUATIONS ON FINITE MORSE INDEX SOLUTIONS OF HIGHER ORDER FRACTIONAL LANE-EMDEN EQUATIONS MOSTAFA FAZLY AND JUNCHENG WEI Abrac. W claify fini Mor indx oluion of h following nonlocal Lan-Emdn quaion u u u for <

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!

More information

Lecture 4: Laplace Transforms

Lecture 4: Laplace Transforms Lur 4: Lapla Transforms Lapla and rlad ransformaions an b usd o solv diffrnial quaion and o rdu priodi nois in signals and imags. Basially, hy onvr h drivaiv opraions ino mulipliaion, diffrnial quaions

More information

Voltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields!

Voltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields! Considr a pair of wirs idal wirs ngh >, say, infinily long olag along a cabl can vary! D olag v( E(D W can acually g o his wav bhavior by using circui hory, w/o going ino dails of h EM filds! Thr

More information

Double Slits in Space and Time

Double Slits in Space and Time Doubl Slis in Sac an Tim Gorg Jons As has bn ror rcnly in h mia, a am l by Grhar Paulus has monsra an inrsing chniqu for ionizing argon aoms by using ulra-shor lasr ulss. Each lasr uls is ffcivly on an

More information

Wave Equation (2 Week)

Wave Equation (2 Week) Rfrnc Wav quaion ( Wk 6.5 Tim-armonic filds 7. Ovrviw 7. Plan Wavs in Losslss Mdia 7.3 Plan Wavs in Loss Mdia 7.5 Flow of lcromagnic Powr and h Poning Vcor 7.6 Normal Incidnc of Plan Wavs a Plan Boundaris

More information

THE SHORT-RUN AGGREGATE SUPPLY CURVE WITH A POSITIVE SLOPE. Based on EXPECTATIONS: Lecture. t t t t

THE SHORT-RUN AGGREGATE SUPPLY CURVE WITH A POSITIVE SLOPE. Based on EXPECTATIONS: Lecture. t t t t THE SHORT-RUN AGGREGATE SUL CURVE WITH A OSITIVE SLOE. Basd on EXECTATIONS: Lcur., 0. In Mankiw:, 0 Ths quaions sa ha oupu dvias from is naural ra whn h pric lvl dvias from h xpcd pric lvl. Th paramr a

More information

Jonathan Turner Exam 2-12/4/03

Jonathan Turner Exam 2-12/4/03 CS 41 Algorim an Program Prolm Exam Soluion S Soluion Jonaan Turnr Exam -1/4/0 10/8/0 1. (10 poin) T igur low ow an implmnaion o ynami r aa ruur wi vral virual r. Sow orrponing o aual r (owing vrx o).

More information

Flow networks. Flow Networks. A flow on a network. Flow networks. The maximum-flow problem. Introduction to Algorithms, Lecture 22 December 5, 2001

Flow networks. Flow Networks. A flow on a network. Flow networks. The maximum-flow problem. Introduction to Algorithms, Lecture 22 December 5, 2001 CS 545 Flow Nework lon Efra Slide courey of Charle Leieron wih mall change by Carola Wenk Flow nework Definiion. flow nework i a direced graph G = (V, E) wih wo diinguihed verice: a ource and a ink. Each

More information

Algorithmic Discrete Mathematics 6. Exercise Sheet

Algorithmic Discrete Mathematics 6. Exercise Sheet Algorihmic Dicree Mahemaic. Exercie Shee Deparmen of Mahemaic SS 0 PD Dr. Ulf Lorenz 7. and 8. Juni 0 Dipl.-Mah. David Meffer Verion of June, 0 Groupwork Exercie G (Heap-Sor) Ue Heap-Sor wih a min-heap

More information

Why Laplace transforms?

Why Laplace transforms? MAE4 Linar ircui Why Lalac ranform? Firordr R cc v v v KVL S R inananou for ach Subiu lmn rlaion v S Ordinary diffrnial quaion in rm of caacior volag Lalac ranform Solv Invr LT V u, v Ri, i A R V A _ v

More information

Introduction to Congestion Games

Introduction to Congestion Games Algorihmic Game Theory, Summer 2017 Inroducion o Congeion Game Lecure 1 (5 page) Inrucor: Thoma Keelheim In hi lecure, we ge o know congeion game, which will be our running example for many concep in game

More information

cycle that does not cross any edges (including its own), then it has at least

cycle that does not cross any edges (including its own), then it has at least W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th

More information

FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS I: Introduction and Linear Systems

FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS I: Introduction and Linear Systems FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS I: Inroducion and Linar Sysms David Lvrmor Dparmn of Mahmaics Univrsiy of Maryland 9 Dcmbr 0 Bcaus h prsnaion of his marial in lcur will diffr from

More information

Does Inequality Lead to Greater Efficiency in the Use of Local Commons? The Role of Strategic Investments in Capacity

Does Inequality Lead to Greater Efficiency in the Use of Local Commons? The Role of Strategic Investments in Capacity Do nqualiy Lad o Grar fficincy in h U of Local Common? Th Rol of Sragic nvmn in Capaciy by Rimjhim Aggaral and Tulika A. Narayan Working Papr No. 00-03 Da of Submiion: March 000 Copyrigh @ 000 by Rimjhim

More information

Revisiting what you have learned in Advanced Mathematical Analysis

Revisiting what you have learned in Advanced Mathematical Analysis Fourir sris Rvisiing wh you hv lrnd in Advncd Mhmicl Anlysis L f x b priodic funcion of priod nd is ingrbl ovr priod. f x cn b rprsnd by rigonomric sris, f x n cos nx bn sin nx n cos x b sin x cosx b whr

More information

SECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.

SECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero. SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain

More information

CS4445/9544 Analysis of Algorithms II Solution for Assignment 1

CS4445/9544 Analysis of Algorithms II Solution for Assignment 1 Conider he following flow nework CS444/944 Analyi of Algorihm II Soluion for Aignmen (0 mark) In he following nework a minimum cu ha capaciy 0 Eiher prove ha hi aemen i rue, or how ha i i fale Uing he

More information

Introduction to SLE Lecture Notes

Introduction to SLE Lecture Notes Inroducion o SLE Lecure Noe May 13, 16 - The goal of hi ecion i o find a ufficien condiion of λ for he hull K o be generaed by a imple cure. I urn ou if λ 1 < 4 hen K i generaed by a imple curve. We will

More information

On the Speed of Heat Wave. Mihály Makai

On the Speed of Heat Wave. Mihály Makai On h Spd of Ha Wa Mihály Maai maai@ra.bm.hu Conns Formulaion of h problm: infini spd? Local hrmal qulibrium (LTE hypohsis Balanc quaion Phnomnological balanc Spd of ha wa Applicaion in plasma ranspor 1.

More information

Selfish Routing and the Price of Anarchy. Tim Roughgarden Cornell University

Selfish Routing and the Price of Anarchy. Tim Roughgarden Cornell University Selfih Rouing and he Price of Anarchy Tim Roughgarden Cornell Univeriy 1 Algorihm for Self-Inereed Agen Our focu: problem in which muliple agen (people, compuer, ec.) inerac Moivaion: he Inerne decenralized

More information

Syntactic Complexity of Suffix-Free Languages. Marek Szykuła

Syntactic Complexity of Suffix-Free Languages. Marek Szykuła Inroducion Upper Bound on Synacic Complexiy of Suffix-Free Language Univeriy of Wrocław, Poland Join work wih Januz Brzozowki Univeriy of Waerloo, Canada DCFS, 25.06.2015 Abrac Inroducion Sae and ynacic

More information

Copyright 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright 2012 Pearson Education, Inc. Publishing as Prentice Hall. Chapr Rviw 0 6. ( a a ln a. This will qual a if an onl if ln a, or a. + k an (ln + c. Thrfor, a an valu of, whr h wo curvs inrsc, h wo angn lins will b prpnicular. 6. (a Sinc h lin passs hrough h origin

More information

Circuits and Systems I

Circuits and Systems I Circuis and Sysms I LECTURE #3 Th Spcrum, Priodic Signals, and h Tim-Varying Spcrum lions@pfl Prof. Dr. Volan Cvhr LIONS/Laboraory for Informaion and Infrnc Sysms Licns Info for SPFirs Slids This wor rlasd

More information

B) 25y e. 5. Find the second partial f. 6. Find the second partials (including the mixed partials) of

B) 25y e. 5. Find the second partial f. 6. Find the second partials (including the mixed partials) of Sampl Final 00 1. Suppos z = (, y), ( a, b ) = 0, y ( a, b ) = 0, ( a, b ) = 1, ( a, b ) = 1, and y ( a, b ) =. Thn (a, b) is h s is inconclusiv a saddl poin a rlaiv minimum a rlaiv maimum. * (Classiy

More information

Part I: Short Answer [50 points] For each of the following, give a short answer (2-3 sentences, or a formula). [5 points each]

Part I: Short Answer [50 points] For each of the following, give a short answer (2-3 sentences, or a formula). [5 points each] Soluions o Midrm Exam Nam: Paricl Physics Fall 0 Ocobr 6 0 Par I: Shor Answr [50 poins] For ach of h following giv a shor answr (- snncs or a formula) [5 poins ach] Explain qualiaivly (a) how w acclra

More information