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CS6840AlgorithmicGameTheorySpring2017Lecture30April14Lecturer19EvaTard


301DataandPriceofAnarchyinRoutingGamesGiventestdataitiseasytodeterminetheactualdelayandgettheoptimalroutingRecalltheoptimalroutingistheroutingofexistingtra14csuchthatweminimizethesumofthedelaysminPdel

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Document on Subject : "CS6840AlgorithmicGameTheorySpring2017Lecture30April14Lecturer19EvaTard"— Transcript:

1 CS6840:AlgorithmicGameTheorySpring2017Le
CS6840:AlgorithmicGameTheorySpring2017Lecture30:April14Lecturer:EvaTardosScribe:RishabGupta 30.1DataandPriceofAnarchyinRoutingGamesGiventestdata,itiseasytodeterminetheactualdelayandgetthe"optimal"routing.Recall,theoptimalroutingistheroutingofexistingtracsuchthatweminimizethesumofthedelays,minPdelay.Furtherrecallthatforagivenedgeewithtracxe,andcostce(xe),thetotalcostalongtheedgeisxece(xe).Thereforethe"optimal"routingobjectivefunctionisminXexece(xe)Ourhopeisthatisthatxece(xe)isaconvexfunction,since,ifso,convexoptimizationcanbeusedtodeterminetheoptimalrouting.Thisisequivalenttoaskingwhether(xece(xe))0=xec0e(xe)+ce(xe)ismonotonicallyincreasing.Ingeneralthesefunctionstendtobeconvex.30.2EmpiricalPriceofAnarchyAllofourtheoremssofararefor ndingupperboundsonthePriceofAnarchy.However,wecanalsoreasonaboutPriceofAnarchyempirically.TheempiricalcostofNashisjustcostNash costoptimalevaluatedonanexample.30.3DataandPriceofAnarchyinAuctionsThedatawecangetfromroutinggamesisdi erentthanwhatwecangetinauctions.Forexample,consideraninternetaddauction,whereplayerihasavalueviforbeingclickedon.Thedatathatwecangetwillcontainlistofauctions,winners,andpayments,butnovi's.Withoutviwecannotevaluatethesocialwelfareoftheauctionandcannot ndtheoptimal.Soourgoalforthissectionistoextractavifromthedata.30.3.1ExtractingvaluesinGeneralizedSecondPriceauctionsTheformofaGSPisasfollowsMultipleaddslots1,2,3,...nSlotihasaclickrateof iPersonjhasarelevanceof j.30-1 30-2Lecture30:April14Ifpersonjgetssloti,P(click)= j iBidsb1;b2;:::;bn.Assumeindexingsuchthatb1 1b2 2:::bn nPaymentperclickforplayericanbecomputedaspi i=bi+1 i+1=)pi=bi+1 i+1

2 i(Assumingb1 1b2 2:::bn
i(Assumingb1 1b2 2:::bn n)Let'sconsidera1-shotfullinformationgame.Inthisgame,everyplayerknowsall j, i,andotherplayer'sbids.Wenowwanttotryandinfervaluesfromthebids.Wecangetaboundonvi'susingtheNashcondition.Weknowplayerj'sutilityinslotiis j i(vj�pj),wheretheoricepjthatjpaysperclickdependsontheslot.Foreaseofnotation,let'ssortplayersbybn n,soplayeriisinsloti.Then,playeri'sutilityinslotiisjust i i(vi�pi).Nowconsiderwhathappensifplayerideviatestoaslotk.Therearetwopossiblecasesfori'snewutility(a)k�i: i k(vi�p0i),wherep0i= k+1bk+1 i,computedbythesameformulausedforslotk.(b)ki: i k(vi� kbk i).Thelasttermisnolongercomputedusingbk+1andplayeriisnowalsoabovek,sotheoriginalkthbidderbecomesthek+1stinthenewordering.UndertheNashcondition,weknowthatforplayeri,beinginslotimaximizeshisutility.Withthisconditionandtheaboveutilities,wecangeneraten�1inequalitiesoftheformutilityslotkutilitysloti.Solvingtheseinequalitieswillgiveusanupperandlowerboundonvi.Speci cally,inequalitieswithk�iconsideredin(a)willgiveuslower-boundsonvi,as i i� k i,theselowerslotso erlessclicksatacheaperprice,whichwouldbepreferableifthevalueviwaslow.Inequalitieswithkiwillgiveusupper-boundsonvi,consideredin(b)willgiveusupper-boundsonvi,as i i k i,thesehigherslotso ermoreclicksatahigherprice,whichwouldbepreferableifthevalueviwashigh.However,withn�1inequalities(fornslots)thiswon'tbeabletopinpointtheexactvaluevi.Anotherissueisthattheseinequalitiesreliedontheassumptionthattheoutcomeofthegameatthisonetimestepisatequilibrium.Itisreasonablethattheplayersarewellenoughinformedtohavetheequilibriumconditionholdateverytimestep