X:2ingoverpaymentoftruthfulmechanisms,thebudgetfeasibilityframeworkena - PDF document

X:2ingoverpaymentoftruthfulmechanisms,thebudgetfeasibilityframeworkenablesdesigningmechanismswithfavorableapproximationguaranteesforbroadclassesofutilityfunctions(seee.g.[Singer2010;Chenetal.2011;Dobzinskietal.2011;Beietal.2011]).Thesecondinformationallimitationisduetotheonlinearrivaloftheagents.Fol-lowingtheliteratureononlinealgorithms,wecanmakethreemaindistinctionsinmodelingthewayinwhichthearrivalsequenceoftheagentsisdetermined.Themostgeneralistheadversarialmodel,whereanadversarychoosesthearrivalsequenceoftheagentsandtheircostinamannerthatyieldstheworstpossibleoutcomeforthemechanism.Althoughinprincipalwewouldlikeourmechanismstoberobusttosuchinput,suchastrongassumptionleaveslittlehopeformakingformalperformanceguarantees.Apopularalternativeisthesecretarymodelwhichassumesthevaluesarechosenadversarially,thoughthearrivalorderischosenuniformlyatrandomfromthesetofallpossiblepermutationsoftheagents.ThisisthemodelconsideredbyDynkin[Dynkin1963]forchoosingtheelementwiththehighestvaluefromanonlinesequence,andhasbeenwidelyusedsincetheninvariousotherproblemdomains(seee.g.[Babaioffetal.2007]).Lastly,aslightlystrongerassumptionisthattheeachagentisindependentlydrawnfromsomeunknowncommondistribution(seee.g.[KleinbergandLeighton2003;Babaioffetal.2011;BesbesandZeevi2009]).Aprocurementmarketcanthereforebedenedintermsoftheobjectivefunctionthebuyerwishestooptimize,andthearrivalmodeloftheagents.Thegoalistode-signtruthfulbudgetfeasiblemechanismsthatmaximizethebuyer'sutility.Therearetwocommonapproachesforagentstoexpresstheircost.Abiddingmechanismlearnsthecostsbysolicitingabidfromeachagentuponitsarrival.Thebiddingmodelisoftenconsideredunnecessarilycomplex,especiallyfromthestandpointofthebidders,andapreferredalternativeisthepostedpricemechanism.Apostedpricemechanismpresentseachagentwitha(possiblydifferent)price,andtheagentcaneitheracceptorrejectthemechanism'soffer.Postedpricemechanismsarecompellingduetotheirsimplicityandarethemostcommonlyusedformofpricing.Assuch,thereislargebodyofalgorithmicresearchonpostedpricemechanisms(seee.g.[Chawlaetal.2010]).Naturally,thebiddingmodelisstrongerthanthepostedpricemodel,sinceamech-anismthatsolicitsbidscanalwayssimulateapostedpricemechanismbydecidingonanofferpricewithoutobservingtheagent'sbid,andpaytheagentonlyifherbiddoesnotexceedtheoffer.Therearecases,however,wherebidscannotbesolicitedandonlypostedpricemechanismscanbeimplemented.Willthemarketsuffer?Thisisthemainquestionwewishtoaddressinthispaper:Arepostedpricesaspowerfulasbiddinginonlineprocurement?1.1.PostedPriceMechanismsinOnlineProcurementInthispaperwestudypostedpricemechanismsinonlineprocurementmarkets.Weaddressthequestionabovebystudyingvariousonlineprocurementmarketscharac-terizedbythebuyer'sutilityandthearrivalorderoftheagents.Itiseasytoshowthatthemostgeneralclassofobjectivesforwhichbudgetfeasiblemechanismscanbeobtainedisthatofsubadditiveutilityfunctions,andwhentakingcomputationlimita-tionsintoaccount,themostgeneralclassknownistheclassofnondecreasingsubmod-ularutilityfunctions.Wewillthereforefocusourattentiononthisclass.Themainresultinthispapershowsthatthereareprocurementmarketswherepostedpricemechanismsarecompetitivewithbiddingmechanisms.ACMJournalName,Vol.X,No.X,ArticleX,Publicationdate:February2012. X:4(2)Thesecretarymodel:Theagentscostsarechosenbyanadversary,thoughtheirarrivalorderisapermutationthatisdrawnuniformlyatrandomfromthesetofallpossiblepermutationsovertheagents.1(3)Thei.i.dmodel:Ateachtimestep,anagentisdrawnfromsomeunknowndis-tribution.Weusethismodelonlyforsymmetricutilityfunctions,i.e.functionsforwhichthevalueonlydependsonthecardinalityoftheset.Insuchcasesthismodelisequivalenttohavingasequenceofcostsdrawnfromanunknowndistribution.Theabovemodelsaredescribedinadecreasingorderofgenerality:thei.i.d.modelisaspecialcaseofthesecretarymodelwhichisaspecialcaseoftheadversarialmodel.2.2.MechanismsAmechanismM=(A;p)isapairthatconsistsofanallocationfunction2A:Rn+!2[n]andapaymentfunctionp:Rn+!Rn+.TheallocationfunctionselectsasubsetofagentsS=A(b)givenasetofbidsb,andweuseAi(b)todenotetheindicatorfunctionthatreturns1ifagentaiisallocatedand0otherwise.Thepaymentfunctionreturnsavectorofpaymentsthatdescribestheagents'compensation,wherepi(b)isthepaymenttoagentaiwhenthebidvectorisb.Inthecasewhereagentsarriveinanonlinefashion,asinourmodel,theinputisreceivedinasequentialmannerandthemechanismdecidesontheallocationandpaymentateverystageiasagentaiappearsandplacesherbid,andthemechanismreturns(Ai(b1;:::;bi);pi(b1;:::;bi)).TheallocationisthesetS=fai:Ai(b1;:::;bi)=1gandthetotalpaymentsarePni=1pi(b1;:::;bi).Forbrevity,whenitwillbeclearfromthecontext,wewillusepitodenotethepaymentforagentai.Inourmodelweseektruthful(incentivecompatible)mechanismswherereportingthetruecostsisadominantstrategyforagents.Formally,amechanismM=(A;p)istruthfulifforeveryai2Nwithcostciandbidbi,andeverysetofbidsb�ioftheNnfaigagents(orfa1;:::;ai�1gagentsintheonlinearrivalcase)wehave:pi(ci;b�i)�ci
Ai(ci;b�i)pi(bi;b�i)�ci
Ai(bi;b�i)Arandomizationovertruthfulmechanismsisauniversallytruthfulmechanism.Animportantpropertywerequireisthatthemechanism'spaymentsshouldnotexceedthebudget:PipiB.Thispropertyiscalledbudgetfeasiblity.Inaddition,ascommoninmechanismdesign,weseeknormalized(ai=2Simpliespi=0),individuallyrational(pici)mechanismswithnopositivetransfers(pi0).Inthispaperwewillbeparticularlyinterestedinpostedpricemechanisms.Suchmechanismsoffereachagentaprice(mayofferdifferentagentsdifferentprices),andtheagentcanacceptorrejectthemechanism'soffer.Sinceagentsareassumedtoberational,anagentacceptsthepriceifhercostisbelowthepriceofferedbythemech-anismandrejectsitotherwise.Toeachagentthatacceptstheoffer,themechanismmustpayatleastthepriceitwasoffered.Wewillalsodiscussthebiddingmodelwherethemechanismsolicitsbidsfromtheagentsastheyarriveandmakesanirrevocabledecisiononwhethertheagentisallocatedandhowmuchsheispaid.Theobjectiveistomaximizetheutilityfunctionunderthebudget,i.e.allocatetothesubsetofagentsSthatyieldsthehighestvaluepossible,underalltheconstraintsdiscussedabove.Wewillcompareourmechanismsagainstthemostdemandingbench- 1ThisassumptiononthearrivalisknownasthesecretarymodelduetoDynkin'scelebrated“secretaryproblem”,wherethisassumptionisintroduced.2Althoughthemechanismprocuresservicesfromagentsweusethetermallocationfunctiontobeconsistentwithtraditionalmechanismdesignterminology.Sincethemechanismallocatesbudgetresourcestotheselectedagents,thistermseemsappropriateinthiscaseaswell.ACMJournalName,Vol.X,No.X,ArticleX,Publicationdate:February2012. X:6 APostedPriceMechanismforunknowndistributions Initializetheconstants=1=10,r=2,a=4000,z=300000. Withprobability1=2: Considerstatesp1;p2;:::;pmwherep1=B n,pi+1=r
pifor1im,andB rzpmB z. Initializestatei=m. TEST(i):Setpricep=pi. 1.Offerpriceptothenextanp Bsellers,oruntilthesellersorbudgetrunsout. Incasemorethana(1+)offersareaccepted,thenstopofferingandmovetostep2. 2.Lettbethenumberofsellerswhoaccept. i.Ifa(1�)ta(1+)thengotoTEST(i). ii.Ift�a(1+)thenupdateitoi�1andgotoTEST(i�1). iii.Ifta(1�)thenupdateitoi+1andgotoTEST(i+1). Withprobability1=2: AllocatetotherstagentwithcostciB Thealgorithmusesstates1;2;:::;mwithassociatedpricesp1;p2;:::;pmthatformageometricprogressionwithcommonratior.TheprogressionstartsatB=n,sothatp1=B=n;B=(z
r)pmB=z,andpi+1=r
pifor0im.Themainloopofthealgorithmisaloopthat,instatei,runsasubroutineTEST(i)whoseoutputisanelementoffi�1;i;i+1g.TheoutputofTEST(i)becomesthenewstate.SubroutineTEST(i)operatesasfollows.Itofferspricep=pitoanp=Bbidders,forsomeconstanta.Ifthenumberofacceptedoffersisevergreaterthana(1+)itquitsthesubroutineimmediatelyandoutputsstatei+1.Ifthenumberofsuccessesislessthana(1�),thenitoutputsstatei�1.Otherwiseitoutputsstatei.Let`betheindexsuchthatB=p`�1n
s(p`�1)andB=p`n
s(p`)wheres(p)istheprobabilitythatasellersellswhileofferingatpricep.(NotethatB=pisadecreasingfunctionofpwhereass(p)isnon-decreasing,hencep`isundenedifandonlyifB=pmn
s(pm).Butinsuchacasethesecondhalfofouralgorithm—whichwithprobability1=2sellstoasingleplayer—isa2r2zapproximationaccordingtoLemma3.1.)LEMMA3.1.E(jOPTj)2r
B=p`PROOF.Consideranyspecicrealizationoftherandomvariablesrepresentingthecostsofeachseller.DeneO1=fijcip`�1andi2OPTgO2=fijci&#x-278;p`�1andi2OPTgO3=fijcip`�1g:WeobtainourupperboundonE(jOPTj)bycomparisonwiththeexpectedcardinalitiesofO1;O2;O3asfollows.E(jOPTj)=E(jO1j+jO2j)E(jO3j)+E(jO2j)=ns(p`�1)+E(jO2j)ns(p`�1)+B p`�1B p`�1+B p`�1=2rB p` ACMJournalName,Vol.X,No.X,ArticleX,Publicationdate:February2012. X:8 ::: ` `+1 `+1 `+2 `+2 ::: `+3 Fig.1.Actualmarkovchainwithiforil+1 ::: 0 1  2  1� :::  1� Fig.2.NewmarkovchainPROOF.Thecouplingiseasytodescribe.WhenthetwoMarkovchainsareinstates(i;k),respectively,ifi2GSwecouplethemarbitrarily,e.g.byupdatingiandkinde-pendentlyusingthetransitionprobabilityoftheirrespectiveMarkovchains.Ifi62GSleti0denotetheneighboringstatethatisclosertoGSthani:i0=i+1ifijforallj2GS,andi0=i�1ifi&#x-333;jforallj2GS.Letidenotethetransitionprobabilityfromitoi0inM1,andletidenotethetransitionprobabilityfromitoitselfinM1.Ourcouplinginstate(i;k)worksasfollows.Withprobabilityweupdateitoi0andktok�1.Withtheremaining1�probability,weupdatektok+1andupdateiasfollows:ittransitionstoi0withprobabilityi�,toiwithprobabilityi,andto2i�i0(theotherneighboringstate)withprobability1�i�i.Itisimmediatefromourdenitionofthecouplingthatif(it�1;kt�1)and(it;kt)aretwoconsecutivestatepairssuchthatit�162GS,then
(it)�
(it�1)kt�kt�1:(1)Thelemmafollowsbysumming(1)overalltimestepsprecedingt. LEMMA3.4.Considerasequenceofphasesinouralgorithm'sexecutionbeginningwithTEST(i)wherei62GSandendingimmediatelybeforetherstsubsequentinstanceofTEST(j)suchthatj2GS.Intotalduringthissequenceofphases,thealgorithmmakesnomorethan(anp`=B)(
(i))offersinexpectationandspendsnomorethana(1+)p`(
(i))ofitsbudgetinexpectation,where(
)isthefunction(k)=r r��(1�)r2
(rk�1):(2)PROOF.Let`0denotetheminimumelementofGS(either`or`�1).Foragivenvalue
,thereareatmosttwostatesisuchthat
(i)=
,namelyi=`+
andi=`0�
.Instate`+
,thealgorithmmakesanpi=B=anp`r
=Boffersinexpectation,anditspendsatmosta(1+)pi=a(1+)p`r
becausesubroutineTEST(i)stopsmakingoffersaftera(1+)offershavebeenaccepted.Instate`0�
thealgorithmmakesanp`0r�
=Boffersinexpectation,anditspendsatmosta(1+)p`0r�
.Thus,foragivenvalueof
,theexpectednumberofoffersmadeandtheexpectedamountspentinstate`+
arebothgreaterthantheircounterpartsinstate`0�
.Accordingly,itsufcestoprovethelemmafori=`+
.Todoso,weusethecouplingprovidedbyLemma3.3.Ouralgorithmproceedsthroughasequenceofstatesi0;i1;:::;iTsuchthatiTistherststateinthese-quencethatbelongstoGS.MeanwhileMarkovchainM2proceedsthroughasequenceACMJournalName,Vol.X,No.X,ArticleX,Publicationdate:February2012. X:10beforethersttimethealgorithmreachesagoodstate.LetE1(resp.E2)denotetheeventthattheactualnumberofoffersmade(resp.amountspent)inthestartupstageexceedstheboundgiveninCorollary3.5byafactorofmorethan10.ByMarkov'sinequality,eachofE1;E2hasprobabilityboundedby0.1.Imaginerunningthealgorithmforaninnitenumberofsteps,disregardingthefactthatiteventuallyspendsmorethanitsbudgetandmakesmorethanitsallottednoffers.DeneagoodphasetobeanexecutionofTEST(i)suchthati2GS.Deneanexcursiontobethe(possiblyempty)setofoffersmadebetweentwoconsecutivegoodphases.Letx=minB 4p`(a+10ac1);B 40(1+)ac1p`andconsidertherstxgoodphasesalongwiththexexcursionsthatoccurimme-diatelyaftereachofthem.Howmanyoffers,inexpectation,aremadeduringthesexphasesandexcursions?Theexpectednumberofoffersduringthegoodphasesisboundedaboveby(anp`=B)
x.AccordingtoLemma3.4theexpectednumberofoffersduringtheexcursionsisboundedaboveby(anp`=B)c1(r�1)andtheexpectedamountspentduringtheexcursionsisboundedabovebya(1+)p`c1(r�1).LetE3(resp.E4)denotetheeventthattheactualnumberofoffers(resp.amountspent)madeduringtheexcursionsdoesnotexceedthisboundbyafactorofmorethan10.Onceagain,byMarkov'sinequality,eachofE3;E4hasprobabilityboundedby0.1.Ourchoiceoftheparametersa;r;z;hasbeendesignedtoensurethat,assumingeventsE1;E2;E3;E4donotoccur,thecombinednumberofoffersmadeuntiltheendofthexthgoodphaseislessthannandthecombinedamountspentuntilthistimeislessthanB.Thus,alloftheoffersmadeduringtherstxgoodphasesactuallytookplaceduringthealgorithm'sexecution,i.e.beforethebudgetwasexpendedorthemaximumnumberofofferswasreached.Inanygoodphase,theexpectednumberofoffersacceptedisatleasta(1�2).LetE5denotetheeventthatfewerthana(1�3)xoffersareacceptedinalloftherstxgoodphasescombined.ApplyingChebyshev'sInequalitytothissumofindependentBernoullirandomvariablesasintheproofofLemma3.2impliesthatPr(E5)0:1,againbyourchoiceofa;r;z;.Bytheunionbound,theprobabilitythatnoneofE1;:::;E5occurisatleast1 2,andwhenthishappensatleasta(1�3)xoffersareaccepted.Thetheoremfollows,becausex= (B=p`). 3.2.ExtensiontoSymmetricSubmodularFunctionsWhenfisasymmetricsubmodularfunction,itmeansthatthereexistsanondecreas-ingconcavefunctiongsuchthatf(S)=g(jSj)forallsetsS.Itturnsoutthattheargu-mentfromSection3.1carriesthroughtothiscasewithveryfewmodications.AsanupperboundonE[f(OPT)],weusethefollowinglemmawhichgeneralizesLemma3.1.LEMMA3.7.E[f(OPT)]2r
g(B=p`)PROOF.Consideranyspecicrealizationoftherandomvariablesrepresentingthecostsofeachseller.DeneO1=fijcip`�1andi2OPTgO2=fijci�p`�1andi2OPTgO3=fijcip`�1g:WeobtainourupperboundonE[f(OPT)]viathefollowingmanipulation,whoserst,second,andlastlinesfollowfromthefactthatgisaconcavefunctionsatisfyingg(0)=ACMJournalName,Vol.X,No.X,ArticleX,Publicationdate:February2012. X:12ThemechanismrunsDynkin'salgorithmwithprobability1=2toallocatetoanagentwithasufcientlyhighvalue[Dynkin1963].Thatis,withprobability1=2themech-anismsamplestherstn=eagentsandthen,fromtheremaining(1�1=e)nagents,allocatestotherstagenta0forwhichf(a0)argmaxi2f1;:::;n=egf(ai).Inexpectationoverthearrivalorderoftheagentsthisguaranteesthatf(a0)(1=e)
argmaxa2Nf(a).Sincethemechanismisarandomizationovertwopostedpricemechanisms,itistruthfulandindividuallyrational.Budgetfeasibilityisimpliedfromtheconditionin(4a)whichveriesthattheremainingbudgetB0isgreaterthanthepayment.Through-outtherestofthispaperwewilluseOPT(N0)todenotetheoptimalvalueoverthesetofagentsN0,N1todenotethesetofagentswhoareinthesampleandN2=NnN1.LetS=OPT(N),S1=S\N1andS2=S\N2.Letv=maxff(a);a2Ng.Considerthecasewhenf(v)f(OPT)=1024.InthiscasethealgorithmrunsDynkin'salgorithmwithprobability1=2andgetsanapproximationratioof2048e.Sofortherestofthesectionwewillassumethatf(a)f(OPT)=1024;8a2N.Toprovethecompetitiveratioofthemechanism,wewillusethefollowinglemma.LEMMA4.2.Whenf(a)f(OPT) 1024;8a2Nthenminff(S1);f(S2)gOPT(N) 4withprobabilityatleastp9 10.PROOF.LetS=fa1;:::;a`gbetheoptimalsolution.Withoutlossofgenerality,assumethata1;:::;a`aresortedaccordingtodecreasingmarginalcontributions,andletwidenotethemarginalcontributionofai.Sincetheagentsareassumedtoarriveinauniformlyrandomorder,andischosenwithaspecicprobabilitydistributionwehavethatthesampledsetN1isauniformlyrandomsetofN.HenceeachagentisinN1withprobability1=2independentlyofotheragents.ConsidertherandomvariablesX1;:::;X`,s.t.XitakesthevaluewiwithifagentibelongstoN1and0otherwise.SinceN1isauniformlyrandomsetthisimpliesthatXitakesvaluewiwithprobability1=2and0withprobability1=2.SuchatrickofchoosingsetN1sothatXi'sareindependentwasrstintroducedby[Kleinberg2005].LetX=P`i=1XiandX=f(S)�X.Observethatf(S1)X,andf(S2)Xbysubmodularity.ToshowthedesiredpropertiesofXandXwewillusetheChernoffbound:THEOREM4.3.(ChernoffBound)LetX1;:::;X`beindependentrandomvariableswhereXitakevaluesin[0;wi]andlet=E[P`i=1Xi].Then,forany�0wehavethat:Prh`Xi=1Xi(1+)i(e (1+)(1+)) maxiwi(4)Prh`Xi=1Xi(1�)ie�2 2
maxiwi(5)Sinceourassumptionthatmaxiwif(S) 1024andthat=f(S)=2theaboveboundimpliesthat:PrhXf(S) 4i=PrhX3f(S) 4i1=20(6)ACMJournalName,Vol.X,No.X,ArticleX,Publicationdate:February2012. X:14whichimpliesthatf(S)isa128approximationoff(S2).Finally,inthecasewhereB0piandB0&#x-278;B=2,wehave:B 2B0pi=B tf(Sj[faig)�f(Sj)B t
f(ai)64B f(S2)
f(ai)64B 256(17)Hereequation17isacontradiction.HencethecasethatB0�B=2neverhappens. TherearetwocasestoshowthatthealgorithmisO(logn)competitiveratio.(1)Iff(v)f(OPT)=1024thenweALG2isa1024eapproximationwhichisrunwithprobability1=2.Inthiscasewegeta2048eapproximationasremarkedabove.(2)Iff(a)f(OPT)=1024;8a2N,weconsidertheapproximationratioofALG1whichisrunwithprobability1=2.Withprobability1=2visincludedinN1(EventT1)andwithprobabilityatleast9=10wehavethatf(S2)f(OPT)=4bylemma4.2(EventT2).Hencebyunionboundboththeseevents(EventsT1andT2)happenwithprobabilityatleast4=10=1�(1=10+1=2).Independentlyoftheseeventstsuchthatf(S2)=64tf(S2)=2ischosenwithprobability1=O(log(n))whichresultsinanapproximationratioof128bylemma4.4.Hencethenalapproximationratiois1 24 101 log(n)1 41 128=1 O(log(n)).ThisgivesusTheorem4.1whichisthemainresultofthissection.5.ACONSTANT-COMPETITIVEMECHANISMINTHEBIDDINGMODELInthissectionwepresentabiddingmechanismfornondecreasingsubmodularmar-ketsinthesecretarymodel.Aseachagentarrives,themechanismcollectsisbidandmustmakeanirrevocabledecisionofwhetherornottheagentshouldbeallocatedandhowmuchtheagentshouldberewarded.Wewillshowthefollowingtheorem.THEOREM5.1.Inthebiddingmodel,foranynondecreasingsubmodularutilityfunction,thereisauniversallytruthfulbudgetfeasiblemechanismwhichisO(1)-competitive.Theresultoftheprevioussectionshowedthatthegapbetweenpostedpricemech-anismsandbiddingmechanismsinnondecreasingprocurementmarketsinthesecre-tarymodelisatmostO(logn).Theabovetheoremandtheorem3.6whichareaspecialcasesofmodelconsideredinsection4,hinttowardsthepossibilitythatthegapintheorem4.1canbereducedtoO(1).Likethemechanismfromtheprevioussection,themechanismherewillalsosampletheagentsanduseathresholdvaluetodecideontheallocation.Intheprocessofestimatingthethreshold,themechanismwillcomputeanapproximationoftheoptimalsolutionofthebidsofthersthalfoftheagents.Amodicationofthegreedyalgorithmthatsortsagentsaccordingtotheirdensity–theirmarginalcontributionnormalizedbytheircost–achievesanapproximationratioofe=(e�1)[Khulleretal.1999;Sviridenko2004]whichisknowntobeoptimal[Feige1998].WeuseA(N1)todenotethevalueofthisalgorithmcomputedoverasubsetofagentsN1N.ACMJournalName,Vol.X,No.X,ArticleX,Publicationdate:February2012. X:16AsinLemma4.2letS=fa1;:::;a`gbetheoptimalsolutionOPT(N)andwithoutlossofgeneralityassumethata1;:::;a`aresortedaccordingtodecreasingmarginalcontributions.Letwidenotethemarginalcontributionofai,S1=S\N1;S2=SnS1andW1=Pai2S1wiandW2=Pai2S2wi.Considerthesecondcasewhere8a2N;f(a)f(OPT)=1024.Insuchacasebylemma4.2withprobability9=10wehavethatminff(S1);f(S2)gOPT(N) 4.Giventhiseventwewillshowthatf(S2) 2tf(S2) 64;(18)RecallthatinLemma4.4,weshowedthatusingathresholdwithproperty(18)toal-locatetoagentsinN2ifandonlyiftheratiobetweenthethresholdandtheirmarginalcontributionexceedstheircosttobudgetratio(asdescribedinstep(4)ofthemech-anism),guaranteesthatinexpectationthesetoftheallocatedagentsisaconstantfactorapproximationoff(S2).Hence,showingproperty(18)abovewouldimplythatwithaconstantprobabilitythevalueofthesetofagentsallocatedbythemechanismisaconstantfactorapproximationoff(S2).FromLemma4.2thisimpliesthemechanismisconstantcompetitive.Tocomputethethresholdt,instep(3)weapplythegreedyalgorithmforsubmodularmaximizationunderabudgetconstraintonthesampleN1.Thisalgorithmisguaran-teedtoprovideatleasta(1�1=e)fractionofOPT(N1).Duetothedecreasingmarginalutilitiespropertyofthesubmodularfunction,wehavethatOPT(N1)f(S1)W1andthatOPT(N2)f(S2)W2.FromLemma4.2weknowthatthereisaconstantprobabilityforwhichOPT(N)=4minff(S1);f(S2)g.Thethresholdtcanbeboundedfromabove:t=A(N1) 8OPT(N1) 8OPT(N) 8f(S2) 2(19)Toboundtfrombelowweuse =e=(e�1):t=A(N1) 8OPT(N1) 8 f(S1) 8 f(S) 32 f(S2) 32 f(S2) 64(20)Hencebylemma4.4andlemma4.2wegetthatitisaO(1)approximation. 6.DISCUSSIONANDOPENQUESTIONSInthispaperweinvestigatedonlinebudget-feasiblemechanismdesign.Ourmainpos-itiveresultsareasfollows.Wepresentaconstant-competitivepostedpricemechanismwhenagentsareidenticallydistributedandthebuyerhasasymmetricsubmodularutilityfunction.Fornonsymmetricsubmodularutilities,undertherandomorderingassumptionwegiveapostedpricemechanismthatisO(logn)-competitiveandatruth-fulmechanismthatisO(1)-competitivebutusesbiddingratherthanpostedpricing.ThemainquestionleftopenbythisworkiswhetherthereexistsaO(1)-competitivepostedpricemechanisminthenonsymmetricsubmodularcase.ReferencesBABAIOFF,M.,BLUMROSEN,L.,DUGHMI,S.,ANDSINGER,Y.2011.Postingpriceswithunknowndistributions.InICS.166–178.BABAIOFF,M.,IMMORLICA,N.,ANDKLEINBERG,R.2007.Matroids,secretaryprob-lems,andonlinemechanisms.InSODA.434–443.ACMJournalName,Vol.X,No.X,ArticleX,Publicationdate:February2012.