Optimal Bidder Selection in Clearing House Default Auctions

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) Optimal Bidder Selection in Clearing House Default Auctions Rodney Garratt; David Murphy; Travis Nesmith; Xiaopeng Wu 2023-033 Please cite this paper as: Garratt, Rodney, David Murphy, Travis Nesmith, and Xiaopeng Wu (2024). “Optimal Bidder Selection in Clearing House Default Auctions,” Finance and Economics Discussion Series 2023-033r1. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2023.033r1. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Optimal Bidder Selection in Clearing House Default Auctions RodneyGarratt* DavidMurphy† TravisNesmith‡ XiaopengWu§¶ August28,2024 Abstract Central counterparties’ ability to hold successful default auctions is critically important to financial stability. However, due to the unique features of these auctions, standard auction theory results do not apply. We present a model of CCP default auctions that incorporates boththevital,butnon-standard,objectiveofminimizingthelikelihooditsuffersreputationally damaging losses and the potential for information leakage to affect CCP members’ private portfoliovaluations. ThisgivesinsightintothekeyquestionofhowCCPsshouldselectauction participants. In particular, we prove that an entry fee, by appropriately incentivizing some membersnottoentertheauction,canmaximizetheprobabilityofauctionsuccess. Theresultis novel,bothinauctiontheoryandasamechanismforCCPauctiondesign. Keywords: Auctions; Central counterparties; CCPs; Default; Derivatives; Entry mechanism; Financialstability;Systemicrisk JEL:D44;D47;G13;G23 ∗ MonetaryandEconomicDepartment,BankforInternationalSettlements,Centralbahnplatz2,4051Basel,Switzerland. Email:rodney.garratt@bis.org † LawSchool,LondonSchoolofEconomicsandPoliticalScience,HoughtonStreet,LondonWC2A2AE,UnitedKingdom. Email:D.Murphy3@lse.ac.uk ‡ Correspondingauthor. QuantitativeRiskAnalysis,FederalReserveBoard,20thStreet&ConstitutionAvenueN.W., Washington,D.C.20551,UnitedStatesofAmerica.Email:travis.d.nesmith@frb.gov § RotmanSchoolofManagement,UniversityofToronto,105SaintGeorgeStreet,Toronto,ONM5S3E6,Canada.Email: xp.wu@rotman.utoronto.ca ¶ TheauthorsthankparticipantsatFRBChicago’s6thAnnualConferenceonCCPRiskManagement;WFEClear:WFE’s 40thClearing&DerivativesConference;FRBChicago’sFinanceseminar;theBoard’sQuantitativeFinanceseminar; andtheEconomicsofPaymentsXIIfortheirhelpfulobservations.Inparticular,wearegratefultoAniketBhanuand AndrewSweetingfortheirthoroughcomments.HaoxiangZhuwasafullcontributortothisworkuntilDecember2021. Thisresearchdidnotreceiveanyspecificgrantfromfundingagenciesinthepublic,commercial,ornot-for-profitsectors. Theanalysisandconclusionssetfortharethoseoftheauthorsanddonotindicateconcurrencebyothermembersofthe researchstaffortheBoardofGovernorsoftheFederalReserveSystem,theBankforInternationalSettlements,orany otherbody. 1

1 Introduction Centralclearinghaslongbeenaubiquitousfeatureofexchange-tradedderivatives. Itisacritical marketmechanismtomitigatecounterpartycreditrisk: alllargefuturesexchangeshaveanaffiliated clearing house, and trades are novated to this central counterparty (or ‘CCP’) following their executionontheexchange. Inresponsetothe2008GlobalFinancialCrisis,centralclearingwas mandatedforover-the-counter(OTC)derivativestoo,withtheG-20declaringthat‘allstandardized OTCderivativecontractsshouldbe...clearedthroughcentralcounterpartiesbyend-2012atthe latest’. OTCderivativesclearingmandateshavenowbeenenactedinmostleadingjurisdictions. Asaresult,majorCCPsarecriticaltothefunctioningoffinancialmarketsandareoftendeemedto besystemicallyimportantfinancialinfrastructures. TheseCCPsintermediatesubstantialamounts ofrisk,asconsiderationofthetotalinitialmargintheyrequireindicates. ThethreelargestCCPsfor exchange-tradedderivativesintheUSandUKrequiredover$300billionininitialmarginatthe beginningof2024,forinstance,whilethemarginrequirementsofthethreelargestOTCderivatives CCPstopped$360billion. Murphy(2012)andKingetal.(2023)discussthesystemicimportance andrisksofcentralclearinginmoredetail. TheroleofaCCPistositbetweenmarketparticipants,guaranteeingtheirperformancetoeach other and acting as the counterparty to all cleared trades. If one of their members defaults, the clearinghousemustclose-outthedefaulter’sportfolio. Thus,theyshouldmakeitfarlesslikelythat afailureofonemarketparticipantcausesdirectlossesatanotherbyactingasabuffer,preventing systemicrunsasinZawadowski(2013). Inordertobeabletodothis,CCPsmusthavesufficient resourcestoabsorbanyplausiblelossesresultingfromthedefaultmanagementprocess. Alossof confidenceinaclearinghousecancauseorincreasealossofconfidenceinthefinancialsystem,as Bernanke(2015)discusses,sothisisanimportantissue. DifferentCCPsusedifferentmethodologiestosizetheseresources,subjecttoapplicableregulatoryminima. Thistopichasreceivedsubstantialattentionintheliterature,withafocusonthe adequacyofresourcesinvarioussituations. See,e.g.,Antinolfietal.(2022),Boisseletal.(2017),Capponietal.(2017),Cerezettietal.(2019),KuongandMaurin(2023),MurphyandNahai-Williamson 2

(2014),Rec(2019a,b). Aparticularissueisthepotentialclusteringofdefaultsandepisodicmarket illiquidity,asinAzizpouretal.(2018),Carlinetal.(2007). Therelatedandimportantquestionof CCP(funding)liquidityriskhasalsobeenstudied: see,e.g.Cont(2017)andKingetal.(2023). In contrast,theclose-outprocess,andparticularlythedesignofCCPdefaultauctions,isrelatively less studied, despite it being crucial to the losses experienced by the clearing house in default management. Defaultactionsarethereforethefocusofthispaper. Whenthedefaulter’sportfolioisrelativelysmallcomparedtoavailablemarketliquidity,the defaultmanagementproblemisstraightforward: inmarketswithcentrallimitorderbooks,the CCP can simply enter orders to liquidate the defaulter’s portfolio. Similarly, in markets which tradelargelyonarequestforquotebasis,quoteswillreadilybeavailable,andhenceanauction maynotbenecessary. The problem can become much more challenging with the default of larger or concentrated portfolios. Suchdefaultscancreatesignificantstress,astheCCP’sabilitytocloseoutthedefaulter nearthemarketpriceisuncertain.1 Moreover,unliketheprobleminmanytradingsituations,the CCPmusttradeaparticularportfolioinarelativelyshorttimeframe: ithasneitherthemandate nor the loss-absorbing resources to bear market risk. Therefore, it is common for the CCP to auctionthedefaulter’sportfolioeitherasis,orafterithasbeenhedged. BothLehmanBrothers’ $9trillioninterestrateswapsportfolioclearedbytheLondonClearingHouse,andtheir$2billion exchange-traded futures and options portfolios cleared by Chicago Mercantile Exchange, were liquidatedviamultipleauctions,forinstance.2 TheanalysisofCCPdefaultauctionsistherefore practicallyimportant. Itisalsotechnicallynon-trivial,astheseauctionshavenovelfeatures. CCPshaveafixedamountoffundedresourcestomanagedefaults,structuredinwhatisknown as their default waterfall. The first tranche of these resources are provided by the defaulter in the form of initial margin. This tranche is followed by the defaulter’s contribution to a layer of 1 AgoodexampleisthestressattheNewZealandclearinghousecausedbythedefaultofStephenFrancisdescribedin Coxetal.(2016). 2 SeeFlemingandSarkar(2014)andLCH.Clearnet(2008)forLCHandValukas(2010)forCME.Forfurtheranalysisof Lehman,seeWigginsandMetrick(2019) 3

mutualizedresources: theCCP’sguaranteeordefaultfund. Thenthereisa(typicallythin)layerof theCCP’sownresources,knownasskininthegame. Afterthiscomestherestoftheguarantee fund. Thus, first the defaulter pays, then the CCP does, then non-defaulting clearing members. Anyresourcesfromthedefaulterthatarenotneededtomanagetheclose-outarereturnedtothe defaulter’sestate. TherearelegalrequirementsinleadingjurisdictionsforCCPstoactreasonably intheirclose-out,3 buttheserequirementsleavesubstantialflexibilityindefaultmanagement. All of this means that CCPs are much more concerned with getting an acceptable bid in a default auction than in the price of that bid. If it cannot completely close-out the defaulter’s portfolio without suffering a loss bigger than the resources that they have provided, not only will it face financialloss,butalsoitwillalmostcertainlysufferreputationaldamage. Obtaininganacceptablebidforthedefaulter’sportfolioisbynomeansguaranteed. Forinstance, inthe2018defaultbyEinarAasattheNasdaqNordicclearinghouse,thefirstattemptatauctioning the portfolio failed, and the second attempt resulted in a loss of €114 million in excess of the resources Aas had provided to the CCP, resulting in losses to both the CCP’s skin in the game anditsguaranteefund. Subsequently,itwasrevealedthatthesecondauctionhadnotliquidated the portfolio, but rather had just established hedges. In both auctions, Nasdaq Nordic limited participationtofourfirmsthatithadselected.4 Standardauctiontheorysuggeststhatrevenueisincreasedbyattractingasmanybiddersas possibletoanauction.5 However,thisisnotnecessarilythebeststrategyinCCPdefaultauctions duetoaphenomenonknownasinformationleakage. ThetriggereventforaCCPauction—the default of one or more clearing members—is public knowledge, as are the market movements associatedwiththedefault. However,beforetheauctionbegins,onlytheCCPhasfulldetailsof 3 SeeBraithwaiteandMurphy(2017)forfurtherdetails. 4 Fordetailsonthedefaultandauctions,seeClancy(2018b),Clancy(2018a),andMourselas(2019). Mourselas(2021) coversthedetailsthatemergedlateraboutthefailureofthesecondauctionandsubsequentregulatoryfine.Also,see BoxA:TwodefaultsatCCPs,10yearsapart,byS.BellandH.Holden(Faruquietal.,2018,pp. 75–76in)formore analysis,andMcConnellandSaretto(2010)foradifferentexampleofauctionfailures. 5 AuctiontheoryiscoveredinMilgrom(2004)andKrishna(2010);bothcoverstandardmodelsandequivalenceresults. 4

theportfoliotobeauctioned. Ofcourse, ithastoprovidethisinformationtobiddersand, once theyknowit,theycaneitherchoosetobid,ortotradeagainsttheCCP.Thus,thereisaconcern that inviting more and more bidders will lead to an adverse change in market prices between the portfolio being revealed and bids being submitted. This period cannot be very short, as a defaulter’sportfoliomaybelarge,andbiddersneedsometimetopriceit. ACCPdefaultauctionis thusanalogoustoapredictabletrade,asdiscussedbyBessembinderetal.(2016),combinedwith fundamentalparticipationquestions. NasdaqNordicexplicitlystatedthatitlimitedparticipation initsauctionAas’positionstoavoidinformationleakage,asMourselas(2019)discusses. Therecent paperpreparedbyregulators,CPMI-IOSCO(2020),emphasizesboththistopicandthequestions ofwhotoincludeinanauctionandhowtoincentivizebidders,aspartofitsbroaderdiscussionof issuesforCCPstoconsiderindefaultauctions. Thatpaperraisesthesetopicswithoutproviding guidanceonhowtosimultaneouslychooseandincentivizebidders,limitinformationleakage,and successfullyconductanauction. Tosummarise,clearinghousedefaultauctionshavetwosalientfeaturesthatsharplydistinguish themfromstandardauctions. First,theCCPwantstominimizetheprobabilityofusingitsownor non-defaultingmembers’resources,nottomaximizeincomefromtheauction. Second,thepossible negative effects of information leakage means that it is not desirable to attract bidders who are unlikelytosubmitcompetitivebids. Thislateraspectbearssomeresemblancetotheliteratureon endogenousauctionparticipation(LauermannandWolinsky,2017,MenezesandMonteiro,2000). Thecombinationofthesetwoaspectsisparticularlynovelfromanauctiontheoryperspective. Ourcontributionistomodelthesalientfeaturesofaclearinghousedefaultauctionandtodesign anoptimalprocessforconductingitwhileallowingendogenousparticipationdecisions. Different CCPswillhavedifferentrulesforconductingauctions,anddifferentstrategiesforrecoveringfrom an unsuccessful auction, so we focus on the common goals of the default management process across CCPs. We show how the opposing effects of information leakage—where having more bidders tends to lower the value of the object and hence leads to lower prices—and the need tomaximizetheprobabilityofasuccessfulbidcanleadtoanoptimumnumberofbiddersthat 5

is strictly less than the number of available bidders. The key aspect from a mechanism design perspectiveistheuseofanoptimallyselectedentryfeetogetgoodbidderstoself-select. Biddersusecommonlyknowninformationaboutthedirectionandlikelysizeofthedefaulted position, and private information about the value to them of adding the position to their own portfolio, to decide whether to pay the entry fee, become more informed about the actual size of thedefaulted position, andthen decide whether or notto submit abid.6 We provethat even withareservationprice,ifthecostofinformationleakageishighenough,aCCPmaximizesits probabilityofreceivingabidaboveitsreservationpricebysettinganon-zeroentryfee. Although theentryfeelimitsparticipation,itdoessoinanequitablefashioninthatitissetbeforethedefault occurs. Furthermore,theentryfeeexcludesbidderswhoarenotlikelytobidhighenoughtowin theportfolioandconsequentlywhosemainimpactistoincreaseinformationleakage. 2 Literature Review Inadditiontothescholarshipcitedabove,variousauthorsexploreaspectsofcentralclearingon marketfunction. Forexample,LoonandZhong(2014)findthatcentralclearingimprovesmarket liquidity. Duffie et al. (2015) and Grothe et al. (2023) find that CCPs’ impact on the demand for collateraldependsonthemarketstructureandthatifmarginisrequiredfornon-centrallycleared trades,thencentralclearinglowersthedemandforcollateral. Adifferentstrandoftheliteraturefocusesmoreonclearinghousedefaultmanagement. Cont (2015)andArmakollaandLaurent(2017)bothfocusontheimpactoflossallocationinCCPdefault waterfalls. Cerezettietal.(2019)isclosertothespiritofthispaperasitlooksathowtooptimize CCPdefaultprocesses,butitfocusesonhedginganddoesnotanalyzeauctionsatall. Koeppletal. (2012)considerstheimpactofdefaultmanagementonthemarket,demonstratinghowconcerns over default at a CCP can harm market liquidity. However, this research looks at the impact of centralclearingpriortoadefaultactuallyoccurring. 6 Thedecisionbyeachpotentialbidderwhethertopaytheentryfeeisakeypartofthisprocess.Apre-committedfee,such asadefaultfundcontributionatriskfromafailuretobid,wouldnotachievethesameeffect. 6

Afterdefaultoccurs,aCCPwilllikelyneedtoliquidatethedefaulter’spositionsandcollateral. Thedefaultofalargeinstitutioncancauselargedisorderlycollateralliquidations,asOehmke(2014) discusses. Given that CCP default management is intended to make managing a default more orderly, this is unhelpful. The other liquidation—that of positions—has received less attention, whichmotivatesourstudyoftheproblem. Thereare,however,threepapersthatdirectlyanalyzeCCPdefaultauctionsofpositions. The closestpapertothisone,Ferraraetal.(2019),considersvariousdesignsofCCPdefaultauctions theoretically, but makes the standard assumption that the CCP seeks to maximize revenue. As we detail below, that assumption ignores crucial features of the CCP’s payoff. Their paper also assumesthatthenumberofbiddersisfixed,soitissilentonthequestionofwhoshouldactual beinvitedtoparticipateintheauction. Theauthorsdoexaminethepossibilitythatpoorbidscan face a negative externality due to low competitiveness in the bidding process, which has some similaritiestotheexternalitydrivenbyinformationleakageinourmodel. However,thereisno endogenousentryintheirframework,sotheimplicationsaremoremuted.7 Inrelatedresearch,Oleschak(2019)considersfirstpricesingleitemCCPdefaultauctionswhere biddershaveprivatevaluesandshareeventuallosseswiththeCCP.Theauthordoeslookatthe impactofbeinginvitedtotheauctionornot,findingthatinvitedbiddersarebetteroffthanthose whoarenotinvitedtotheauction. However,thismechanismdependsontheCCPbeingableto pick bidders with high private values. The inability to do so is exactly what we seek to study: whetheraCCPcanincludebidderswithhighvaluationsandconverselyexcludethosewithlow valuationswithoutknowingorevenhavingasignalaboutprivatevaluations. Lastly,therearesimilaritiesbetweenthispaperandHuangandZhu(2024),whichbuildson DuandZhu(2017). However,toavoidpriceimpacts,theformerpaperassumesthatbiddersare 7 Thisnegativeexternalityisenoughhowevertobreaktherevenueequivalencebetweenfirstandsecondpriceauctions; theauthorsfindthatasecondpriceauctionwithlosssharing,ratherthanfirstpriceauctionswithorwithoutpenalty, increasestheliquidationvalueoftheportfolio.Thispaperalsousesasecondpriceauctionframework,butfocuseson designinganeffectivemechanismtomaximizethechanceofsuccesswithstrategicparticipation,ratherthantheimpact expostlosssharingwhentheauctionisnotassuccessfulunderfixedparticipation. 7

infinitesimal and, similar to the other two papers on CCP auctions, focuses on analyzing what happensifanauctionresultsinlossesthatmustbeabsorbedbytheguaranteefund. Itexamines thebenefitsofjuniorizationofguaranteefundcontributionsinsuchcircumstances. Interestingly, all three papers focus on the impact of loss sharing among clearing members whenthedefaultauctiongoespoorly. Butineachcase,thereisnoreputationalcosttotheCCPof suchapoorauctionoutcome. Insharpcontrast,wearguethatCCPsaresostronglymotivatedby thereputationalcosts,whichwouldresultfromneedingtoapportionlossestoitsmembers,that avoidinguncoveredlossesistheirprimaryauctionobjective. Postulatingloss-avoidancerather than revenue maximization as the auction objective leads to a novel auction problem, not just relativetothescantliteratureonCCPauctions,butalsointhebroaderauctionliterature. Besidesspecifyingarealisticauctionobjective,wefurthertackletherelatedquestionofhowto constructthepoolofbidders. ThisisakeyissueraisedinCPMI-IOSCO(2020),anditischallenging duetotheconcernsraisedaboutinformationleakage. Asmentioned,thereisaliteraturearound predictabletrades. AdmantiandPfleiderer(2015)arguesforapositiveviewhere. Thepertinent concernsaboutinformationleakageare,however,muchmoreconsistentwithanegativeview,asin BrunnermeierandPedersen(2005)andCarlinetal.(2007). Indeed,theworse-casescenarioisthat theliquidationofadefaulter’sportfoliotakesonthecharacteristicsofafiresale: seeCovaland Stafford(2007)andKuong(2020). Furthermore,limitedparticipationisnotmerelyatheoretical concernasitseemstobeacommonfeatureoftherelativelyfewCCPauctionsobserved. Besides thepriordiscussionofNasdaqNordic,wheretheCCPselectedfourbiddersonly,itwasreported in Sourbes (2015) that LCH’s auctions of Lehman portfolios had (depending on the currency concerned)aroundfiveparticipants. There also exists a related market microstructure literature on how information leakage, or front-running, affects trading. Burdett and O’Hara (1987) model how an institutional investor constrainsthenumberofdealersitapproachestoexecutealargetradeduetoinformationleakage. InHendershottandMadhavan(2015),thelevelofinformationleakagedeterminesthevenuethat adealerutilizesforatrade. Morerecently,BaldaufandMollner(2024)endogenizetheimpactof 8

informationleakageinthesubsequenton-markettrading. Similarlytothislastpaper,wefocuson informationthatisavailabletoauctionparticipants. TherearecriticaldifferencesbetweenthisliteratureandtheCCPproblem,however. First,inthe paperscited,theinvitationtobeinanauctionrevealsinformationtothoseinvited,whileaCCP’s needtoconductanauctionispublicknowledge. Second,inpreviousinformationleakagepapers, therearenoconsequencestoanauctionfailing,whileforCCPsanauctionfailurecausesmaterial lossandwouldsignificantlydamageitsreputation,potentiallytothepointthattheCCPwouldno longerremainviable. Third,inaCCPmemberdefault,thedirectionthemarketmovedtocausethe clearingmemberdefaultrevealsthedirectionofthedefaultingportfolio. Consequently,theoptimal strategyinBaldaufandMollner(2024)ofrequestingtwo-sidedbidsisnotavailable. Asthefocus hereisonthepriceimpactofinformationleakageontheauctionitself,weexogeneouslyspecify thepriceimpact,whichresemblesBaldaufandMollner(2024)butdonotmodelthemechanism throughwhichitoccursliketheydo. Insummary,themicrostructureliteraturefocusesonhow andwheretosearchforbestexecutionofregularandrepeatedtradestominimizetheimpactof informationleakageontransactioncosts. Incontrast,wefocusonhowtooptimallyendogneize bidders’participationtominimizetheimpactofinformationleakageonthesuccessoftheextremely irregular,butcriticallyimportant,auctionofadefaulter’sportfolio. TheinstitutionalcharacteristicsofCCPdefaultauctionsmeanthatouranalysisisnon-standard andarguablynovel. Butitdoesconnecttocertainstrandsofthebroaderauctionliterature. Lou etal.(2013)findsthatevenintheliquidTreasurymarket,announcedauctionscancausevariations invaluationsduetodealers’risk-capacity. LevinandSmith(1994)endogenizesentryandfindsthat theresultsoftheauctioncandivergefromthosepredictedbyananalysisthatignorestheentry question. FurthermoreLauermannandWolinsky(2017)andMenezesandMonteiro(2000)both provethatincircumstanceswithendogenousentryrevenuecandecreasewithanincreaseinthe numberofparticipants. Similarly,inourpaper,entrytakesacentrestageinlinewiththediscussion in CPMI-IOSCO (2020). Our model explicitly allows that more competition is not necessarily desirable. ThisisconsistentwiththeempiricalfindingsinHongandShum(2002)(althoughwe useprivatevaluationsratherthancommonones)andthemodelofGlebkinandKuong(2023). A 9

keydifferenceisthathereendogenousentryisconsideredinanauctionwheretheCCP’sobjective isnotmaximizingrevenue. Inaddition,potentialauctionparticipantsarerisk-averseratherthan risk-neutral. OursetupalsorelatestoPinkseandTan(2005),inthatinformationleakagecreates affiliationamongstindependentvaluations,althoughinasecond-priceauctionratherthanfirst priceasintheiranalysis. Finally,MilgromandWeber(1982)examinecompetitionandentryfees, raisingthepossibilitythatmonotonicequilibriamightnotexist. Landsberger(2007)extendsthis analysis and finds that such existence becomes increasingly unlikely as the number of bidders grows. Ouranalysisisinthesamevein;ourstructureimpliesnon-monotonicity,soaddingmore biddersisnotoptimal. 3 Derivatives Markets and CCPs ModernderivativesmarketsarecharacterizedbytheclearingofstandardizedcontractsatCCPs whileother,potentiallybespoke,contractsareclearedbilaterally.8 Dealers,andperhapssomeother marketparticipants,aredirect,or‘clearing’membersofCCPs. Thusadealer’snetriskpositionis composedofitsclearedpositionanditsbilateralone,andonlytheformeristypicallyknowntothe CCP. CCPs require that their members post initial and variation margin at least daily. Variation marginoneachclearedportfolioor‘account’isdeterminedbasedonthecurrentmark-to-market, soitcanbethoughtofassettlingthevalueoftheportfolioeveryday. Initialmarginisbasedonthe riskoftheportfolio: itisintendedtocoveritspotentialchangeinvalueoverafixedliquidation horizon, known as the margin period of risk, to a high degree of confidence. Regulation sets minimumstandardsforthemarginperiodofriskandtheconfidencelevelofmargin. ForOTC derivatives,initialmarginisrequiredtocoveratleastthe99th percentileofpotentialchangesin portfoliovalueoverafivedayperiod. Figure1onthefollowingpageillustratestheidea. The margin period of risk is intended to be long enough that the non-defaulting party can determinethataneventofdefaulthasoccurred,begindefaultmanagement,hedgethedefaulter’s 8 Suchcontractsareproperlyconsideredtobenon-centrallycleared,butareoftenreferredtoasuncleared. 10

Figure1: Therolesofinitialandvariationmargin portfolioifnecessary,andsellit. Inourcontext,‘sellingit’meansconductinganauction,including determining who to invite to the auction, communicating the portfolio to them, receiving bids, decidingonawinner,andnovatingthedefaulter’sportfoliotothem. Bidderswillusethemarketvalueoftheportfolioatthepointofbiddingasabasisfortheirbids. Thus,theCCPismostatriskfromlossesindefaultmanagementwhenthevalueoftheportfolio has fallen between the last successful variation margin call with the defaulter, at t = 0 say, and whenbidsaresubmitted,att = T say. Wewillmodelamodernfuturesmarketsubjecttoinitialandvariationmargin,reflectingthese features. Thismeansthatmarketparticipants,whentheyenterintotrades,onlypayinitialmargin. Changesinvalueoftheirportfoliosaresettledday-by-daythroughvariationmargin. Beforewe develop our model of the derivatives market, the remainder of this section provides a concrete exampleofthissituation. Example. SupposewearedealingwiththefrontmonthLondonMetalExchange(‘LME’)copper futures,andassumethatapartybought10lotsofthisfutureon19thMay2020. Onthisdaythe futurespricewas $5,314. TheLMEcopperfutureistherighttoreceive25tonnesofcopperand is priced in US dollars per tonne. Therefore, this party locked in a price of $5,314×10×25 = $1,328,500for250tonnesofcopperattheexpiryofthefuture. Theclosingpriceofthisfutureon Friday19thJune2020ofthefrontmonthwas$5,855.50,meaningthatsomeonewhoboughtthis future on that day, locked in a price of 25×$5,855.50 = $146,387.50 for 25 tonnes of copper at 11

expiry. SupposethatourpartydefaultedatthecloseofbusinessonFriday19thJune. TheCCP willhavepaidacumulative($5,855.5−$5,314)×25×10 = $135,375ofvariationmargintothe defaulter. Notethatifthefutureexpireswiththislevel,5,855.50asitssettlementprice,longspay thisamountpertonneandshortsreceivethisamount: thereasonthatthedefaulterlockedinthe lowerlevelof$5,314pertonneisthattheyhavereceivedthedifferenceasvariationmargin. Theuseofvariationmarginmeansthatifthefuturespricegoesup,longpositionholdersreceive money;ifitgoesdown,theypaymoney. Itisonlyatexpirythattheobligationarisestopaythe settlementpriceandreceivethecommodity. Nowsupposethattheinitialmarginwas500pointsor$12,500perlot;thedefaulterwillhave paidinitialmarginof$125,000. Furthersupposethattheirguaranteefundcontributionwas$5,000 andthattheydefaultonMonday22ndJune,beforethemarketopens. TheCCPneedstoauction the right to pay $5,855.50 per tonne for 250 tonnes of copper. If the bid is say a per-lot price of $5,200, then the CCP has lost ($5,855.50−$5,200)×25×10 = $163,875 and it only has initial margin of $125,000 plus defaulter’s guarantee fund contribution of $5,000 so there is a loss of $163,875−$125,000−$5,000 = $33,875tobeallocatedfirsttotheCCP’sskininthegame(‘SITG’) then, if that is inadequate, to non-defaulter’s guarantee fund contributions. The CCP’s desired per-lotauctionpriceisatleast$5,855.50− $125,000+$5,000 = $5,335.50,asthedefaulter’sresources 25×10 aresufficienttocoverlossesifthewinningbidisatorabovethislevel. Notefinallythattheprofitsorlossesoftheauctionarerealizedinthevariationmargincallatthe endofthedayoftheauction. IfthesuccessfulbidinanauctiononMonday22ndJulywasaper-lot priceof$5,200foralongpositionin10lotsoffutures,andthefutureclosesat$5,800(closetothe previousFriday’sclose),thentheCCPwillpaythesuccessfulbidder($5,800−$5,200)×25×10 = $150,000. Thereisnocashflowintheauctionitself—justliketheinitialpurchase,whatisbeing agreedisnotapricetopaybutalevelfromwhichtobasefuturevariationmarginpayments. Similarly,ifthedefaulterwasshort10lots,thenbidderswouldrationallybidabovethecurrent priceof$5,855.50. Ifthewinningbidwas,say$5,950,andthefutureclosesontheeveningofthe auctionat$5,870,theCCPwouldpaythebidder($5,950−$5,870)×25×10 = $20,000. 12

4 A Model of a Derivatives Market Thissectionintroducesthemodelofaderivativesmarketthatwewillusefortherestofthepaper. Wewillmodelasingleriskfactor,tobethoughtofasthepriceofacommodityfuture. There are two key features to this model. First, we assume that market participants are risk averse. Thisismodelledbyaprivatevalueforpositionwhichreducestheirvaluetotheholderthe biggertheyare. Second,themodelcapturesbothOTCforwardsandclearedfuturespositions,so theCCPdoesnotknowthenetriskpositionofanymarketparticipant.9 Withoutthis,theproblem ofselectingauctionmarketparticipantsistrivial,asitsimplyinvitesthosewithpositionsclosest toandoppositeinsignfromthedefaulter. ItisalsorealistictoassumethatOTCpositionscanbe significant,andcanhaveamaterialeffectontheexchange-tradedmarket.10 4.1 Positions Thenetriskpositionofeachclearingmemberisdefinedintermsofasingleriskfactorthatcan takepositiveandnegativeintegralvalues. Theriskfactortradesbothasaclearedfutureandas unclearedforwards,sotheCCPdoesnotknowanyclearingmember’snetposition. Thepositionis expressedasafuturesequivalent. Wewillwrites forthepositionofclearingmemberi ∈ I,where i s < 0denotesashortpositionands > 0denotesalongposition. Becausewearedealingwitha i i derivativesmarket,thesumofthelongsequalsthesumoftheshorts: ∑ ∑ S := max(s ,0) = − min(s ,0), (1) i i i∈I i∈I whereSdenotesthesizeofthemarket. 9 TheOTCmarketcanbeseveraltimesthesizeoftheexchange-tradedmarket,sothisisarealisticassumption.SeeOliver Wyman(2023)foranexampleofthissituation. 10 See,forinstance,LME’sConsultation22/145,2022,whichnotesthat“RecenteventsintheLMENickelmarkethave demonstratedtheeffectsthatOTCactivitycanhaveonthewiderLMEmarket.” 13

4.2 TheFuturesPriceandVariationMargin WewillwriteV(t)forthequotedpriceofonelotattimet: thinkofthisasthefuturesprice. Because ofvariationmargin,thepublicmark-to-marketofallpositionsclearedbynon-defaultersattheend ofeachdayisalwayszero. Lett = 0denotetheendofoneday. Ifthemarketparticipantacquiresa newpositionofsattheclose,andt = 1denotestheendofthenextday,thenthetotalvariation marginpaidtothemarketparticipantwillbes(V(1)−V(0))ifthisispositive,orfromthemarket participantifitisnegative. Afteradefault,theCCPhastocontinuetopayorreceivevariationmarginontheotherside ofthedefaulter’sclearedposition. Wedenotethedefaulterby Danditsclearedpositionbys .11 D Withoutlossofgenerality,denotethelasttimevariationmarginwasexchangedpriortodefaultby t = 0. Then if the per-lot pricechanges from V(0) to V(T) at the time T when the CCPnovates thepositiontoanauctionparticipant,themid-marketprofitorlossfortheCCPonthedefaulter’s positionis(V(T)−V(0))s . D 4.3 FundedResourcesandaSuccessfulAuction Wewilldenotetheinitialmarginpostedbyclearingmember i by M, theirguaranteefundconi tribution by G, and the CCP’s skin in the game by SITG. The total guarantee fund G is ∑ G. i i∈I i Theresourcescontributedbythedefaulterare M +G . Afteradefault,theCCPcanusethese D D resourcestocoveranylossesitincursinclosingoutthedefaulter’sposition. Figure 2 on the next page shows an example path of the mark-to-market of the defaulter’s portfoliothroughthemarginperiodofrisk. Itstartsatzerobydefinitionasweassumeasuccessful variationmargincallatt = 0. Weassumethatatt = TtheCCPtransfersthepositionatapricebwhichcouldbeeitherpositive ornegative: positivebrepresentscashcomingintotheCCPfromthenextvariationmargincalland negative,cashleaving,asusual. TheCCP’stotalprofitorlossonthedefaulter’sportfolio,after 11 Theremainderofitsportfolio,thebilaterallyclearedpart,wouldbemanagedduringbankruptcyasidefromtheCCP auction,soitisnotrelevanthere. 14

Figure2: Thevalueofthedefaulter’sportfoliothroughthemarginperiodofriskandtheuseofresourcesin thedefaultwaterfalltoabsorblossesonit usingtheirresources,andincludingvariationmargin,istherefore: P = (V(T)−V(0))s +b. (2) D IftheCCPhastopayoutbintheauction,thentheprofitorlossisbbelow(V(T)−V(0))s . The D totalfundedresourcesare M +SITG+G. Ingeneral(V(T)−V(0))s couldtakeeithersign,but D D theauctionismoredifficultwhenitisnegative,sothissituationisillustratedasfollows. If the CCP does not make a loss on closeout, P ≥ 0, it is obliged to return the defaulter’s resources to the administrator of the estate. On the other hand, if it makes a loss, P < 0, it can absorb that loss with the available resources.12 The CCP’s total profit or loss on the defaulter’s portfolioafterusingthedefaulter’sresourcesistherefore: P if P ≥ 0, 0 if−M −G ≤ P < 0, (3) D D P+M +G otherwise. D D Thiscanbesimplifiedto max(P,min(P+M +G ,0)). (4) D D 12 WeassumethattheCCPdoesnotmakeanyfurtherrecoveriesinexcessoftheavailablecollateral. 15

TheCCP’sownresourcesareatriskifthecostofdefaultmanagementislargerthan M +G . D D Forsimplicity,weassumethattherearenoothercosts—suchashedgingcosts—andthatanauction succeedsiftheCCPcanliquidatetheportfoliowithoutpayingmorethanthis. Thepayofffunctioninequation(4)isoneofthethingsthatmakesCCPdefaultauctionsnovel. Astandardassumptioninauctiontheoryisthatthesellerseekstomaximizerevenue,andmost analysisofdifferentauctioncharacteristicsfocusesontheimpacttoexpectedrevenue(Krishna, 2010). Butinadefaultauction,theCCPdoesnotactuallyprofitfrombetterbidsbwithintherange −M −G ≤ P. (5) D D There is therefore little or no incentive for a CCP to care about increasing the bid within this range. Incontrast,thereisaconsiderablenegativereputationalimpactassociatedwithadefault auctionwhicheatsintoskininthegame. Furthermore,althoughitvariesacrossCCPs,skininthe gameisgenerallyarelativelythinlayerofresources. IftheCCP’sskininthegameisexhausted, theCCPstartsmutualizinglossesamongtheremainingclearingmembers. Whileunlikely, this situationcouldsoseverelydamagetheCCP’sreputationthatavoidingitisparamounttotheCCP. Consequently,aCCPismuchmoreconcernedwithmaximizingtheprobabilitythatitwillreceivea winningbidthatishighenoughtoensurethatlossesarecoveredbythedefaulter’sresourcesthan withmaximizingauctionrevenue. Putanotherway,theCCPcanbeviewedaslargelyindifferentto thelevelofrevenuegeneratedbytheauctionabovethethresholdillustratedinFigure2,butitfaces adiscontinuouslossbelowit. Thisproducesaparticularformofriskaversion,sothatCCPsfocus onminimizingthedownsideriskratherthanseekingtomaximizerevenuesasasellernormally does. Ratherthanspecifyingsomeformofriskaversionthatwouldincorporatesthesecomplicated thresholdandreputationaleffects,weassumethattheCCPseekstomaximizetheprobabilitythat itslossontheportfolioiscoveredbythedefaulter’sresources M +G . D D 16

4.4 PrivateValues Followingtheset-upinDuandZhu(2017),supposethatmarketparticipantshaveanaversionto risk,whichaffectstheirprivatevalueofpositions. Inparticular,theprivatemark-to-marketofa positionofsizes ∈ R is −βs2 (6) forsomepositive β(i.e.,thebiggerpositionsbecome,themoreholders(quadratically)discount them). Foreasewewillassumeβisconstantforallparticipantsandthisisknown. Thisformulation is consistent with the findings of Lou et al. (2013) that dealer’s risk capacity affects valuations aroundUSTreasuryauctions. 4.5 PrivateValuesforWinningBidders Thedefaultingclearingmemberhasapositions . Ifaclearingmemberwinstheauctionforthis D positionatapriceb,itwillbenettedwiththeirexistingposition. Ifs wastheclearingmember’s i oldposition,theoriginalpositionhasaprivatevalueof−βs2,andthenewpositionisprivately i worth−β(s +s )2. Hence,theclearingmemberisindifferentbetweenbuyingthepositions for i D D bandnotbuyingitwhen −β(s +s )2−b = −βs2. i D i Wewillwriteb˜ forthisrationalbidthresholdforclearingmemberi: i b˜ = −β(2s s +s2 ). i i D D Clearlyitisirrationaltobidabovethislevel,as−β(s +s )2−b < −βs2 whenb > b˜ . i D i i 4.6 DesirableandUndesirableBidders Suppose the defaulter’s position is big, s ≫ 1, and all the bidders are on the same side of the D marketasthedefaulterandbigtoo,s ≫ 1foralli ∈ I. Then(2s s +s2 )isalwayspositiveand i i D D large. Hence,unless βisverysmall,foralli ∈ I,b willbenegativeandlarge,andtheCCPwillnot i 17

getabidaboveitsthreshold−M −G . Thus,onlyhavingbiddersintheauctiononthesameside D D ofthemarketasthedefaulterisdamagingfortheCCP’sobjectiveofhavingasuccessfulauction. Conversely, if there is a bidder i with position opposite to the defaulter, sgn(s ) ̸= sgn(s ), i D and bigger than it in size, |s | > |s |. Then 2s s is negative and s2 > s2 , so (2s s +s2 ) < s2, i D i D i D i D D i andhencewinningtheauctionfreesupprivatevalue. Forthisbidder,b˜ > 0. Clearly,ifinstead i s = −s /2,then2s s +s2 = 0andsob˜ = 0. i D i D D i Followingonfromthis,ifthenumberofauctionparticipantsislargeands issymmetrically i distributed, then there will be some bidders for whom s and s are of opposite signs. Any of i D these participants who have |s | > |s | will make positive bids, so unless P < −M −G , the i D D D probabilitythattheCCPwillgetabidaboveitsthresholdapproaches1asthenumberofbidders increases. Inthissetting,invitingmoreparticipantstotheauctionisalwaysbetter. Thisresultisnot surprising. Evenwiththechangetotheseller’sobjective,thefolktheoremresultthataddingmore biddersgenerallyincreasesrevenueintuitivelysuggeststhataddingmorebidderswillincreasethe probabilitythattheCCPreceivesabidaboveitsthreshold. 4.7 TheAuctionandInformationLeakage Itisreasonabletoassumethatmarketparticipantsknowtheoveralldirectionofthedefaulter’s portfolio s —long or short—but not its precise size. The direction is revealed because the the D directionofmarketmovescanbeeasilycomparedtothetimeofdefault,whichisknown. Thesize isrevealedtoauctionparticipantsbytheCCPjustbeforethebiddingprocessopens. Ofcourse, actual portfolios are more complicated, and in a default auction the actual positions would be revealedtoparticipants;revealingsizestandsasagoodproxyfortherevealedinformationinthe simplermarketmodel. Thusfar,themarketvalueofdefaulter’sportfolioatthetimeoftheauctionhasjustdepended on the drift in the futures price V(t). We assume this movement is determined by a standard normalrandomvariable Z ∼ N(0,1)sothat V(T) = V(0)(1+σZ), 18

whereσisavolatility-likescalingparameter. Inaddition,wewillassumethatinformationleakage creates a risk that auction participants trade outside the auction against the defaulted position, whichhurtsitsvalue.13 Asnotedabove,regulatorsrecommendthatCCPsshouldbalancetherisk ofinformationleakageandtheaimofobtainingacompetitivepricewhendecidingonthemost appropriateexecutionmethod,highlightingtheimportanceofmodelinginformationleakage. Inordertodothis,weassumethathaving N bidderswithcertaintyreducesV(0)byanamount γ(N−1)Q forfixedγ > 0and Q. Theconstantsγ > 0and Qareknownandabstractlyaccount forthecostofinformationleakage. Weassumethattheimpactofinformationleakageincreases non-linearly in the number of auction participants. So for N bidders, the futures price at the momentbidsaresubmittedis: (cid:16) (cid:17) V(T) = V(0) 1+σZ−γ(N−1)Q . Note that the price move against the CCP depends on the number of (potential) bidders, so we are assuming that having more bidders means more information leakage. In their model of client trading, Baldauf and Mollner (2024) show how the possibility of information leakage, or front-running,canreducedealer’scompetitionfortheclient’strade. Althoughtheirmechanism endogenizestheimpactofmovingfromonedealertotwo,theactualpriceimpactisexogenously assumed. Asthefocushereisonthepriceimpactofinformationleakageontheauctionitself,we exogeneouslyspecifythepriceimpactforageneralincreaseinthenumberofauctionparticipants. TheproblemtheCCPnowfacesisthattherearetwocompetingpressures: havingmorebidders increasestheprobabilityofreceivingagoodbid(competitioneffect),butitalsoincreasesthesize ofthepricemovingagainsttheCCP(informationleakageeffect). Thesimplestrategyofincluding everyoneintheauctionisnolongeroptimal. Inthenextsection,weconsidertheCCP’sauction 13 Theneedtosendportfolios,whichpotentiallyconsistoftensofthousandsofinstruments,tomultiplemarketparticipants andallowthoseparticipantstimetopricetheportfoliosanddeterminetheimpactwinningthemwouldhaveontheirrisk, capitalandliquidityinevitablymeansthattheminimumtimebetweenparticipantsreceivingdetailsofthedefaulter’s portfolioandbidsbeingdueismeasuredinhours.Thisisampletimeforinformationleakagetooccur. 19

strategy in this situation. The CCP would rather like to be discriminating in who it invites. As the CCP does not have the information available to fully discriminate, we examine whether it canuseanentryfeetoendogenouslyencouragetherightparticipantsanddiscouragethewrong participantssothatitcaninducetheparticipantstobalancethecompetingpressures. 4.8 BidderValues Biddervaluesattime T dependuponthepriceofthefutureV(T),thenumberofbidders N,and thedefaulter’sposition s . WeassumethattheCCPtruthfullyreveals s toallthebiddersthat D D entertheauction. Bidderi’smaximumvalueofbiddingv thenis i (cid:16) (cid:17) v ≡ s V(T)−βs (2s +s ) = s V(0) 1+σZ−γ(N−1)Q −βs (2s +s ). i D D i D D D i D Wecantakev torepresentbidderi’sprivatevalue. Interestingly,informationleakagehasmade i bidders’ private values affiliated, so that competition is less intense than a bidder would have thoughtbeforetheauction;themechanismisdifferentbuttheresultissimilartothatinPinkseand Tan(2005). Despite the unusual event in the benchmark US oil futures market in March 2020 when the frontmonthfuturespricebrieflyturnednegative,weassumenotjustthatV(t) > 0butalsothat V(t)−βs > 0foralltandanys ,i.e.,longpositionsalwayshavepositivevalues. i i 5 Self-Selecting Mechanism Withoutlossofgenerality,wetakeV(0) = 0,i.e.,allmark-to-marketfluctuationsontheportfolio aresettledcontinuouslybeforetheauction. Themaximumwillingnesstopay,orvalue,ofbidderi forthedefaultportfolioisthen v ≡ −βs (2s +s ). (7) i D i D 20

Thetimelineofthemodelisasfollows: ■ TheCCPsetstheentryfeee ≥ 0exante,beforeanydefault.14 ■ Defaulthappens. TheCCPannouncesthesideofthedefaultportfolio,sgn(s ),butnotits D magnitude. Bidders believe (correctly) that the magnitude of s is distributed according D toadistributionwithsupportfrom[0, ∞).15 Biddersdecidewhethertheywouldpayeto observes . EntrydecisionsaremadesimultaneouslyandonlydisclosedtotheCCP. D ■ The CCP announces the number of bidders K who have paid the entry fee, entered the auction,andthusobserveds . D ■ Bidderswhoentertheauctionrealizethattheirper-unitvalueofthedefaultportfoliodrops by −γ(K−1)Q, (8) reflectingthecostofinformationleakage(ascapturedbyγ > 0andQ). Putdifferently,while thefairmarketvalueofthedefaultportfolioremainsV = 0giventhepublicinformation set,theprivatevaluesofeveryoneintheauctiondropbyγ(K−1)Q.16 Bidderswhodonot entertheauctiondonotseeK andthereforedonotobservetheresultingdropinvalue. Let R ≡ M +G > 0betheresourcestheCCPhasfromthedefaulter. TheCCPconductsa D D second-priceauctionwithreservationpriceequalto−(R+Ke). TheCCP’sobjectivefunctionis max Prob(A), whereA = {P+R+Ke ≥ 0}, (9) e andK isarandomvariablethatdetermineshowmanybiddersentertheauctionconditionalonthe entryfeee. Theobjectiveseekstosettheentryfeetomaximizetheprobabilitythatitavoidsaloss 14 Settingtheentryfeeexanteensuresthattheentryfeerevealsnoinformationaboutthesizeofthedefaulter’sposition. 15 ThesupportcouldberestrictedtohaveanupperlimitsmallerthanS;thelargersupportjustsimplifiesthenotationas thepointwherethecumulativedistributionreachesonedoesnotaffecttheresults. 16 Anothermechanismthatcouldresultinsuchreducedprivatevalueswouldbecommunicationamongsttheparticipants; AgranovandYariv(2018)demonstrateexperimentallythatcommunicationcanreducebids. Theimpactwouldbe capturedbycostsincreasingasthenumberofparticipantsrises. 21

fromtheauction. Avoidingalossisequivalenttoensuringthat P(thecashflowfromtheauction including any variation margin on the defaulting portfolio paid at the end of the auction day) plus R(theavailableresourcesfromthedefaulter)plusKe(theentryfeespaidbytheK bidders) is non-negative. Importantly, P declines as the number of bidders increases due to information leakage. Wecansolveforthebidders’entrydecisions. Conjecturethatv isthecutoffvalue,correspond- 0 ingtoacutoffinventorylevels sothat 0 v ≡ −βs (2s +s ). (10) 0 D 0 D Thebidderwhoisatthecutoffshouldbeindifferentbetweenenteringandnotenteringtheauction, sotheexpectedprofitofenteringisequaltotheentryfeee. Note that besides the non-standard objective function, our model has two features that are absentintheconventionalauctionmodels. First,entrydecisionsaremadewithoutobservings , D buttheactualbiddingdependsonlearnings afterentry. Second,eachentrant’svalueislowerif D morebiddersenter. Thesetwofeaturesrepresenttwoshockstobidderswhoentertheauction. Thethresholdbidderwhoisindifferentwinsifandonlyifsheistheonlyoneintheauction, inwhichcasethereisnocostofinformationleakage,i.e.,−γ(K−1)Q = −γ(1−1)Q = 0. Inthe second-priceauction,thewinningpriceforthesolebidderistheCCP’sreservationvalue,whichis −(R+e),i.e.,theCCPgives R+etothesolebidder,whoisthecutoffbidder,inreturnforexiting thedefaulter’sposition. Thesolebidder’sexpostprofit,ifshebids,is v +(R+e) = −βs (2s +s )+R+e. (11) 0 D 0 D Ifs turnsouttobeverypositive,thenthis“profit”canbenegative,sotheoptimalactionforthe D soleentrantisnottobidinthatcase. Therefore,theindifferenceconditionforthecutoff-typebidder is e = (1−F(s ))N−1E[max(−βs (2s +s )+R+e,0) | e], (12) 0 D 0 D 22

where F(·)isthecumulativedistributionfunctionofs . Becausetheentryfeeeissetinadvance,it i doesnotreveals inequilibrium. D The CCP’s objective is to minimize the probability that the auction fails, i.e., it receives no bids above the reservation price. Obviously, the auction fails if no one enters, which happens withprobability (1−F(s ))N. Theauctionmayalsofailifbidderswhoenterrefusetobidafter 0 observingtherealizationsofs andK. D Ifonlythecutoff-typebidderenters,theauctionsucceedsifandonlyif −βs (2s +s )+R+e ≥ 0. (13) D 0 D NotethatfromtheCCP’sperspective,onceeisset,theleft-handsideofthisconditionisdeterministicbecauseedeterminess byequation(12). 0 We now proceed to develop a series of building blocks (stated as lemmas) that we use to establish our main result that characterizes the optimal CCP auction. Our first building block establishesanupperlimitontheentryfee. Lets bethelowestinventoryleveloftheK bidders. min Thenwecanstatethefollowing: Lemma1. Iftheentryfeeeissufficientlyhigh,thennobidderentersandtheauctionfails. Theresultisnottrivialsincethewinningbiddermaybecompensatedinpartbytheentryfee(s). Itisthereforenecessarytoprovethats isdecreasingine,aswedoinAppendixA.Thisresultshows 0 thatincreasingtheentryfeewilleventuallypushs belows ,atwhichpointProb(s ≤ s ) = 0, 0 min min 0 andnobidderenters. Ingeneral,ifK ≥ 1biddersenter,theauctionsucceedsaslongas max v − γ(K−1)Qs ≥ −(R+Ke), (14) i D i∈K (cid:124)(cid:123)(cid:122)(cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) bidder’s costofinformation reservation value leakage value orequivalently, −βs (2s +s )−γ(K−1)Qs +R+Ke ≥ 0. (15) D min D D 23

Therefore,theauctionsucceedsifandonlyifahigh-valuebidder,whohasahighwillingnessto pay,entersandalsobids,i.e.,theCCPsolvesthefollowing: max Prob(s ≤ s and −βs (2s +s )−γ(K−1)Qs +R+Ke ≥ 0), (16) min 0 D min D D e where: ■ s isdeterminedbyequation(12),i.e.,theindifferencecondition. 0 ■ K is a random variable, determined by the rule that entry happens if a potential auction participant’sinventoryislowerthans . 0 ■ edependsonthedistributionsofs D ands j for∀j ∈ K,where{s j } j∈K representstheinventory levelsoftheparticipants,butdoesnotdependontherealizationofs .17 D ItwillbehelpfultorewritethetwoinequalitiesintheProb(·)ofexpression(16)as (cid:18) R+Ke−βs2 −γ(K−1)Qs (cid:19) s ≤ min s , D D . (17) min 0 2βs D Thisformulationmakesclearthatthesmalleroneofthetwotermsinmin(·,·)wouldbebinding. Intuitionsuggeststhatwewantthetwotermstobe“close”tomaketheprobabilitylarge. Inone extremewithe = 0,equation(12)impliesthat F(s ) = 1,i.e.,s istheupperboundoftheinventory 0 0 distribution. Asaresult,everyonewouldentertheauction. However,inthiscasethesecondterm becomes R−βs2 D −γ(N−1)QsD (asK = N),whichissmallifQislargeandγisnottoosmall. Intheother 2βsD extremeofsettingetobehigh,thenK wouldbesmall,so R+Ke−βs2 D −γ(K−1)QsD islessbindingbuts 2βsD 0 wouldbequitebinding. Therefore,thereshouldbesomeintermediatevaluesofethatmaximize theprobabilityofauctionsuccess. Moreformally,wefirstneedtoestablishtheexistenceofanoptimalentryfeee∗. Thisresultis statedinthefollowinglemma: 17 RecallthatthepotentialforalargeOTCmarketmeansthattheCCPcannotuseitsinformationaboutthedistributionof clearedpositionstoreliablyinferthedistributionofs. j 24

Lemma2. Thereexistsanoptimalentryfeee∗,possibly0,whichmaximizestheCCP’sobjectivegivenin expression(16). TheproofofthislemmaisprovidedinAppendixB. ItisnotsurprisingthattheoptimalityofnoentryfeecannotberuledoutinLemma2. Optimality ofazeroentryfeewouldbethestandardresultwithoutinformationleakage,astheCCPwould wanttomaximizeauctionparticipation. Whatwewishtoshowisthatundersomecircumstancesa positiveentryfeewouldbeoptimal,namelye∗ > 0. Todoso,weneedtoderivetheanalyticalexpressionofProb(·)inexpression(16). Notethatthe randomvariable K hasabinomialdistributionwith probabilityofsuccess F(s ). Let k denotea 0 realizationofK. Clearly,theconditions ≤ s isequivalenttok ≥ 1,i.e.,atleastonebidderhas min 0 inventorybelows . Inaddition,conditionalonk ≥ 1biddersenteringtheauction,s islower 0 min thans . Stillconditioningonk,foranyrealvaluex ≥ s ,Prob(s < x | s < s ) = 1;forx < s , 0 0 min min 0 0 wehave Prob(s < x,s < s ) Prob(s < x) Prob(s < x | s < s ) = min min 0 = min min min 0 Prob(s < s ) Prob(s < s ) min 0 min 0 1−Prob(s ≥ x) 1−(1−F(x))k = min = , (18) 1−Prob(s ≥ s ) 1−(1−F(s ))k min 0 0 whereinthelaststepweusethefactthatifkbiddersentertheauction,thentheminimuminventory of the N potential bidders is equal to the minimum inventory of the k bidders who have the k lowestinventoriesandwhoactuallyenter. Theauctionsuccessprobability,conditionalons ,is D writtenas Prob(s ≤ s , −βs (2s +s )−γ(K−1)Qs +R+Ke ≥ 0 | s ). (19) min 0 D min D D D BecauseK isbinomial,thisprobabilitycanbemadeexplicit. Inparticular,let R+Ke−βs2 −γ(K−1)Qs ϕ(s ,K) := D D . (20) D 2βs D 25

Thenwehavethefollowing Prob(s ≤ s , −βs (2s +s )−γ(K−1)Qs +R+Ke ≥ 0 | s ) min 0 D min D D D    R+Ke−βs2 −γ(K−1)Qs  = ProbK ≥ 1,s < D D | s   min D 2βs  D  (cid:124) (cid:123)(cid:122) (cid:125) :=ϕ(sD,K) N ∑ = Prob(K = k)Prob(s < ϕ(s ,k) | K = k | s ) (21) min D D k=1 = ∑ N (cid:18) N (cid:19) F(s )k(1−F(s ))N−k 0 0 k k=1 (cid:20) 1−(1−F(ϕ(s ,k)))k(cid:21) · 1(ϕ(s ,k) ≥ s )+1(ϕ(s ,k) < s ) D , D 0 D 0 1−(1−F(s ))k 0 where 1(·)denotesanindicatorfunctionthattakesthevalueoneiftheconditioninthebracketsis satisfied,andzerootherwise. Note,ϕ(s ,k)isadecreasingfunctionofs as D D ∂ϕ(s ,k) βs2 +R+ke D = − D < 0. (22) ∂s 2βs2 D D The final step to obtain the unconditional probability is to integrate the above conditional probabilityovers . LetG(·)denotethecumulativedistributionfunctionofs ,thentheprobability D D ofauctionsuccess,denotedasα,is α = ∑ N (cid:18) N (cid:19) F(s )k(1−F(s ))N−k 0 0 k k=1 (cid:90) ∞(cid:20) 1−(1−F(ϕ(s ,k)))k(cid:21) · 1(ϕ(s ,k) ≥ s )+1(ϕ(s ,k) < s ) D dG(s ). (23) 0 D 0 D 0 1−(1−F(s 0 ))k D The function ϕ(s ,k) is decreasing in s , so ϕ(s ,k) ≥ s if and only if s ≤ ζ(s ), for some D D D 0 D 0 26

functionζ(s )thatisdecreasingins .18 Thenwecanwritetheprobabilityofauctionsuccessas: 0 0 α = ∑ N (cid:18) N (cid:19) F(s )k(1−F(s ))N−k 0 0 k k=1 · (cid:20)(cid:90) ζ(s0 ) g(s )ds + (cid:90) ∞ 1−(1−F(ϕ(s D ,k)))k g(s )ds (cid:21) . (24) 0 D D ζ(s0 ) 1−(1−F(s 0 ))k D D Basedonequation(24),wehavethefollowinglemma,whichisthekeyconstructiveresult: Lemma3. Ifγissufficientlyhigh,asmallincreaseofentryfeefromzeroraisestheprobabilitythatthe auctionsucceeds. TheproofofthislemmaisprovidedinAppendixC. Wecannowstateourmainresultasthefollowingproposition,whichfollowsimmediatelyfrom Lemmas1through3(thefirstpartfollowsfromLemmas1and2andthesecondpartfollowsfrom Lemma3): Proposition1. Thereexistsanoptimalentryfeee∗thatmaximizestheCCP’sobjectivegiveninexpression (16). Iftheimpactofinformationleakageissufficientlyhigh,thentheoptimalentryfeeisstrictlypositive. Ourmainresultshowsthat,inthefaceofinformationleakage,apositiveentryfeecanbean effectivemechanismtoencouragepotentialbidderstoself-selectwhethertoenteranauctionornot sothattheprobabilityofasuccessfulauctionismaximizedfromtheCCP’spointofview. Toour knowledgethisisanovelmechanism. ManyCCPsrequirepotentialauctionparticipantstoincur 18 Thefunctionζ(s )is 0 (cid:113) −γ(k−1)Q−2βs + (cid:2) γ(k−1)Q+2βs (cid:3)2+4β(R+ke) 0 0 ζ(s )= >0, 0 2β andsothefirst-orderderivativewithrespecttos is 0 dζ(s ) γ(k−1)Q+2βs 0 =−1+ (cid:113) 0 <0, ds 0 (cid:2) γ(k−1)Q+2βs (cid:3)2+4β(R+ke) 0 whichmeansthatζ(s )isdecreasingins . 0 0 27

exantecosts,forexamplebyparticipatingindefaultexercisesexante,whichcouldbeviewedas anentryfee,butsuchimplicitentryfeesdonothelptheCCPmaximizetheprobabilityitavoids needingtouseitsskininthegameornon-defaulters’resources. The advantage of the entry fee mechanism developed here is that it is effective with low informationrequirements. Inparticular,theCCPdoesnotneedtoknowthecompletepositionsof eachpotentialbidder. Ifitdid,amoreefficientmechanismlikelycouldbeconstructed. Butwithout suchinformation,theentryfeemechanismavoidstheCCPneedingtouseitsownjudgmenton whotoinviteandsubsequentlybeingopentocriticismforitsdecision,asNasdaqNordicwas,after theauction.19 Itshouldbeclearthattheresultdependsonanassessmentofhowcostlyinformationleakage willbe. Themodelofinformationleakagecostissimple,butthekeycharacteristicisthatprivate valuesdecreaseasthenumberofparticipantsincreases. Thischaracteristicseemsintuitiveandthe resultsarelikelytoholdforothermodelsofinformationleakagethatmaintainthisfeature. Inour set-up,asthemarginalcostofincludingparticipantsintheauctionincreases,theoptimalentryfee alsoweaklyincreases. Westatethisasthefollowingproposition: Proposition2. Theoptimalentryfeee∗ isweaklyincreasingintheimpactofinformationleakage. TheproofofthispropositionisprovidedinAppendixD. Thedependenceonthecostofinformationleakagedoes,however,implythatwhetherornot anentryfeeisaneffectivemechanismmayvaryfromCCPtoCCP.Forexample,CCPsclearing exchange-traded products might be less concerned about information leakage; more generally, information leakage costs are likely higher in less liquid markets.20 But ignoring the potential impactofinformationleakage,andtheconsequentdifficultquestionofwhotoinvitetoadefault 19 SuchcriticismwasreportedinClancy(2018b)andMourselas(2019). 20 TheimpactofmarketliquidityisalsoapparentintheanalysisoftransactioncostsonCCPhedgingstrategiesduringa close-outinCerezettietal.(2019)andinthegeneralpriceimpactoflargetradeseveninliquidequitymarkets(Bouchaud etal.,2009,Eisleretal.,2012). 28

auction,wouldbeinconsistentwithCPMI-IOSCO(2020),wheretheregulatorsdevotesignificant attentiontotheseissues. 6 Numerical Example Inthissection,wepresentanumericalexamplethatillustratestheresultsofourmodel. Weassume that{s } followstheLaplacedistribution,i.e.,theprobabilitydensityfunctionsuchthatfor j j=1,2,...,N λ > 0ands ∈ (−∞ , ∞), F j  f(s ) = 1 λ e −λF |sj | =    1 2 λ F e−λFsj, if s j ≥ 0 . (25) j F 2    1λ eλFsj, if s < 0 2 F j Inaddition,weassumethat s followstheexponentialdistribution,i.e.,theprobabilitydensity D functionsuchthatforλ > 0ands ∈ [0, ∞), G D g(s ) = λ e −λGsD. (26) D G Thealgorithmfornumericallycalculatingtheprobabilityofauctionsuccessasafunctionofeis detailedasfollows: ■ Set R = 1, β = 0.1,γ ∈ {0.1,0.3,0.5,0.7,0.9}, N = 10,Q = 2,andλ = λ = 1. F G ■ Pickanon-negativee. Wechoosethegridsuchthate ∈ [0,0.01,0.02,...,0.99,1]. ■ Givene,numericallysolvefors fromequation(12).21 0 ■ Draws ,s ,...,s independentlyfrom F(·)(theLaplacedistributionabove). ThenumberK 1 2 N issettobethenumberofbidderswhoseinventorylevels islowerthans thatwehavejust j 0 21 Bypluggingtheexponentialdensityfunction(26)intoequation(12),solvingequation(12)isequivalenttosolvingthe followingequation: (cid:104) (cid:105) e=(1−F(s ))N−1 2βs e−b(b+1)−2βs −2β+βe−b(b2+2b+2)+(R+e)(1−e−b) , 0 0 0 √ whereb≡ −2βs0 + 4β2s2 0 +4β(R+e) .ThederivationisdemonstratedinAppendixE. 2β 29

solvedinthepreviousstep. Thesebiddersentertheauctionandpaytheentryfeee. Sets min tobethelowestinventoryofthesebidders. ■ Draws fromthedistributionG(·)(theexponentialdistributionabove). D ■ CheckifbothinequalitiesinProb(·)inexpression(16)hold. Ifbothofthemhold,thenthe auctionissuccessful. Otherwise,theauctionfails. ■ Repeattheaforementionedthreestepsmultipletimes,settobe100,000times,toobtainthe probabilityofauctionsuccessαˆ foreache: 100000 (cid:16) (cid:17) αˆ = 1 ∑ 1 sm ≤ s ,−βsm(2sm +sm)−γ(Km−1)Qsm +R+Kme ≥ 0 , 100000 min 0 D min D D m=1 wherethesuperscriptmdenotesthem-throundofsimulationand 1(·)denotestheindicator function. ■ Plottheprobabilityofauctionsuccessasafunctionofe. The numerical result for the case when γ = 0.5 is shown in Figure 3 on the following page. Clearly,iftheentryfeeeincreasesabitfromzero,theprobabilitythattheauctionsucceedsincreases. ThisfindingconfirmsourideathatimposingapositiveentryfeeisaneffectivemechanismforCCP defaultauctionsthroughendogenizingpotentialbidders’entrydecisions. Figure 4 on page 32 presents simulation results for different marginal costs of information leakage. ConsistentwithProposition2,weseethattheoptimalentryfeeincreasesasγincreases. Atthesametimewealsoseethatincreasingγreducestheprobabilityofauctionsuccess. Thisisa generalpropertyofthemodelthatfollowsfromthefactthatforanyfixede,throughequation(12) s isalsofixedandinvarianttochangesinγandsothefirstconstraintinexpression(16)isfixed. 0 Consequently,theprobabilitythatthesecondconstraintinexpression(16)holds,i.e., −βs (2s +s )−γ(K−1)Qs +R+Ke ≥ 0, (27) D min D D isdecreasinginγ(andQ).22 22 Foragivene,ifthissecondconstraintisbinding,theprobabilityofauctionsuccesswillstrictlydecreaseatthatefor 30

% Figure3: Thisfigureillustratestheprobabilityofauctionsuccessasafunctionoftheentryfeee. Parameter values: R =1,β =0.1,γ =0.5,N =10,Q =2,λ = λ =1. F G 7 Conclusion CCPdefaultauctionsarecriticallyimportant. ACCP’smaintaskismanagingitsrisksuchthata defaultdoesnotspread. Itisnearlyatautologytosaythatifadefaultauctionissuccessful,the CCP’sriskmanagementissuccessful. Conversely,assuggestedbyClancy(2018a),anunsuccessful defaultauctioncancastdoubtnotonlyontheindividualCCPbutoncentralclearingmorebroadly. Nevertheless,CCPdefaultauctionshavereceivedrelativelylittleattentionintheliterature. WehavedevelopedasimplebutrealisticmodeloftheclearedmarketandthechallengesaCCP facesinconductingadefaultauction. Wepaidparticularattentiontowhatinformationwouldbe known to both the CCP and its clearing members at various points in the default management process. Inparticular,theoccurrenceofadefault,theresultingneedtoconductanauction,andthe directionofthemarketmoveassociatedwiththedefaultareassumedtobecommonknowledge. The CCP is assumed to know members’ cleared positions, but not their total market exposures. This assumption, and the associated opacity of clearing members’ risk preferences, is realistic. higherγ. 31

% Figure4:Thisfigureillustratestheprobabilityofauctionsuccessasafunctionoftheentryfeee,withdifferent valuesofγ. Parametervalues: R =1,β =0.1,γ ∈ {0.1,0.3,0.5,0.7,0.9},N =10,Q =2,λ = λ =1. F G Consequently,theCCPlackstheinformationtojustspecifywhoshouldbeinvitedtoanauction withanydegreeofreliability. Withinthismodel,wehaveaddressedseveraluniqueaspectsofCCPdefaultauctions. First, thatrevenuemaximizationisnotareasonableobjectiveforCCPs. Second, thatoneofthemost commonandfundamentalquestionsCCPsfaceiswhotoincludeintheauction. Third,wehave explicitlyincorporatedinformationleakage,whichmeansinvitingeveryoneisnotoptimal. These characteristicstakentogetherresultinahighlynon-standardauctionproblem,sostandardresults inauctiontheorydonotapply. Nevertheless,weareabletoobtainaconstructiveanalyticalresult. IntheCCPframework,thereservepriceandtheentryfeeperformdifferentfunctions,contrary to their equivalence in the standard auction model. The impact of this is that a positive entry feecanbeoptimalpreciselybecauseofitseffectonendogenousentrydecisions. Totheauthors’ knowledge,thisresultisnovel. ItfocusesattentiononthekeyquestionofhowCCPsdecidewhich marketparticipantstoinvitetoanauction,andprovidesaneffectivemechanismforresolvingit. 32

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Appendices A Proof of Lemma 1 Proof. Weformallyprovethats decreasesaseincreases,i.e., ds0 < 0. First,weneedtoobtainthe 0 de mathematicalexpressionforequation(12). Weknowthatthefirstterminthemax(·,·)operatoris thefollowing: −βs (2s +s )+R+e = −2βs s −βs2 +R+e. D 0 D 0 D D Therefore,itisusefultosolvethefollowingquadraticequationwithrespecttos : D βs2 +2βs s −(R+e) = 0, (28) D 0 D where β > 0and R+e > 0. Sincethediscriminantofequation(28)is ∆ = (2βs )2−4β[−(R+e)] = 4β2s2+4β(R+e) > 0, 0 0 thisquadraticequationhastwodifferentroots,denotedby aandb,whichare (cid:113) −2βs − 4β2s2+4β(R+e) 0 0 a = , 2β and (cid:113) −2βs + 4β2s2+4β(R+e) 0 0 b = . 2β Hence, b > a. Since −β < 0, R+e > 0, and the quadratic function of s , i.e., h(s ) ≡ −βs2 − D D D 2βs s +(R+e),intersectswiththeverticalaxisatthepoint(0,h(0)) = (0,R+e),wehave 0 D a < 0 < b, andthus, −βs (2s +s )+R+e ≥ 0, s ∈ [a,b], D 0 D D 38

−βs (2s +s )+R+e < 0, s ∈ (−∞ ,a)∪(b,+∞). D 0 D D Inaddition,becausewerequirethats ≥ 0,wethenhave D −βs (2s +s )+R+e ≥ 0, s ∈ [0,b], D 0 D D and −βs (2s +s )+R+e < 0, s ∈ (b,+∞). D 0 D D Asaresult,equation(12)canbeexpressedas e = (1−F(s ))N−1E[max(−βs (2s +s )+R+e,0) | e] 0 D 0 D (cid:90) +∞ = (1−F(s ))N−1 max(−βs (2s +s )+R+e,0)g(s )ds 0 D 0 D D D 0 (cid:90) b = (1−F(s ))N−1 (−βs (2s +s )+R+e)g(s )ds . (29) 0 D 0 D D D 0 Next,takingthefirst-orderderivativeswithrespecttoeonbothsidesofequation(29),wecan obtain ds (cid:18)(cid:90) b (cid:19) 1 = −(N−1)(1−F(s ))N−2f(s ) 0 (−βs (2s +s )+R+e)g(s )ds 0 0 D 0 D D D de 0 (cid:20) ∂ (cid:90) b +(1−F(s ))N−1· (−βs (2s +s )+R+e)g(s )ds 0 D 0 D D D ∂e 0 (cid:18) ∂ (cid:90) b (cid:19) ds (cid:21) + (−βs (2s +s )+R+e)g(s )ds 0 . (30) D 0 D D D ∂s de 0 0 BytheLeibnizintegralrule,wefirstget ∂ (cid:90) b (−βs (2s +s )+R+e)g(s )ds = D 0 D D D ∂e 0 db (cid:90) b (−βb(2s +b)+R+e)g(b) + g(s )ds 0 D D de (cid:124) (cid:123)(cid:122) (cid:125) 0 =0 = G(b)−G(0) = G(b) ∈ (0,1). 39

Wealsoget ∂ (cid:90) b (−βs (2s +s )+R+e)g(s )ds = D 0 D D D ∂s 0 0 db (cid:90) b (−βb(2s +b)+R+e)g(b) + −2βs g(s )ds 0 D D D ds (cid:124) (cid:123)(cid:122) (cid:125) 0 0 =0 (cid:90) b = −2β s g(s )ds , D D D 0 where,inbothequations, g(·)andG(·)denotetheprobabilitydensityfunctionandthecumulative distributionfunction,respectively. Therefore,equation(30)canbewrittenas 0 < 1−(1−F(s ))N−1G(b) 0 (cid:18)(cid:90) b (cid:19) ds = −(N−1)(1−F(s ))N−2f(s ) (−βs (2s +s )+R+e)g(s )ds 0 0 0 D 0 D D D de 0 (cid:18)(cid:90) b (cid:19) ds −2β(1−F(s ))N−1 s g(s )ds 0 . (31) 0 D D D de 0 Weknowthat (cid:82)b(−βs (2s +s )+R+e)g(s )ds > 0and (cid:82)b s g(s )ds > 0,so 0 D 0 D D D 0 D D D (cid:18)(cid:90) b (cid:19) −(N−1)(1−F(s ))N−2f(s ) (−βs (2s +s )+R+e)g(s )ds < 0, 0 0 D 0 D D D 0 and (cid:18)(cid:90) b (cid:19) −2β(1−F(s ))N−1 s g(s )ds < 0. 0 D D D 0 Asaresult,thecondition ds 0 < 0 de shouldholdinordertomaketheright-handsideofequation(31)beequaltotheleft-handsideof equation(31),i.e.,1−(1−F(s ))N−1G(b),whichispositive. 0 B Proof of Lemma 2 Proof. From Lemma 1, without loss of generality we can set an upper limit for the entry fee at e¯ such that Prob(s ≤ s ) = 0 at e¯. Existence of an optimal entry fee is immediate from the min 0 40

ExtremeValueTheorem,bynotingthattheoptimumisoveraclosedandboundedset[0,e¯],and thatexpression(16)isacontinuousfunctionbecauseprobabilitiesarecontinuous. Sinceweare onlyinterestedinthemaximum,e¯canberuledoutasacandidate,andwehavetheexistenceof e∗ ≥ 0. C Proof of Lemma 3 Proof. We want to show that if we increase e from 0 to something positive, then α increases. Equivalently, we can show that α is decreasing in s for sufficiently large s , i.e., if we reduce s 0 0 0 ∞ from tosomethingsmaller,αincreases. Werepeatequation(24),whichspecifiestheauctionsuccessprobability,hereforconvenience, α = ∑ N (cid:18) N (cid:19) F(s )k(1−F(s ))N−k 0 0 k k=1 · (cid:20)(cid:90) ζ(s0 ) g(s )ds + (cid:90) ∞ 1−(1−F(ϕ(s D ,k)))k g(s )ds (cid:21) . (32) 0 D D ζ(s0 ) 1−(1−F(s 0 ))k D D (cid:124) (cid:123)(cid:122) (cid:125) :=H k (s0 ) Takingthefirst-orderderivativewithrespecttos ,wehave 0 dα =F(s )NH ′ (s )+NF(s )N−1f(s )H (s )+NF(s )N−1(1−F(s ))H ′ (s ) ds 0 N 0 0 0 N 0 0 0 N−1 0 0 +[N(N−1)F(s 0 )N−2(1−F(s 0 ))−NF(s 0 )N−1]f(s 0 )H N−1 (s 0 ) + N ∑ −2(cid:18) N (cid:19) F(s )k(1−F(s ))N−kH ′(s ) (33) k 0 0 k 0 k=1 N ∑ −2(cid:18) N (cid:19) + k k=1 (cid:104) (cid:105) · kF(s )k−1(1−F(s ))N−k−(N−k)F(s )k(1−F(s ))N−k−1 f(s )H (s ). 0 0 0 0 0 k 0 BytheLeibnizintegralrule,wecanshowthat k(1−F(s ))k−1 (cid:90) ∞ H ′(s ) = − 0 f(s ) [1−(1−F(ϕ(s ,k)))k]g(s )ds < 0. (34) k 0 [1−(1−F(s 0 ))k]2 0 ζ(s0 ) D D D 41

Therefore,toprovethatdα/ds isnegativeass issufficientlylarge,itissufficienttoshowthatall 0 0 termsnotinvolving H′(s )arenegative,i.e.,wewanttoshowthat k 0 (cid:34) 0 > f(s ) NF(s )N−1H (s ) 0 0 N 0 −NF(s 0 )N−1H N−1 (s 0 )+N(N−1)F(s 0 )N−2(1−F(s 0 ))H N−1 (s 0 ) (cid:35) + N ∑ −2(cid:18) N (cid:19) [kF(s )k−1(1−F(s ))N−k−(N−k)F(s )k(1−F(s ))N−k−1]H (s ) (35) 0 0 0 0 k 0 k k=1 for sufficiently large s . Because density f(s ) is positive, we only need to show that the entire 0 0 termwithinthesquarebracketisnegative. Butnotethatass becomeslarge,(1−F(s ))N−k and 0 0 (1−F(s ))N−k−1 bothgotozero. Sothesufficientconditionfordα/ds becomesthat 0 0 s l 0 i → m ∞ [H N−1 (s 0 )−H N (s 0 )] > 0. (36) Ass → ∞ ,wehaveζ(s ) ↓ 0as dζ(s0 ) < 0,sothesufficientconditionisthat 0 0 ds0 (cid:90) ∞ (cid:90) ∞ (1−F(ϕ(s ,N)))Ng(s )ds > (1−F(ϕ(s ,N−1)))N−1g(s )ds . (37) D D D D D D 0 0 Thesufficientconditionfortheaboveconditionisthat (cid:90) ∞ Γ(k) ≡ (1−F(ϕ(s ,k)))kg(s )ds (38) D D D 0 isstrictlyincreasingink. Ignoringtheintegralconstraintonkandtreatingitasarealnumber,a sufficientconditionisthat Γ′(k) > 0fork ∈ [N−1,N]. Ate = 0,weneed (cid:90) ∞ 0 < Γ′(k) = [1−F(ϕ(s ,k))]kln[1−F(ϕ(s ,k))]g(s )ds D D D D 0 γkQ(k−1)Q−1 (cid:90) ∞ + [1−F(ϕ(s ,k))]k−1f(ϕ(s ,k))g(s )ds , (39) D D D D 2β 0 thatis, (cid:82)∞ kQ(k−1)Q−1 γ > − 0 (cid:82) [ ∞ 1−F(ϕ(s D ,k))]kln[1−F(ϕ(s D ,k))]g(s D )ds D . (40) 2β [1−F(ϕ(s ,k))]k−1f(ϕ(s ,k))g(s )ds 0 D D D D 42

∞ Note that the left-hand side goes to as γ becomes large. On the right-hand side, the term −[1−F(ϕ(s ,k))]kln[1−F(ϕ(s ,k))] in the numerator is bounded above regardless of γ; label D D this upper bound M > 0. For the denominator, we want a lower bound. Choose a finite ϵ k independentofk. Then (cid:90) ∞ [1−F(ϕ(s ,k))]k−1f(ϕ(s ,k))g(s )ds D D D D 0 (cid:90) ϵ > [1−F(ϕ(s ,k))]k−1f(ϕ(s ,k))g(s )ds D D D D 0 (cid:90) ϵ > [ min f(x)] [1−F(ϕ(s ,k))]k−1g(s )ds . (41) D D D x∈[0,ϵ] 0 Note that as γ → ∞ , ϕ(s ,k) → −∞ . Thus, there exists an ϵ′ < ϵ and γ¯ such that for all γ > γ¯, D s ∈ [ϵ′,ϵ],andk ∈ [N−1,N],(1−F(ϕ(s ,k)))k−1 > 1/2. Sothelowerboundofthedenominator D D becomes (cid:90) ϵ 1 [ min f(x)] g(s )ds . (42) D D x∈[0,ϵ] ϵ′ 2 Therequiredinequalitybecomes kQ(k− 2β 1)Q−1 γ > [min max f k ( ∈ x [N )] − (cid:82) 1, ϵ N] 1 M g( k s )ds , (43) x∈[0,ϵ] ϵ′ 2 D D whichholdsforsufficientlylargeγ. D Proof of Proposition 2 Proof. Wefirstprovethatforanyentryfeee,theprobabilityofauctionsuccess(weakly)decreases inγ. Foranyfixede,throughequation(12), s isalsofixedandinvarianttochangesinγ,sothe 0 firstconstraintinexpression(16)isfixed. Consequently,theprobabilitythatthesecondconstraint inexpression(16)holds,i.e., −βs (2s +s )−γ(K−1)Qs +R+Ke ≥ 0, (44) D min D D 43

isdecreasinginγ(andQ)andallothervariablesarefixed. Inaddition,foragivene,ifthissecond constraintisbinding,theprobabilityofauctionsuccesswillstrictlydecreaseatthateforhigherγ. Next,lete∗ maximizeexpression(19). Ifγincreases,thenbasedontheaforementionedresult, theprobabilityofauctionsuccessdecreasesate∗. Therearetwopossibilities. Ifthefirstconstraint binds,thene∗ isstilltheoptimumand ∂e∗ = 0. Alternatively,thesecondconstraintbinds. Then,we ∂γ cantakeK tobefixedforsmallchangesinγ. Wethencanrestatethesecondconditionas 1 (cid:16) (cid:17) e ≥ βs (2s +s )+γ(K−1)Qs −R . (45) D min D D K (cid:124) (cid:123)(cid:122) (cid:125) :=C(γ) Defineγ∗ suchthat e ∗ = C(γ ∗). (46) Thisconditionmeansthatasγincreasestoγ∗,thesecondconstraintinexpression(19)bindsate∗. Inaddition,define ∆ γ > 0ande∗∗ suchthat ∂e∗ e ∗∗ := e ∗+∆e = e ∗+ ∆ γ, (47) ∂γ and ∂C(γ) (K−1)Qs C(γ ∗+∆ γ) = C(γ ∗)+ ∆ γ = C(γ ∗)+ D∆ γ. (48) ∂γ K Sincethecondition(45)holds,weshouldhavee∗∗ ≥ C(γ∗+∆ γ). Therefore, ∂e∗ (K−1)Qs e ∗∗ = e ∗+ ∆ γ ≥ C(γ ∗+∆ γ) = C(γ ∗)+ D∆ γ (49) ∂γ K ∂e∗ (K−1)Qs =⇒ ∆ γ ≥ C(γ ∗)−e ∗+ D∆ γ, (50) ∂γ K (cid:124) (cid:123)(cid:122) (cid:125) =0 andso ∂e∗ (K−1)Qs ≥ D > 0, (51) ∂γ K fors heldconstant. D 44

E Derivation of the Analytic Expression for e in Sec. 6. Numerical Example Proof. Wewanttoderivetheanalyticalexpressionforequation(12)whens followstheexponential D distribution. Theoriginalequationis e = (1−F(s ))N−1E[max(−βs (2s +s )+R+e,0) | e]. 0 D 0 D BasedontheresultsdemonstratedinSectionAoftheAppendix,weknowthat −βs (2s +s )+R+e ≥ 0, s ∈ [0,b], D 0 D D and −βs (2s +s )+R+e < 0, s ∈ (b,+∞). D 0 D D Thus,weareabletocalculatetheexpectationterm. Wehave E[max(−βs (2s +s )+R+e,0) | e] D 0 D (cid:90) +∞ = max(−βs (2s +s )+R+e,0)g(s )ds D 0 D D D 0 (cid:90) b (cid:90) +∞ = (−βs (2s +s )+R+e)g(s )ds + 0×g(s )ds D 0 D D D D D 0 b (cid:90) b = (−βs (2s +s )+R+e)g(s )ds D 0 D D D 0 (cid:90) b = (−βs (2s +s )+R+e)λ e −λGsD ds D 0 D G D 0 = 2βs e−λGb(λ G b+1)−1 −β 2−e−λGb(λ2 G b2+2λ G b+2) +(R+e)(1−e −λGb). 0 λ λ2 G G 45

Whenλ = 1,then G E[max(−βs (2s +s )+R+e,0) | e] D 0 D = 2βs [e −b(b+1)−1]−β[2−e −b(b2+2b+2)]+(R+e)(1−e −b) 0 = 2βs e −b(b+1)−2βs −2β+βe −b(b2+2b+2)+(R+e)(1−e −b). 0 0 Therefore,inordertonumericallysolvefors fromequation(12),weneedtosolvethefollowing 0 equation: e = (1−F(s ))N−1 0 (cid:104) (cid:105) · 2βs e −b(b+1)−2βs −2β+βe −b(b2+2b+2)+(R+e)(1−e −b) , 0 0 √ whereb ≡ −2βs0 + 4β2s2 0 +4β(R+e) . 2β 46