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Informational Efficiency of Decision Markets with Risk-Averse Insiders and Uncertain Manipulation

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Title: Informational Efficiency of Decision Markets with Risk-Averse Insiders and Uncertain Manipulation
Physical Description: Book
Language: English
Creator: Twinam, Tate A.
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2010
Publication Date: 2010

Subjects

Subjects / Keywords: Prediction Markets
Market Microstructure
Information Aggregation
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: I examine the informational efficiency of prices in low-volume speculative markets under a variety of conditions involving price manipulation. I construct a market microstructure model based on the one-period batch-clearing framework of Kyle (1985), probabilistically incorporating a manipulator with preferences over the deviation of an asset price from a privately-known target as well as $N$ profit-maximizing risk-averse traders each receiving a noisy signal of the asset value. I find that the price error is normally distributed with mean zero. For a variety of plausible parameter values, I find that 1) the informed traders bid more aggressively in the presence of manipulation despite the increased risk penalty, 2) the variance of the price error is monotonically increasing in the level of risk aversion and the degree of manipulation, 3) the effectiveness of manipulation is highly sensitive to the size of the market, 4) the introduction of informed traders into the market is subject to a period of increasing returns followed by a period of decreasing returns, and 5) prices aggregate traders' private information even in the presence of a high degree of manipulation when the market is sufficiently thick.
Statement of Responsibility: by Tate A. Twinam
Thesis: Thesis (B.A.) -- New College of Florida, 2010
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: Elliot, Catherine

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Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2010 T97
System ID: NCFE004340:00001

Permanent Link: http://ncf.sobek.ufl.edu/NCFE004340/00001

Material Information

Title: Informational Efficiency of Decision Markets with Risk-Averse Insiders and Uncertain Manipulation
Physical Description: Book
Language: English
Creator: Twinam, Tate A.
Publisher: New College of Florida
Place of Publication: Sarasota, Fla.
Creation Date: 2010
Publication Date: 2010

Subjects

Subjects / Keywords: Prediction Markets
Market Microstructure
Information Aggregation
Genre: bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: I examine the informational efficiency of prices in low-volume speculative markets under a variety of conditions involving price manipulation. I construct a market microstructure model based on the one-period batch-clearing framework of Kyle (1985), probabilistically incorporating a manipulator with preferences over the deviation of an asset price from a privately-known target as well as $N$ profit-maximizing risk-averse traders each receiving a noisy signal of the asset value. I find that the price error is normally distributed with mean zero. For a variety of plausible parameter values, I find that 1) the informed traders bid more aggressively in the presence of manipulation despite the increased risk penalty, 2) the variance of the price error is monotonically increasing in the level of risk aversion and the degree of manipulation, 3) the effectiveness of manipulation is highly sensitive to the size of the market, 4) the introduction of informed traders into the market is subject to a period of increasing returns followed by a period of decreasing returns, and 5) prices aggregate traders' private information even in the presence of a high degree of manipulation when the market is sufficiently thick.
Statement of Responsibility: by Tate A. Twinam
Thesis: Thesis (B.A.) -- New College of Florida, 2010
Electronic Access: RESTRICTED TO NCF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE
Bibliography: Includes bibliographical references.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The New College of Florida, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Local: Faculty Sponsor: Elliot, Catherine

Record Information

Source Institution: New College of Florida
Holding Location: New College of Florida
Rights Management: Applicable rights reserved.
Classification: local - S.T. 2010 T97
System ID: NCFE004340:00001


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InformationalEciencyofDecisionMarketswith Risk-AverseInsidersandUncertainManipulation TateA.Twinam AThesis SubmittedtotheDivisionofNaturalSciences andtheDivisionofSocialSciences NewCollegeofFlorida inpartialfulllmentoftherequirementsforthedegree BachelorofArtsinEconomicsandMathematics UnderthesponsorshipofDr.CatherineS.Elliott Sarasota,Florida May,2010

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Abstract Iexaminetheinformationaleciencyofpricesinlow-volumespeculative marketsunderavarietyofconditionsinvolvingpricemanipulation.Iconstructamarketmicrostructuremodelbasedontheone-periodbatch-clearing frameworkofKyle,probabilisticallyincorporatingamanipulator withpreferencesoverthedeviationofanassetpricefromaprivately-known targetaswellas N prot-maximizingrisk-aversetraderseachreceivinga noisysignaloftheassetvalue.Indthatthepriceerrorisnormallydistributedwithmeanzero.Foravarietyofplausibleparametervalues,Ind that1theinformedtradersbidmoreaggressivelyinthepresenceofmanipulationdespitetheincreasedriskpenalty,2thevarianceofthepriceerror ismonotonicallyincreasinginthelevelofriskaversionandthedegreeof manipulation,3theeectivenessofmanipulationishighlysensitivetothe sizeofthemarket,4theintroductionofinformedtradersintothemarket issubjecttoaperiodofincreasingreturnsfollowedbyaperiodofdecreasingreturns,and5pricesaggregatetraders'privateinformationeveninthe presenceofahighdegreeofmanipulationwhenthemarketissuciently thick.

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Acknowledgments Thesuccessfulcompletionofthisworkwouldhavebeenimpossiblewithout thesupportofmyfriends,family,andcolleagues. Ioweagreatdealofthankstomycommittee,Dr.CatherineS.Elliott, Dr.TarronKhemraj,Dr.PatrickMcDonald,andDr.DavidMullins.They haveallconsistentlysupportedmeinmyacademicendeavorssincetherst dayIarrivedatNewCollege.IoweaspecialthankstoDr.Elliott.In additiontoprovidingvaluablefeedbackonmultipledraftsofmythesis, sheencouragedmetopursueextensiveindependentresearchearlyinmy thirdyear.Withoutthisfreedomtoexploremanydirectionsfarinadvance ofactuallywritingthethesis,Iwouldhavebeenfarlesspreparedforthe challengesIencountered. Myfriendshavebeenaconstantsourceofsupport,mostnotablyLuca. Thesupportweprovidedeachotheraswebothwentthroughthisexperience togetherwasanindispensablesourceofstrengthandcomfort.Luca,you cannotbegintoimaginehowmuchIappreciateyou. Imustalsothankmyfamily,especiallymyparentsCindi&Mark.Ican reasonablysaythat90%ofthehardworkindeliveringthisthesistothis pointwasdonebytheminmakingmethepersonIam.Mom&Dad,Ihope Icanbeasgoodaparentformychildrenasyouhavebeenforme. ii

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Contents Acknowledgmentsii 1Introduction1 1.1PredictionMarkets........................1 1.2RealWorldPerformance.....................2 1.3MicrostructureofTypicalPredictionMarkets.........4 1.4PotentialApplicationsofDecisionMarkets..........7 1.5Manipulation...........................8 1.5.1AnIllustrativeExample.................9 1.5.2ManipulationinTheoryandPractice.........10 1.5.3TheGoalofThisThesis.................10 1.6OutlineoftheThesis.......................12 2LiteratureReview14 2.1FinancialMarkets:Theory...................15 2.1.1InformationAggregationinFinancialMarkets....15 2.1.2MarketMicrostructureModels.............18 2.2PredictionMarkets:Theory...................20 2.2.1InformationalEciency.................20 2.2.2InterpretingPredictionMarketPrices.........22 2.2.3ManipulationandStrategicBehavior..........24 2.3PredictionMarkets:ExperimentalandEmpiricalFindings..27 2.3.1ExperimentalMarkets..................27 2.3.2Real-WorldMarkets:Public-AccessMarkets......32 2.3.3Real-WorldMarkets:CorporateExperiments.....36 2.4Conclusion............................38 3TheModel39 3.1Introduction............................39 iii

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Contentsiv 3.2OutlineoftheModel.......................40 3.3DiscussionofAssumptions....................44 3.3.1Risk-NeutralMarketMakerandExogenousLiquidity44 3.3.2InformationStructure..................46 3.3.3TraderUtilityFunctionsandRiskPreferences.....48 3.4EquilibriumStrategies......................51 3.5Game-TheoreticPropertiesoftheEquilibrium.........56 3.6Conclusion............................57 4ComparativeStaticsandAnalysis58 4.1DerivationoftheComparativeStatics.............58 4.2AnalysisoftheEquilibrium...................62 4.3NumericalExamples.......................65 4.3.1PriceError,ManipulatorCharacteristicsandMarket Thickness.........................65 4.3.2DirectEectsofRiskAversion.............70 4.3.3EciencyoftheEquilibriumPrice...........70 4.4Conclusion:ImplicationsforDecisionMarketDesign.....71 5Conclusion73 5.1DirectionsforFurtherResearch.................75 Glossary77 Bibliography82

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ListofFigures 4.1asafunctionof q for N =1,10,and20.Baselinecurves arethe q =0variancelevelsforthecorresponding N .....66 4.2asafunctionof 2 t for N =1 ; 10 ; 20.Baselinecurvesare 2 t =1variancelevelsforthecorresponding N .........67 4.3 asafunctionof 2 t for N =1 ; 10 ; 20..............68 4.4 @ @N underfourmanipulationscenarios..............69 4.5asafunctionof for N =1 ; 10 ; 20.Baselineisrisk-neutral case.................................71 v

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Chapter1 Introduction 1.1PredictionMarkets Predictionmarkets",alsoknownasinformationmarkets"orideafutures",arelow-volumespeculativemarketswheretradersexchangecontracts whosefuturepayosdependupontheoutcomeofoneormoreuncertain events.Eachcontractisdesignedsothatitspayohasaknown,deterministicrelationshipwiththeoutcomeoftheuncertaineventcontractedupon; whenthereisnouncertainlyabouttradersolvency,themarketpriceofa contractshoulddependonlyontraders'beliefsabouttherelativelikelihood ofthepossibleoutcomes.Bytradingthesesecurities,marketparticipants pushthepriceofeachcontracttowardanequilibriumvaluethatshouldreectaconsensusofallofthemarketparticipantsonthecorrectvalue.From thisvalueandthepayostructureofthecontract,onecaninferthemarket 1

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1.2.RealWorldPerformance2 estimateoftheprobability 1 ofeachoutcomecontractedupon. 2 Thus,the marketpriceforacontractwhosepayodependsupontheoutcomeofan event E aggregatesthecombinedbeliefsofallmarketparticipantsregardingthelikelihoodofeachpossibleoutcomeof E ;inotherwords,thisprice shouldbeasucientstatisticfortraders'beliefsabout E 3 Unliketraditionalnancialmarkets,predictionmarketsareemployedexclusivelyforthe purposeofquicklyaggregatinginformation. 1.2RealWorldPerformance Financialmarketsindevelopednationstendtobeecient.Inlarge,highly liquidnancialmarketsitisdiculttondinformationnotalreadycompoundedintopricesMalkiel2005.Becauseagentsincorporatetheirexpectationsaboutfuturepricesintotheirtradingdecisions,currentmarketprices aggregatetraders'beliefsaboutfuturepricesand,implicitly,aboutfuture eventsthatmayaectthoseprices.Thisresultsintheuncannyabilityof nancialmarketpricestopredictfutureevents,relativetootherforecastingmethods.Forinstance,pricesinorangejuicefuturesmarketsimprove onNationalWeatherServicetemperatureforecastsforcentralFloridaRoll 1984.FinancialmarketssingledoutthermresponsiblefortheChallenger accidentbeforethedayended,andmonthsbeforethesameverdictwas reachedbyinvestigatorsMaloneyandMulherin2003.Pricesinhighlyliquidnancialmarketsareextremelyresponsivetonewinformationaswell. 1 Probability"isusedhereinthesubjectivesenseasdescribingdegreesofbelief. 2 Undercertainconditions,seesection2.2.2. 3 Seetheglossaryfollowingchapter5fordenitionsoftechnical/economicterminology.

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1.2.RealWorldPerformance3 Pricesusuallyincorporatenewinformationsecondsafteritbecomespublic. Itwaslargelytheremarkablesuccessoftraditionalnancialmarketsin quicklyaggregatinginformationthatspurredtheinitialdevelopmentofpredictionmarkets.Itremainstobeseentowhatextenttheeciencycharacteristicsoflarge,liquidnancialmarketsdesignedforcapitalallocationand hedgingcanbereplicatedinsmall,relativelyilliquidmarketsdesignedspecificallyforinformationaggregation.Theevidencesofarisencouraging.The IowaElectronicMarketsIEMhaveconsistentlybeatenpollsinforecasting theoutcomeofUSpresidentialelectionsBergetal.2008.TheHollywood StockExchange,anonlinepredictionmarketoperatedbyCantorFitzgerald, hasgeneratedaccuratepredictionsofboxocenumbersaswellasOscar winnersWolfersandZitzewitz2004.SiemensAustriaimplementedanexperimentalmarkettopredictwhetherornotasoftwaredevelopmentproject wouldbecompletedonscheduleand,ifnot,howlongitwouldbedelayed. Themarketsmoothlyandsuccessfullyincorporateddispersedinformation longbeforeocialannouncements,accuratelypredictingatwoweekdelay inprojectcompletionthreemonthsbeforethedeadlineOrtner1998.InternalpredictionmarketsimplementedbyHewlett-PackardCorporationto forecastproductsalesbeatthecompany'socialforecastssixoutofeight times,despiteaverythinmarketandtheavailabilityofthenalmarket pricestothosesettingtheocialforecasts.Additionally,theactualsales outcomeswereconsistentwiththeprobabilitydistributionsgeneratedby themarketsChenandPlott2002.

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1.3.MicrostructureofTypicalPredictionMarkets4 1.3MicrostructureofTypicalPredictionMarkets Mostexistingpredictionmarketsareeitherlarge-scalemarketsopentothe generalpublic,orsmall-scalemarketsoperatingwithinrmsinwhichonly employeescanparticipate.Mostlarge-scalemarketsarecontinuousdouble auctionsinwhichtraderspostbuyandsellordersthatarecontinuously matchedaccordingtoorderprecedencerulesbasedonordersubmissiontime andprice.Real-moneymarketstendtolimittheamountonecaninvest; forexample,theIEMhasa$500capontraderinvestment.Short-sellingis typicallyprohibited,limitingtheabilityoftraderstopushpricesdownward. Sincepredictionmarket"isanumbrellatermdescribinganumberof highlyrelatedtypesofmarkets,itisnecessarytoestablishmoreprecise denitions. 4 Predictionmarketsarecharacterizedby 1.thetypesofcontractsoered, 2.theirpurposee.g.,entertainment,decisionsupport,andsoon, 3.theexpressivenessofthebettinglanguage,and 4.thetradingmechanism. Themostcommontypesofcontractsfoundinpredictionmarketsare: Winner-take-all:Contractpays$ X ifandonlyifevent Y occurs. X typicallyequals$1; 5 whenitdoes,thepriceofthecontractcanbe 4 Whilepredictionmarketssharemanysimilaritieswithtraditionalnancialmarkets, thereareimportantdierences.Forexample,unlikethestockstradedintraditional nancialmarkets,predictionmarketcontractshaveatrue"valuethatdoesnotdepend upontrader'sexpectations.Inotherwords,predictionmarketpricesarenotsubjectto theKeynesianbeautycontest"phenomenon. 5 Seeforinstance,Intrade.comandtheIEM.

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1.3.MicrostructureofTypicalPredictionMarkets5 roughlyinterpretedasanestimatedprobabilitythatevent Y occurs. Index:Contractpays$ Y ,where Y isthevalueofsomediscreteor continuousvariableofinterest.Thepriceforthiscontractcanbe interpretedas E [ Y ]. 6 Conditional:Canbeanyoftheabove,butisonlybindingwhenan event Z alsooccurs.Forexample,awinner-take-allcontractconditionalon Z wouldpay$ X ifandonlyif Y occurs,conditionalon Z alsooccurring.If Z doesnotoccur,thecontractisnullandpayments foritarerefunded.Pricesforconditionalcontractsconveyinformation abouthowthestateof Z aectstheprobabilityof Y Interlude:DecisionMarkets Whenpredictionmarketsareexplicitlyusedtoinformdecisionmaking, theyarecalleddecisionmarkets.Decisionmarketsoftenmakeextensiveuse ofconditionalcontracts,wherethevariablesofinterestareconditionedon decisionoptions.Theadvantageofthisisthatthedistributionsgenerated bythemarketcanprovideinsightintohowdierentcoursesofactionmay aectthevalueofrelatedvariables.Considerthefollowingexample:Traders cantradetwocontracts,A=ProductXsells100,000+unitsinquarter 4,conditionalonfeatureYbeingaddedtoProductXinquarter4"and B=ProductXsells100,000+unitsinquarter4,conditionalonfeatureY not beingaddedtoProductXinquarter4".Fromtherelativepricesof AandBthedecisionmakercandeterminehowthemarketbelievesadding 6 Whentradersareriskneutral;seesection2.2.2fordetails.

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1.3.MicrostructureofTypicalPredictionMarkets6 featureYwillaectsalesofproductX.Typically,thosewhowishtousea markettoimprovedecisionmakingwillbetheoneswhoestablishand,if necessary,subsidizethemarket. 7 Thebettinglanguageofapredictionmarketdeterminestheformbets overcontractsmaytake.Afullycombinatorialpredictionmarketallows traderstomakebetsonalllogicalvariablevaluecombinationsHanson 2003a.Combinatorialpredictionmarketsaggregatebeliefsovervariable interactionsintoacompletesetofjointprobabilitydistributionsoverall variables.Afullycombinatorialpredictionmarketseemsliketheobvious choiceformostsituations;however,thestatespaceisexponentialinthe numberofvariables,socomputationallimitationsmakefullycombinatorial marketsimpossiblewhenthenumberofvariablesislargeChenetal.2008. Forexample,just5binaryvariablesresultsin2 2 5 =4 ; 294 ; 967 ; 296possible contractswheneverypossiblelogicalbetisallowed. Toaddressthisproblem,anumberofrestrictedbettinglanguageshave beendeveloped.TheseincludesubsetandpairbettingaswellascombinatorialmethodsfortournamentsbasedonBayesiannetworksChenetal. 2007,2008.Atthispoint,nohighlyexpressiveandgenerallytractablei.e., polynomial-timeupdatinglanguagehasbeenfound. Traditionalnancialmarkets,liketheNewYorkStockExchange,conductmosttradingthroughacontinuousdoubleauctionmechanismwhere potentialbuyersandsellersasynchronouslysubmitorderswhicharethen 7 Iwillrefertothemasthemarketpatron"whennecessary.WhileIammostinterested inaddressingmanipulationindecisionmarkets,Iwillfrequentlyusethetermprediction marketwhenspeakingmoregenerally.

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1.4.PotentialApplicationsofDecisionMarkets7 matchedaccordingtopre-setpriceandorderprecedencerules.Thisauctionframeworkwasadoptedbytherstpredictionmarkets,includingthe IowaElectronicMarkets.However,ifthistypeofmarketistoremainsucientlyliquid,theremustbemanymoretradersthancontracts;otherwise, orderswillrarelybematchedandtraderswillloseinterest.Unfortunately, mostpredictionmarketshavefewtradersandmanycontracts. 8 Toaddressthisproblem,Hansonaconstructsamarketscoring rule",ageneralizationoftraditionalscoringrulesthatessentiallybecomes anautomatedmarketmakerwhosepotentiallossescanbeboundedChen andPennock2007.Thelogarithmicversionofthemarketscoringrulehas manydesirabletheoreticalproperties,hasperformedwellexperimentally seesection2.3.1,andisbecomingquitepopularChenetal.2008,Hanson 2003a,2007a. 1.4PotentialApplicationsofDecisionMarkets Predictionmarketsareaworthwhilesubjectofinquirybecauseoftheirapparentreal-worldsuccessandthevastspaceofpotentialapplications.The shortcomingsofstandarddecisionmakingproceduresandtheinabilityof organizationstocaptureandutilizetheinformationpossessedbytheirmembersiswellknown.Groupdiscussionscanamplifythecognitiveerrorsof members;groupthinkcancausecommitteestoreachanobviouslyincorrect consensus;committeescanfailtoaggregateinformationbyfocusingheavily oncommonknowledge;subordinatesmayhaveincentivestohidevaluable 8 Thisisespeciallytrueofintraorganizationalmarkets.

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1.5.Manipulation8 informationfromtheiremployersHahnandTetlock2006.Predictionmarketsmayamelioratesomeofthesefailures,makingthemaninvaluabletool toanyorganization,publicorprivate. Manypossibleapplicationsfordecisionmarketshavebeenproposedin theliterature.DecisionmarketscouldbedesignedtoestimatethelikelyeffectsofUSforeignpolicychangesongeopoliticaltrendsinunstableregions. 9 BergandRietzsuggestthat,hadtheRepublicanpartyconsultedthe conditionalcontractstradedontheIowaElectronicMarkets,theycould havepredictedthatDolewouldbeaweakcandidateagainstClintoninthe 1996USpresidentialelection.GaspozandPigneursuggeststhat predictionmarketscouldbevaluableaidsinmanagingR&Dportfolios.DecisionmarketscouldbeusedtoidentifyandreunderperformingCEOs Hanson2006.Hansonbgoessofarastosuggestthatdecisionmarketscouldfunctionasaformofgovernmentwhichhedubsfutarchy", directlysettingpublicpolicyatthelocal,state,andnationallevel. 1.5Manipulation Ifinformationgeneratedbyapredictionmarketisusedtoinformdecision making,thosewhohaveastakeinthedecisionbeingmademayattempt tomanipulatepricesinthemarketawayfromtheirinformationallyecient valuessoastoinuencethedecisionmakingprocess.Thisconcernisespeciallyrelevantfordecisionmarkets,wheretheroleofpricesinthedecision 9 SeeHansonbforadiscussionofPolicyAnalysisMarket,apredictionmarket developedwithintheDefenseAdvancedResearchProjectsAgencyforjustthispurpose. TheprojectwascanceledinNovember2003amidstamediarestorm,shortlybeforethe marketwouldhaveopened.

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1.5.Manipulation9 makingprocessismadeexplicit.Ifeective,thistypeofmanipulationcould signicantlyreducethevalueofdecisionmarkets,especiallyinhigh-stakes environments,wheresuchmarketsmightbemostuseful. 1.5.1AnIllustrativeExample Apharmaceuticalcompanyisattemptingtodevelopanewdrug X .Alice istheleadscientistinchargeofthedevelopmentofdrug X .Thereisa greatdealofuncertaintyregardingthelikelihoodoftheproject'ssuccess, andinternaldiscussionshavefailedtoyieldaconsensusaboutwhetheror notthedrugwillbereadyforclinicaltrialswithinareasonabletimeframe. Thedrugdevelopmentprocessisexpensive,anditisonlyprotableforthe companytocontinuetheprojectifthereisatleastaprobability p ofsuccess bydate d Toestimatethisprobability,thecompanydecidestoimplementadecisionmarket,allowingemployeestotradecontractswhoseterminalpayois $1ifdrugXisreadyforclinicaltrialsbydate d and$0otherwise,witha distinctcontractforeachchosen d .Thecompanyintendstousethepredictionmarketprices,aswellasotherinformation,toestimate p foreach d ; thisestimate,alongwithknowncostsandprojectedprots,willinformthe company'sdecisiontocontinueorscraptheproject. Aliceisnotrmlycondentthattheprojectwillsucceed.Unfortunately forher,Aliceisaspecialistinthemethodsparticulartothedevelopmentof thisspecicdrug.Iftheprojectiscanceled,Alicebelievesshewillbered. Topreventthisoutcome,shedecidestobuyaggressivelyinanattemptto inatethedecisionmarketpricesabovethelevelthatherbeliefssuggestis

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1.5.Manipulation10 justied. 1.5.2ManipulationinTheoryandPractice Thereareseveraldistincttypesofmanipulativebehaviorinpredictionmarketsaswellasnancialmarketsingeneral.Thebehaviordescribedin theaboveexampleisreferredtoaspricemanipulation"becauseitsgoal istoforcepricesawayfromtheirinformationallyecientvalues,anditis achievedwithintheconnesofthemarket.Intheaboveexample,theattemptedpricemanipulationisaimedataectingaprice-contingentdecision. Anagentmayalsoattempttomanipulatepricestomisleadthemarketin theshortruninordertoprotfromalaterpricecorrection.Whilethisis alsocalledpricemanipulation,itisnotinitiatedtoinuenceeventsoutside ofthemarketandsoitisdistinctfromthetypeofmanipulationIfocuson here.Thereisalsothepossibilityofoutcomemanipulation,whereagents takeactionsoutsideofthemarketthataecttheoutcomeofaneventcontractedupon. 10 1.5.3TheGoalofThisThesis CanAlicesucceedinmanipulatingpricesandsavingherjob?Moreprecisely,underwhatconditionscanAlicesucceed?Existingmodelsandexperimentsaimedatunderstandingtheconditionsnecessaryforsuccessful manipulationhavefailedtoincorporateanumberofelementscommonly foundinrealworldmarkets.Forinstance,mostmodelsassumethattraders 10 Thistopicisbeyondthescopeofthisthesis,butseesection2.2.3forabriefdiscussion oftherelevantliterature.

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1.5.Manipulation11 limittheirordersizesonlytodampenthepriceeectoftheirtrade.This isclearlyunrealisticwhentradersareriskaverse,sincetheywillnottake largepositionswhentheoutcomeishighlyuncertain.Thus,inthepresence ofriskaversion,itisconceivablethatwell-informedtraderswouldbeunable totradeinsucientvolumetocorrecttheeectofdistortionarytradesperpetratedbyahighlymotivatedpricemanipulator.Thisisapossibilitythat mustbeaddressed. Akeyfailureofvirtuallyallmodelsandexperimentsisalackofuncertaintyaboutthepresenceofmanipulatorsinthemarket.Sincemanipulators tradeagainsttheirinformation,theirtradesrepresentaprotopportunity toinformedtraderswhentheycanbedetected.Thus,manyoftheresults showingrobustnessofpricestomanipulationmaybereversedifboththe presenceandintendeddirectionofmanipulationisunknown. Iintendtoutilizeamarketmicrostructuremodelthatincorporatessome keyfeaturesoftypicaldecisionmarketstoexploretheeectsofpricemanipulationontheaccuracyofprices.Ifocusonarelativelysmallmarket populatedbyrisk-aversetraderseachpossessingnoisyinformationabout thevalueoftheassetforwhichthemarketwasconstructed.Iwillconsider thesituationwheretradersareuncertainaboutboththepresenceandintentionsofamanipulator:Inthemodel,themanipulatorisintroducedinto themarketwithaxedprobability,andtradershaveonlynoisyinformation aboutthemanipulator'stargetprice. Therearetwochannelsthroughwhichgreateruncertaintyregardingthe intentionsofthemanipulatorcanaecttheaccuracyofprices:

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1.6.OutlineoftheThesis12 Themanipulatorservesasaliquiditytradersoagreatervariancein hertradesimplies, ceterisparibus ,asmallerpriceimpactfortrades i.e.,amoreliquidmarketthatmayencourageinformedtradersto trademoreaggressively.Thismayleadtomoreaccurateprices. Increasingthevarianceofthemanipulator'stradeincreasesthevarianceofprices,whichmaycauserisk-aversetraderstobidlessaggressively,resultinginlessecientprices. Thus,themodelwillallowmetoanalyze 1.howriskaversionlimitstheabilityofinformedtraderstocorrectmispricingduetoamanipulator, 2.therelativeimportanceofmarketsizeversusleveloftraderriskaversionindeterminingtheabilityofthemarkettomaintainecientprices inthepresenceofmanipulation,and 3.foravarietyofparametervaluesandtheirresultingequilibrium,the valueofmarketpricesasaggregatorsofinformation. 1.6OutlineoftheThesis Inchapter2,Ireviewtherelevantliteratureonnancialeconomics,market microstruturetheory,andpredictionmarkets.Iconsiderboththetheoretical workonpredictionmarketeciencyaswellasnumerousempiricaland experimentalstudies.Inchapter3,Idevelopthemodelthatformsthecore ofthethesisandderivetheoptimalstrategiesforthemarketparticipants.

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1.6.OutlineoftheThesis13 Inchapter4,Iderivecomparativestaticsfortheequilibriumstrategiesand explorethegeneralpropertiesofthestrategiesthroughnumericalexamples.

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Chapter2 LiteratureReview Beforemovingontotheconstructionofthemodel,Ireviewtheprevious contributionstotheliteratureuponwhichmycurrentworkisbased.In section2.1,Idiscussthefoundationalworkontheeciencypropertiesof nancialmarketsaswellassomespecicmarketmicrostructuremodels thatIintendtobuildupon.Insection2.2,Ireviewsomeofthetheoretical modelsthathavebeendevelopedtoaddressconcernsabouttheeciency ofpredictionmarketsinparticular.Insection2.3,Idiscussavarietyof studiesaimedattestingthepredictionmarketconceptbothexperimentally andinreal-worldsettings.Thepatternsandbehaviorsobservedinsome oftheseactualmarketsmotivatethespecicdesignfeaturesIincorporated intothemodeldevelopedinthenextchapter,sotheyillustratethecontext andrelevanceofthemodel. 14

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2.1.FinancialMarkets:Theory15 2.1FinancialMarkets:Theory 2.1.1InformationAggregationinFinancialMarkets Theideathatmarketpriceseectivelyaggregate,summarize,andtransmit theinformationheldbymarketparticipantswasadvancedmostprominently byHayekinthecontextofthedebatesurroundingthefeasibilityof anorganizedeconomyinasocialiststatei.e.,thepossibilityofcalculating pricesintheabsenceofmarkets.Hayekarguedthatthemajorobstacle tocentralplanningwasalackofinformationabouttheconstantlychanginglocalcircumstancessurroundingeacheconomicactivity.Inmarket economies,participantsonlyhavedetailedknowledgeoftheirimmediate surroundings,butrelevantinformationaboutrelativescarcityandvalue thatisdispersedthroughouttheeconomyissummarizedandcommunicated tothembymarketpricesforallgoodsandservices.Thus, [the]wholeactsasonemarket,notbecauseanyofitsmembers surveythewholeeld,butbecausetheirlimitedindividualelds ofvisionsucientlyoverlapsothatthroughmanyintermediaries therelevantinformationiscommunicatedtoall.Themerefact thatthereisonepriceforanycommodity{orratherthatlocalpricesareconnectedinamannerdeterminedbythecostof transport,etc.{bringsaboutthesolutionwhichitisjustconceptuallypossiblemighthavebeenarrivedatbyonesinglemind possessingalltheinformationwhichisinfactdispersedamong allthepeopleinvolvedintheprocess.Hayek1945,pg.526

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2.1.FinancialMarkets:Theory16 Thisinsighthasbeenextendedtonancialmarkets.Grossmanand Stiglitzanalyzetwosimplemodelsofinformationrevelationandaggregationinamarketsetting.Intherstmodel,therearetwohomogeneous classesoftraders,theinformedandtheuninformed,wheretheinformedare thosewhohaveexpendedcostlyeorttoacquireasignalofthevalueofa riskyasset.Theyndaninteriorsolutionfortheproportionoftraderswho choosetogatherinformation,andtheyshowthatpriceswillrevealsome, butnotall,oftheinformedtraders'information.Inthesecondmodel,they consideraspotmarketandafuturesmarketforaparticularcommodity, populatedbyagroupofheterogeneouslyinformedtraders.Theyndthat theequilibriumspotmarketpriceperfectlyaggregatestraders'information, buttheyraisethepossibilitythatthismayleadtoamarketbreakdownin apreliminarydiscussionofwhatwouldbecomeknownastheGrossmanStiglitzparadoxdiscussedbelow. Inthemodernnanceliterature,pricesaresaidtobeinformationallyefcientiftheyfullyandcorrectlyreectallrelevantinformation.Economists distinguishbetweenthreeformsofinformationaleciency.Ifthepricereectsonlythehistoryofpricesandreturns,itissaidtobeweaklyecient. Ifthepricereectsonlypubliclyavailableinformation,itissaidtobesemistronglyecient.Ifthepricereectsallpubliclyknownandprivatelyheld information,itissaidtobestronglyecientBrunnermeier2001. GrossmangeneralizesGrossmanandStiglitzbyallowing dierentinformedtraderstohavedierentpieces"ofinformationabout therealizedvalueofariskyasset.Moreprecisely,thereare N informed traderseachreceivinganoisysignal y i ofthetruevalue v ,where y i =

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2.1.FinancialMarkets:Theory17 v + i with i N )]TJ/F15 10.9091 Tf 5 -8.836 Td [(0 ; 2 forall i .Hendsthattheequilibriumprice p eectivelyaggregatesalloftheinformedtraders'information,andisa sucientstatisticfor v i.e.,E[ v j p;y 1 ;:::;y n ]=E[ v j p ]. Theideathatpricesmayrevealtoomuch"information,sothattraders havenoincentivetocollectinformationintherstplace,wasformalized inGrossmanandStiglitz.Theyshowthatacompetitiveequilibriumisimpossiblewhen1informationischeapandthereisnonoise;or 2informationisperfect.Ineitherofthesecases,thepresenceofinformed traderswouldresultinstronglyecientprices,whichisnotanequilibrium situation.Equilibriumisonlypossiblewhenpricesfailtorevealfullyall oftheavailableinformation.ThisisknownastheGrossman-Stiglitzparadox,anditistypicallyresolvedinmarketmicrostructuremodelsbythe inclusionofanexogenousrandomliquiditytrade.Thisisanalogoustothe situationinreal-worldnancialmarketswheremanytradesareconducted fornon-informationreasons,i.e.,hedging,investment,gambling,andsoon. Arelatedresulttiesamarketbreakdowntotheindividualrationality oftheparticipatingtraders.MilgromandStokeymodelamarket populatedbyrisk-aversetraderswithconcordantbeliefswhoeachreceive adierentprivatesignalofthevalueofariskyasset.Startingfroma Pareto-optimalallocation, 1 theyshowthat,whentradershaverationalexpectationsandthisfactiscommonknowledge,noneofthemwillbewilling totrade.Thiscanbeexplainedintuitivelyasfollows:Ifatraderiswillingto acceptatrade,thatwillingnessrevealssomethingaboutthetrader'sprivate information.Thisisessentiallyamarketbreakdownduetoextremeadverse 1 Inotherwords,agentshavenoexogenousnon-speculativemotivetotrade.

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2.1.FinancialMarkets:Theory18 selection.Thepresenceofliquiditytradersamelioratesthisproblem. 2.1.2MarketMicrostructureModels Untilthe1980s,mosttheoreticalworkonnancialmarketsignoredthe mechanismsthroughwhichtradesactuallyoccurred.Theinteractionof tradersandthepriceformationprocesswereeitherignoredordealtwithin ahighlyabstractway;traders'actionsdidnotaectthebehaviorofother tradersandpricesweresetbyaWalrasianauctioneer.Thisapproachwas fruitful,buteventuallyitbecameclearthatcertainproblems,suchasthe bid-askspread,certainintra-dayvolatilitypatterns,andthemanipulation ofstockprices,couldnotbedealtwithinthisframework.Thisledtothe developmentofmodelsthatmadethetradingmechanismexplicit,andthese cametobeknownasmarketmicrostructure"models. Kylewasthersttoanalyzethestrategicbehaviorofinformed tradersinamarketmicrostructuremodel.Hedevelopedastaticframework wherearisk-neutralinformedtraderwhoknowsthenalvalueofariskyassetwithcertaintyandagroupofrisk-neutraluninformedliquiditytraders bothsubmitorderstoacompetitive,risk-neutralmarketmaker.Theinformedtradermaximizesexpectedprotsandtheliquiditytraders'orders aregivenbyanexogenouslydeterminedrandomvariablethatiscommon knowledgewithinthemodel.Themarketmakerobservesonlytheaggregateorderowandsetsapriceequaltotheexpectedvalueoftheasset conditionedontheparticularorderowobserved. 2 Thus,thepricewillbe 2 Kylenotesthatitisneveroptimalfortheinformedtraderorthemarketmakerto implementamixedstrategy.

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2.1.FinancialMarkets:Theory19 semi-stronglyecient.Inthisstaticsetting,Kyleprovestheexistenceof arationalexpectationsequilibriumwherepricesandquantitiesarelinear functionsofobservations.Heshowsthatonlyhalfoftheinformedtrader's informationisrevealed,andthatpriceeciencyisunaectedbythevariance ofliquiditytrading. Kyleextendsthismodeltoan N periodsequentialauctionandderives auniquelinearequilibriumgivenbyasystemofdierenceequations.The limitcasewheretradingisconductedcontinuouslyisalsoderivedandKyle showsthattheuniqueequilibriumispreservedasasystemoflineardierentialequations.Inthecontinuous-timeversionofthemodel,Kyleshows thatthevolatilityofpricesovertimeisconstant,theinformedtrader'sinformationisincorporatedintopricesataconstantrate,andpricesconverge totheirstronglyecientvalues. Duetotheirversatilityandtractability,boththestaticanddynamicversionsofKyle'smodelhavebecomestandardtoolsforanalyzingthestrategic behavioroftraders.Importantshortcomingsofthemodelincludetheexogeneityofinformationgatheringandthelackofstrategicbehavioronthe partofuninformedtraders.Thebasicmodelhasbeenextendedinnumerousdirections,andthemodelIhaveconstructedisbasedonthestaticKyle framework. SubrahmanyamgeneralizesthestaticKylemodelbypositingmultipleinformedtradersandallowingthemtoberiskaverseusingthestandard negativeexponentialutilityfunction.Hendsthattheequilibriumsolution ofthemarketmaker'spricingproblemistherootofaquinticpolynomial, andviaageometricargumentheshowsthatapositiverealsolutionmustex-

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2.2.PredictionMarkets:Theory20 ist;comparativestaticsarederivedviatheimplicitfunctiontheorem. 3 The keyresultastheyrelatetothemodelinchapter3isthatpriceeciencyis decreasinginthevarianceofliquiditytradingandinthelevelofriskaversion oftheinformedtraders.Thismakessenseintuitively:thehighervariance ofliquiditytradingincreasesthevarianceofpricesandhenceincreasesthe riskinessofagiventrade.ThisisareversaloftheresultsofKyle, wheregreaterliquiditytradinghadnoeectonpriceeciency. 2.2PredictionMarkets:Theory 2.2.1InformationalEciency Thetheoreticalliteratureontheallocativeandinformationaleciencyof traditionalnancialmarkets,alongwithitsempiricalsupport,initiallylent legitimacytotheconceptofpredictionmarkets.Researchontheeciency ofpredictionmarketsinparticularisstilllimited,howeversomefoundationalworkinextendingtraditionalmodelstothepeculiarcharacteristics ofpredictionmarketshastakenplace.Thisliteratureisreviewedhere,with particularemphasisonmodelsthataddressinformationaleciencyandthe eectsofmanipulation. TetlockandHahnanalyzethesituationwhereapredictionmarketcouldprovidevaluableinformationtoadecisionmaker.Inamarket consistingofonlyinformed,rationaltraders,theno-tradetheoremofMilgromandStokeywouldapply.However,TetlockandHahnshow 3 Itisnoteworthythatsuchaseeminglysimplegeneralizationleadstoasubstantially morecomplicatedsolution.Thisisacommonthemeinthemarketmicrostructureliterature,anditisaproblemthatwillbeencounteredinchapter4.

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2.2.PredictionMarkets:Theory21 thatadecisionmakerwillprovideliquiditytoanilliquidmarketpopulated entirelybyrationalinformedtradersiftheliquiditysubsidynecessaryto achieveagivenlevelofpriceinformativenessislesscostlythanthepotential gainsinallocativeeciencyresultingfromthepresenceofaninformative price.Inotherwords,thedecisionmakerwillacceptlossesinthemarket toinduceinformationacquisitionbyothertraders.Theyshowthatthis liquiditysubsidyalwaysimprovesexpectedsocialwelfarethroughenhanced allocativeeciency;however,itwillnotinducetheoptimallevelofinformationacquisitionbytheothertraders.Theseresultsareinterestingbecause, inadditiontoprovidingarigorousjusticationfortheexistenceofdecision markets,theyshowthatdecisionmarketscanoperateintheabsenceofliquidity/noisetraders;themarketcanfunctionaslongasthereisamarket makerwillingtooperateatalossseesection3.3.1. Inasecondmodel,TetlockandHahnaddadecisionstakeholderwhose payodependsontheactiontakenbythedecisionmakerafterobservingthe marketprice.Thisdecisionstakeholdershouldactasapricemanipulator. Inthismodel,thetradingpopulationconsistsofaperfectlyinformedrationaltrader,anuninformedmanipulator,andacompetitivemarketmaker. Notethataprot-seekingmarketmakercanoperateinthismarketdueto theexistenceofthemanipulator,whocanbeviewedasanoisetraderi.e., atraderwhosetradeswillbeuncorrelatedwiththetruevalueoftheasset. Theyndthattheexpected 4 extentofmanipulationhasnoeectonprice informativeness;thisisunsurprisingduetothepresenceofaperfectlyinformedrationaltraderwithunlimitedtradingresources.Thus,Tetlockand 4 Thisreferstotheexpectationsoftheothertraders.

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2.2.PredictionMarkets:Theory22 Hahnsuggestthatthepresenceofamanipulatorshouldhelpthedecision makerbyprovidingfreeliquidity.Itisquestionablewhetherornotthis resultwillholdwhenthereisuncertaintyabouttheexistenceofthemanipulatorandinformedtradershaveonlynoisysignalsorareconstrainedin theirabilitytorespondduetoriskaversion.Theseissueswillbeaddressed inchapter3. OttavianiandSrensenarguethatheterogeneouspriorsshould betypicalamongtradersinreal-worldpredictionmarkets,sincethesemarketsareusuallyconstructedtoprovidepredictionsabouttheoutcomesof non-recurringevents.Theymodelapredictionmarketoverabinaryoutcomeeventpopulatedbyrisk-neutraltraderswithlimitedbudgetsandheterogeneouspriorbeliefswhoreceiveaprivatesignal.Inthemodel,traders' priorbeliefsandthedistributionofthesignalsarebothcommonknowledge. Theyndthatthemarketunderreactstonewinformation,resultingina favorite-longshotbiasi.e.,contractswhoseoutcomeisfavoredbythemarketareunderpricedwhilecontractswhoseoutcomeisconsideredalongshot areoverpriced. 5 2.2.2InterpretingPredictionMarketPrices Amongmostresearchersandpractitionerswhodealwithpredictionmarkets, itiscommonlyacceptedthatthepredictionmarketpricesforanall-ornothingcontractthatpays$1ifandonlyifaspecicoutcomeoccursreects amarketconsensusontheprobabilityofthatspecicoutcomeoccurring. 5 Theirresultsarerobusttoarelaxationofthebudgetconstraintaswellastheintroductionofdecreasingabsoluteriskaversionpreferences.

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2.2.PredictionMarkets:Theory23 Moregenerally,itisassumedthatamarket-estimatedprobabilityofanevent occurringcanbeinferredfromthepricesofcontractswhosepayosdepend upontheoutcomeofthatevent. Manskihighlightstheabsenceofarmtheoreticalgroundingfor thispracticeandattemptstorefutethenotionthatpricesrepresentmarketestimatedprobabilities.Heemphasizesthatpredictionmarketpricesreect notjustthebeliefsoftraders,butalsotheirriskpreferencesandbudget constraints.Hendsthatpricesdonotequalthemeanbeliefsoftraders orconveyinformationaboutthedispersionoftraderbeliefs.Hesuggests thatscoringrulesandopinionpoolsi.e.,averagesofscoringruleresponses wouldbeamoreeectivemethodofcreatingconsensuspredictionsthat wouldnotdiscardinformationaboutthedispersionofbeliefs. 6 InresponsetothiscritiqueofManksi,WolfersandZitzewitzprovidetheoreticalgroundingfortheinterpretationofpredictionmarketprices. Theyndthatwhentradershavelogutilityfunctionsandbudgetsareuncorrelatedwithbeliefs,marketpricesareequaltothemeanoftraderbeliefs. Theyfurthergeneralizethemodeltoavarietyofriskpreferencesandutility functions,ndingthatpredictionmarketpricesdivergefromthemeanof traderbeliefsonlyslightly. 7 TheyconrmandelaborateonsomeofManski'sndings,namelythedependenceofpriceaccuracyontraderutility functions,riskpreferences,budgetconstraints,andbeliefdispersion.They ndthatthemostsignicantdeviationsofpricefrommeanbeliefsoccurat 6 NotetheexperimentalndingsofLedyardetal.thatscoringrules/opinion poolsunderperformedpredictionmarketswhenbeliefswerewidelydispersed. 7 Interestingly,theyndthemostextremedivergencewhenadoptingManski'smodel, whichheclaimedrepresentsabestcase"forecientaggregation.

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2.2.PredictionMarkets:Theory24 veryhighandverylowprices,perhapsreectingthelongshotbiasobserved inmanyreal-worldbettingmarkets.Theyinterprettheirresultsasprovidingamicrofoundationfortheclaimthatpredictionmarketsapproximately ecientlyaggregatebeliefs"WolfersandZitzewitz2006,pg.2. 2.2.3ManipulationandStrategicBehavior OttavianiandSrensendescribetherstformalmodelofoutcome manipulation,withspecicreferencetocorporatepredictionmarkets.They deneoutcomemanipulationtomeanactionstakenbytraderstoaectthe likelihoodofthepotentialoutcomesforwhichthemarketsupportscontracts. TheymodelamarketsupportingtwoArrow-Debreusecuritiesthatcovera binaryevent.Tradersareriskaverseandhaveheterogeneouspriorsaswell asprivatesignalsofvalue.Toavoidthedistortionscausedbywealtheects, 8 theyconsideronlyconstantabsoluteriskaversionpreferencesmanifested throughthestandardexponentialutilityfunction.Intheirmodel,every traderisabletomanipulateoutcomesandthisiscommonknowledge. Intherationalexpectationsequilibriumtheyanalyze,pricesarefully revealing,everytraderwithanon-zeronetpositionhasanincentivetomanipulate,andthereistypicallynon-negligibleaggregatemanipulationi.e., theupwardmanipulationsofoptimistsandthedownwardmanipulationsof pessimistsrarelycanceleachotheroutexactly 9 HansonandOpreapresentanelaboratemodelofpricemanipulationinasmallscalepredictionmarket.BecauseIincorporateelementsof 8 See,forexample,OttavianiandSrensen. 9 Notethatcostlymanipulativeactionswhoseneteectiszeroaresociallywasteful Tullock1967.

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2.2.PredictionMarkets:Theory25 itsstructureintothemodelIconstructinchapter3,Iprovideadetailed outlineoftheirmodelhere.HansonandOpreautilizeasingle-periodKyle frameworkseesection2.1.2withacompetitive,risk-neutralmarketmaker tradingasingleassetwhosetruevalueisdrawn 10 as v N v;S v .Theyassumeanexogenousliquiditytrade l N l;S l theirresultsholdif S l =0 and T risk-neutraltraders,labeled i = f 1 ; 2 ;:::T g ,whoeachgainatrading prot i x i = x i v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p where x i isthequantityoftheassetpurchasedand p istheprice.Inaddition, thereisaspecialtraderwhosetradingprotisgivenby 0 x 0 = x 0 v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p )]TJ/F18 10.9091 Tf 10.909 0 Td [(k t )]TJ/F18 10.9091 Tf 10.909 0 Td [(p 2 = x 0 v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p )]TJ/F18 10.9091 Tf 10.909 0 Td [(k v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p 2 + wp )]TJ/F15 10.9091 Tf 11.194 2.879 Td [(^ k where w =2 k t )]TJ/F15 10.9091 Tf 10.035 0 Td [( v and ^ k = k t 2 )]TJ/F15 10.9091 Tf 10.035 0 Td [( v 2 .Thisspecialtrader'spayodepends inquadraticfashionontheextenttowhichthepricedeviatesfromatarget value t ;thus,thistraderhasanincentivetomanipulatepricetowardthe targetvalue.Thestrengthoftheincentivetomanipulateisgivenby k whichisassumedtobecommonknowledge.Thebias" w equivalently, thetargetvalue t isprivateinformationandtheothertradersknowonly thatitwasdrawnas w N w;S w .Asubsetofsize N ofthe T traders canbecomeinformedbyacquiringacostlysignalofthetruevalueofthe asset.Itisassumedthatthemanipulatorisuninformedaboutthetrueasset 10 Thenotation x N x;S x referstoanormallydistributedrandomvariable x with mean x andvariance S x .

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2.2.PredictionMarkets:Theory26 valueandcannotacquireasignal.Tocomputeanequilibrium,eachtrader isassumedto 1.privatelychoosetheaccuracyoftheirassetvaluesignal, 2.observetheirprivateinformation,and 3.chooseamarketorder x i Themarketmakerobservesthetotalorderquantity y = l + T X i =0 x i andsetsthemarketprice p = E [ v j y ]+ ,where N ;S describes errorintheprice-settingprocesstheirresultsholdwhen S =0. Theyndthatthedierencebetweentheasset'spriceanditstruevalue p )]TJ/F18 10.9091 Tf 11.532 0 Td [(v isdistributedwithmean0andavariancedependingonlyonthe varianceoftheinformedtraders'signals,thenumberofinformedtraders, thevarianceoftheassetvalue,andthevarianceofthepricesettingerror. Thus,onaverage,thereisnobiasinpriceandthemanipulatorcanonly aectpricesbyaectingtheinformedtraderschoiceofeortinacquiring signals.Furthermore,changesinthemeanbias w havenoeectonprices whileanincreasein S w lowerspriceerror.Theyinterpretthisndingas suggestingthattraderswillattempttoacquiremoreinformationandtrade moreaggressivelywhentheyexpectastrongermanipulationattempt;the manipulatoractsverymuchlikeanoisetrader,andtheresultingimpacton pricesisminimizedbecauseinformedtradersincreasethevolumeoftheir

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2.3.PredictionMarkets:ExperimentalandEmpiricalFindings27 tradestotakeadvantageoftheprotopportunitypresentedbythemanipulator. Thereareseveralassumptionsinthemodelthatfailtocapturesomeof thesituationsonewouldexpecttoseeinareal-worldpredictionmarket.The informedtraders'responsetothebehaviorofthemanipulatorislimitedonly bythedepthofthemarket.However,iftheinformedtraderswereriskaverse orhighlyuncertainaboutthepresenceofthemanipulator,theirresponse mightbelimitedseverelyenoughtoallowsignicantmanipulation. 2.3PredictionMarkets:ExperimentalandEmpiricalFindings 2.3.1ExperimentalMarkets Experimentalmarketsprovideanexcellentsettingforexaminingtheinformationaggregationpropertiesofpredictionmarkets.Thelabsettinggives theresearchersubstantialcontroloverthetypeoftradingmechanism,the informationstructure,andeachtrader'spayostructure.Sincethelabsettingallowstheresearchertoconstructanasset,theresearchercanfully controltheinformationthateachindividualtraderreceivesabouttheasset. Thus,theresearcherknowsexactlywhatinformationthemarketasawhole possesses,allowingthecalculationoftheperfectlyrevealingrationalexpectationsequilibriumprice,whichcanbeusedasacomparisonbenchmark. Ledyardetal.describesaseriesofexperimentsperformedin20022003undertheauspicesoftheDefenseAdvancedResearchProjectsAgency

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2.3.PredictionMarkets:ExperimentalandEmpiricalFindings28 insupportoftheFutureMAPPolicyAnalysisMarketinitiativethataimed totesttherelativespeedandaccuracyofavarietyofpredictionmarketsand sometraditionalinformationaggregationmechanisms.Sixmechanismswere tested:asimplepredictionmarketimplementedasacontinuousdouble auction,twocombinatorialpredictionmarketsoneimplementedthrougha callmarketwithbatchclearing,theotherwithalogarithmicmarketscoring rulegivenauniformprior,aproperscoringruletowhicheachindividual reported,andtwoopinionpoolsonelinear,onelogarithmic.Theresults indicatedthat,inasimpleenvironmentwiththreebinaryvariables,the marketscoringrulesignicantlyoutperformedtheothermechanisms,and thedoubleauctionsignicantlyunderperformedtheothermechanisms.In amorecomplexenvironmentwitheightbinaryvariables 8 =256possible states,themarketscoringruleandthetwoopinionpoolswerecomparable andallsignicantlyoutperformedtheothermechanisms.Ledyardetal. interpretthesendingsassuggestingthat 1.non-combinatorialmechanismsarerelativelyineectivewhenvariables arestronglyrelated,and 2.opinionpoolsperformrelativelywellwheninformationisevenlydistributed. Thefailureofthedoubleauctionmechanismisunsurprisinggiventhesize ofthestatespacerelativetothesizeofthetraderpool;thesuccessofthe marketscoringruleinlightofthissizedisparityisespeciallyinteresting. Akeyndingofthispaperisthatthemarketscoringruleaggregatesinformationveryquickly;onaverage,themarketscoringruleachievedpeak

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2.3.PredictionMarkets:ExperimentalandEmpiricalFindings29 accuracywithinveminutesandremainedstablearoundthatpriceforthe remainingtenminutesofthesession. ManipulationinExperimentalMarkets Hansonetal.examinedirectlytheeectmanipulatorshaveoninformationaleciencythroughaseriesofexperimentsconductedin20032004.Thesubjectstradedcontractswhoseterminalpayowasdrawnfrom acommonknowledgesetofthreepossiblevalues,witheachvaluehaving aprobability 1 3 ofbeingdrawn.Ineachsession,everysubjectreceiveda signalaboutthetruevalueoftheterminalpayo;eachwastoldoneofthe twopossiblevaluesthatthepayowould not take.Thus,thesubjectsas awholehadenoughinformationtoidentifythetrueassetvalue,butnone hadenoughinformationindividually,andthesubjectswerenotpermittedto communicate.Inaddition,asubsetofthesubjectsineachgroupweregiven astrongincentivetomanipulatepricesupward.Theexistence,strength, anddirectionofthemanipulationincentiveswerecommonknowledge. Theauthorsfoundthat,whilemanipulatorsdidpersistentlyattempt todistortpricesupward,theyconsistentlyfailedtoreducepriceaccuracy regardlessoftheactualstatecomparedtoacontrolgroupwithnomanipulators.Theresultsindicatedthat,whentraderswereawareofthe presenceofmanipulators,theyrealizedtheprotopportunityandactively tradedagainstthem.Thekeyweaknessofthisexperimentistheextent towhichmanipulationattemptsaretransparent.Inthedecisionmarket environmentIamfocusingon,itispossiblethattherewillbesubstantial uncertaintyaboutthepresenceofmanipulatorsinthemarket.

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2.3.PredictionMarkets:ExperimentalandEmpiricalFindings30 Opreaetal.reportonaseriesofexperimentsanalyzingtheability ofmanipulatorstomisleadmarketobservers.Intheirexperimentaldesign, subjectseachreceiveanoisysignalabouttheterminalpayoofanasset theassettakesavalueinthecommonknowledgeset f 0 ; 100 g andare thenallowedtotradetheirendowedstockoftheassetinastandarddoubleauctionframework.Aftertradingends,agroupofuninformedsubjects whoobservedthetradingaswellasthenalmarketpriceareaskedto forecasttheassetvalueandarepaidbasedontheaccuracyoftheirforecast.Abaselinetreatmentdesignedasdescribedaboveiscomparedtoan experimentaltreatmentthatisidenticalexceptthathalfofthetradersreceiveadditionalcompensationbasedonhowclosetheforecastedvalueisto aprivatelyknowntargetvalue.Thus,thesetradershavepreferencesover theforecastsmadebytheuninformedobservers,givingthemanincentiveto manipulatepricessoastomisleadtheobservers.Whileboththeobservers andnon-manipulatingtradersareawareofthepresenceofthemanipulators, themanipulators'targetpriceisunknown.Thisisakeydierencebetween thisexperimentandthatofHansonetal..Inthatstudy,traders couldanticipatethemanipulators'tradesandtakeadvantageofthatknowledgebyrefusingtoaccepthighbuy/selloersandtheevidenceshowsthat theydid.Here,however,tradershadtocombinetheirsignalswithobserved marketbehaviortoformbeliefsaboutthedirectioninwhichmanipulators wereattemptingtoswayprices. Opreaetal.ndthattraderswithpreferencesoverforecastsdoattempt tomanipulatepricesand,whentheirtargetedvalueishigh,theysucceedin

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2.3.PredictionMarkets:ExperimentalandEmpiricalFindings31 raisingpricesbyasignicantmargin. 11 However,theyfailtomanipulate observerforecasts.Theauthorsspeculatethatthismaybeduetoltering" bytheobservers,whoinfactproduceforecaststhataresuperiortothose thatwouldbemadebysimplyinterpretingpricesasecientprobability estimates"Opreaetal.2007,pg.16.Thissuggeststhatjudgmentmay beasignicantfactorininterpretingandutilizingpredictionmarketprices fordecisionmaking.Whilethemarketfailedtocompletelycorrectforthe noiseintroducedbythemanipulators,theeortsoftraderscombinedwith thejudgmentofthemarketobserverswereabletomaintaintheaccuracy oftheforecastsdespitethefactthathalfofthetraderswereattemptingto distortthem. Theexperimentalevidenceindicatesthatmarketscaneectivelycounteracttheeortsofmanipulatorswhentradersareawareoftheirpresence. However,theexperimentalstudiesconductedthusfarhavefailedtoexplore thevarietyofwaysinwhichtradersmaybeuncertainaboutthepresence andcharacteristicsofamanipulator. 12 Theyhavealsofailedtoaddress adequatelythepossibilitythatnon-manipulatorsmayberelativelymore constrainedthanmanipulatorsintheirabilitytotradeduetoriskaversion. 11 Attemptstolowerpricefail;itislikelythatthisisduetothedesignofthemarket which,likemanyrealworldpredictionmarkets,prohibitsshortselling,makingpricesless exibledownward. 12 Arelateddrawbackofthistypeofexperimentistheforcedimpositionofacommon prioramongthetraders.Byconstructingawhollyarticialasset,theresearcherensuresthatnoneoftheexperimentalsubjectswillhavedistinctpriorinformationorbeliefs abouttheasset.Thisimpliesthatmodelsassumingheterogeneouspriorscannotbetested experimentally.

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2.3.PredictionMarkets:ExperimentalandEmpiricalFindings32 2.3.2Real-WorldMarkets:Public-AccessMarkets MuchoftheinterestinpredictionmarketsasaforecastingtoolinitiallyresultedfromtheapparentsuccessesoftheIowaElectronicMarketsinpredictingtheoutcomesofpresidentialelections.SinceIam,ultimately,interested intheapplicationofpredictionmarketstosolvingrealforecastingproblems, itisworthwhiletoconsiderindetailtheperformanceofrealworldmarkets. Camererdiscussesaninterestingeldexperimentthatattempted tomanipulatepricesinapari-mutuelbettingmarketforthoroughbredracing.Bymakinglargebetsonrandomlychosenhorseslargeenoughtovisiblychangestheodds,heattemptedtomisleadothertradersintobelieving thattherewasatraderinthemarketwithvaluableprivateinformation. 13 Eventhoughthebetswererelativelylargeonaverage7%ofthewinpool, therewerenosystematicresponsesbyothertradersandnostatistically signicanteectsontheodds.Hedrawstheconclusionthat thesemarketssimplyaggregateinformationremarkablywell,and accordingly,bettorsknowenoughtoignorealargebetthatis madefarbeforeposttimeandisnotbackedupbyasteadyow ofmoney,whichkeepstheheavilybethorse'soddsdown...the inabilityoftheselargebetstomovethemarketsystematicallyis ablowtothebeliefsofthosewhothinkthatmarketsareeasily androutinelymanipulatedbylargeinvestors.Camerer1998, pg.480 13 Thebetswouldbecanceledbeforetheracebegan.

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2.3.PredictionMarkets:ExperimentalandEmpiricalFindings33 TheIowaElectronicMarkets TheIowaElectronicMarketsareasetofreal-money,publicaccessprediction marketsadministeredbytheUniversityofIowa.Createdin1988asan educationaltool,themarketsbeganbyoeringArrow-Debreucontractsfor eachcandidateinthe1988USpresidentialelection.Sincethen,theyhave expandedtocovercongressionalelections,economicindicators,andeven movieboxocereturns.Traderscanhavebetween$5and$500USdollars atstakeineachmarket,andshortsellingofcontractsisprohibited.A signicantliteraturehasdevelopedaddressingtheextenttowhichprices inthesemarketsimproveonotherpredictorse.g.,electionpolls.This literaturewillbereviewedhere. Forsytheetal.examinetheprecursortotheIEM,theIowaPresidentialStockMarket.Thisdoubleauctionmarketallowedtraderstopurchaseandexchangecontractswhosepayowouldbedeterminedbythe fractionofthepopularvoteeachcandidatereceived.Assumingastraightforwardinterpretationofpricesseesection2.2.2,acandidate'sexpected voteshareisequaltothepriceofthecontractforthatcandidatedividedby $2.50.Theauthorsfoundthatthemarketpricesdidnotreactsignicantly topolldata,polldataweresubstantiallymorevolatilethanmarketprices, andmarketpricesoutperformedthepollsaspredictorsdespiteevidenceof systematicbiasesintraderbehavior.Forsytheetal.claimthatthesystematicbiasesinthejudgmentofaveragetraderswerecorrectedbyrational marginaltraders.

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2.3.PredictionMarkets:ExperimentalandEmpiricalFindings34 OlivenandRietzpresentadetailedanalysisoftraderbehavior inanattempttoexplaintheaccuracyofIEMpredictions.Theyndthat manytradersarepronetoerror,andthatthesetradersoftenpursuebuyand-holdstrategiesandplacemarketorders,thustradingatpricessetby others.Additionally,theyndthattradersbehavemorerationallywhen actingasmarketmakers,i.e.,submittingat-marketlimitorders.Theyargue thatrationaltradersself-selectintotheroleofmarketmakers,andprices aredeterminedbythetermsofthepostedlimitorders.Thus,pricesare determinedbythemorerationaltradersinthemarket. MoststudiesexaminingtheeciencyoftheIEMcompareitselection eveforecastswithelectionevepollsandtheactualoutcomes.Bergetal. looksatalongertimehorizon,comparingpredictionmarketforecasts withpollsasfaras100daysbeforetheelection.Usingmarketpricesfrom veUSpresidentialelections-2004and964polls,theyndthatthe marketsgenerallyoutperformpolls.Themarkettendstobeatthepolls between68and84percentofthetime,andthesuperiorityofthemarketsis moresignicantfartherawayfromtheelection.Theresultsareverysimilar ifmarketpricesareinsteadcomparedwithave-pollmovingaverage.More than100daysoutfromtheelection,thepollerroraveraged4.49percentage pointswhilethemarketerroraveraged2.65percentagepoints.Withinve daysoftheelection,pollerroraveraged1.62percentagepointswhilemarket erroraveraged1.11percentagepoints.Additionally,Bergetal.found thatmarketpricestendedtodisplaysignicantlylessvariance,anddidnot reactirrationallytopartyconventions. 14 14 Pollsfrequentlyshowaconventionbounce"whereacandidate'spopularityrisesand

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2.3.PredictionMarkets:ExperimentalandEmpiricalFindings35 Theevidencecitedsuggeststhatpredictionmarketsaresuperiortopolls aselectionforecastingtools.However,itisnaturaltoaskhowwellprediction marketswouldperformintheabsenceofpolldata,i.e.,howimportantis polldatainformingthebeliefsofthemarketparticipantswhomustchoose thepricesatwhichtheyarewillingtotrade?Verylittlehasbeendoneto addressthisveryinterestingquestion. RhodeandStrumpftakesaninterestingapproachtoaddressing thisquestionusingahistoricalcasestudy.Usingrecordsfromeightmajor newspapers,theyexaminetheinformationaleciencyofbettingmarkets organizedaroundUSpresidentialelectionsbetween1868and1940.They ndthattherewereusuallylarge,well-organizedbutoftenillegalmarkets runbybookmakersinmostmajorcities,withoverhalfofthetradingoccurringinNewYorkCity.Inthe15electionsbetween1884and1940,the mid-Octoberbettingfavoritewoneleventimes,arelativelongshotonce,and intheremainingthreeracestheoddswereroughlyeventheseraceswere particularlyclose.Typically,intheelectionsonwhichtheoddswerevery close,victorymarginswerenarrow.Themarketsweregenerallysuccessful inpickingthewinnerearlywhentheelectionwasdecidedbywidemargin. Usingdatasetssynthesizedfromnewspaperreports,theauthorsnd thattherewereusuallynoarbitrageopportunitieswithinorbetweencities, butthatthearbitrage-freeconditionwasviolatedonsomeoccasions.Additionally,theauthorsestimateasimpleregressionofcontractpricesontheir one-periodlaggedvalue,ndingthattheycannotrejectthenullhypothethenquicklyfallsimmediatelyfollowingthepartyconventionatwhichthecandidateis ociallyendorsed.

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2.3.PredictionMarkets:ExperimentalandEmpiricalFindings36 sisthatpricesfollowarandomwalkindictingthatthesemarketsmaybe weaklyecient.Giventherelativelylowfrequencyofthedata,itsquestionablequality,andthelackofthorougheconometricanalysis,theseresults arehighlypreliminary.However,thisisanintriguingrststepinexploring thedependenceofpredictionmarketpricesonpollingdata. 2.3.3Real-WorldMarkets:CorporateExperiments AnearlycorporatepredictionmarketexperimentwasrunbySiemensAustriainApril1997.Themarketwascreatedtoforecastwhetherornota largesoftwaredevelopmentprojectwouldbecompletedontimeand,ifnot, howlongitwouldbedelayed.Themarketranforsevenmonthswithabout 50activetraderswhowereemployeesworkingontheproject.Duetoa restructuringthatoccurredthreemonthsintotheproject,therewereactuallytwomarkets:therstmarketwasliquidatedthreemonthsin,anda secondmarketwascreatedimmediatelyafterwardtotakeitsplace.Ortner foundthatbothmarketsarrivedatastablepricequicklywithin onemonthandincorporatedinformationrapidlyandsmoothlyi.e.,there werenopricespikesfollowingocialannouncements.Morethanthree monthsbeforethescheduledcompletiondeadline,marketpricessuggested a2-3weekdelay.Theactualdelayturnedouttobe13days. Cowgilletal.reportontheplay-moneypredictionmarketestablishedbyGoogleinApril2005,documentinganumberofinteresting ndingsoverthesampleperiodof2005Q2to2007Q3.Themarketswere opentoallactiveemployeesaswellassomecontractorsandvendors,and inthetimeframestudied1,463peopleplacingatleastonetrade.Over

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2.3.PredictionMarkets:ExperimentalandEmpiricalFindings37 thesampleperiod,therewere25-30marketscreatedeachquarteronevents whoseoutcomeswouldbeknownbytheendofthatquarter,andtradeswere conductedviaacontinuousdoubleauction. 15 Eachemployeewasgivena newendowmentofarticialcurrencyeachquarter,andthiscurrencywas convertedintoraeticketsredeemableforprizesattheendofeachquarter. TheauthorsfoundthatGoogle'smarketswerereasonablyecient,but theyalsofoundevidenceoffourpersistentbiases:anoverpricingoffavorites, anunderpricingoflongshots,anoptimismbias,andanaversiontoselling securities.Thersttwobiasesareespeciallyinteresting,sincetheyarethe reverseofthetypicalfavorite-longshotbiasobservedinmanyotherpredictionandbettingmarkets.Optimismbias"referstoageneraloverpricing underpricingofsecuritieswhosepayoispositivelynegativelycorrelated withanoutcomeconsideredfavorableunfavorabletoGoogle.Thesecuritiesaectedmostseverelybythisbiaswerethosethatpertainedtoevents mostdirectlyunderthecontrolofGoogleemployeese.g.,aproductcompletiondate,andtheoptimisticbiasintradingwasampliedondayswhen Google'sstockappreciated.Iknowofnocompellingexplanationforthe aversiontoselling.Interestingly,thebiasesinindividualtraderbehavior diminishedsubstantiallyastheygainedmoreexperienceinthemarket,suggestingthattradersdolearnfromtheirmistakesandcorrecttheirbiasesto someextent. 15 Itisnotablethatsometraderscreatedautomatedtradingprogramsthatactedas marketmakers.

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2.4.Conclusion38 2.4Conclusion ThetheoreticalmodelsdiscussedinthischapterprovidethenecessarycontextforthemodelIpresentinthenextchapter.Theempiricalevidence presentedlendscredencetothepossibilitythatpredictionmarketsmaybe avaluabletoolforforecastinganddecisionmakinginhigh-stakesenvironments,thusjustifyingtheeortinvestedinthedevelopmentofthemodel.

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Chapter3 TheModel 3.1Introduction Inthischapter,Idevelopaformalframeworkthatwillallowforananalysisoftheinformationaleciencyofpricesinthepresenceofuncertain manipulation.Themodelisbasedonthesingle-periodbatch-clearingauctionframeworkdevelopedinKyle.Iextendtheoriginalframework toreectmanyofthecharacteristicsonewouldexpecttoseeinadecision market,incorporatingsomeofthefeaturesconsideredinHansonandOprea andSubrahmanyam. Inthemodel,thereisasinglemarketforasingleriskyassetandariskfreezerorateofreturnasset.Thereare N risk-averseinformedtraderswho receivenoisysignalsofthetruevalueoftheassetbeforechoosingexpected utilitymaximizingorderquantities.Thereisalsoarisk-neutraluninformed trader,themanipulator",whoisintroducedintothemarketwithaknown probability.Thistraderhaspreferencesoverthenalpriceoftheasset, 39

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3.2.OutlineoftheModel40 independentofitsactualvalue.Shetradessoastomovethenalpriceof theassettowardatargetpriceandawayfromtheecientprice,theasset's truevalue.Theinformedtradershaveonlyimperfectinformationabout themanipulator'stargetprice. Alltraders'orders,alongwithanexogenouslydeterminedliquiditytrade, areaggregatedandsenttoacompetitive,risk-neutralmarketmakerwho observesonlytheaggregateorderowbeforesettingasinglemarket-clearing price.Iexaminetheaccuracyofthisequilibriummarketpriceasafunction of1traders'beliefsaboutthepresenceofthemanipulator,2traders'beliefs aboutthemanipulator'stargetedprice,and3thedegreeoftheinformed traders'riskaversion. Insection3.2Ioutlinethestructureoftheformalmodel.Adetailed discussionandjusticationofthespecicationsandassumptionsunderlying themodelfollowsinsection3.3.Insection3.4,Ideriveamarketequilibriumforthecasewherethemanipulatorispresentwithcommonknowledge probability q .Adetailedanalysisofthepropertiesoftheequilibriumfollows inchapter4. 3.2OutlineoftheModel Iconsiderasinglemarketwithaxedpopulationoftraders,oneofwhomis acompetitive,risk-neutralmarketmaker.Thereisasingleriskyasset,and traders'commonpriorbeliefabouttheassetvaluetakestheformofanormal distributionwithmean v andnitevariance 2 v v N v; 2 v 1 Thereis 1 Unlessexplicitlystated,allrandomvariablesareindependentlynormallydistributed withanitemeanandanite,nonzerovariance.Thisassumptionisstandardinthe

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3.2.OutlineoftheModel41 alsoarisk-free,zerorateofreturnasset.Thereare N risk-averseinformed tradersindexedby i i 2f 1 ;:::N g ,whofacethefollowingexpectedutility maximizationproblem maximize x E[ U i x i ]=E h )]TJ/F15 10.9091 Tf 8.485 0 Td [(e )]TJ/F19 7.9701 Tf 6.586 0 Td [( x i v )]TJ/F19 7.9701 Tf 6.587 0 Td [(p i where U i istrader i 'sutilityfunction, 2 x i istrader i 'sdemandfortherisky asset, isthecommonknowledgeriskaversioncoecientforall i traders, and p isthepriceoftheriskyasset. Inadditiontotheinformedtraders,thereisanuninformed,risk-neutral traderthemanipulator"whoseexpectedutilitymaximizationproblemis maximize x 0 E[ U 0 x 0 ]=E x 0 v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p )]TJ/F18 10.9091 Tf 10.909 0 Td [(k t )]TJ/F18 10.9091 Tf 10.909 0 Td [(p 2 where x 0 isthemanipulator'sdemandfortheriskyassetand k> 0is interpretedasthestrengthofherpreferencesoverthedeviationofprice fromatarget t .Themanipulatorknowsthetruevalueof t ,butthebeliefsofthemarketmakerandtheinformedtradersabout t areoftheform t N t; 2 t .Theprobabilitythattheinformedtradersattachtothemanipulator'sactualpresenceinthemarketis q .Inadditiontotheseexplicitly modeledtrades,Iallowforanexogenousliquiditytrade, 3 l N l; 2 l Henceforth,thecommonpriorbeliefsofalltraderswillbesummarizedby K = l N l; 2 l ;q;t N t; 2 t ;v N v; 2 v andallexpectationswill marketmicrostructureliteratureO'Hara1997. 2 Iassumeanegativeexponentialformfortheinformedtraders'utilityfunctions,asthis isthestandardtechniqueinthemarketmicrostructureliteratureformodelingrisk-averse tradersVives2008.Seesection3.3.3fordetails. 3 Seesection3.3.1foranovelinterpretationofthistrade.

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3.2.OutlineoftheModel42 beconditionedon K aswellastherelevantparticularinformation. Beforesubmittinganorder,theinformedtradersobserveanoisysignal ofthetruevalueoftheriskyasset. 4 Thissignalisgivenby S = v + ,with N ; 2 Aftertheinformedtradersobservestheirsignals,alltraderschoosetheir optimumorderquantity.Theseordersaresummedintoanaggregateorder ow y y = l + N X i =0 x i .1 Ifthereisnomanipulatorinthemarket, x 0 0.Themarketmakerobserves onlythisaggregatedorderowbeforesettingazeroexpectedprotprice 5 p =E[ v j y;K ]= v + y E[ v y j K ] E[ y y j K ] .2 Sincealloftherandomvariablesinthemodelarenormallydistributed,the lastequalityholdsbytheTheoremofProjectionforNormalDistributions DeJongandRindi2009,pg.40. 6 Iaimtoexaminetheeectofuncertaintyregardingthepresenceand intentionsofthemanipulatorontheinformationaleciencyofprices.Itis importanttonotethattherearetwomechanismsthroughwhichamanip4 Iassumethatthemanipulatorhasnospecialinformationaboutthetrueassetvalue, i.e.,shedoesnotupdatefromthecommonpriorbeliefthat v N v; 2 v 5 Foranyrandomvariable x x x )]TJ/F47 8.9664 Tf 9.215 0 Td [(E[ x j K ]. 6 Thetheoremstatesthat,foranytwonormalrandomvariables x and y E[ x j y ]=E[ x ]+ x )]TJ/F47 8.9664 Tf 9.216 0 Td [(E[ x ] Cov[ x;y ] Var[ y ] .3 and Var[ x j y ]=Var[ x ] )]TJ/F47 8.9664 Tf 10.411 5.809 Td [(Cov[ x;y ] 2 Var[ y ] .4

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3.2.OutlineoftheModel43 ulatorcanaectprices.Clearly,thetradesofthemanipulatorwill, ceteris parabus ,causepricestobelessinformative.However,sincethemanipulator istradingfornon-informationreasons,hertradeisnotcorrelatedwiththe assetvalueandhencerepresentsaprotopportunityforinformedtraders. Thisprotopportunitymayaecttheorderchoicemadebytheinformed traders. Traders'optimalstrategiesaregivenby x 0 =max x 0 2 R U 0 x 0 E[ U 0 x 0 j t;x 0 ;K ] .5 x i =max x i 2 R U i x i E[ U i x i j S;x i ;K ] .6 Inthemodel,thetimingofmovesisasfollows: 1.Nature"moves: l k t ,and v areset;actualpresenceofthemanipulatorisdecided. 2.Theinformedtradersrealizetheirsignal,whichisprivateinformation. 3.Alltraderssubmittheirorderquantity,whichisprivateinformation foreachtrader. 4.Themarketmakerobservestheaggregateorderowandsetstheprice equaltotheexpectedvalueoftheassetconditionedontheobserved orderow. andIlookforequilibriumstrategiesthatareanefunctionsoftraderin-

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3.3.DiscussionofAssumptions44 formationandpreferences: 7 p = + y .7 x = + S .8 x 0 = 0 + 0 t .9 The N informedtradersaresymmetric,sotheirstrategieswillbesymmetric. Beforeexplicitlyderivingtheequilibriumstrategies,Idiscussthereasoning behindthestructureadoptedforthemodelaswellastheassumptionsimposed. 3.3DiscussionofAssumptions 3.3.1Risk-NeutralMarketMakerandExogenousLiquidity Thismodelassumestheexistenceofacentralizedmarketmaker,which mayormaynotbecontrolledbyamarketpatron.Notallpredictionmarketshaveacentralizedmarketmaker,butsmallermarketsfrequentlydo, andlargermarketsusuallyhavemanytradersthatactasmarketmakers. 8 Hansonadiscussesthenumerousadvantagesofacentralized,patron controlledmarketmaker,especiallyforthesmallerdecisionmarketsthat 7 Theassumptionofanestrategiesisstandardinthemarketmicrostructureliterature. Indeed,verylittleisknownaboutthepropertiesof,oreventheexistenceof,equilibriain caseswheretraderstrategiesarenonlinearfunctionsoftheirexpectations.Futurework couldpartiallyaddressthisproblembytestingforthestabilityoflinearequilibriasubject tosmallperturbations. 8 Cowgilletal.documentshowtradersinGoogle'sinternalmarketscreated automatedtradingprogramsthatactedasmarketmakersinasubstantialproportionof trades.

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3.3.DiscussionofAssumptions45 arethefocusofthiswork. Intypicalmarketmicrostructuremodels,marketmakersarereferredto ascompetitive"when,presumablyasaresultofcompetition,theyearn zeroexpectedprots.Whenmarketmakersarecompetitive,priceswill equaltheconditionalexpectedvalueoftheassetO'Hara1997.Sinceprice accuracyisanexplicitgoalwhenpredictionmarketsareimplementedwithin organizations,itisreasonabletoassumethatanycentralizedmarketmaker controlledbytheorganizationwilloperateatzeroexpectedprot.Therisk neutralityofthemarketmakerisanunrealisticassumptioninlargereal worldnancialmarketssee,forinstance,theevidenceonforexmarkets inLyons,butitisreasonableinthisenvironmentduetothesmall stakesandthepossibilityofanexplicitboundonthemaximumabsolute possiblelossofapatroncontrolledmarketmakerChenandPennock2007. TheinclusionofanexogenousliquiditytradeistypicalofmarketmicrostructuremodelsO'Hara1997.GrossmanandStiglitzfound that,intheabsenceofuninformedtrades,marketpricesrevealalloftheinformationpossessedbymarketparticipants.Thisallowstraderstodeduce allavailableinformationfrommarketprices.However,ifinformationacquisitioniscostly,thisimpliesthatnotraderwouldhaveanincentivetoengage ininformationacquisitionandthuspriceswouldnolongerbeinformative. ThisistheGrossman-Stiglitzparadox"describedinsection2.1.1,andit isresolvedbyintroducingnoisetraderswhoseuninformedtradesresultin pricesthatdonotfullyrevealinformedtraderinformation. Sincemostmarketmicrostructuremodelsareconstructedwithtraditionalnancialmarketsinmind,exogenousliquiditytradesaretypically

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3.3.DiscussionofAssumptions46 interpretedastradesinitiatedbythosewhowishtoinvest,hedgeagainst risks,orgambleHarris2003.Thesekindsoftradesmaynotbepresent inapredictionmarket.Sincepredictionmarketsimplementedinsideorganizationsaretypicallycreatedtoenhancethecapabilitiesoftheorganization,theorganizationshouldbewillingtosubsidizethemarket.Thus,in thismodel,onecaninterprettheexogenousliquiditytradeasarandom, negativeexpectedprottradesubmittedbythemarketpatrontoencourageparticipationandadditionalinformationacquisitionbyotherpotential traders.Thisis,asfarasIamaware,anovelinterpretation. 3.3.2InformationStructure Iassumethatallvariablesarenormallydistributedandindependent,which resultsinaneconditionalexpectationsthatsimplifythecomputations. WhilethisisobviouslylessgeneralthanIwouldprefer,therearenovariable pairsinthemodelthatintuitively should bedependent. Iassumethatboththeinformedtradersandthemarketmakerareuncertainabout1thepresenceofthemanipulator,and2themanipulator's targetedprice.Thisisverylikelytobethecaseinmediumtolargescale predictionmarkets.Whileitisveryunlikelythatcertainaspectsoftraders' preferencese.g., k and wouldbecommonknowledgeinarealmarket, Imaketheassumptionherethattheyaretosimplifythecomputations. Addingnoisetotheinformedtraders'beliefsabout k wouldberedundant giventheuncertaintyaround t andtheuncertaintyregardingthepresenceof themanipulator.Sincethemanipulatorisriskneutral,uncertaintyregardingothertraders'riskpreferenceswouldprobablynotsubstantiallyaect

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3.3.DiscussionofAssumptions47 herstrategy. Forconvenience,Isummarizethestateofknowledgeofeachtraderat thetimeofordersubmissionintable3.1. TradertypeVariableStateofknowledge ll N l; 2 l Marketmaker vv N v; 2 v kk tt N t; 2 t ll N l; 2 l Manipulator vv N v; 2 v kk tt ll N l; 2 l Informedtraders v E[ v j S ]= S 2 v 2 v + 2 kk tt N t; 2 t Table3.1: Stateofknowledgeforeachtraderattimeofordersubmission. Inthismodel,thequalityoftheinformedtraders'signalisexogenously determined.Inotherwords,theinformedtradershavenocontroloverthe precisionofthesignaltheyreceive.Whilethisassumptionmakesthemodel substantiallymoretractableanalytically,itignoresthefactthattradersin realworldmarketsdoexertcostlyeortstoobtaininformation.Endogenousinformationacquisitionmaycausemarketstobemoreecientinthe presenceofamanipulator,iftheprotopportunitypresentedbythemanipulatorspursothertraderstogathermoreinformationthisistheresult obtainedinHansonandOprea.Thus,futureworkshouldexamine

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3.3.DiscussionofAssumptions48 theeectsofallowingcostlyinformationacquisition. 3.3.3TraderUtilityFunctionsandRiskPreferences Iassumethatthemanipulatorisriskneutralprimarilytoallowabestcase" formanipulation.Implementingapredictionmarketwithinanorganization requiresasubstantialinvestmentinhardwareandtraining.Predictionmarketsmustalsoovercomealegitimatepreferenceforexisting,well-understood forecastingmethodsandpossiblyanirrationalstatusquobiasonthepartof potentialmarketpatrons.Thus,muchofthetheoreticalandexperimental researchonpredictionmarketshasbeendesignedtostresstest"theconcept.SinceIbelievethatthisapproachisworthwhile,Ihaveadheredtoit intheconstructionofmymodel. Thespecicquadraticfunctionalformchosentorepresentthemanipulator'sutility U 0 x 0 = x 0 v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p )]TJ/F18 10.9091 Tf 10.909 0 Td [(k t )]TJ/F18 10.9091 Tf 10.909 0 Td [(p 2 wasintroducedbyHansonandOpreaandisageneralizationofthe linearformintroducedbyKumarandSeppi.Ithasseveralproperties thatmakeitintuitivelyappealing.The k coecientallowsforawiderange ofchoiceinselectingthemanipulator'srelativepreferenceformanipulation vs.wealth.Themanipulatorclearlydoesnotdiscriminatebetweenaprice beloworabovehertarget;onlythemagnitude,andnotthesign,ofthedierenceisrelevant.Thismaynotberealisticwithcertaincontractstructures, butitnonethelessseemsareasonablestart.Since d d t )]TJ/F19 7.9701 Tf 6.586 0 Td [(p k t )]TJ/F18 10.9091 Tf 9.355 0 Td [(p 2 =2 k t )]TJ/F18 10.9091 Tf 9.355 0 Td [(p theutilitypenaltyforpricedeviationincreasesnonlinearlywiththemagni-

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3.3.DiscussionofAssumptions49 tudeofthedeviation.Thisisreasonablesincemanipulationfornon-market reasonsimpliesthatlargerdeviationswillhaveamuchmoresubstantial impactuponthemanipulator'slargergoalsthansmallerdeviations. Theassumptionthattheinformedtradersareriskaverseservesseveral purposes.Itaddsadditionalrealismtothemodelsincemanyrealworld tradersareriskaverse. 9 TheresultsofHansonandOprea,described insection2.2.3,wereobtainedundertheassumptionthatalltraderswere riskneutral.However,itispossiblethattheseresultswillfailtoholdwhen tradersareriskaverse. 10 Perhapsmoreimportantly,riskaversionservesasanadditionalconstraintontraders'ordersizes.Inmostmarketmicrostructuremodels,includingthatofHansonandOprea,riskneutralityimpliesthattraders limittheirordersizeonlybecausemarketdepthisnite.Here,thereisan additionalconstraintoninformedtraders'ordersthatcanbealteredexogenously,allowingforaninterestingrangeofcomparativestatics.Thiswill shedlightontheabilityofmanipulatorstoaectpriceswhenthecapacity ofinformedtraderstorespondislimited.Theeectsofriskaversionwill beespeciallyinterestingbecauseofthesubstantialuncertaintysurrounding themanipulator'sstrategy. Thespecicfunctionalformassumedfortheinformedtraders,anegative exponentialutilityfunction U x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(e )]TJ/F19 7.9701 Tf 6.587 0 Td [( x v )]TJ/F19 7.9701 Tf 6.587 0 Td [(p ,isthestandardformused tomodelthebehaviorofrisk-aversetraders.Thisutilityfunctionexhibits constantabsoluteriskaversion,i.e.,theriskaversioncoecient isnot 9 See,e.g.,theevidencecitedinLengwiler. 10 See,forinstance,KyleandSubrahmanyam.

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3.3.DiscussionofAssumptions50 afunctionofwealth.Thus,atrader'sriskpreferenceisinsensitivetohis absolutelevelofwealth.Whileitislikelythatthisassumptionwillnothold acrosslargedierencesinwealth,itseemsareasonableapproximationina predictionmarketsettinggiventherelativelysmallpositionsatstake. Theadvantageofthenegativeexponentialformisthesimplicityofthe resultingoptimizationproblem,whichresultsfromthefollowingfact: Fact1. Foranynormalrandomvariable x ,withmean andvariance 2 andany t 2 R ,E e tx = e t + t 2 2 2 Proof. Notethat x = + z where z N ; 1.Thus, E e tx =E h e t + z i =e t E e tz =e t 1 p 2 Z 1 e tz e )]TJ/F20 5.9776 Tf 7.782 3.259 Td [(z 2 2 dz =e t 1 p 2 Z 1 e 2 tz )]TJ/F20 5.9776 Tf 5.757 0 Td [(z 2 2 dz =e t 1 p 2 Z 1 e )]TJ/F17 5.9776 Tf 7.782 4.025 Td [( z )]TJ/F20 5.9776 Tf 5.756 0 Td [(t 2 2 + t 2 2 2 dz =e t e t 2 2 2 1 p 2 Z 1 e )]TJ/F17 5.9776 Tf 7.782 4.025 Td [( z )]TJ/F20 5.9776 Tf 5.756 0 Td [(t 2 2 dz =e t e t 2 2 2 =e t + t 2 2 2 ThiswillproveusefulwhenIsolvethemodelinsection3.4.

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3.4.EquilibriumStrategies51 3.4EquilibriumStrategies Ibeginbyderivingthemarketmaker'spricingrule.Byequation.2, p = v + y E[ v y j K ] E[ y y j K ] = v + y N 2 v 2 l + N 2 2 v + 2 + q 2 2 0 2 t Thus,marketdepthisgivenby )]TJ/F16 7.9701 Tf 6.586 0 Td [(1 = h N 2 v 2 l + N 2 2 v + 2 + q 0 2 2 t i )]TJ/F16 7.9701 Tf 6.587 0 Td [(1 and = v .Themarketmaker'spricingstrategyisananefunctionofthe observedorderquantityand,byequation.1,thisobservedorderquantity isananefunctionofthesumoftraders'ordersandtherandomliquidity trade.Traders'orderstrategiesareanefunctionsoftherandomvariables t and S ,andso p willbeafunctionofasumofnormalrandomvariables. 11 Thisimpliesthat p isanormalrandomvariable.Andsince v isanormal randomvariable, x i v )]TJ/F18 10.9091 Tf 10.91 0 Td [(p isanormalrandomvariable.This,combined withfact1,derivedinsection3.3.3,allowsmetorewritetheinformed traders'optimizationproblem maximize x i E[ U i x i j K ]=E h )]TJ/F15 10.9091 Tf 8.485 0 Td [(e )]TJ/F19 7.9701 Tf 6.586 0 Td [( x i v )]TJ/F19 7.9701 Tf 6.587 0 Td [(p j K i as maximize x i E[ U i x i j K ]= )]TJ/F15 10.9091 Tf 8.485 0 Td [(e )]TJ/F19 7.9701 Tf 6.587 0 Td [( E[ i j K ] )]TJ/F20 5.9776 Tf 7.782 3.693 Td [( 2 Var[ i j K ] 11 Ishowbelowthat,giventhateachtraderconjecturesthat.7,.8,and.9will hold,.7,.8,and.9isaBayesianNashequilibrium.

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3.4.EquilibriumStrategies52 whichcanbewrittenmoresimplyas maximize x i ^ i x i =E[ i j K ] )]TJ/F18 10.9091 Tf 12.105 7.38 Td [( 2 Var[ i j K ] where i = x i v )]TJ/F18 10.9091 Tf 10.923 0 Td [(p and^ i isinformedtrader i 'srisk-adjusted"expected protfunction. Isolvefortraders'optimalstrategiesusingthesimpliedformsderived above x 0 =max x 0 2 R U 0 x 0 E[ U 0 x 0 j t;x 0 ;K ] x i =max x i 2 R [^ i x i E[ U i x i j S;x i ;K ]] startingwith x i Letting K x i = f S;x i ;K g denotetheinformedtrader'sinformationset atthetimeofordersubmission,therst-orderconditionon^ i x i isgiven by d dx i ^ i x i = d dx i h E[ i j K x i ] )]TJ/F18 10.9091 Tf 12.104 7.38 Td [( 2 Var[ i j K x i ] i = d dx i h x i E[ v j K x i ] )]TJ/F15 10.9091 Tf 10.909 0 Td [(E[ p j K x i ] )]TJ/F18 10.9091 Tf 12.105 7.38 Td [( 2 x 2 i Var[ v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p j K x i ] i =0 Letting = 2 v 2 v + 2 and = 2 v 2 2 v + 2 ,itfollowsthat E[ v j K x i ]= v + S

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3.4.EquilibriumStrategies53 and E[ p j K x i ]= v + E[ y j K x i ] = v + [ N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 S + x i )]TJ/F15 10.9091 Tf 10.909 0 Td [(E[ x i j K ]] ThecalculationofVar[ v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p ]isabitmorecomplicated: Var[ v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p j K x i ]=E h v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p )]TJ/F15 10.9091 Tf 10.909 0 Td [(E[ v j K x i ] )]TJ/F15 10.9091 Tf 10.909 0 Td [(E[ p j K x i ] 2 j K x i i =E h v 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 vp )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 v E[ v j K x i ]+2 v E[ p j K x i ]+ p 2 +2 p E[ v j K x i ] )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 p E[ p j K x i ]+E[ v j K x i ] 2 +E[ p j K x i ] 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2E[ v j K x i ]E[ p j K x i ] i =E v 2 j K x i )]TJ/F15 10.9091 Tf 10.909 0 Td [(2E[ vp j K x i ] )]TJ/F15 10.9091 Tf 10.909 0 Td [(E[ v j K x i ] 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(E[ p j K x i ] 2 +E p 2 j K x i +2E[ v j K x i ]E[ p j K x i ] butthenalresultis Var[ v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p j K x i ]= )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 + 2 2 l + N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 2 + q 2 2 0 2 t Withtheseresults,therst-orderconditionon^ i x i canbeexplicitlycom-

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3.4.EquilibriumStrategies54 puted: d ^ dx i = d dx i h x i )]TJ/F18 10.9091 Tf 10.909 0 Td [( N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 S )]TJ/F18 10.9091 Tf 10.909 0 Td [(x i x i )]TJ/F18 10.9091 Tf 12.104 7.38 Td [( 2 x 2 i )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 + 2 i = )]TJ/F18 10.9091 Tf 10.909 0 Td [( N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 S )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x i + E[ x i j K ] )]TJ/F18 10.9091 Tf 10.909 0 Td [(x i )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 + 2 =0 where = 2 l + N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 2 + q 2 2 0 2 t Thisimpliesthat x i = )]TJ/F18 10.9091 Tf 10.909 0 Td [( N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 S + E[ x i j K ] 2 + i = )]TJ/F18 10.9091 Tf 10.909 0 Td [( N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 + i S where i =Var[ v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p j K x i ]andthesecondequalityholdssinceE[ x i j K ]= 0.Thiscanbeveriedeasily: E[ x i j K ]=E )]TJ/F18 10.9091 Tf 10.909 0 Td [( N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 S + E[ x i j K ] 2 + i j K = E[ x i j K ] 2 + i whichimpliesthatE[ x i j K ] + i =0.Suppose + i =0.Then, i = )]TJ/F19 7.9701 Tf 6.586 0 Td [( .But ;> 0impliesthat i < 0,whichisacontradictionsince i =Var[ v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p j K x i ].Thus,E[ x i j K ]=0.Ihavenotyetshowningeneral that > 0,butthisistheonlyeconomicallysensiblecase,andIshowin

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3.4.EquilibriumStrategies55 chapter4thatthisholdsforavarietyofplausibleparametervalues.The second-orderconditionissatisedaswell: d 2 ^ i dx 2 i = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F18 10.9091 Tf 10.909 0 Td [(' i < 0 wheretheinequalityfollowsfromthefactthat > 0, > 0,and i > 0the inequalitiesarestrictbecauseIassume N 6 =0.Thus, x i isamaximum. Inowderivetheoptimaltradingstrategyforthemanipulator, x 0 .At thetimeofordersubmission,themanipulator'sinformationsetis K x 0 = f K;t;x 0 g .Therstorderconditionon U 0 x 0 isgivenby d U 0 dx 0 = d dx 0 x 0 v )]TJ/F18 10.9091 Tf 10.909 0 Td [(x 0 E[ p j K x 0 ] )]TJ/F18 10.9091 Tf 10.909 0 Td [(kt 2 +2 kt E[ p j K x 0 ] )]TJ/F18 10.9091 Tf 10.909 0 Td [(k E p 2 j K x 0 andbecause E[ p j K x 0 ]= v + x 0 and E p 2 j K x 0 = v 2 +2 v x 0 + 2 x 2 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 x 0 E[ x 0 j K ]+ E[ x 0 j K ] 2 itfollowsthat d U 0 dx 0 = )]TJ/F18 10.9091 Tf 5 -8.836 Td [( +2 k 2 E[ x 0 j K ] )]TJ/F18 10.9091 Tf 10.909 0 Td [(x 0 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 +2 k 2 +2 k t )]TJ/F15 10.9091 Tf 11.325 0 Td [( v =0 Fromthisitiseasytoshowthat E[ x 0 j K ]=2 k t )]TJ/F15 10.9091 Tf 11.324 0 Td [( v

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3.5.Game-TheoreticPropertiesoftheEquilibrium56 andsoitfollowsthat x 0 =2 k t )]TJ/F15 10.9091 Tf 11.324 0 Td [( v + k 1+ k t Thesecondorderconditionon U 0 isgivenby d 2 U 0 dx 2 0 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 k 2 < 0 andthus x 0 isamaximum. 3.5Game-TheoreticPropertiesoftheEquilibrium Theconjecturedstrategiesforthemarketmaker,informedtradersandmanipulator p = + y x = + S x 0 = 0 + 0 t ledtotheresultingequilibriumstrategies 12 p = v + N 2 v 2 l + N 2 2 v + 2 + q 0 2 2 t y x = )]TJ/F18 10.9091 Tf 10.909 0 Td [( N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 + 2 S .10 x 0 =2 k t )]TJ/F15 10.9091 Tf 11.325 0 Td [( v + k 1+ k t 12 Recallthat = 2 v 2 v + 2 = 2 v 2 2 v + 2 ,and = 2 l + N )]TJ/F47 8.9664 Tf 9.215 0 Td [(1 2 2 + q 0 2 2 t .

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3.6.Conclusion57 whichfullledtheconjectureswith = N 2 v 2 l + N 2 2 v + 2 + q 0 2 2 t = )]TJ/F18 10.9091 Tf 10.909 0 Td [( N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 + 2 .11 0 = k 1+ k Foragivenvectorofparameters 2 2 l 2 t 2 v k q N ,and ,anysetof valuesof ,and 0 thatsimultaneouslysatisfy.11willeachyield,since theconjecturesarecorrect,abest-responsestrategyforeachagentgiventhe strategiesemployedbytheotheragents.Thus,the ,and 0 satisfying .11willmake.10aBayesianNashequilibrium. 3.6Conclusion Theexpected-protmaximizingstrategiesforthemarketmaker,manipulator,andinformedtradershavebeenderived,andIhaveshownthatthese strategiesformaBayesianNashequilibrium.Inchapter4,Iderivecomparativestaticresultsandexaminethepropertiesofthesolutionforavariety ofplausibleparametervalues.

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Chapter4 ComparativeStaticsand Analysis Inthischapter,Iusetheequilibriumstrategiesderivedinchapter3to studytheeciencyofmarketpricesaswellastheeectsofchangesin keyparametersonpriceeciency.Priceeciencyischaracterizedbythe distributionofthepriceerror v )]TJ/F18 10.9091 Tf 11.364 0 Td [(p ,i.e.,E[ v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p j K ]andVar[ v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p j K ]. IdeterminetheequilibriumvaluesofE[ v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p j K ]andVar[ v )]TJ/F18 10.9091 Tf 10.91 0 Td [(p j K ]and examinehowtheyvarywithchangesinthenumberoftradersinthemarket, thecharacteristicsofthemanipulator,andthelevelofriskaversionofthe informedtraders. 4.1DerivationoftheComparativeStatics Anexplicitsolutionfortheendogenousvariables ,and 0 intermsof theexogenousparameters 2 2 l 2 t 2 v k q N ,and isnot,ingeneral, 58

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4.1.DerivationoftheComparativeStatics59 feasible.Eveninthesimplestcase, N =1,thesolutionfor intermsofthe exogenousvariablesistherootofasepticpolynomial.However,comparative staticderivativesof ,and 0 withrespecttotheexogenousparameters arestillpossibleif.11denesasetofimplicitfunctions: = f 1 )]TJ/F18 10.9091 Tf 5 -8.837 Td [( 2 ; 2 l ; 2 t ; 2 v ;k;q;N; = f 2 )]TJ/F18 10.9091 Tf 5 -8.836 Td [( 2 ; 2 l ; 2 t ; 2 v ;k;q;N; .1 0 = f 3 )]TJ/F18 10.9091 Tf 5 -8.836 Td [( 2 ; 2 l ; 2 t ; 2 v ;k;q;N; Let = 2 v + 2 and I = 2 ; 2 l ; 2 t ; 2 v ;k;q;N; .Then,iftheequations F 1 ;; 0 ; I = h 2 l + N 2 + q 0 2 2 t i )]TJ/F18 10.9091 Tf 10.909 0 Td [(N 2 v F 2 ;; 0 ; I = 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 + 2 )]TJ/F18 10.9091 Tf 10.909 0 Td [( + N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 F 3 ;; 0 ; I = 0 + k )]TJ/F18 10.9091 Tf 10.909 0 Td [(k areall C 1 withrespectto 0 2 2 l 2 t 2 v k q N ,and ,and thereexistsapoint I 0 satisfyingallthreeequationssimultaneouslywhere theJacobiandeterminantof F = F 1 ;F 2 ;F 3 j J j = @F 1 @ @F 1 @ @F 1 @ 0 @F 2 @ @F 2 @ @F 2 @ 0 @F 3 @ @F 3 @ @F 3 @ 0

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4.1.DerivationoftheComparativeStatics60 isnonzero,thentheimplicitfunctiontheoremguaranteestheexistenceofa neighborhoodofthepoint I 0 wherethesetofimplicitfunctions.1exists andeachequationis C 1 withrespectto 2 2 l 2 t 2 v k q N ,and .Since F 1 F 2 ,and F 3 arepolynomialsineachofthevariables,theyarenotonly C 1 but C 1 .Takingtherelevantpartialderivatives @F 1 @ = 2 l + N 2 + q 0 2 2 t @F 1 @ =2 N 2 )]TJ/F18 10.9091 Tf 10.909 0 Td [(N 2 v @F 1 @ 0 =2 q 2 0 2 t @F 2 @ =2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+2 + N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 @F 2 @ =2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ 2 +2 2 2 N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 + N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 @F 2 @ 0 =2 2 q 2 0 2 t @F 3 @ = k 0 @F 3 @ =0 @F 3 @ 0 =1+ k thedeterminantoftheJacobianof F is j J j = 2 l + N 2 + q 0 2 2 t 2 N 2 )]TJ/F18 10.9091 Tf 10.909 0 Td [(N 2 v 2 q 2 0 2 t 2 +2 + N )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 + + N )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 2 q 2 0 2 t k 0 01+ k where = )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ 2 +2 2 2 N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 and =1 )]TJ/F18 10.9091 Tf 8.815 0 Td [( N )]TJ/F15 10.9091 Tf 10.909 0 Td [(1. Ingeneral,thesignof j J j isindeterminant.Iproceedformallyandderive

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4.1.DerivationoftheComparativeStatics61 thegeneralformsforthecomparativestaticderivatives,andtheninsection 4.3Isolveforthemnumericallyatspecicpointssatisfyingtheconditions oftheimplicitfunctiontheorem.Thecomparativestaticderivativesof ,and 0 withrespectto 2 t q N ,and cannowbederived: 2 6 6 6 6 6 6 6 6 6 6 6 6 4 @F 1 @ @F 1 @ @F 1 @ 0 @F 2 @ @F 2 @ @F 2 @ 0 @F 3 @ @F 3 @ @F 3 @ 0 3 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 4 @ @ 2 t @ @ 2 t @ 0 @ 2 t 3 7 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F18 10.9091 Tf 9.681 7.38 Td [(@F 1 @ 2 t )]TJ/F18 10.9091 Tf 9.681 7.38 Td [(@F 2 @ 2 t )]TJ/F18 10.9091 Tf 9.681 7.38 Td [(@F 3 @ 2 t 3 7 7 7 7 7 7 7 7 7 7 7 7 5 .2 2 6 6 6 6 6 6 6 6 6 6 6 6 4 @F 1 @ @F 1 @ @F 1 @ 0 @F 2 @ @F 2 @ @F 2 @ 0 @F 3 @ @F 3 @ @F 3 @ 0 3 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 4 @ @q @ @q @ 0 @q 3 7 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F18 10.9091 Tf 9.681 7.381 Td [(@F 1 @q )]TJ/F18 10.9091 Tf 9.681 7.38 Td [(@F 2 @q )]TJ/F18 10.9091 Tf 9.681 7.38 Td [(@F 3 @q 3 7 7 7 7 7 7 7 7 7 7 7 7 5 .3 2 6 6 6 6 6 6 6 6 6 6 6 6 4 @F 1 @ @F 1 @ @F 1 @ 0 @F 2 @ @F 2 @ @F 2 @ 0 @F 3 @ @F 3 @ @F 3 @ 0 3 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 4 @ @N @ @N @ 0 @N 3 7 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F18 10.9091 Tf 9.68 7.38 Td [(@F 1 @N )]TJ/F18 10.9091 Tf 9.68 7.38 Td [(@F 2 @N )]TJ/F18 10.9091 Tf 9.68 7.38 Td [(@F 3 @N 3 7 7 7 7 7 7 7 7 7 7 7 7 5 .4

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4.2.AnalysisoftheEquilibrium62 2 6 6 6 6 6 6 6 6 6 6 6 6 4 @F 1 @ @F 1 @ @F 1 @ 0 @F 2 @ @F 2 @ @F 2 @ 0 @F 3 @ @F 3 @ @F 3 @ 0 3 7 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 6 4 @ @ @ @ @ 0 @ 3 7 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F18 10.9091 Tf 9.681 7.38 Td [(@F 1 @ )]TJ/F18 10.9091 Tf 9.681 7.38 Td [(@F 2 @ )]TJ/F18 10.9091 Tf 9.681 7.38 Td [(@F 3 @ 3 7 7 7 7 7 7 7 7 7 7 7 7 5 .5 WhentheJacobianisinvertible,thesesystemshaveauniquesolution. Theindividualcomparativestaticderivativescanbeextractedmosteasily usingCramer'srule,e.g., @ @ 2 t = J 1 2 t j J j where J 1 2 t istheJacobianof F withtherstrowreplacedbythesolution vectorofequation.2. 4.2AnalysisoftheEquilibrium Icannowexplicitlycalculatetheeciencyofpricesforanyvectorofparametervalues I = 2 ; 2 l ; 2 t ; 2 v ;k;q;N; andcompareittothebenchmark casewiththesamevectorofparameters sans themanipulator q =0.I canalsocomputetheeectsofchangesin 2 t q N ,and ontraders'strategiesand,thus,theireectsonthekeyvariablesofinterest,theexpected priceerrorE[ v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p j K ]andthevarianceofpriceerrorVar[ v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p j K ].The distributionofthepriceerrorcanbecomputedinfullgenerality. Theorem1. Forall 0 2 2 l 2 t 2 v k q N 2 R + satisfying

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4.2.AnalysisoftheEquilibrium63 .11 ,thepriceerror v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p isnormallydistributedwith E [ v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p j K ]=0 and Var [ v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p j K ]= 2 h 2 l + N 2 )]TJ/F18 10.9091 Tf 5 -8.836 Td [( 2 v + 2 + q 2 2 0 2 t i )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 N 2 v + 2 v Proof. Notethat v )]TJ/F18 10.9091 Tf 11.569 0 Td [(p isananefunctionof y andthenormalrandom variable v ,and y isaanefunctionofthenormalrandomvariables t and S .Thus, v )]TJ/F18 10.9091 Tf 11.116 0 Td [(p isanormalrandomvariable.Suppose 0 2 2 l 2 t 2 v k q N 2 R + andsimultaneouslysatisfy.11.Then, E[ v )]TJ/F18 10.9091 Tf 10.91 0 Td [(p j K ]=E[ v )]TJ/F18 10.9091 Tf 10.909 0 Td [( y j K ]=E[ v j K ] )]TJ/F18 10.9091 Tf 10.909 0 Td [( E[ y j K ] 0 and Var[ v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p j K ]=E v )]TJ/F18 10.9091 Tf 10.909 0 Td [(p 2 j K =E 2 y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 v y + v 2 j K = 2 E y 2 j K )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 E[ v y j K ]+E v 2 j K = 2 h 2 l + N 2 )]TJ/F18 10.9091 Tf 5 -8.837 Td [( 2 v + 2 + q 2 2 0 2 t i )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 N 2 v + 2 v Thus, p isanunbiasedestimatorof v ,andthecharacteristicsofthe manipulatorhavenoeectontheexpectedpriceerror.Inowhaveanexplicit

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4.2.AnalysisoftheEquilibrium64 formulaforthevarianceofthepriceerrordenotedbyhereafter,and usingthecomparativestaticsderivedabove,Icancomputethederivative ofwithrespecttoanyoftheexogenousparameters. Let 1 = 2 l + N 2 2 + q 0 2 2 t .Thederivativeofwithrespectto 2 t is @ @ 2 t =2 @ @ 2 t 1 + 2 2 @ @ 2 t N 2 +2 q 2 0 @ 0 @ 2 t 2 t + q 2 2 0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 N 2 v @ @ 2 t + @ @ 2 t Thederivativeofwithrespectto q is @ @q =2 @ @q 1 + 2 2 @ @q N 2 +2 q 2 0 2 t +2 q 2 0 @ 0 @q 2 t )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 N 2 v @ @q + @ @q Thederivativeofwithrespectto is @ @ =2 @ @ 1 + 2 2 @ @ N 2 +2 q 2 0 @ 0 @ 2 t )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 N 2 v @ @ + @ @ Andthederivativeofwithrespectto N is @ @N =2 @ @N 1 + 2 2 N 2 +2 N 2 @ @N +2 q 2 0 @ 0 @N 2 t )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 v @ @N N + @ @N N + Unfortunately,thecomplexityoftheseexpressionsmakesgeneralstate-

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4.3.NumericalExamples65 mentsabouttheirpropertiesinfeasible.However,theirvaluescanbeestimatednumericallyataspecicpointaslongasthedeterminantofthe Jacobianisnonzeroatthatpoint.Iexamineavarietyofequilibriawitha rangeofreasonableparametervaluesinordertodevelopageneralideaof whatonemightexpecttoseeinadecisionmarketwiththecharacteristics ofthemodeldevelopedabove. 4.3NumericalExamples Inallofthefollowingcalculations,Iassumethat 2 = 2 l = 2 v =1and k =10. 1 Theassumptionofunitvariancesiscommonand k =10implies arelativelystrongdesiretomanipulategiventhesmallvaluesthat v )]TJ/F18 10.9091 Tf 11.493 0 Td [(p canbeexpectedtotake. 2 Unlessstatedotherwise,Iassume =2;this impliesthatthevarianceofatrader'sgainsweighsasheavilyinhisutility functionastheexpectedvalueofthosegains,andsoitreectsarelatively highdegreeofriskaversion. 4.3.1PriceError,ManipulatorCharacteristicsandMarket Thickness Irstconsiderhowthevarianceofthepriceerrorchangesas q ,the probabilitythatamanipulatorisinthemarket,varies.Iassumethat 2 t =1 andcomputefor q 2 [0 ; 1]andmarketsfeaturingone,ten,andtwenty informedtraders. 1 AllofthecomputationswereperformedinMATLAB. 2 v )]TJ/F48 8.9664 Tf 8.911 0 Td [(p hasmeanzeroand,asshownbelow,itsvariancetendstobemuchsmallerthan 10.

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4.3.NumericalExamples66 Figure4.1: asafunctionof q for N =1,10,and20.Baselinecurves arethe q =0variancelevelsforthecorresponding N Theresultsareshowningure4.1.Theyarestrikingforseveralreasons. Theyrevealtheimportanceofthesizeofthemarketindeterminingtheaccuracyofprices.Withonlyteninformedtraders,amarketfacingacertain attemptatmanipulationisstillnotablymoreecientthanamarketwith onlyoneinformedtraderandaguaranteedabsenceofmanipulation.Infact, thedierenceinvariancesisalmost10percentofthevarianceoftheasset valueitself.Increasingthenumberofinformedtraderstotwentymorethan doublesthisadvantage.At q = 1 2 ,themarketwithtwentyinformedtraders isjustasecientasthemarketwithtentradersandnomanipulator.The graphalsohighlightsthenonlinearrelationshipbetweenand N ;thedeviationoffromitsbaselineissmallerinthe N =1and N =20casesthanit isinthe N =10case.Interestingly,thepriceresultingfromthemarketwith oneinformedtraderandamanipulatorhasavarianceofapproximately1.

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4.3.NumericalExamples67 Thus, p N v; 1 S N v; 1,i.e.,thepricecapturestheinformed trader'sinformationdespitethepresenceofthemanipulator. Figure4.2: asafunctionof 2 t for N =1 ; 10 ; 20.Baselinecurvesare 2 t =1variancelevelsforthecorresponding N Letting q = 1 2 ,gure4.2plotsagainst 2 t for 2 t 2 1 5 ; 4 i.e.,Iallow 2 t tovaryfromclosetozerotofourtimesaslargeasthevarianceofthe assetvalue.Thereareseveralinterestingpatternstobeobservedingure 4.2.ContraHansonandOprea,anincreaseinthevarianceofthe manipulator'stargetpricedoesnotincreasepriceeciency.Thiscouldbe duetothefactthat,inthismodel,informationacquisitionisexogenous. 3 Thiscouldalsoresultfromtheriskaversionoftheinformedtraders;unlike therisk-neutralcase,introducinggreatervarianceintothedeterminationof thenalprice, ceterisparibus ,reducestheincentiveforrisk-averseagents 3 Inotherwords,theinformedtradersinthismodelcannotaecttheprecisionofthe signaloftheassetvaluethattheyreceive.

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4.3.NumericalExamples68 totrade.However,ascanbeseeningure4.3,informedtradersactually trademoreaggressivelywhen 2 t islarger. Figure4.3: asafunctionof 2 t for N =1 ; 10 ; 20. Thiscanbeexplainedbytheeectof 2 t on ;sincethemanipulatoris aliquiditytrader,alargervalueof 2 t correspondstoalargervalueof )]TJ/F16 7.9701 Tf 6.587 0 Td [(1 i.e.,amoreliquidmarket.Thus,informedtraderssuerlessofaprice penaltyforlargerordersandtheyscaleuptheirtradesaccordingly.This increasein partlyreversestheeectof 2 t on ,butitisclearfromgure 4.2that isstilllargeenoughtocausepricestoreecttheincreasein 2 t Theformalmodelallowsmetodisentanglethedirecteectsof 2 t on and aswellasitsindirecteectson mediatedthrough .Inthesenumerical examples,itisclearthattheincreasein inducedbyanincreasein 2 t is insucienttocompensateforthenoiseintroducedintothepricedirectlyby thechangein 2 t .

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4.3.NumericalExamples69 Usingthecomparativestaticderivativesderivedaboveaswellasthe varianceformulaforthepriceerror,wecanexaminethemarginalimpact ofaddingadditionalinformedtradersintothemarketunderavarietyof conditions.Figure4.4showsthemarginalimpactofanadditionalinformed traderonthevarianceofthepriceerror )]TJ/F19 7.9701 Tf 6.922 -4.541 Td [(@ @N for N 2f 1 ; 2 ;:::; 20 g under fourdierentscenarios. Figure4.4: @ @N underfourmanipulationscenarios. Theresultsshowthatinitiallytheintroductionofinformedtradersis subjecttoincreasingreturns,i.e.,themarginalreductioninthevariance ofthepriceerrorislargerforthe i thtraderthanforthe i )]TJ/F15 10.9091 Tf 11.42 0 Td [(1thtrader. 4 Atsomepoint,dependentuponthecharacteristicsofthemanipulator,it appearsthatthistrendreversesanddiminishingreturnssetin.Itappears thatthereversalpointislargerwhenthemanipulationattemptisstronger, 4 Thereisnoobviousintuitivereasonforthis,whichfurthervalidatestheuseofaformal modelinthisinquiry.

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4.3.NumericalExamples70 anditmaynotevenexistinsomescenarios;furthernumericaltestingisnecessarytoestablishthispointmoresoundly.Overall,itappearsthatthereis nopointatwhichtheadditionofmoreinformedtradersbecomessubstantiallylessvaluable.Inotherwords,thereisnooptimalmagicnumber"of informedtraders. 4.3.2DirectEectsofRiskAversion Figure4.5showsasafunctionofthelevelofriskaversionformarkets withone,ten,andtwentyinformedtraders.Comparedtothebaseline scenarioofuniversalriskneutrality,highlevelsofriskaversiondoresultin asubstantiallossofeciency.Theseverityofthisproblemissensitiveto thenumberoftradersinthemarket.Ahighernumberofinformedtraders resultsinsubstantiallylower;amarketwithonerisk-neutralinformed traderismarginallymoreecientthanamarketwithteninformedtraders with =5,eventhoughthisrepresentsanimplausiblyhighlevelofrisk aversion.Italsoappearstobethecasethat,when N islarge,theimpact ofariseinthelevelofriskaversionisdampened. 4.3.3EciencyoftheEquilibriumPrice Finally,itisworthwhiletoassessthevalueofthepredictionmarketprice tooutsidersasasourceofinformationaboutthetruevalueoftheasset v Recallthat p N v; .Inalloftheequilibriadiscussedabove, < 1 andthus p issuperiorinthosecasesto S asanestimateof v .Recall thatinthemarketanalyzedbyKylecontainingoneinsiderandan exogenousliquiditytrade,pricesonlyincorporatedone-halfoftheinsider's

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4.4.Conclusion:ImplicationsforDecisionMarketDesign71 Figure4.5: asafunctionof for N =1 ; 10 ; 20.Baselineisrisk-neutral case. information.Ineachofthescenariosanalyzedabove,thisideal"price eciencyisnotreached.However,evenwitharelativelysmallmarket N = 20andacertainmanipulationattempt, 3 4 2 v when 2 t =1.Doubling 2 t stillresultsinpricesthatimproveuponindividualtraderinformationby20 percent.Thissuggeststhatpredictionmarketpricescouldbevaluableeven inthepresenceofnontriviallevelsofriskaversionandmanipulation,however theimprovementoverexistingsourcesofinformationmaybemarginal. 4.4Conclusion:ImplicationsforDecisionMarket Design Myresultssuggestthatmanipulationmaysubstantiallyreducetheinformationcontentofpricesundercertainmarketconditions.However,the eectsofmanipulationarecontingentuponthesizeofthemarket.Infact,

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4.4.Conclusion:ImplicationsforDecisionMarketDesign72 theresultsofmyanalysissuggeststhatthereisnoproblemthatcannotbe solved"byintroducingmoreinformedtradersintothemarket.Thisisconsistentwiththeexperimentalandempiricalliteratureonpredictionmarkets andnancialmarketsmoregenerally.Whenthemarketisstillsomewhat small, N =20,evenarelativelyhighlevelofriskaversioncoupledwitha seriousmanipulationattemptstillyieldsapricethatimprovesupontraders' individualinformationbymorethan10percent.Doublingthesizeofthe marketleadstoapricewhosevarianceisa25percentimprovementon traders'individualinformation.Thus,toensureecientprices,potential marketpatronswoulddowelltoimplementmarketsonlywhenthereare areasonablylargenumberofpotentialtraderswhohaveaccesstorelevant information. Additionally,itisclearthatahighdegreeofriskaversionamongthe informedtradersinthemarketmakesthemanipulationofpricessubstantiallyeasier.Thismayprovideaneciencyrationalefortheinvestment constraintsimposedbymanyreal-worldpredictionmarkets.Iftradersare limitedtosmallstakes,itisunlikelythatriskaversionwillbeaseriousproblemsincethelevelofriskaversionnecessarytosubstantiallyaectsmall stakestradingwouldimplyhighlyunusualpreferences.Thereis,however, apotentialdownsidetoinvestmentcaps;limitingtheabilityoftradersto protfromtheirprivateinformationmayreducetheincentivetogather informationwheninformationacquisitionisendogenousandcostly,thus loweringtheoveralleciencyofthemarket.

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Chapter5 Conclusion Inthemodeldevelopedinchapter3,Ibuildupontheexistingtheoretical literatureonpredictionmarketsbyconsideringamarketforariskyasset populatedby N risk-aversetraderswhoareuncertainaboutboththepresenceandintentionsofapricemanipulator.Themanipulatorisintroduced intothemarketprobabilisticallyandattemptstomovetheassetpriceaway fromitsinformationallyecientvalueandtowardaprivatelyknowntarget. MyapproachdiersfromprevioustheoreticalworkinthatIallowthe nitenumberofinformedtraderstoberiskaverse,whereasHansonand OpreaandTetlockandHahnallowedonlyrisk-neutraltraders. Also,Igeneralizethenatureoftheinformedtraders'uncertaintyregarding themanipulatorbyintroducingthemanipulatorprobabilistically.These changesallowmetoanalyze 1.howtheadditionalconstraintofriskaversionontheabilityofinformed traderstocorrectpricemanipulationaectstheeciencyofprices, 73

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74 and 2.howriskaversionanduncertainmanipulationinteracttoaectprice eciency. Ideriveoptimaltraderstrategiesinthismodelandshowthat,whenan economicallysensibleequilibriumexists,itisaBayesianNashequilibrium. Ishowthattheequilibriumdoes,infact,existforavarietyofplausible parametervalues. Indthatthepriceerror v )]TJ/F18 10.9091 Tf 10.823 0 Td [(p isnormallydistributedwithmeanzero. Thekeyresultsfromnumericaltestsofthemodelusingavarietyofplausible parametervaluesindicatethat 1.theinformedtradersbidmoreaggressivelyinthepresenceofmanipulationdespitetheincreasedriskpenalty, 2.thevarianceofpriceerrorismonotonicallyincreasinginthelevelof riskaversionandthedegreeofmanipulation, 3.theeectivenessofmanipulationishighlysensitivetothesizeofthe market, 4.theintroductionofinformedtradersintothemarketissubjecttoa periodofincreasingreturnsfollowedbyaperiodofdecreasingreturns, and 5.pricesaggregatetraders'privateinformationeveninthepresenceofa highdegreeofmanipulationwhenthemarketisrelativelythick.

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5.1.DirectionsforFurtherResearch75 Ialsondthat,foralloftheparametervaluesIexamine,informedtraders submitnonzeroorders.Sincetheycouldabstainfromtradingbysubmitting anullorder,thisimpliesthatthereisanincentivetoparticipateinthe marketandsoendogenizingentry/exitofinformedtradersshouldnotspur anexodusofinformedtradersandanassociatedunravelingofthemarket. Myresultssuggestthatprospectivepredictionmarketpatronsshould implementmarketsonlywhenthereareareasonablylargenumberofpotentialtraderswhohaveaccesstorelevantinformation.Theyalsosuggest thatinvestmentcapscouldimprovetheeciencyofpricesifriskaversionis aseriousproblemandtheinformationavailabletoinformedtradersisnot dependentontheirabilitytoprotfromitinthemarket. 5.1DirectionsforFurtherResearch Thereareanumberofimportantelementsofrealworldpredictionmarkets notincorporatedintomyanalysisthatcould,andshould,beincludedin futurework.Budgetconstraintsareimposedbymanypredictionmarkets, andwhentheyarenottheimperfectionsofcapitalmarketsaswellasthe basicresourceconstraintsthatareaubiquitousfeatureofhumanaairs servetolimittraders'possibleinvestment.Thisisanadditionalnon-riskbasedconstraintontheactionsofmanipulatorsandnon-manipulatorsalike, andthedistributionofwealthbetweenmanipulatorsandnon-manipulators or,inthecaseofinvestmentcaps,thedistributionoftradersbetweenthe twogroupscouldseriouslyaecttheaccuracyofmarketprices.Additionally,endogenizinginformationacquisitionandtraderentry/exitwouldadd

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5.1.DirectionsforFurtherResearch76 substantialrealismtothemodel. Giventheanalyticaldicultiesencounteredwhilesolvingthisrelatively simplemodel,itseemslikelythatanapproachutilizinganagent-basedcomputationalmodelwouldbemoreproductivethanthetheoreticalapproach followedhere.Thistypeofmodelcouldincorporate1budgetconstraints, 2heterogeneousriskpreferences,3endogenousinformationacquisition, 4endogenousentry/exitoftraders,5agreaternumberandvarietyof manipulators,and6learningbyboundedlyrationaltraders.Sincereal predictionmarketsarelikelytoincludeallofthesecharacteristics,aricher computationalmodelmaybethebestapproachforunderstandinghowthese marketsbehave.

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Glossary Arbitrage Atradeorcombinationoftradesiscalledanarbitrageifitguarantees apositive,risk-freeprot. Arrow-Debreusecurity Assetthatpaysoneunitofthenumeraireifandonlyifacertainstate oftheworldisreached. Arrow-Prattcoecientofabsoluteriskaversion Measureofriskaversiondenedas A w )]TJ/F18 10.9091 Tf 21.195 7.38 Td [(u 00 w u 0 w ,where u w isan agent'sutilityfunctionand w istheagent'swealth. Atmarket Anorderissaidtobeatmarketifthespeciedexecutionpriceisclose tothecurrentmarketprice;whatexactlyclose"meansdependson theparticularmarket. BayesianNashequilibrium ABayesianNashequilibriumspecies,foreachplayer,astrategyproleandbeliefsaboutthecharacteristicsoftheotherplayersthatmaximizestheplayer'sexpectedpayogiventheirbeliefsabouttheother players'characteristicsandthestrategiestheywillplay. Commonknowledge Afact F iscommonknowledgeinapopulation P ifeverymemberof P knows F ,everymemberof P knowsthateverymemberof P knows F ::: ,everymemberof P knowsthat n everymemberof P knows F ::: andsoonforall n 77

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Glossary78 Concordantbeliefs Thebeliefsoftraders i and j aresaidtobeconcordantif P i x j y = P j x j y 8 x;y ,i.e.,tradersagreeontheconditionaldistributionof x Constantabsoluteriskaversion PropertyofautilityfunctionwhoseArrow-Prattcoecientdoesnot dependonwealth,i.e. A 0 w =0.Riskpreferenceisinsensitivetothe absolutelevelofwealth. Decreasingabsoluteriskaversion PropertyofautilityfunctionwhoseArrow-Prattcoecientisdecreasinginwealth,i.e. A 0 w < 0.Anincreaseinwealthreducesabsolute riskaversion. Depth SeetheentryunderLiquidity. Doubleauction Anauctionwherepotentialbuyerssubmittheirdemandschedulesand potentialsellerssimultaneouslysubmittheirsupplyschedulestoan auctioneer,whothenmatchesbuyersandsellersaccordingtopre-set rules.Iftraderscansubmitordersasynchronously,andordersare matchedcontinuously,itisacontinuousdoubleauction. Favorite-longshotbias Empiricallyobservedphenomenonwherebettorsundervaluethemost probableevents`favorites'whileovervaluingmoreimprobableevents `longshots'. Incentivecompatible Ascoringruleissaidtobeincentivecompatibleifanagentmaximizes herexpectedpayobyreportinghertruebeliefs. Increasingabsoluteriskaversion PropertyofautilityfunctionwhoseArrow-Prattcoecientisincreasinginwealth,i.e. A 0 w > 0.Anincreaseinwealthincreasesabsolute riskaversion.

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Glossary79 Limitorder Alimitorderisanordertobuysellacertainquantityofasecurity atnomorenolessthanaspecicprice.Thetraderhascontrolover thepriceatwhichthetradeisexecuted;however,theordermaynever beexecuted. Liquidity Theliquidityofamarketreferstothespeedwithwhichatradecan bearrangedimmediacyandthesizeofanordernecessarytomove pricesbyagivenamountdepth.Ifamarketishighlyliquid,trades canbearrangedquicklyandthepriceimpactoftradesisnegligible. Inmymodel,thedepthofthemarketisgivenby )]TJ/F16 7.9701 Tf 6.587 0 Td [(1 ,whichmeasures theeectonpricesofaunitchangeinorderow.SeeHarris fordetails. Marketmaker Traderwhoquotesbothbidandaskpricesforanassetandstands readytotradewithanyoneatthoseprices.Acompetitivemarket makeroperateswithzeroprots. Marketmicrostructuremodel Amarketmodelwherethetradingmechanism,therulesthatgovern it,andthepriceformationprocessaremadeexplicit. Marketorder Amarketorderisabuyorsellordertobeexecutedimmediatelyat currentmarketprices.Amarketorderguaranteesexecutionbutmay executeatanunfavorableprice. Marketthickness Thethicknessofamarketisthenumberofeectiveparticipants. Noisetrader Anoisetraderisanytraderwhotradesforreasonsnotbasedonprivate informationaboutthefuturevalueofanasset. Opinionpool Anaverageofscoringruleresponses.Alinearopinionpoolcorresponds toanarithmeticmeanwhilealogarithmicopinionpoolcorresponds toageometricmean.

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Glossary80 Paretoeciency AnallocationofgoodsamongagentsissaidtobeParetoecientif theredoesnotexistanexchangeofgoodsamonganysetofagents thatwouldleavenoagentwithlowerutilitywhileresultinginatleast oneagenthavinghigherutility. Patron Amarketpatronisanpersonororganizationthatestablishesorsubsidizesapredictionmarket. Predictionmarket Lowvolumespeculativemarketforsecuritieswhoseterminalpayo isdeterminedinaxedmannerbytheoutcomeofawell-dened uncertainevent. Pricemanipulator Apricemanipulatortradessoastomovepricestowardaspecic target,possiblyawayfromthecorrecti.e.,informationallyecient price.Forexample,ifamanipulatorprefershigherpricesshemaybuy aggressivelysoastoforcepricesupwards. Scoringrule Afunctionthattakesaprobabilityoravectorofprobabilitiesas inputandproducesasoutputanumberrewardbasedonthedifferencebetweentheactualoutcomeanditsassignedprobability.A scoringruleiscalledproper"ifitisincentivecompatible.Example:Supposeaweatherforecasteristaskedtoassignprobabilitiesto theevents rain j =1and no )]TJ/F18 10.9091 Tf 11.409 0 Td [(rain j =2forthenext n days. Considerthefunction SR = 1 n P 2 j =1 P n i =1 f ij )]TJ/F18 10.9091 Tf 10.909 0 Td [(E ij 2 where f ij isthe probabilitytheforecasterassignedtotheeventthat j wouldoccuron day i and E ij takesthevalue1ifitrainedonday i and0otherwise. Brierclaimsthatthisscoringruleisincentivecompatible;Seltenprovesthatitisanddemonstratesthatithasanumberof desirableproperties. Shortselling Sellingassetsthathavebeenborrowedfromathirdpartyusuallyfor afee,withtheintentionofbuyingidenticalassetsbackatalaterdate

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Glossary81 andreturningthemtothelender.Theshortsellergainslosesifthe pricehasdeclinedrisenbetweenthesaleandtherepurchase. Sucientstatistic Astatistic T issaidtobesucientforaparameter ifthedistribution P x 1 ;::;x n j T x 1 ;::;x n isnotafunctionof Utilityfunction Suppose X isacollectionofgoods.Afunction u assigningnumerical valuestomembersof X suchthat u x >u y i x ispreferredto y iscalledautilityfunction. Walrasianauctioneer AWalrasianauctioneerisahypotheticalpricesettingagentwhohas completeknowledgeofthedemand/supplyschedulesofallofthe agentsinamarketandwhousesthisinformationtosetaperfect marketclearingpriceforanynumberofgoods.Implicitinthisformulationisanabsenceoftransactioncosts.

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