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PAGE 1 InformationalEciencyofDecisionMarketswith Risk-AverseInsidersandUncertainManipulation TateA.Twinam AThesis SubmittedtotheDivisionofNaturalSciences andtheDivisionofSocialSciences NewCollegeofFlorida inpartialfulllmentoftherequirementsforthedegree BachelorofArtsinEconomicsandMathematics UnderthesponsorshipofDr.CatherineS.Elliott Sarasota,Florida May,2010 PAGE 2 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. PAGE 3 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 PAGE 4 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 PAGE 5 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 PAGE 6 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 PAGE 7 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 PAGE 8 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. PAGE 9 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. PAGE 10 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$1; 5 whenitdoes,thepriceofthecontractcanbe 4 Whilepredictionmarketssharemanysimilaritieswithtraditionalnancialmarkets, thereareimportantdierences.Forexample,unlikethestockstradedintraditional nancialmarkets,predictionmarketcontractshaveatrue"valuethatdoesnotdepend upontrader'sexpectations.Inotherwords,predictionmarketpricesarenotsubjectto theKeynesianbeautycontest"phenomenon. 5 Seeforinstance,Intrade.comandtheIEM. PAGE 11 1.3.MicrostructureofTypicalPredictionMarkets5 roughlyinterpretedasanestimatedprobabilitythatevent Y occurs. Index:Contractpays$ 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. PAGE 12 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. PAGE 13 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. PAGE 14 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. PAGE 15 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 PAGE 16 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. PAGE 17 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: PAGE 18 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. PAGE 19 1.6.OutlineoftheThesis13 Inchapter4,Iderivecomparativestaticsfortheequilibriumstrategiesand explorethegeneralpropertiesofthestrategiesthroughnumericalexamples. PAGE 20 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 PAGE 21 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 PAGE 22 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 = PAGE 23 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. PAGE 24 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. PAGE 25 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- PAGE 26 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. PAGE 27 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. PAGE 28 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. PAGE 29 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. PAGE 30 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. PAGE 31 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 . PAGE 32 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 PAGE 33 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 PAGE 34 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 PAGE 35 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. PAGE 36 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 PAGE 37 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. PAGE 38 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. PAGE 39 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. PAGE 40 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 PAGE 41 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. PAGE 42 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 PAGE 43 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. PAGE 44 2.4.Conclusion38 2.4Conclusion ThetheoreticalmodelsdiscussedinthischapterprovidethenecessarycontextforthemodelIpresentinthenextchapter.Theempiricalevidence presentedlendscredencetothepossibilitythatpredictionmarketsmaybe avaluabletoolforforecastinganddecisionmakinginhigh-stakesenvironments,thusjustifyingtheeortinvestedinthedevelopmentofthemodel. PAGE 45 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 PAGE 46 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 PAGE 47 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. PAGE 48 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 PAGE 49 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- PAGE 50 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. PAGE 51 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 PAGE 52 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 PAGE 53 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 PAGE 54 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- PAGE 55 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. PAGE 56 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. PAGE 57 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. PAGE 58 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 PAGE 59 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- PAGE 60 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 PAGE 61 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 PAGE 62 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 . PAGE 63 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. PAGE 64 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 PAGE 65 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 PAGE 66 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 PAGE 67 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 PAGE 68 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 PAGE 69 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 PAGE 70 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- PAGE 71 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. PAGE 72 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. PAGE 73 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. PAGE 74 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 . PAGE 75 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. PAGE 76 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 PAGE 77 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, PAGE 78 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. PAGE 79 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 PAGE 80 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. PAGE 81 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 PAGE 82 5.1.DirectionsforFurtherResearch76 substantialrealismtothemodel. Giventheanalyticaldicultiesencounteredwhilesolvingthisrelatively simplemodel,itseemslikelythatanapproachutilizinganagent-basedcomputationalmodelwouldbemoreproductivethanthetheoreticalapproach followedhere.Thistypeofmodelcouldincorporate1budgetconstraints, 2heterogeneousriskpreferences,3endogenousinformationacquisition, 4endogenousentry/exitoftraders,5agreaternumberandvarietyof manipulators,and6learningbyboundedlyrationaltraders.Sincereal predictionmarketsarelikelytoincludeallofthesecharacteristics,aricher computationalmodelmaybethebestapproachforunderstandinghowthese marketsbehave. PAGE 83 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 PAGE 84 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. PAGE 85 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. PAGE 86 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 PAGE 87 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. PAGE 88 Bibliography Berg,JoyceE.andThomasA.Rietz.Predictionmarketsasdecision supportsystems. InformationSystemsFrontiers 5 ,pp.79{93. Berg,JoyceE.,ForrestD.Nelson,andThomasA.Rietz.Prediction marketaccuracyinthelongrun. InternationalJournalofForecasting 24 ,pp.283{298. Brier,GlennW..Vericationofforecastsexpressedintermsofprobability. MonthlyWeatherReview 78 ,pp.1{3. Brunnermeier,MarkusK.. AssetPricingunderAsymmetricInformation:Bubbles,Crashes,TechnicalAnalysis,andHerding .NewYork: OxfordUniversityPress. Camerer,ColinF..Canassetmarketsbemanipulated?Aeldexperimentwithracetrackbetting. JournalofPoliticalEconomy 106 pp.457{482. Chen,YilingandDavidM.Pennock.Autilityframeworkfor bounded-lossmarketmakers.In Proceedingsofthe23rdConferenceon UncertaintyinArticialIntelligence ,pp.49{56. Chen,Kay-YutandCharlesR.Plott.Informationaggregationmechanisms:Concept,designandimplementationforasalesforecastingproblem.CaliforniaInstituteofTechnology,SocialScienceWorkingPaper 1131. Chen,Yiling,LanceFortnow,NicolasLambert,DavidM.Pennock,and JenniferWortman.Complexityofcombinatorialmarketmakers. In Proceedingsofthe9thACMConferenceonElectronicCommerce ,pp. 190{199,ACMNewYork,NY,USA. 82 PAGE 89 Bibliography83 Chen,Yiling,LanceFortnow,EvdokiaNikolova,andDavidM.Pennock .Bettingonpermutations.In Proceedingsofthe8thACMConferenceonElectronicCommerce ,pp.326{335,ACMNewYork,NY,USA. Chen,Yiling,SharadGoel,andDavidM.Pennock.Pricingcombinatorialmarketsfortournaments.In Proceedingsofthe40thAnnualACM SymposiumontheTheoryofComputing ,pp.305{314,ACMNewYork, NY,USA. Cowgill,Bo,JustinWolfers,andEricW.Zitzewitz.Usingprediction marketstotrackinformationows:EvidencefromGoogle.Dartmouth CollegeWorkingPaper. DeJong,FrankandBarbaraRindi. TheMicrostructureofFinancial Markets .NewYork:CambridgeUniversityPress. Forsythe,Robert,ForrestNelson,GeorgeR.Neumann,andJackWright .Anatomyofanexperimentalpoliticalstockmarket. TheAmerican EconomicReview 82 ,pp.1142{1161. Gaspoz,CedricandYvesPigneur.PreparinganegotiatedR&Dportfoliowithapredictionmarket.In Proceedingsofthe41stHawaiiInternationalConferenceonSystemScience Grossman,SanfordJ..Ontheeciencyofcompetitivestockmarkets wheretradershavediverseinformation. JournalofFinance 31 ,pp. 573{585. Grossman,SanfordJ.andJosephE.Stiglitz.Informationandcompetitivepricesystems. TheAmericanEconomicReview 66 ,pp.246{ 253. .Ontheimpossibilityofinformationallyecientmarkets. TheAmericanEconomicReview 70 ,pp.393{408. Hahn,RobertW.andPaulC.Tetlock. InformationMarkets:ANew WayofMakingDecisions .Washington,DC:AEIPress. Hanson,Robina.Combinatorialinformationmarketdesign. InformationSystemsFrontiers 5 ,pp.107{119. b.Shallwevoteonvalues,butbetonbeliefs.GeorgeMason UniversityWorkingPaper. PAGE 90 Bibliography84 .You'rered. Forbes 38 a.Logarithmicmarketscoringrulesformodularcombinatorialinformationaggregation. JournalofPredictionMarkets 1 ,pp. 1{15. b.Thepolicyanalysismarket:Athwartedexperimentinthe useofpredictionmarketsforpublicpolicy. Innovations 2 ,pp.73{88. Hanson,RobinandRyanOprea.Amanipulatorcanaidprediction marketaccuracy. Economica 76 ,pp.304{314. Hanson,Robin,RyanOprea,andDavidPorter.Informationaggregationandmanipulationinanexperimentalmarket. JournalofEconomic BehaviorandOrganization 60 ,pp.449{459. Harris,Larry. TradingandExchanges:MarketMicrostructurefor Practitioners .NewYork:OxfordUniversityPress. Hayek,FriedrichAugustvon.Theuseofknowledgeinsociety. The AmericanEconomicReview 35 ,pp.519{530. Kumar,PraveenandDuaneJ.Seppi.Futuresmanipulationwith cashsettlement". TheJournalofFinance 47 ,pp.1485{1502. Kyle,AlbertS..Continuousauctionsandinsidertrading. Econometrica 53 ,pp.1315{1335. Ledyard,John,RobinHanson,andTakashiIshikida.Anexperimentaltestofcombinatorialinformationmarkets. JournalofEconomicBehaviorandOrganization 69 ,pp.182{189. Lengwiler,Yvan. MicrofoundationsofFinancialEconomics:AnIntroductiontoGeneralEquilibriumAssetPricing .Princeton,NJ:Princeton UniversityPress. Lyons,RichardK.. TheMicrostructureApproachtoExchangeRates Cambridge,MA:TheMITPress. Malkiel,BurtonG..Reectionsontheecientmarkethypothesis: 30yearslater. TheFinancialReview 40 ,pp.1{9. Maloney,MichaelT.andJ.HaroldMulherin.Thecomplexityof pricediscoveryinanecientmarket:Thestockmarketreactiontothe Challengercrash. JournalofCorporateFinance 9 ,pp.453{480. PAGE 91 Bibliography85 Manski,CharlesF..Interpretingthepredictionsofpredictionmarkets. EconomicsLetters 91 ,pp.425{429. Milgrom,PaulandNancyStokey.Information,tradeandcommon knowledge. JournalofEconomicTheory 26 ,pp.17{27. O'Hara,Maureen.. MarketMicrostructureTheory .Malden,MA: Wiley-Blackwell. Oliven,KennethandThomasA.Rietz.Suckersarebornbutmarkets aremade:Individualrationality,arbitrage,andmarketeciencyonan electronicfuturesmarket. ManagementScience 50 ,pp.336{351. Oprea,Ryan,DavidPorter,ChrisHibbert,RobinHanson,andDorinaTila .Canmanipulatorsmisleadpredictionmarketobservers.George MasonUniversityWorkingPaper. Ortner,Gerhard.Forecastingmarkets:Anindustrialapplication. Mimeo,TechnicalUniversityofVienna. Ottaviani,MarcoandPeterN.Srensen.Outcomemanipulationin corporatepredictionmarkets. JournaloftheEuropeanEconomicAssociation 5 -3,pp.554{563. .Aggregationofinformationandbeliefs:assetpricinglessons frompredictionmarkets.DiscussionPapers09-14,UniversityofCopenhagen,DepartmentofEconomics. Rhode,PaulW.andKolemanS.Strumpf.Historicalpresidential bettingmarkets. JournalofEconomicPerspectives 18 ,pp.127{142. Roll,Richard.Orangejuiceandweather. TheAmericanEconomic Review 74 ,pp.861{880. Selten,Reinhard.Axiomaticcharacterizationofthequadraticscoring rule. ExperimentalEconomics 1 ,pp.43{62. Subrahmanyam,Avanidhar.Riskaversion,marketliquidity,and priceeciency. ReviewofFinancialStudies 4 ,pp.417{441. Tetlock,PaulC.andRobertW.Hahn.Optimalliquidityprovision fordecisionmakers.SSRNWorkingPaperSeries. Tullock,Gordon.Thewelfarecostoftaris,monopoly,andtheft. WesternEconomicJournal 5 ,pp.224{232. PAGE 92 Bibliography86 Vives,Xavier. InformationandLearninginMarkets:TheImpactof MarketMicrostructure .Princeton,NJ:PrincetonUniversityPress. Wolfers,JustinandEricW.Zitzewitz.Predictionmarkets. TheJournalofEconomicPerspectives 18 ,pp.107{126. .Interpretingpredictionmarketpricesasprobabilities.SSRN WorkingPaperSeries. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||