A Game Model of Irreversible Investment under Abst

AGameModelofIrreversibleInvestmentunderUncertaintyPauliMurtoSystemsAnalysisLaboratoryHelsinkiUniversityofTechnologyP.O.Box1100,02015HUT,Finlande-mail:[e-mail]IndustrialandOperationsEngineeringUniversityofMichigan1205BealAvenue,AnnArbor,MI,48109-2117,USAe-mail:[e-mail]ratureonrealoptionsconsiderstheoptimaldecisionofafirminisolationfromcompetitors.Inreality,however,theactionsofcompetingfirmsoftenaffecteachother’sinvestmentopportunities.Wedevelopagamemodelwheremanyfirmscompeteforasingleinvestmentopportunity.Whenoneofthefirmstriggerstheinvestmenttheopportunityiscompletelylostfortheotherfirms.ThevalueoftheprojectforthefirmsisassumedtofollowageometricBrownianmotion.Themodelcombinesgametheoryandthetheoryofirreversibleinvestmentunderuncertainty.WecharacterizetheresultingNashequilibriumunderdifferentassumptionsontheinformationthatthefirmshaveabouteachother’svaluationsfortheproject.Asanexample,wepresentacaseofbuildingatelecommunicationsnetwork.Keywords:realoptions,irreversibleinvestment,gametheory,uncertainty,telecommunication21.IntroductionInrecentyears,theliteratureonrealoptionshasimprovedconsiderablyourunderstandingoftheirreversiblecapitalinvestmentproblemsunderuncertainty.Thisliteraturestressesthesimilaritybetweenafinancialcalloptionandanopportunitytoinvestinarealasset.TheinvestmentproblemcanbeseenasatypicalvaluationproblemofanAmericanoptioninwhichthetheoryofoptimalstoppingtimes[e.g.KaratzasandShreve(1988)]isused.AnexcellentsurveyofthemaintheoryisgiveninDixitandPindyck(1994).Importantcontributionsinclude,e.g.,McDonaldandSiegel(1986),Pindyck(1988),andDixit(1995).AnothersurveyondifferentmodelsisTrigeorgis(1996).Inamarketwithnolargeinvestors,thevalueoftherealoptionisequaltothenetpresentvalueoftheinvestmentafterallcostsplusthetimevalueoftherealoption.Theentrytimeisselectedsothatthevalueoftheoptionismaximized.Inotherwords,investmentismadeatsuchamomentwhenthetimevalueiszeroandnetpresentvalueisstrictlypositive.However,inthecaseoflargeinvestors[seee.g.KeppoandLu(1999)]theproblemismuchmorecomplicatedbecausewehavetoconsidertheimpactoftheinvestmentsonthenetpresentvalues.Thisleadstoaninvestmentgamebetweenthefirms.Mostofthemodelspresentedintheliteraturedonottakeintoaccountthecompetitiveaspects.Thisisoftenaseriouslimitationifthepurposeofthemodelingistogetmoreunderstandingonfunctioningofsomeindustry,butalsoifthepurposeofthemodelingistohelpdecisionmakersmakebetterinvestmentdecisions.DelSolandGhemawat(1999)arguethattypicaloptionvaluationmodelstendtorecommendwaitingtoolongbeforeinvesting,becausetheyfailtorecognizethecompetitionforalimitednumberofbusinessopportunities.3Theexistingliteraturethattakescompetitiveaspectsintoaccountcanbedividedintotwoclasses.First,therearepapers,whichstudythemarketequilibriumundertheassumptionofperfectcompetitionandfreeentry.LucasandPrescott(1971)showedthesocialoptimalityoftheequilibriuminadiscrete-timeMarkovchainmodel.Leahy(1993)discoveredthattheequilibriumentrytimeunderfreeentryisthesameastheoptimalentrytimeofamyopicfirmwhoignoresfutureentrybycompetitors.BaldurssonandKaratzas(1996)generalizetheresultutilizingsingularstochasticcontroltheory.Grenadier(1999)enrichestheanalysesbyincludingconstructiondelays.AninterestingapplicationisdevelopedinTvedt(1999).Asecondandmorerecentstreamofliteraturetakesthegametheoreticapproach.Inthesemodels,thenumberoffirmsisexogenousandthereforeindependentontheequilibriumresult.Someofthemodelsaresetinatwoperiodframework,e.g.,KulatilakaandPerotti(1998),whichanalyzethestrategicinteractionofinvestmentsthatinducecostadvantageoverrivals.Incontinuoustime,Williams(1993)andBaldursson(1998)areexamplesofmodelswherefirmscanadjustcapacitycontinuously.Ontheotherhand,Grenadier(1996),Lambrect(1999),andJoaquinandButler(1999)presentmodelswherecompetingfirmshaveopportunitiestoinvestindiscreteinvestmentprojectsandwherethegameisplayedonthetimingoftheseinvestments.Thispaperaddstothissecondstreamofliterature.Weanalyzeaparticularkindofsetting,wheretheinvestmentofafirmcompletelyeliminatescorrespondinginvestmentopportunitiesoftheotherfirms.Theagentsmustfirstpaythefixedsunkcostbeforethenewinvestmentcanstart.Thus,atthebeginningoftheoptimizationperiodtheagentsholdAmericanoptionsand,therefore,weemploytheoptimalstoppingtheory.However,thestrategiceffectscomplicatethesituation.Evenifthemodelitselfisdynamic,thestrategicinteractionbetweentheplayersresultsinaone-shotgameininvestmentstrategies.Theequilibriumstrategiesofthefirmsdependcruciallyontheinformationthat4thefirmspossessaboutthevaluationofthebusinessopportunitybytheircompetitors.WefindtheNashequilibriumunderdifferentassumptionsontheinformationstructure.Undertheassumptionthatfirmsdonotknoweachother’svaluationfortheproject,weproposethatfirmsusesimpleassessmentsonthelikelihoodthattheircompetitorswillinvestwithinthenext‘timeinstant’.Wecharacterizetheequilibriumthatresultsfromthistypeofassessments.Asanexampleofthemodelwepresentacaseofbuildinganewtelecommunicationsnetwork.Ifweassumethatthenewnetworkwil