ABGgeb2016

Contents lists available atScienceDirect

Games

and

Economic

Behavior

www.elsevier.com/locate/geb

Project

selection:

Commitment

and

competition

Vidya Atal

a

,

Talia Bar

b

,

∗

,

Sidartha Gordon

c

a_{DepartmentofEconomicsandFinance,MontclairStateUniversity, 1NormalAvenue,SchoolofBusiness531,Montclair,NJ07043,}

United States

b_{DepartmentofEconomics,UniversityofConnecticut, 365FairfieldWay,OakHallRoom335,Storrs,CT06269,UnitedStates}

c_{Laboratoired’EconomiedeDauphine(LEDa),ParisDauphineUniversity,PSLResearchUniversity,PlaceduMaréchaldeLattredeTassigny,}

75016Paris,France

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received26July2014 Availableonline1February2016

JELclassification: L10

L13 D21

Keywords: Projectselection Search Commitment

Markovperfectequilibrium

Weexamineprojectselectiondecisionsoffirmsconstrainedinthenumberofprojectsthey canhandleatonce. Anew projectopportunityarises every period.Takingonaproject requiresacommitmentofuncertainduration,preventingthefirmfromselectinganother projectinsubsequentperiodsuntilthecommitmentends.Inourdynamicgame,whentwo firmsarefreeofcommitment,theymovesequentially inrandomorder.Symmetricpure strategyMarkovperfectequilibriaalwaysexist.Inequilibrium,thefirstmoverstrategically rejectssomeprojectsthatarethenselectedbythesecondmover,evenwhenthevalueof theprojectisthesameforbothfirms.Amonopolistrejectsmoreprojects,andadoptsones ofhigher averagequality comparedto theduopolist. Duopolists selecttoofew projects comparedtotheirjointlyoptimalbehavior.Weextendthemodeltoallowforexternalities, asymmetry,andn

>

2 firms.

©2016ElsevierInc.All rights reserved.

1. Introduction

1.1. Motivation

Firms,researchersandgovernmentagenciesrepeatedlyselectprojects(e.g.,researchanddevelopmentprojects,clientsto

serve,acquisitionofstart-ups,andnewproducts).Inmanysituations,firmsareconstrainedinthenumberofprojectsthey

can work onat once,andprojects canlast fora numberof periods.Committing limitedresources (e.g.,researchers’and

otherskilledemployees’time,physicalspace,labequipment)tooneprojectmayresultintheneedtoforgofuture,perhaps

moreprofitableopportunities.Inastrategicenvironment,firmscompeteforprojects;onefirm’sdecisiontoselectaproject

canaffectitsrival’sopportunitiesatpresentbecausetheselectedprojectisnolongeravailable,andinthefuture,because

the firmcommitsits resources.Inthispaper,westudyprojectselectionofstrategicfirmsthatfaceuncertaintyaboutthe

durationoftheprojecttowhichtheycommitandaboutthevalueoffutureprojects.

Weanalyzeadynamic,discretetime,infinitehorizongame.Everyperiod,anewprojectopportunityarisesandperishes

ifitwasnotselected.Firmslearntheexpectedreturnofthecurrentproject,andknowthedistributionfromwhichfuture

projectreturnsare drawn.Randomprojectduration iscapturedbyafixedprobability thatcommitmentwillendandfree

the firmto selecta newprojectinthefollowingperiod.Whentwocompetingfirmsare freeofcommitment,weassume

*

Correspondingauthor.

E-mailaddresses:[email protected](V. Atal),

[email protected]

(T. Bar),

[email protected]

(S. Gordon). http://dx.doi.org/10.1016/j.geb.2016.01.011

theyconsiderprojectssequentially(andthedecisionnottoselectaprojectisirreversible).Aperiod’sleader—thefirstfirm

toconsidertheproject—ischosenatrandom.Theperiod’sfollowercanselecttheprojectonlyiftheleaderdidnot.

Sequentialproject selection decisions occur insome markets. For example,pharmaceutical companies repeatedly face

project opportunities presented by small biotech firms who negotiate with the pharmaceutical companies sequentially.1

Editorsofcompetingjournalschoosingwhichsubmissionstoacceptforpublicationareapproachedbyauthorssequentially.

Service providers (e.g.,consulting firms,contingent fee lawyers, contractors) are sequentially approached by clients, and

need todecide whethertotake the client,knowing thataccepting toserve himrequires commitmentandthat rejecting

himmaypushhimtoa rivalfirm.ANovember 2000headline intheChicagoSun-Timesread, “Cokebacksoffbidtobuy

QuakerOats:Company’sdecisioncouldopendoorforrivalPepsiCo.”PepsilateracquiredQuakerOatsforitsGatoradesports

drink.2

Inourgame,apurestrategysymmetricMarkovperfectequilibriumalways exists.Ifthecommitmenttoa projectonly

lastedone period,inasymmetricgame,a leader wouldtake onanyprojectwitha positivereturnanda followerwould

nevertakeonanyproject.But,whenprojectscanlastlongerthanoneperiodandrequirecommitmentofresources,positive

return projects maybe rejected by both firms.The leader ina periodwould selectany projectthat hasa high enough

return.Interestingly,we show that thereare projects withintermediatelevels ofreturn thatare rejectedby theperiod’s

leader, yet selected by the follower. This is true even though both firms havethe same value from the currentproject,

andtheleadershippositionisonlyguaranteedforthecurrentperiod.Tounderstandtheintuitionforthisresult,consider

the lowestvalue project that thefollowerwould select–the followeris indifferentbetweenselectingand rejecting this

marginal project.The leader hasthe same benefitas thefollower fromselecting this project,buta higher benefitfrom

rejectingitbecauseiftheleaderrejects,thefollowerwouldselectand(unlesstheprojectends)thefollowerwillthen be

busyinthenextperiodguaranteeingtheleader theopportunity toselectnext period.Thus, abenefitfromhavingabusy

rivalcanarisebecausefirmscompeteoverthesameprojectopportunities.

Toexamine theeffectsofcompetition onprojectselection behavior intheenvironment we describe,we comparethe

outcome of thegame withtwo benchmarks: adecision maker witha capacityofone project,thisanalysis issimilar to

earlierworkandisincludedforeasycomparison3;andadecisionmakerwithatwo-projectcapacity.Wefindthatproject

selectionthresholdsarelowerundercompetition.Thatis,aduopolistselectsprojectsthatitwouldhaverejectedabsenta

competitorinthemarket.Oneexplanationforthisisthattheduopolistexpectsaninferior selectionofprojectscompared

withthemonopolist(becausehisrival willselectsomeofthehighvalueprojects). Thisreducesthevalueofwaiting and

makes the duopolist more likely to select the current project.We also compare equilibriumselection withthe optimal

choice ofa jointventure (maximizingsumof profitswitha two-project capacity).We find that induopoly competition,

firmsrejecttoomanyprojectscomparedwiththejointlyoptimalbehavior.Intuitively,thejointventureismoreflexiblein

usingthecapacitythantheduopolyandthebenefitfromabusyrival,whichisimportantintheduopolysituation,isnota

considerationforthejointventure.

Ourmain modelassumesthat the leader ineach periodis chosen atrandom.This assumptionis naturalwhenfirms

areex-antesymmetric.It illustratesthebenefitfromhavingabusyrival,inasettingwhereitseemsmostsurprisingthat

one firm wouldaccepta projectrejectedby theother.However, insome cases,oneof thefirmsmayhavean advantage

(perhapsit isbetter known,ormore easily accessible).We studya version ofthe gameinwhich one firm isthe leader

in every period. Inthis asymmetric game,the follower hasa lower thresholdfor acceptingprojects than the leader for

an additionalreason: projectopportunities forthefollowerarise froman inferior (right censored)distribution ofproject

returns,becausewhenfree,theleaderselectsthemostpromisingprojects.

Ourresultsarerobusttotheinclusionofpayoffexternalities.Consideringtheeffectofexternalities onprojectselection

thresholds(in thecaseoflong durationprojects),we showthat firmshavelowerthresholds(select moreprojects)under

positivepayoffexternalitiesthanwithoutexternalities,andhigherthresholdswithnegativeexternalities.Onemightexpect

positiveexternalities tocreatean incentivetorejectmoreprojectswhentheother firmisfree.However,thebenefitfrom

havingabusyrivalisreducedwithpositiveexternalities,andtheconcernaboutcommittingresourcesislower,sothatthe

firmwouldhavelessreasontorejectprojects.Wealsoextendtheduopolygametoanoligopoly.

1.2. Backgroundliterature

Ourpapercontributestotheliteratureoninvestmentinstrategicenvironments.Inadeterministicsetting,Fudenbergand

Tirole (1985)showthatstrategicfirmsadoptanewtechnologywithdecreasingcostssoonerthantheywouldinamonopoly

marketstructure. Intheirmodel,firmsadoptpreemptivelytoprevent ordelay adoptionbytheir opponents,while inour

model,afirmcouldbenefitwhenitsopponentselectsaproject.Weeds (2002),Grenadier (2002),Pawlina and Kort (2006),

1 _{See,forexample,}

_{Kolchinsky (2004)}

_{page56onnegotiationagreements,and}

_http

_{://www.secinfo.com/d12MGs.1n4.htmforanewsreleasedescribing} MicrologixsBiotechInc.enteringanexclusivenegotiationperiodwithapharmaceuticalcompany.

2 _{Lazare,L.ChicagoSun-Times,November22,2000.}

_http

_{://www.highbeam.com/doc/1P2-4568251.html.InformalconversationwithaformerCocaCola} employeesuggestedthatintroducinganewdrinkrequiressignificantcommitmentofresourcesonthepartofthecompany(e.g.,marketresearch,bottle designandmarketingefforts).

Mason andWeeds (2010) offerreal options modelsof strategicinvestment underuncertainty. The firsttwo papers have

symmetric models.PawlinaandKort (2006)includeasymmetryininvestmentcosts, andinMasonandWeeds (2010)the

leader andthefollowerhavedifferentflowprofits.In ourmodel,thefirmsareidentical,butsequentialmoveseffectively

addsomeasymmetry.

Weeds’(2002) duopolypatentracehastwo sourcesofuncertainty:first,thevalueofthepatentevolvesstochastically;

second,successinresearchhasaPoissonarrival.Irreversibleinvestmentanduncertaintycreateanoptionvaluefordelayin

investment,butsinceonlythefirsttosucceedwinstheprize,preemptioncreatesanincentivetoinvestearly.Inherpaper,

thepositionofafirstmoverisendogenouslydetermined.Inourmodel,atemporaryleaderpositionarisesexogenously,the

advantage tomovingfirstresultsfromtheabilitytoselecthighreturnprojectsandtocommittonotpursuelowerreturn

ones.InWeeds’(2002)model,onlyasymmetricequilibriaexistforsomeparametervalues,whileinourmodelasymmetric

equilibriumalwaysexists,butasymmetricequilibriadonotnecessarilyexist.

Grenadier (2002) introducesamodelinwhicheach firmhasanoptiontoinvestinexpandingcapacityatanypointin

time.Marketpricedependsonastochasticdemandshockandonthetotalquantityfirmsproduce.Grenadier (2002)shows

thatcompetitionleadsfirmstoinvestsoonerasthefearofpreemptionerodestheoptionvalueofinvestment.Whatdrives

this isthat afirm’s profitsdecline withthe quantity produced by its competitors, creatingan incentive to invest before

others. In equilibrium, all firmsinvest at the sametime anddo so sooner than a monopolist would. In our model too,

the thresholdsforprojectselection arelower intheduopolygame thanwithamonopoly. Thisisbecausefirmscompete

for thesame projects.In Grenadier’smodel, afirm becomes worseoff when itsopponent invests, whilein ourmodel a

firm benefitswhenitsopponenthasinvestedandhisresourcesarecommitted.Also,inGrenadier’smodel,investinginthe

currentperioddoesnot precludeafirm frominvestingagaininthefuture.While inourmodel,whenafirm invests,itis

committedandwillbeunabletoselectfutureprojectsuntilthecommitmenttothecurrentprojectends.

Our model also relates to a surprisinglysmall number of strategic search models. Reinganum (1982, 1983a)

consid-ers a gamewhere sequential search fora technology isundertaken simultaneously. Once all firms havecompleted their

search, they compete inthe goods market. The dates at which thefirms stop searchingplay no role. Some authors

of-fered variations and extensions to Reinganum’s strategic search models (e.g., Lippman and Mamer, 1993; Taylor, 1995;

DaughetyandReinganum,2000; Hoppe,2000).Inthesemodels,theagentssearchindifferentpools.Theinteractioncomes

from the effect that the other agent’s search outcome has in a subsequent stage. Ourmodel differs from thisin that a

firm’sprojectselectionstrategyaffectsthedistributionofprojectsfacedbytheotherfirm.Additionally,ourfirmsalternate

betweenperiodsofsearch andperiodsofdevelopment.Thisisacommonsettinginjobsearchmodels,but,tothebestof

ourknowledge,ithasnotyetbeenanalyzedinastrategicsearchgame.4

Inthemanagementliterature,CassimanandUeda (2006) studythedecisionofanestablishedfirmthatgenerates

inno-vationstocommercializeeitherinternally,orinanexternalventure.Wecompareaversionofourmodelwiththeirpaper;

inthatversion,thereisadominantfirm(thatalwayshaspriorityinselection).Oneoftheirfindingsisthattheinnovations

commercialized by the established firm have “lower profitability than innovations commercialized through external

ven-tures.”Incontrast,ourdominantfirmselectshighervalueprojects.Thedifferenceinresultsaredrivenbymanydifferences

inourmodels,forexample,inCassimanandUedatheestablishedfirmhasmoreresourcesanddifferentcommercialization

capacitywhileinourmodelaprojectgeneratesthe samevaluewhetheritisselectedby thedominantfirmortheother

firm. Anotherimportantdifferenceisthat theyconsiderNashbargaining, whichgivestheestablishedfirman incentiveto

raisetotalsurplus,whilewetakeanon-cooperativegameapproach.AsCassimanandUedanote,relatedempiricalevidence

is limited. Gompers andLerner (2000)compare corporateventure capitalinvestments (which one mightinterpret as

in-vestmentbydominantfirms)toinvestmentsbytraditionalprivateventurecapital.Examiningempiricalevidence,theyfind

that “farfrombeingfailures,corporateventureinvestmentsinentrepreneurialfirmsappeartobeatleastassuccessful[...]

as thosebacked by independent ventureorganizations.” Thisseems to be more consistent withourprediction, although

additionaloralternativeexplanationsmayberesponsiblefortheobservationsintheirempiricalstudy.5

The rest ofthis paper proceeds asfollows. In Section 2,we describe the basic model with a single decision maker.

Section 3 develops the game andanalyzes the strategic interaction between two firms. In Section 4, we discuss some

extensions, includingadominantfirm, an oligopolywithn firms,andexternalities. Section 5concludes.Proofs ofall the

propositionsareprovidedinAppendix A.

2. Basicmodel

Webeginbydescribingthemodelinthecontextofadecisionmaker–asinglefirmthatrepeatedlydecideswhetherto

selectprojectsthatarisesequentially.ThiscaseservesasabenchmarkforcomparisonwiththegameinSection3.Consider

a discretetimeinfinitehorizonmodel.Thefirmmaximizesthediscountedsumofexpectedpayoffswithadiscountfactor

4 _{Othermodelsofinvestmentunderuncertaintyincludepatentraces(e.g.,}

_Loury,

_{1979; LeeandWilde,1980; Reinganum,1983b),searchintensityis} themainstrategicvariableinthesemodels.

Thomas

(2014)analyzesamodelofstrategicexperimentationwithanegativeexternality.Ourresultsarenot directlycomparableasthemodelsdiffersignificantly.

0

< δ <

1.Anewprojectopportunityariseseveryperiod.Ifthefirmisnotcurrentlycommitted,itdecideswhethertoselect

thecurrentproject.Aprojectrequiresacommitmentofresources,whichpreventsthefirmfromworkingonmorethanone

projectatatime.6 Bindingcommitmentstoaprojectcanariseduetoagreementswithclients,employees,orsuppliers,or

becausethefirmcannotsearchfornewopportunitieswhileworkingonthecurrentproject.Itisalsopossiblethatprojects

requireasunkcostthatmakesabandoningaproject,evenforabetterone,notworthwhile.

Theduration ofprojectsisuncertain.Ineachperiod,thecommitmentto aprojectwillendbythe nextperiodwitha

probability p

∈

[0

,

1].Forexample,foraserviceprovider,projectdurationcanbethetimeittakestocompletetheservice;

forafirm engagedinacquisitions, thetimeit takesto transferknowledgefromtheinnovatorto thefirm,todevelop the

product,andtocomeupwithamarketingstrategy.

Eachprojecthasarandomlydrawn returnv,whichistheexpecteddiscountedpresentvalueofnetbenefitsfromthe

projectatthetimeitisselected.Project returnsareidenticallyandindependentlydrawn froma knowndistributionwith

acumulativedistributionfunction F

(

v

)

onafinitesupport

[

v

,

v

]

,suchthat v

<

0,andv

>

0.Forall v, F isdifferentiable

withafinitedensity f

(

v

) >

0.Theexpectedvalueofreturns ispositive,

_vvv f

(

v

)

dv

>

0.The payoffinaperiodinwhich

noprojectisselectediszero.

We assumed the payoff v is obtainedimmediately when the project is selected. With our assumption of a binding

commitment,thisvaluecouldstandforanexpectedvalueofreceivingtheprizeattheendofthecommitmentperiod(e.g.,

afirm canreceive paymentfora serviceonlywhenitiscompleted).7 Or itcanbe thepresentvalueofflow profits(e.g.,

profitsfromlaunchinganewproductonceitsdevelopment iscomplete),andflowcoststhatareincurredfortheduration

oftheproject’scommitment.

2.1. Adecisionmaker’sprojectselection

Denoteby V0 thevaluefunction whenthedecisionmakerisnotcommitted, beforerealizationoftheproject’s return,

anddenoteby V1 thevaluefunctionforthecommitteddecisionmaker.Whenthedecisionmakeriscommitted,hecannot

selectanotherproject.Thus,

V1

=

δ

[pV0

+

(

1

−

p

)

V1]

.

(1)

Whenheisfree,hechoosestoselectornottoselectsoastomaximize:

max

⎧

⎪

⎨

⎪

⎩

v

+

δ

[pV0

+

(

1

−

p

)

V1

]

payoff if select

,

δ

V0

payoff if reject

⎫

⎪

⎬

⎪

⎭

.

NotethatthecontinuationvalueifthefirmselectsisequaltoV1.Thus,aprojectisselectedifv

≥

v0where

v0

=

(δ

V0

−

V1

) .

(2)

Thevalueinthestatewithoutcommitmentis:

V0

=

v0

v

δ

V0f

(

v

)

dv

+

v

v0

(

v

+

V1

)

f

(

v

)

dv

.

(3)

Ifprojectsdonotrequirecommitment

(

p

=

1

)

,theselectionthresholdisv0

=

0.Butwhenprojectsrequirecommitment

(

p

<

1

)

,somepositivereturnprojectsarerejected, v0

>

0.

Proposition1.(i)Thereexistsauniquesolutiontothesystem

(1)–(3)

,withathresholdvalueforprojectselectionv0

∈

(

0

,

v

)

.(ii) The thresholdv0 ishigherwhenthecommitmentrequiredforeachprojectisexpectedtolastlonger(lowerp);andwhenthedecision makerismorepatient(higher

δ

).(iii)Thethresholdv0isatleastashighforadistributionofreturnsthateitherfirstorderstochastically dominatesanotherorisamean-preservingspreadofanother.

By Proposition 1, the returnfrom a selectedproject is expectedto be higherin an industry in which firmstypically

committoprojects thattake alongtime tocomplete (low pin ourmodel).Inthepharmaceutical industry,forexample,

it takes about ten years to bring a drug to the market (see Nichols, 1994). The threshold for selection is higher for a

dominatingdistributionbecausethereis ahigherprobabilitythat abetter projectwillariseinthefollowing periods.The

thresholdforselection ishigherforthespreadbecausethefirm canenjoythehigherreturnprojectwhilerejectingthose

withlowerreturn.

6 _{Capacityconstraintscouldariseduetoalimitednumberofskilledscientistsandengineers,limitedphysicalspacetorunexperiments,etc.Capacity} constraintsratherthanbudgetconstraintsareusedalsointheliteratureonrationalinattention,see

Sims (2010)

.

2.2. Two-projectcapacityconstraint

We nowconsideradecisionmakerwhocanworkontwoprojectsatatime.Wecompareprojectselectionofthisless

constrained firm withthesingle projectcapacityfirm in Section 2.1. Weuse thetwo-projectcapacitymodel later, when

comparingaduopolywithajointventure.

We denotethevaluefunctionsofthedecisionmaker witha two-projectcapacitywith Wi,todistinguishfromthatof

the singleprojectcapacityvalues Vi.Theindexi

∈ {

0

,

1

,

2

}

in Wi refers tothenumberofprojectsto whichthedecision

makeriscommitted.Instate2,thedecisionmakercannotselectaproject.Theoptimalchoicesinstates0 and1 aregiven

bythresholdsofselection w0 andw1,respectively.Thevalueinstate2(whenthefirmiscommittedtotwoprojects)is:

W2

=

δ

p2W0

+

2p

(

1

−

p

)

W1

+

(

1

−

p

)

2W2

.

(4)

Instate1,thethresholdlevelsatisfies:

w1

=

δ (

pW0

+

(

1

−

p

)

W1

)

−

W2

.

(5)

Thevaluefunctioninstate1 is:

W1

=

v

w1

(

v

+

W2

)

f

(

v

)

dv

+

F

(

w1

) δ (

pW0

+

(

1

−

p

)

W1

) .

(6)

Instate0,thethresholdlevelsatisfies:

w0

=

δ (

1

−

p

) (

W0

−

W1

) .

(7)

Thevaluefunctioninstate0 is:

W0

=

v

w0

[v

+

δ (

pW0

+

(

1

−

p

)

W1

)

]f

(

v

)

dv

+

F

(

w0

) δ

W0

.

(8)

Equations (4)–(8) define thesolution tothe optimaldecisionofthe firm who canwork on atmosttwo projectsata

time. Ina modelwithout commitment (p

=

1), the selection thresholds would be w1

=

w0

=

0.When p

<

1, selecting

projects requirescommitment.Committingto aprojectismore costlytothe firm whenithasalreadycommittedtoone

project before,andits thresholdwouldbehigher. Moreover,comparing firmswithdifferentcapacityconstraints,we find

that the single projectdecisionmaker’s thresholdis higherthanboth thresholds ofselection ofthetwo-project capacity

firm. Itisevenhigherthanthethresholdthetwo-projectcapacityfirmuseswhenitisalreadycommittedtooneproject.

Intuitively,forthetwo-projectcapacityfirm,whenallitsresources arecommitted,theexpectedtime untilatleastoneof

thetwocommitmentsisrelievedisshorterthanforthecommittedsingleprojectcapacityfirm.

Proposition2.Supposep

<

1.Foradecisionmakerwhocanworkonatmosttwoprojects,theselectionthresholdishigherwhenone projectisunderwaythanwhenitisnotcommitted,andthesethresholdsarelowerthantheselectionthresholdofthesingleproject capacityfirm:v0

≥

w1

>

w0

>

0.Thefirstinequalityisstrictifp

>

0.

AnimplicationofProposition 2isthatafirmthathasamorestringentcapacityconstraintselectslessprojectsandthat

theprojectsitselectsareofhigheraveragereturn,comparedtowhatthelessconstrainedfirmdoes.Italsospendsahigher

fractionofitstimewaitingforaprojecttoselect.Theconstrainedfirmdoesnotselectsomeofthelowreturnprojectsthat

thelessconstrainedfirmwouldhaveselected.

3. Strategicprojectselection

3.1. Thegame

We considercompetitionbetweentwo firms A and B.Everyperiod,anewproject opportunitywithavalue v drawn

from thedistribution F

(

v

)

arises. Ifone firm is committedto an earlierproject andtheother isfree, thefree firm can

decidewhethertoselecttheproject.Ifbothfirmsarefree,oneischosenatrandom(withaprobability 1₂)tobethatperiod’s

leader- or tobe thefirstfirm tomakeadecisiontoselecttheprojectornot.Iftheperiod’sleaderselectstheproject,the

followercannottakeonaprojectinthatperiod.Iftheperiod’sleaderrejectstheproject(whichisanirreversibledecision),

thefollowercandecidewhethertoselectit.Thus,whenbothfirmsarefree,theyplaysequentially.8 Ifneitherfirmselected

theproject,itdisappears andanewprojectopportunityarisesinthe nextperiod.9 As weassumedinSection 2.1, afirm

canworkonatmostoneprojectatatimeandthecommitmentofresourcesendseachperiodwithaprobability p.

OursolutionconceptisaMarkovperfectequilibriumwhichrulesoutnon-crediblethreats,andrestrictstostrategiesthat

dependonlyon“payoff-relevant”history(seeMaskinandTirole,1988).10_{TheMarkovperfectequilibriumisastandard}

solu-tionconceptfordynamicgames.Astateinourgameischaracterizedby

(

i

,

j

)

whichdescribecommitment:i

=

0 andi

=

1

indicatewhetherthefocalfirm(thefirmcurrentlydeciding,orwhosevalueisbeingdefined)isfreeorcommitted

respec-tively,and j

=

0 orj

=

1 indicateiftheotherfirmisfreeorcommitted.Additionally,kindicatestheidentityofthefocalfirm

(

k

=

A orB

)

andl (whichisonlymeaningfulinstate

(

0

,

0

)

)indicateswhetherthefirm isaleader(l

=

1) orthefollower

(l

=

2).Forsimplicity,wewillreferto

(

i

,

j

)

asthestateomittingtheidentityofthefirmwhenthisisclearfromthecontext.

We use dynamic programmingto find an equilibrium. The value function (capturing the presentdiscounted value of

profitsbeforerealizationoftheperiod’sproject’sreturn) inanystate isdenotedby Vk_i,j.Wefocusoncharacterizing

sym-metricequilibria. Symmetricequilibriaaresimpler. Weshow inProposition 3that apure strategy symmetricequilibrium

exists.When asymmetric equilibriumexists, itseems tobe anaturalsolutionconcept fora symmetricgame.We briefly

discussasymmetric equilibriaatthe endofSection 3.2,andprovide moredetails in theonlineappendix. In thissection

wefocusonsymmetricequilibria,thus,V_iA_,j

=

V_iB_,j

=

Vi,j.Symmetricequilibriumstrategiesarecharacterizedbythresholds

v0,1

,

v1_0,0

,

v2_0,0

,where v0,1 isthethresholdforthenon-committedfirminastate

(

0

,

1

)

,v1_0,0 andv2_0,0 arethethresholds

fortheperiod’sleader(thefirstmover)andfollower.Wesaythatathresholdisinteriorifitisintherange

v

,

v

,sothat

thelowestvalueprojectisrejected,whilethehighestvalueprojectisaccepted.

3.2. Analysis

Wederivetheconditionsthatdefinetheequilibriumthresholdsandthevalues.Thevalueinstate

(

1

,

1

)

isgivenby

V1,1

=

δ

p2V0,0

+

p

(

1

−

p

)

V1,0

+

p

(

1

−

p

)

V0,1

+

(

1

−

p

)

2V1,1

.

(9)

Instate

(

0

,

1

)

,thefreefirmchoosestoselectortorejecttheprojectsoastomaximize:

max

⎧

⎪

⎪

⎨

⎪

⎪

⎩

v

+

δ

p2V0,0

+

p

(

1

−

p

)

V1,0

+

p

(

1

−

p

)

V0,1

+

(

1

−

p

)

2V1,1

select

, δ

pV0,0

+

(

1

−

p

)

V0,1

reject

⎫

⎪

⎪

⎬

⎪

⎪

⎭

.

Note that thecontinuation value whenthe firm selects a projectis equal to V1,1.Thus, an interiorthresholdreturn for

selectionsatisfies:

v0,1

=

δ

pV0,0

+

(

1

−

p

)

V0,1

−

V1,1

.

(10)

Usingthethresholdv0,1,thevaluefunctionsinstates

(

0

,

1

)

and

(

1

,

0

)

are:

V0,1

=

v

v0,1

v

+

V1,1

f

(

v

)

dv

+

F

v0,1

δ

pV0,0

+

(

1

−

p

)

V0,1

,

(11)

V1,0

=

v

v0,1

V1,1f

(

v

)

dv

+

F

v0,1

δ

pV0,0

+

(

1

−

p

)

V1,0

.

(12)

Instate

(

0

,

0

)

,oneofthefirmsischosenatrandomtobetheleaderwhocanconsidertheprojectfirst.Ifthisfirmdoes

notselecttheproject,thenthefollowerfacesthechoice:

max

⎧

⎪

⎨

⎪

⎩

v

+

δ

pV0,0

+

(

1

−

p

)

V1,0

select

, δ

V0,0

reject

⎫

⎪

⎬

⎪

⎭

.

(13)

Ifthethresholdisinterior,itsatisfies:

v2_0,0

=

δ (

1

−

p

)

V0,0

−

V1,0

.

(14)

9 _{In“reallife,”afirmmightbeabletoreconsideraprojectthatithadrejectedinanearlierperiod.Inthecontextofourmodel,wecouldthinkofthe} projectarrivingagainatalaterdate.Fortractability,however,weabstractfromtheeffectthatrejectedprojectsmighthaveonthedistributionofquality offutureprojects.

Returningtotheleader’sdecision,ifv

≥

v2_0,0,sothatthesecondfirmwillselectifithastheopportunitytodoso,thenthe

firstfirmfacesthechoice:

max

⎧

⎪

⎨

⎪

⎩

v

+

δ

pV0,0

+

(

1

−

p

)

V1,0

select

, δ

pV0,0

+

(

1

−

p

)

V0,1

reject⇒rival selects

⎫

⎪

⎬

⎪

⎭

.

(15)

Thefirmwouldselectthisprojectifv

≥

v where

v

=

δ (

1

−

p

)

V0,1

−

V1,0

.

(16)

Ifv

<

v2

0,0,sothatthefollowerwillnotselecttheprojecteveniftheleaderrejectedit,thentheleaderfacesessentiallythe

same choiceasthatofthefolloweraftertheleader rejecteda project.Thus,for v

<

v2

0,0,neitherfirm selects.Combining

theresults,theleaderselectsifv

≥

v1

0,0where

v1_0,0

=

max

v2_0,0

,

v

≥

v2_0,0

.

(17)

If

v

>

v2_0,0,thenthereisarangeofprojectreturns v₀2_,0

≤

v

<

v1_0,0 sothattheleader rejects,butthefollowerselects.The

equationthatdefinesthevaluefunctioninstate

(

0

,

0

)

isgivenby

V0,0

=

1 2

v

v2 0,0

v f

(

v

)

dv

+

F

v2_0,0

δ

V0,0

+

1

−

F

v2_0,0

δ

pV0,0

+

(

1

−

p

)

V0,1

+

V1,0 2

.

(18)

Proposition3.(i)Forthegamedescribedinthissection,thereexistsapurestrategyMarkovperfectequilibriumwithinterior thresh-olds

v0,1

,

v1_0,0

,

v2_0,0

∈

v

,

v

3.

(ii)Forlongdurationprojects

(

p

=

0

)

,thesymmetricequilibriumisunique.

Existenceofasymmetricpure strategyequilibriumfollowsfromBrouwer’sfixedpointtheorem.Givenourassumptions

on the parameters of the model, in any symmetric pure strategy equilibrium, all the thresholds must be interior. That

is, when a firm is free to select, it does not reject all projects nor doesit accept all projects. Uniqueness is not always

guaranteed, butwecan show thatthe equilibriumis uniqueatleastforcertain parametervaluesincludingthe case p

=

0, where a firm can only select one project (an assumption often madein literature on irreversible investment). When

p

=

0, the system of equilibriumequations becomes simpler asa committed firm expects no additional payoffs V1,1

=

V1,0

=

0, and the free firm in state

(

0

,

1

)

faces the same problemas that of a single decision maker, therefore v0,1

=

v

=

v0 whichwe haveshownto be unique.We are onlyleft withfinding onethreshold, v20,0, whichwe verifymust be

unique.

For p

=

0,theobservationswe madeaboveonthevaluesandthresholdswhenatleastonefirm iscommittedaretrue

evenifweallowforasymmetricequilibria.Thus,theonlypotentialsourceofasymmetryintheequilibriumstrategiescould

be inthethresholds v₀k,_,2₀.Withan additionalassumptiononthedistributionfunction

(

f

(

x

)

≥

0

)

,thereisnoasymmetric

equilibrium.11

3.3. Strategicrejectionofprojects

Whennocommitmentofresourcesisneeded

(

p

=

1

)

,theequilibriumthresholdsarev0,1

=

v1_0,0

=

v2_0,0

=

0.Inthiscase,

afollowerwillneverselectaprojectthattheleaderrejected.Forp

<

1,weshowthatv2_0,0

<

v1_0,0 sothatthereisarangeof

intermediatevalueprojects,v

∈

v2_0,0

,

v1_0,0

,thattheleaderrejectsandthefollowerselects.Thisresultmightseeminitially

surprisingasone mightexpectthe leaderto selectallthe“good enough”projects leavingonly projectsthat thefollower

would notwant either.Tounderstand whytheleader’s threshold ishigher,consider thetrade-off betweenselecting and

rejectinginstate

(

0

,

0

)

forthefollower,givenin(13)andfortheleadergivenin(15).Thepayofffromselectingtheproject

wouldbethesamefortheleaderandthefollower, v

+

δ

pV0,0

+

(

1

−

p

)

V1,0

.Forthefollower,rejectingwouldresultin

a continuationvalue

δ

V0,0 sincenofirmwouldselecta projectthisperiod.Butfortheleader,rejectinga projectthat the

followerwouldselectprovidesahighercontinuationvalue

δ

pV0,0

+

(

1

−

p

)

V0,1

sincewithprobability

(

1

−

p

)

theleader

11 _{Moredetailsaboutthederivationsofasymmetricequilibriumconditionsareavailableinanonlineappendix.Inanasymmetricequilibrium(whenone} exists),ifvB,0,02<v

A,2

wouldhaveabusyrivalinthefollowingperiod,guaranteeingthattheleadercouldselecttheprojectinthenextperiod,if

ahighvalueprojectshouldarise.

Proposition4.Forp

<

1,inasymmetricMarkovperfectequilibrium,thethresholdsforselectioninstate

(

0

,

0

)

satisfyv1

0,0

>

v20,0; intherangev1

0,0

>

v

≥

v20,0,theleaderrejectstheprojectbutthefollowerselectsit.Additionally,v10,0

≥

v0,1(withastrictinequality forp

∈

(

0

,

1

)

),sothatafirmrejectsmoreprojectswhenitsrivalisfree.

Next, we compare the equilibrium withselection patterns in two benchmarks. First, in Section 3.4, we compare the

behaviorofa firmwithaone-projectcapacityconstraintwiththat ofanequallyconstrainedmonopolist(asingleproject

decisionmaker).Then, inSection 3.5,we comparetheoutcome ofthenon-cooperative gamewiththatoftwo firmsthat

maximizejointpayoffs.

3.4. Competitionandprojectselection

Avast literature ineconomics debates therelation betweenmarketstructure andinnovation (see Gilbert, 2006, fora

survey).Weexaminetheeffectsofcompetitiononprojectselectioninthecontextofourmodel.Wecomparetheselection

strategiesintheduopolygame(Section3.1) withtheselectionbehaviorofthedecisionmaker(Section2.1).Weshowthat

the thresholdreturnfora projectto be selectedis lower whentwo firms compete thanwhen thereis a single decision

makerwhoisconstrainedtoselectoneproject.Thus, (assumingprojectopportunitiesariseatthesamerateregardlessof

marketstructure)moreprojectsareselectedinaduopolymarketthaninamonopolymarket.However,sincethethreshold

ofselectionishigherforthemonopolist,themonopolistwilltendtoworkonprojectsthathaveahigherexpectedreturn.

Proposition5.Forp

∈

(

0

,

1

)

,selectionthresholdsarelowerinduopolycompetitionthanforamonopolist:v1_0,0

<

v0.

ThecomparisoninProposition 5isbasedontheassumptionthattheprojectreturnisdrawnfromthesamedistribution

inthesingledecisionmaker’sproblemasinthegame.Ifthefirmenjoysahigherreturnfromanyprojectwhenitisalone

inthemarket,thedistributionofreturns inthe decisionmaker’s problemmaydominatethatintheduopolycase. Aswe

haveshowninProposition 1,thiswouldimplyanevenhigherthresholdofselection.SotheresultofProposition 4holds,

evenifthemonopolistearnsmorefromeachprojectcomparedwiththeduopolist.

3.5. Jointdecisionversusthenon-cooperativegame

Thetwo-projectcapacityversionweanalyzedinSection2.2allowsustocompareprojectselectiondecisionsofstrategic

non-cooperative firmswiththedecisionof ajointventure. We areinterested in twocomparisons. First,we comparethe

selectionthresholdswhenonefirmiscommittedandoneisfree,thatis,wecomparethethresholdofthejointventurethat

iscommittedtooneprojectbutcanstillselectanother

(

w1

)

,totheselectionthresholdofthenon-cooperativefirmthatis

notcommittedandhasarivalwhoiscommitted

v0,1

.Second,wecomparethethresholdofthejointventurewhenitis

freeofcommitments

(

w0

)

,withtheselectionthresholdabovewhichatleastoneofthecompetingfirmsselectstheproject

whentheyarebothfree

v2

0,0

.

Proposition6.For p

∈

(

0

,

1

)

,anon-cooperativelycompetingduopolisthashigherselectionthresholdscomparedwiththejointly optimaldecisionmaker,w1

<

v0,1andw0

<

v2_0,0.

Toprovethisproposition,wefirstarguethat thejointdecisionmakercanobtain atleastashighavalue instate0 as

thesumofvaluesofboth firmsinthegameinstate

(

0

,

0

)

,i.e., W0

≥

2V0,0.Thisistruebecausethejointdecisionmaker

canmimic theequilibriumselection strategiesinthegame.Wethen derivethegiveninequalitiesonthresholdsfromthe

systemofdynamicprogrammingequations.Proposition 6suggeststhatcompetitionbetweenfirmswithcapacityconstraints

onprojectselectionresultsinthesefirmssettingtoohighabarforselectionandthusrejectingtoomanyprojectscompared

withwhatwouldbeoptimalfortheminajointdecision.

Wecompareourresultsinthecasep

=

0 withthoseinWeeds (2002).InWeeds (2002),whenasymmetricequilibrium

arises(which holdsonly forsome rangeof parameters),then strategic interactionincreases the time to firstinvestment

compared with a joint venture. This result is similar to our finding in Proposition 6, which shows that the thresholds

arelowerinthejointoptimalstrategy. AsWeedsnotes,“thisisinstarkcontrasttotheusualpresumptionthat thefearof

preemptionspeedsupinvestment.”InWeeds’model,thethresholdforthesecondinvestmentinthejointoptimalstrategyis

higherthanthatinthenon-cooperativesymmetricequilibrium,whileinourmodel,thethresholdforthesecondinvestment

islower.The intuition fordelay infirst investmentin Weeds (2002)is that strategicfirmsfear settingoff apatentrace.

Inourmodel,the jointventurerejectslessprojects becauseofitsgreater flexibility inusingcapacity,andbecausewhen

committedtotwoprojects,thejointventureexpectstobefreedfromatleastone commitmentsooner thanacommitted

non-cooperative firm.In Weeds’model,sometimes onlyan asymmetricequilibrium existsandwhen thisisthecase, the

4. Extensions

Weconsideranumberofpossiblegeneralizationsofourmodel.

4.1. Oligopoly

Consider an oligopolywithn firms.Letusindicatestatesby

(

i

,

j

)

wherei

∈ {

0

,

1

}

indicatesifthefocal firmis freeor

busy,and j

∈ {

0

,

1

, ..,

n

−

1

}

isthetotalnumberofotherfirms(notincludingthefocalfirm)thatarecommittedatthetime

thefirmisdecidingonaproject.Weassumeonlylongdurationprojectsinthissub-section,i.e., p

=

0.12

Forcommittedfirmsinanystate

(

1

,

j

)

thevaluesare:

V1,j

=

0 forj

=

0

, . . . ,

n

−

1

.

In state

(

0

,

n

−

1

)

, when all other firmsare committed, the free firm faces the sameproblem asthe decision maker in

Section2.1.Thethresholdandvalueare:

v0,n−1

=

δ

V0,n−1andV0,n−1

=

v

v0,n−1

v f

(

v

)

dv

+

F

v0,n−1

δ

V0,n−1

.

Instate

(

0

,

j

)

, jfirmsarebusyandn

−

jfirmsarefree.Theprojectisconsideredsequentiallybythefreefirmsinarandom

order, until one selectsit or all reject it. If no other firm selected theproject earlier, the thresholdfor the last firm to

considertheproject,the

(

n

−

j

)

thfreefirm,is:

vn₀−_,jj

=

δ

V0,j.

We argue that all the other firms in the sequence of decision makers that consider the project earlier have the same

threshold,whichishigherthanthatofthelastfirm.Thefirstfirmtoconsidertheprojectrejectssomeprojectsthatthelast

firmselects.Ifv

<

v₀n−_,jj,thenthelastfirmwillrejecttheproject.Knowingthis,thesecondtolastmover,the

(

n

−

j

−

1

)

th

firm, also faces thechoice between v ifitselects and

δ

V0,j ifit rejects. Thus,the second to last firm willalsoreject. If

v

≥

vn_0,−_jj,thelastfree firmwillselecttheprojectifitwillbe rejectedby allother firms.The secondto lastfirm hasthe

choicemax

v

, δ

V0,j+1

.Thus,itsthresholdis:

vn₀−_,jj−1

=

max

δ

V0,j+1

,

vn₀−_,jj

=

max

δ

V0,j+1

, δ

V0,j

.

Goingto the

(

n

−

j

−

2

)

thmover, ifv

<

vn₀−_,jj, thisfirmknowsthat noone willaccept it,anditalso rejects.If v

≥

vn₀−_,jj,

ifthe

(

n

−

j

−

2

)

thmoverrejects,one oftheothertwo firmswill accepttheproject.The

(

n

−

j

−

2

)

thmoverthussolves

max

v

, δ

V0,j+1

whichis thesame astheproblemthat the

(

n

−

j

−

1

)

thfirm had, andthus theirthresholds are equal:

v₀n−_,jj−1

=

v₀n−_,jj−2.A similar argument works forevery remaining leading firm. Hence, when j firmsare committed, the

last mover’s threshold is v₀n_,−_jj

=

δ

V0,j, andfor all but the last mover, the thresholds are equal: v10,j

=

. . .

=

v

n−j−1 0,j

=

max

δ

V0,j+1

, δ

V0,j

≥

vn₀−_,jj

=

δ

V0,j.Thevalueinstate

(

0

,

j

)

for j

=

0

,

. . . ,

n

−

1 is:

V0,j

=

1 n

−

j

v

vn₀−_,jj

v f

(

v

)

dv

+

1

−

F

vn₀−_,jj 1

−

1 n

−

j

δ

V0,j+1

+

F

vn₀−_,jj

δ

V0,j.

Proposition7.Whenp

=

0,thereexistsauniquesymmetricequilibriumtotheoligopolygame.Inequilibrium,thevalueofafreefirm islargerthemorerivalfirmsarebusy,V0,j+1

>

V0,j.Selectionthresholdssatisfy:v1_0,j

=

. . .

=

vn₀−_,jj−1

>

v₀n−_,jj,forj

=

0

,

. . . ,

n

−

1.13

Thus, asintheduopolycase, alsointheoligopolymodel,thefirmsthat moveearly rejectsome projectsthat thelast

moveraccepts.However,asthenumberoffreefirmsn

−

japproachesinfinity,almostalltheprojectsareeitherselectedby

thefirst-movingfirmornotselectedatall.Theprooffollowsfromthederivationsabove,andusesinductiononthenumber

offirms.

4.2. Adominantfirm

Asymmetry betweenfirms could be anotherreason whyfirms sometimesreject projects that their rivals then select.

Obviously,iftheleadervaluesaprojectlessthanthefollower,thefollowermightselectaprojectthattheleader rejected.

Eveniftwofirmsagreeonthevalueofaproject,onefirmmightbemorelikelytoevaluateprojectsbeforetheotherwhen

botharefree.Forexample,onefirmmaybebetterknown,oreasiertoapproach,orprojectideasmaystartwithinthisfirm.

We consider herethe extreme casewhere one firm is always the firstto consider projects (and there are no payoff

externalities). The dominant firm, firm A, has no benefit from having a busy rival, asit is always first to choose. The

dominant firm maximizes payoffs by ignoring its opponent, and thus using the threshold that maximizes the decision

maker’spayoffs(derivedinSection2.1)wheneveritisfree.FirmB,thefollower,willbestrespondtothisstrategy.FirmB’s

valuesandthresholdsinabestresponseto theleader’sstrategy solvethesystem(9)–(18) thatwe derivedinSection 3.2

forthesymmetricgame,onlywithminormodifications:weusetheleader’sthresholdasv0 inequation(12)defining V1,0

(now V₁B_,0)andwereplace(18),thevalueinstate

(

0

,

0

)

,withthefollowing:

V₀B_,0

=

(

1

−

F

(

v0

)) δ

pV₀B_,0

+

(

1

−

p

)

V₀B_,1

+

v0

vB,_0,02

v

+

V₁B_,1

f

(

v

)

dv

+

F

v₀B_,,₀2

δ

V₀B_,0

.

(19)

Proposition8.Forp

<

1,inaMarkovperfectequilibrium,instate

(

0

,

0

)

,theleaderselectsprojectsofreturnshigherthanv0(the uniquesolutiontothesystem

(1)–(3)

inSection

2.1

)andrejectsprojectsoflowervalue.Thereexistsathresholdv₀B_,,₀2

<

v0sothatin state

(

0

,

0

)

,intherangev0

>

v

≥

vB0,,02,thedominantfirmrejectstheproject,butthefollowerselectsit.

Inthe symmetricmodel,we showedthatthe benefitfromhavinga busyrival provides anincentive forthe leaderto

reject projectsthat the followerwouldaccept. Inthe perfectdominance casedescribed in thissubsection,thedominant

firmdoesnotbenefitfromabusyrival,butthefollowerstillselectssomeprojectsthattheleaderrejectsbecausehefaces

an inferiorselectionofprojects.Moregenerally,iffirm A hasan advantageintheformofaprobability

β

∈

1 2

,

1

toget

priority,bothmotivescanplayaroleincreatingawedgebetweentheleader’sandthefollower’sthresholds.

4.3. Payoffexternalities

Sofarweassumedthatafirmthatselectsaprojectobtainsthepayoffv anditsopponentdoesnotobtainanimmediate

payoff.Wenowaddressthepossibilityofapayoffexternality.Weassumethatforaprojectwithreturnv totheselecting

firm, thereturn tothe other firm is

γ

v, where

γ

∈

(−

1

,

1

)

, i.e., theexternality issmaller inmagnitudethan the return

fortheselectingfirm. Negativeexternalities

(

γ

<

0

)

mayexist,forexample,ifthefirmscompete inthemarketplace and

theprojectgivestheselectingfirma costadvantage.Positiveexternalities

(

γ

>

0

)

mayexist,forexample,whenthereare

technologyspillovers.Payoffsv and

γ

v canalsorepresentStackelbergpayoffswiththeselectingfirmbecomingthemarket

leaderinanewproductmarket.TheanalysisofthemodelwithexternalitiesfollowsinthesamewayasthatinSection 3.

Thesystemofequilibriumequationsremainsthesameexceptforthefollowingchanges.Accountingforpayoffexternalities,

instate

(

1

,

0

)

thevaluefunctionwhichwasgivenin(12)becomes

V1,0

=

v

v0,1

γ

v

+

V1,1

f

(

v

)

dv

+

F

v0,1

δ

pV0,0

+

(

1

−

p

)

V1,0

.

(20)

Instate

(

0

,

0

)

,theleader’sthreshold

v in(16)foracceptingprojectsthatthefollowerwouldselectisgivenby

v

=

δ (

1

−

p

)

(

1

−

γ

)

V0,1

−

V1,0

,

andthevaluefunctioninstate

(

0

,

0

)

becomes

V0,0

=

1

+

γ

2 v

v2 0,0

v f

(

v

)

dv

+

δ

F

v2_0,0

V0,0

+

δ

1

−

F

v2_0,0 pV0,0

+

(

1

−

p

)

V0,1

+

V1,0 2

.

TheresultsofProposition 4continue tohold. Clearly,however,theequilibriumthresholdsdependontheexternality

γ

.

Ontheonehand,instate

(

0

,

0

)

,forprojectvaluesthatwouldbeselectedbythefollowerifrejectedbytheleader,alarger

γ

increasestheimmediatevaluefromrejecting,

γ

v.Thissuggestsapositiveeffecton v1_0,0.Ontheotherhand,alarger

γ

reducestheincentivetohavingabusyrivalandprovidesaforcethatreducesthresholdsthroughitseffectonfuturevalues.

v1_0,0

=

1 1

−

γ

immediate effect↑

δ (

1

−

p

)

V0,1

−

V1,0

future value effect↓

=

δ (

1

−

p

)

(

1

−

γ

)

(

1

−

γ

)

_vv

0,1v f

(

v

)

dv

1

−

F

v0,1

δ (

1

−

p

)

=

δ (

1

−

p

)

_vv

0,1v f

(

v

)

dv

1

−

F

v0,1

δ (

1

−

p

)

.

When p

=

0,thethresholdv0,1isconstantwithrespectto

γ

(sincetherivalwillneverbeabletoselectanotherproject),

thetwoeffectsonv1

0,0exactlycanceloutsothatv10,0remainsconstantaswell.Butthethresholdv20,0 declines.Whenp

>

0

butstillsmall,weshowthatallthresholdsdecline.

Proposition9.Thereexists0

<

p

≤

1sothattheequilibriumthresholds

v0,1

,

v20,0

,

v

arecontinuouslydifferentiableinp andin

γ

forall

(

p

,

γ

)

∈

R

= [

0

,

p

) ×

(

−

1

,

1

)

.InR,dv

2 0,0

dγ

<

0;also,ddγv

≤

0and dv0,1

dγ

≤

0withastrictinequalityforp

∈

0

,

p

.

Proposition 9impliesthat(for

(

p

,

γ

)

intherangeR)inthepresenceofpositiveexternalities,thethresholdsofselection

are lower (moreprojects areselected) thanwithoutexternalities. Intuitively,withpositiveexternalities, firmsbenefitless

fromhavingacommittedrival,andwouldbelessconcernedaboutcommittingtheirownresources.

Lastly,we commentonthepolarcases

|

γ

| =

1.If

γ

=

(−

1

)

,we havea zerosumgame,V0,0

=

V1,1

=

0, V0,1

= −

V1,0,

andallthethresholdsareequal: v1

0,0

=

v

=

v20,0

=

v0,1.If

γ

=

1,V0,1

=

V1,0.Instate

(

0

,

0

)

,wheneverthefollowerwould

accept a project

v

>

v2_0,0

,the leaderisindifferentbetweenacceptingandrejecting. Thereare many(payoffequivalent)

equilibria.Inoneofthem,theequalityv₀1_,0

=

v2_0,0 holds.

5. Concludingremarks

Projectsoftenrequirefirmstocommitlimitedresources, preventingthem fromselectingother projectswhilethey are

committed. Sincemorepromisingprojectscouldarise duringthetime afirmiscommittedtoaprojectitselectedearlier,

constrainedfirmsrejectsome profitableprojects.Inastrategicenvironment,projectselectionby onefirmcan changethe

profitability ofarival,aswell asthe rival’sopportunityto takeonprojects.In thesymmetricgame,weshow thatafirm

sometimes rejects aprojectthat will thenbe selectedby arival, asthiscanlessen futurecompetitiononprojects. Inan

asymmetricgame,thefollowermayacceptlowervalueprojectsthantheleaderalsobecauseitfacesaninferiordistribution

ofprojects.

We consider the relation between market structure and innovations, in an environment with perpetual selection of

projects.Weshowthataduopolisthaslowerselectionthresholdsthananidenticalfirmwhooperatesasamonopoly.Thus,

competitioninduces moreprojectstobeselected,buttheaveragequalityofprojectsselectedbythemonopolistishigher.

If howeverthetwo firmsare able tojointlymake selection decisions,then theselection thresholds ofthe jointdecision

makerarelowerthaninthenon-cooperativeequilibrium.Thejointdecisionmakercanbettermanagecapacity.

Inattemptingtomaintaintractability,wemadecertainsimplifyingassumptions.Inourmodel,ifthefirmiscommitted,

itcannotselectanotherprojectuntilthecommitmentends.Inreality,itislikelythatfirmscansometimesbereleasedfrom

a commitmentat some cost.In ourmodel,the costof abandoninga projectis highsothat the commitmentisbinding.

If thecost is not toohigh, then whena new projectofhighenough return arises, thefirm might findit worthwhile to

abandonanold project.We expecttheresultstobequalitativelysimilar,withperhapslowerselection thresholdsbecause

the commitmentis lessbinding.14 _{Anadditional interesting butchallenging directionforfuturework isallowing for the}

possibilitythat theduration ofaproject

(

p

)

iscorrelatedwiththesizeoftheprize

(

v

)

.Ifhighprizesareassociatedwith

long commitments,itmightnot bepossible tocharacterizestrategiesasthresholdsofselection. Formalanalyses ofthese

extensionsareleftforfuturework.

In analyzingstrategicinteractions,ourmodelassumessequentialdecisions.We arguethatinmanyeconomic

environ-ments this assumption is reasonable, e.g., clients likely approach service providers sequentially. However, there may be

some marketsinwhichfirmssimultaneouslydecideonprojectselection,orwheretheorderofsequentialmovesmightbe

endogenous. If, ineachperiod,firmssimultaneouslydecide onselection,andeachgets theprojectwithequalprobability

when they both attempt toselect, thereis a rangeof intermediate value project returnsfor whichfirms randomizethe

decisiontoselect.

Ouranalysisisfocusedonthestrategicbehaviorofthefirmsthatselectprojects.If,however,projectopportunitiesarise

when independent innovatorspropose them, they mightalsoact strategically soasto extract surplusfromthe selecting

firms. The game played in each period might take the form of an auction or a bargaining game. These extensions are

interestingdirectionsforfuturework.