A nold, Pa ick; Mölle , Ma c; Roux, Ca he ine
Wo king Pape
The las shall be i s : Inno a ion as a head- o-head ace
Discussion Pape s, No. 25-05
P o ided in Coope a ion wi h:
Depa men o Economics, Uni e si y o Be n
Sugges ed Ci a ion: A nold, Pa ick; Mölle , Ma c; Roux, Ca he ine (2025) : The las shall be i s :
Inno a ion as a head- o-head ace, Discussion Pape s, No. 25-05, Uni e si y o Be n, Depa men o
Economics, Be n
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Economics
The Las Shall Be Fi s : Inno a ion as a
Head- o-Head Race
Pa ick A nold, Ma c Mölle , Ca he ine Roux
25-05
Sep embe , 2025
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DISCUSSION PAPERS
The Las Shall Be Fi s : Inno a ion as a
Head- o-Head Race*
Pa ick A nold†Ma c M¨
olle ‡Ca he ine Roux§
Abs ac
Unce ain y abou he alue o a con es ed inno a ion induces leade s and lagga ds o
upda e hei expec a ions in opposi e di ec ions. We cha ac e ize si ua ions in which
i ms ha ha e ob ained an ini ial ad an age a e no he mos likely o achie e inal suc-
cess. In spi e o ampli ying a leade ’s ad an age, g ea e con es in ensi y acili a es his
e ec , challenging he iew ha lagga ds equi e suppo o emain compe i i e.
Keywo ds: Inno a ion con es s; lea ning; compe i i e balance; leap ogging.
JEL classi ica ion: C72, D82.
1. In oduc ion
While Neil A ms ong’s iconic wo ds—“Tha ’s one small s ep o man, one gian leap o
mankind”—a e e ched in o his o y, i is easily o e looked ha only a ew yea s ea lie , he
USSR had es ablished a signi ican lead in he Space Race. A e he success ul launch o
Vos ok, which made Yu i Gaga in he i s human o o bi ou plane , P esiden Kennedy
acknowledged ha “[. . . ] he So ie Union gained an impo an ad an age by secu ing hese
la ge boos e s” (Logsdon 2010). Howe e , ins ead o discou aging U.S. e o s, he Vos ok
*This wo k was suppo ed by he Swiss Na ional Science Founda ion [g an numbe 100018 207874].
†Uni e si y o Basel. E-mail: [email p o ec ed]
‡Uni e si y o Be n. E-mail: [email p o ec ed], co esponding au ho
§Uni e si y o Basel. E-mail: [email p o ec ed]
1
ma ked a u ning poin in Kennedy’s suppo o he space p og am. A e yea s o downplay-
ing i s impo ance, Kennedy made NASA he la ges ci ilian agency in his o y which helped
he U.S. secu e ul ima e ic o y in he ace o he Moon.1
Examples whe e an inno a o ’s success spa ks he R&D e o s o i s i als a e ubiqui-
ous, anging om he design o win ai c a -engines by Boeing and Douglas (Su on 1998)
and comme cial main ame compu e s by Reming on Rand and IBM (Khanna 1995), o he
mo e ecen de elopmen o ouchsc een phones by Apple and Samsung o con e sa ional
a i icial in elligence by OpenAI and Google. Le ne (1997) p o ides u he his o ical e i-
dence sugges ing ha , in echnology-in ensi e indus ies, ca ch-up beha io is he no m and
i ms delibe a ely inc ease hei R&D expendi u es when ailing behind. This obse a ion
con as s wi h he common conce n ha a lagga d’s incen i e o inno a e migh be mu ed by
a i al’s lead, e en i ha lead is small (ϵ-p eemp ion) and many s eps emain o be comple ed
(discou agemen e ec ). In his pape , we a gue ha i is an inno a o ’s abili y o obse e a
i al’s ou pu s (e.g, echnological ad ancemen s, pa en s) bu no inpu s (e.g., R&D spend-
ing, e o s), ha induces he in ense compe i ion o inno a ion we o en obse e.
Ou a gumen is based on he idea ha , in a ace o an inno a ion o unce ain alue,
leading se es as a signal o a la ge in es men om a leade mo i a ed by high expec a ions,
whe eas lagging indica es a small in es men by a lagga d, likely o be he consequence o a
mo e pessimis ic iew. Leade s and lagga ds hus upda e hei belie s in opposi e di ec ions,
and a lagga d can ha e a s onge incen i e o in es , despi e a leade ’s ini ial echnological
ad an age. We show ha , as a esul , inno a ion can become a head- o-head ace, in which
lagga ds a e equally o e en mo e likely han ini ial leade s o achie e inal success.
Ou main con ibu ion is o cha ac e ize he en i onmen s mos conduci e o making in-
no a ion a head- o-head ace. Such en i onmen s o e su icien ewa ds o in e media e
echnological ad ances, o example, in he o m o easily iled pa en s, and hey elimina e
noise om he de e mina ion o he inal winne . Pe haps su p isingly, and con as ing wi h
exis ing esul s (Wang 2010), noise is no sui able o a head- o-head ace, because al hough
noise makes a leade ’s echnological ad an age less decisi e, i also obs uc s he in o ma-
ional e ec ha induces a lagga d o ca ch-up. A policy implica ion is ha inno a ion con-
es s should ewa d con es an s o in e media e ad ances and o e clea -cu speci ica ion o
he desi ed p oduc .2
Ou indings challenge a long-held iew ha lagga ds equi e ex e nal suppo o main-
ain compe i i e balance in inno a i e indus ies (Fudenbe g e al. 1983; Ha is and Vicke s
1. A i s peak, NASA employed up o 400,000 people and i s spending inc eased o 4.4% o ede al budge
(Kan o and Whalley 2025).
2. An example mee ing hese equi emen is he XPRIZE Ca bon Remo al Compe i ion which o e ed
$1million o he 15 mos ad anced con es an s a e he i s yea and a hi y page compe i ion guideline
speci ying how o measu e each con es an ’s pe o mance wi h g ea p ecision.
2
1987; Lippman and McCa dle 1987). This iew is in line wi h he mo e gene al idea o “le -
eling he playing ield” in compe i i e en i onmen s and he associa ed call o a i ma i e
ac ion (M¨
olle 2012; Chowdhu y, Es e e-Gonz´
alez, and Mukhe jee 2023). We p o ide a
coun e ailing a gumen by showing ha inno a ion aces may be mo e likely o be won by
lagga ds han by leade s, e en wi hou ex e nal suppo , sugges ing he exis ence o a sel -
co ec ing mechanism. This mechanism complemen s leap ogging (Fudenbe g e al. 1983;
Aoki 1991), he oppo uni y o a lagga d o o e ake a leade h ough a su icien ly la ge
echnological jump, in secu ing compe i i e balance. Ra he han being s a egic, ou mech-
anism is in o ma ional. We he eby con ibu e o a mo e ecen li e a u e highligh ing he
ele ance o lea ning e ec s in dynamic inno a ion con es s (Choi 1991; Do aszelski 2003;
Halac, Ka ik, and Liu 2017; Awaya and K ishna 2021).3A dis inguishing ea u e o ou
se ing, c ucial o he insigh ha lea ning imp o es compe i i e balance, is ha con es ou -
comes induce leade s and lagga ds o upda e hei belie s in opposi e di ec ions. Al hough
his ea u e is p esen in M¨
olle and Beccu i (2025), a head- o-head ace canno a ise. In hei
bes -o -N-ba le con es , a lagga d migh in es mo e han a leade bu is ne e mo e likely
o win.
2. The model
Two homogeneous, isk-neu al playe s engage in a wo-s age inno a ion con es ha awa ds
a p ize V ≥0 o he winne o s age ∈ {1,2}. We le ν=V1
V2deno e he alue- a io o winning
he ini ial s age (being he i s in space) ela i e o winning he inal s age (being he i s
on he Moon). The p obabili y ha playe i∈ {A,B}wins s age is gi en by he gene alized
Tullock (1980) con es success unc ion:
pi, (ei, ,ej, )=e
i,
e
i, +e
j,
,(1)
whe e ei, deno es playe i’s R&D expendi u e, o e o , in s age and ∈[0,1] is a pa ame e
de e mining he sensi i i y o he con es ou come wi h espec o playe s’ e o s as opposed
o noise. We deno e he i s -s age winne and he i s -s age lose as he leade (L) and he
ollowe (F), espec i ely.
Assump ion 1. The leade wins s age 2 wi h p obabili y pL,2(1
αeL,2,eF,2), whe e 1
α>1de-
no es a echnological ad an age he leade ob ains by winning s age 1.
3. Lea ning e ec s also play a ole in a ela ed li e a u e on en y- and exi -dynamics in compe i i e indus-
ies (e.g., Awaya and K ishna 2021; Chen, Ishida, and Mukhe jee 2023; Ce emen and Ma ga ia 2024).
3
A key ea u e o ou model is ha playe s a e p i a ely in o med abou he unce ain
alue o inno a ion. Following M¨
olle and Beccu i (2025), we model his unce ain y by
pos ula ing he exis ence o a “good” s a e S=gand a “bad” s a e S=b, equally likely and
such ha V1,V2>0 i and only i S=g, and we make he ollowing assump ion:
Assump ion 2. Each playe i p i a ely obse es a signal si∈ {B,G}. Playe s’ signals a e
independen wi h P(si=G|S=g)=1and P(si=B|S=b)=σ∈(0,1).
No e ha signals a e in o ma i e and ha a bad signal is e en conclusi e, since si=B
can only be ecei ed i S=b. Toge he wi h he ac ha inno a ion has no alue in he
bad s a e, his assump ion g ea ly simpli ies he analysis because i implies ha upon he
obse a ion o si=B, playe iwill choose ei,1=ei,2=0.
In addi ion o hei p i a e signals, playe s obse e con es ou comes, i.e., who won and
who los each s age, bu canno obse e hei i al’s e o s. This assump ion is impo an as
i implies ha playe s will lea n abou hei i al’s signals only h ough hei obse a ion o
con es ou comes. All unce ain y is esol ed a he end o s age 2 when alues V1and V2
ma e ialize.4
Playe i’s payo is gi en by he di e ence be ween his p ize ea nings and his agg ega e
R&D expendi u e, ei,1+ei,2. Ou se ing cons i u es a dynamic Bayesian game, wi h playe s’
ypes gi en by hei signals. We hus use Pe ec Bayesian equilib ium as ou solu ion con-
cep . As playe s choose ze o e o condi ional on a bad signal, a symme ic equilib ium is
comple ely cha ac e ized by a iple (e∗
1,e∗
L,e∗
F) deno ing playe s’ i s -s age e o , leade ’s
e o , and ollowe ’s e o , all upon eceip o a good signal.
3. Lea ning
How do playe s lea n abou hei i al’s signal? Conside a playe who ecei ed a good
signal. Le βdeno e he playe ’s p io belie ha his i al’s signal is also good:
β≡P(sj=G|si=G)=
P(si=sj=G)
P(si=G)=1+(1 −σ)2
2−σ∈(0,1).(2)
A e obse ing he s age 1 ou come, he playe upda es his belie as ollows. A e losing,
he playe concludes ha his i al exe ed posi i e e o in s age 1 which only happens
a e he i al ecei ed a good signal. Hence, he ollowe upda es his belie upwa ds o
4. In an ex ension, we allow V1and V2 o be co ela ed and assume playe s lea n he alue o V1be o e
s age 2. Resul s emain quali a i ely unchanged unless p izes a e pe ec ly co ela ed.
4
β∗
F=1> β. In con as , a e winning s age 1, he leade upda es his belie downwa ds o
β∗
L=
1
2β
1
2β+1−β< β, (3)
because, in equilib ium, he would ha e de ea ed a i al wi h a good signal only wi h p ob-
abili y one hal . No e ha inno a ion is expec ed o ha e posi i e alue only when bo h
playe s’ signals a e good and ha his alue is
VG
≡E[V|si=sj=G]=V
1+(1 −σ)2.(4)
I hus ollows om β∗
F> β > β∗
L ha lea ning induces playe s’ expec a ions abou he alue
o inno a ion o mo e in opposi e di ec ions. In pa icula , lea ning induces he ollowe ’s
expec a ion abou he alue o inno a ion o be highe han he leade ’s: β∗
FVG
> β∗
LVG
.
4. Head- o-head aces
Wi h he aim o iden i y lea ning abou he unce ain alue o inno a ion as a sel -co ec ing
mechanism, his sec ion de i es he condi ions unde which inno a ion becomes a head- o-
head ace. We o malize he no ion o a head- o-head ace as ollows:
De ini ion 1. Inno a ion cons i u es a head- o-head ace i , condi ional on he alue o in-
no a ion being posi i e, he ollowe ’s p obabili y o winning he ace is a leas as high as
he leade ’s:
p∗
F≡(αe∗
F)
(e∗
L) +(αe∗
F) ≥(e∗
L)
(e∗
L) +(αe∗
F) ≡p∗
L.(5)
The ac ha we condi ion on he inno a ion ha ing posi i e alue cap u es he idea
ha a “winne ” o an inno a ion ace would no be obse ed as such i he had ob ained an
inno a ion wi h no alue.
To de i e he condi ions unde which (5) holds, we now cha ac e ize he equilib ium
(e∗
1,e∗
L,e∗
F). In s age 2 i has o hold ha
e∗
F∈a g max
eF≥0
(αeF)
(e∗
L) +(αeF) β∗
FVG
2−eFand e∗
L∈a g max
eL≥0
e
L
e
L+(αe∗
F) β∗
LVG
2−eL.(6)
The unique e o le els sa is ying hese condi ions a e
e∗
L=VG
2β∗
L (αβ∗
L)
(α +(β∗
L) )2and e∗
F=VG
2 (αβ∗
L)
(α +(β∗
L) )2,(7)
5
and hey imply ha
p∗
F
p∗
L
=α
(β∗
L) and p∗
F≥p∗
L⇔β∗
L≤α. (8)
This means ha o a head- o-head ace o exis , lea ning has o be su icien ly s ong ela i e
o he leade ’s echnological ad an age. Howe e , condi ion (8) is only necessa y bu no
su icien o a head- o-head ace o exis . Remembe om Sec ion 3 ha lea ning equi es
ha playe s condi ion hei s age 1 e o s on hei p i a e in o ma ion. Mo e speci ically, in
s age 1, a playe should exe posi i e e o i and only i he playe has obse ed a good
signal.
To de e mine he condi ion unde which e∗
1>0, le U∗
Land U∗
Fdeno e he playe s’
con inua ion alues o en e ing s age 2 as he leade o he ollowe , condi ional on ha ing
ecei ed a good signal. We ha e
U∗
L−U∗
F=(β∗
L)2 +1−α2 −(1 −β∗
L)(αβ∗
L) (1 − )
[α +(β∗
L) ]2VG
2,(9)
and he equilib ium e o in s age 1 has o sa is y
e∗
1∈a g max
e1≥0βe
1
e
1+(e∗
1) U∗
L−U∗
F+VG
1−e1=⇒e∗
1=
4βU∗
L−U∗
F+VG
1.(10)
No e ha , o he equilib ium pa h, he leade ’s con inua ion alue ULdepends on s age 1
e o s h ough he upda ed belie βL=e1β
e1β+(1−β)(e1+e∗
1), bu (10) can be ob ained easily by use
o he en elop heo em.5Also no e ha i p∗
F≥p∗
L, hen (8) implies U∗
L−U∗
F≤0. Hence,
o inno a ion o be a head- o-head ace, he p ize o winning s age 1 has o be su icien ly
la ge. Only when VG
1>U∗
F−U∗
L, playe s exe posi i e e o in s age 1 condi ional on ha ing
ecei ed a good signal, making he con es ’s s age 1 ou come in o ma i e. We a e now eady
o o mula e ou esul :
P oposi ion 1. The e exis a non-emp y in e al (σ, σ)⊂(0,1) o in o ma ion s uc u es o
which lea ning induces inno a ion o be a head- o-head ace i and only i
1
α≤√2and 1+1
ν
=V1+V2
V1
<4
(1 −α)(2 − )+1,(11)
ha is, when bo h he echnological ad an age o winning he ini ial s age and he ac ion
o inno a ion alue ese ed o he winne o he inal s age a e no oo la ge.
5. Equilib ium exis ence can be shown along he lines o M¨
olle and Beccu i (2025).
6
P oposi ion 1 cha ac e izes he condi ions unde which inno a ion becomes a head- o-
head ace. Lea ning e ec s ha e o be su icien ly s ong o o e come he leade ’s echno-
logical ad an age. This equi es he playe s’ in o ma ion o be nei he oo p ecise no oo
imp ecise, because o σ→0 o σ→1, he i al’s signal adds no addi ional in o ma ion
o a playe ’s own signal. Howe e , in an in e media e ange o σ, losing s age 1 induces he
ollowe o exe mo e e o han he leade in s age 2, e en ually o e coming he leade ’s
echnological ad an age and equalizing bo h playe s’ chances o claim inal ic o y. P opo-
si ion 1 shows ha his equi es playe s o be able o secu e a su icien ly la ge pa o he
alue o inno a ion up on . I sugges s ha en i onmen s, whe e inexpensi e and apidly
iled pa en s o in e media e echnological ad ances a e a ailable, a e conduci e o head- o-
head compe i ion.
How does noise in luence whe he inno a ion is head- o-head? Ou model sheds ligh on
his ques ion, because an inc ease in he pa ame e o he con es success unc ion makes
con es ou comes less noisy and mo e sensi i e o playe s’ e o s. In ui i ely, one migh ex-
pec ha inc easing il s he ace owa ds he leade by making his echnological ad an age
mo e ele an . Howe e , his in ui ion neglec s he ac ha an inc ease in also aises play-
e s’ incen i es o p o ide e o in he ini ial s age o he con es , which, as a gued abo e,
cons i u es a p e equisi e o lea ning. The ollowing esul shows ha his indi ec in o ma-
ional e ec is s ong enough o o e come he di ec echnological e ec , making a educ ion
in noise conduci e o head- o-head compe i ion.
Co olla y 1. Reducing he noisiness wi h which he inno a ion con es de e mines i s win-
ne s is sui able o inducing a head- o-head ace, i.e., an inc ease in elaxes (11).
In inno a ion con es s, in e media e and inal winne s a e de e mined in acco dance wi h
e alua ion c i e ia speci ied by he con es ’s o ganize . Co olla y 1 sugges s ha such c i e ia
should lea e li le inde e minacy o p omo e he head- o-head na u e o inno a ion.
5. Concluding ema k
Admi edly, he p esen model is s ylized, bu i enhances ou unde s anding o a basic lea n-
ing mechanism h ough which inno a ion can become a head- o-head ace. Ou main esul
highligh s ha , o p omo e a balanced ace, echnological ad ances should be su icien ly
c edi ed and a ibu ed wi h minimal e o . While he basic implica ions o pa en policy a e
s aigh o wa d, u u e esea ch should ex end he model o explo e how lea ning a ies wi h
he numbe o compe i o s and he leng h o he ime ho izon.
7
