Optimal sequential contests

Hinnosaa , Toomas
A icle
Op imal sequen ial con es s
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Hinnosaa , Toomas (2024) : Op imal sequen ial con es s, Theo e ical Economics,
ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol. 19, Iss. 1, pp. 207-244,
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Theo e ical Economics 19 (2024), 207–244 1555-7561/20240207
Op imal sequen ial con es s
Toomas Hinnosaa
School o Economics, Uni e si y o No ingham and CEPR
I s udy sequen ial con es s whe e he e o s o ea lie playe s may be disclosed
o la e playe s by na u e o by design. The model has many applica ions, in-
cluding en seeking, R&D, oligopoly, public goods p o ision, and agedy o he
commons. I show ha in o ma ion abou o he playe s’ e o s inc eases he o al
e o . Thus, he o al e o is maximized wi h ull anspa ency and minimized
wi h no anspa ency. I also show ha in addi ion o he i s -mo e ad an age,
he e is an ea lie -mo e ad an age. Finally, I de i e he limi s o la ge con es s
and discuss he limi o pe ec ly compe i i e ou comes unde di e en disclosu e
ules.
Keywo ds. Con es design, oligopoly, public goods, en -seeking, R&D.
JEL classi ica ion. C72, C73, D72, D74, D82.
1. In oduc ion
Many economic in e ac ions ha e con es -like s uc u es, wi h payo s ha inc ease in
playe s’ own e o s and dec ease in he o al e o . Examples include oligopolies, pub-
lic goods p o ision, agedy o he commons, en seeking, R&D, ad e ising, and spo s.
The li e a u e ypically assumes ha e o choices a e simul aneous. Simul aneous con-
es s ha e con enien p ope ies: he equilib ium is unique, is in pu e s a egies, and is
ela i ely easy o cha ac e ize.
In his pape , I s udy con es s whe e he e o choices a e no necessa ily simul a-
neous. In many eal-li e si ua ions, some playe s can obse e hei compe i o s’ e o s
and espond app op ia ely o hose choices. Howe e , ea lie mo e s can also an icipa e
hese subsequen esponses and, he e o e, in luence he beha io o la e mo e s. Each
Toomas Hinnosaa : [email p o ec ed]
I am g a e ul o S e ano Ba bie i, Fede ico Bo a, Je Ely, Alex F ankel, And ea Gallice, Dan Ga e , Dino
Ge a di, Ma i Hinnosaa , Johannes Hö ne , Ma in Jensen, Kai Kon ad, Dan Ko enock, Nenad Kos, Ignacio
Monzón, Pe e No man, Ma co O a iani, Mallesh Pai, Alessand o Pa an, Alex Possajenniko , Deb aj Ray,
Ch is ian Seel, Ma co Se ena, Ron Siegel, Andy Sk zypacz, Ma in Szydlowski, Jean Ti ole, and Ashe Wolin-
sky as well as semina pa icipan s a Bocconi Uni e si y, Collegio Ca lo Albe o, Uni e sidad Ca los III de
Mad id, Uni e si y o Be n, Rice Uni e si y, Uni e si y o No h Ca olina a Chapel Hill, Uni e si y o B is ol,
Uni e si y o Bonn, Humbold Uni e si y o Be lin, Canadian Economic Theo y Con e ence, Con e ence on
Economic Design, Con es s: Theo y and E idence Con e ence, SAET, PET, S ony B ook Game Theo y Fes-
i al, SITE a S an o d, EARIE, Midwes Economic Theo y Mee ing, Lancas e Game Theo y Con e ence,
Ma ke s wi h In o ma ional Asymme ies wo kshop (Tu in), IIOC, SEA, Lingnan Wo kshop (Guangzhou),
QuaGaTCo i ual con e ence, and China Mee ing o he Econome ic Socie y o hei commen s and sug-
ges ions. I would also like o hank he Kellogg School o Managemen a No hwes e n Uni e si y and he
Collegio Ca lo Albe o whe e some o his wo k was ca ied ou .
©2024 The Au ho . Licensed unde he C ea i e Commons A ibu ion-NonComme cial License 4.0.
A ailable a h ps://econ heo y.o g.h ps://doi.o g/10.3982/TE5536
208 Toomas Hinnosaa Theo e ical Economics 19 (2024)
addi ional pe iod in a sequen ial con es adds complexi y o he analysis, which migh
explain why p e ious s udies ha e ocused mainly on simul aneous models. I cha ac e -
ize equilib ia o a gene al class o sequen ial con es s and analyze how he in o ma ion
abou o he playe s’ e o s in luences he equilib ium beha io .
Con es s may be sequen ial by na u e o by design. Fo example, in en -seeking
con es s, i ms lobby he go e nmen o achie e ma ke powe . One ool ha egula-
o s can use o minimize such en -seeking is a disclosu e policy. A non anspa en dis-
closu e policy would lead o simul aneous e o choices, bu a ull anspa ency policy
would lead o a ully sequen ial con es . The e may be po en ially in e media e solu ions
as well, whe e he in o ma ion is e ealed only occasionally. O e he las ew decades,
many coun ies ha e in oduced new legisla ion egula ing anspa ency in lobbying ac-
i i y. This lis includes he Uni ed S a es (Lobbying Disclosu e Ac , 1995; Hones Lead-
e ship and Open Go e nmen Ac , 2007), he Eu opean Union (Eu opean T anspa ency
Ini ia i e, 2005), and Canada (Lobbying Ac , 2008). Howe e , he e a e signi ican c oss-
coun y di e ences in egula ions. Fo example, lobbying e o s in he US mus be e-
po ed qua e ly, whe eas in he EU, epo ing occu s annually and on a mo e olun a y
basis.
Ano he classic example o a con es is esea ch and de elopmen (R&D), whe e he
p obabili y o a scien i ic b eak h ough is p opo ional o agen s’ esea ch e o s. The
ques ion is how o bes o ganize he disclosu e ules o maximize agg ega e esea ch e -
o s. In some academic ields, i is common o p esen ea ly indings in wo king pape s
and con e ences. In o he ields, hese e o s a e kep con iden ial un il he wo k has
been e ed and published in a jou nal. Simila ly, when announcing an R&D con es ,
he o ganize can choose a anspa ency le el: whe he o use a public leade boa d o
pe haps keep he en ies sec e un il he deadline.
To add ess such ques ions, I s udy a model o sequen ial con es s. Fi s , I cha ac-
e ize all equilib ia o any gi en sequen ial con es , i.e., o any ixed disclosu e ule.
The s anda d backwa d-induc ion app oach equi es inding bes - esponse unc ions
e e y pe iod and subs i u ing hem ecu si ely. This solu ion me hod is no gene ally
ac able o e en easible. Ins ead, I use an al e na i e app oach, in which I cha ac e ize
bes - esponse unc ions by in e se unc ions. This me hod pools all he op imali y con-
di ions in o one necessa y condi ion and sol es he esul ing equa ion jus once. I p o e
ha o any con es he equilib ium exis s and is unique. Impo an ly, he cha ac e iza-
ion heo em shows how o compu e he equilib ium.
The main esul o he pape shows ha he in o ma ion abou o he playe s’ e o s
s ic ly inc eases he o al e o . Consequen ly, he op imal con es is always one o he
ex emes. When e o s a e desi able (as in R&D compe i ions), he op imal con es is
one wi h ull anspa ency. When he e o s a e undesi able (as in en -seeking), he
op imal con es is one wi h hidden e o s. The in ui ion behind his esul is simple.
While playe s’ e o s could be s a egic subs i u es o complemen s, I show ha e o s
a e s a egic subs i u es su icien ly close o he equilib ium. The e o e, ea lie -mo ing
playe s ha e an addi ional incen i e o exe e o o discou age la e playe s’ e o s. I
he discou agemen e ec we e s ong enough o educe he o al e o , his would o e
p o i able de ia ions o some playe s. The e o e, he discou agemen e ec is less han
Theo e ical Economics 19 (2024) Op imal sequen ial con es s 209
one- o-one. I inc eases ea lie -mo e s e o s mo e han i educes la e -mo e s e o s,
he e o e inc easing o al e o . While he e could be indi ec e ec s ha change he
conclusions, I show ha (again, nea he equilib ium) e o s a e highe -o de s a egic
subs i u es and, he e o e, he esul s ill holds.
The in o ma ion abou o he playe s’ e o s is impo an bo h quali a i ely and
quan i a i ely. Fo example, he sequen ial con es wi h 5 playe s ensu es a highe o-
al e o han he simul aneous con es wi h 24 playe s. The di e ences become e en
la ge wi h la ge con es s. Fo example, a con es wi h 14 sequen ial playe s achie es a
highe o al e o han a con es wi h 16,000 simul aneous playe s. The e o e, he in o -
ma ion abou o he playe s’ e o s is a leas as impo an as o he cha ac e is ics o he
model, such as he numbe o playe s.
I also gene alize he i s -mo e ad an age esul by Dixi (1987), who showed ha
a playe who p e-commi s chooses a g ea e le el o e o and ob ains a highe pay-
o han his ollowe s. This leade exploi s wo ad an ages: he mo es ea lie and has
no di ec compe i o s. Wi h he cha ac e iza ion esul , I can u he explo e his ques-
ion and compa e playe s’ payo s and e o le els in sequen ial con es s. I show ha
he e is a s ic ea lie -mo e ad an age—ea lie playe s choose g ea e e o s and ob-
ain highe payo s han la e playe s.
Finally, I p o ide insigh s o la ge con es s. I de i e an app oxima ion esul o
con es s wi h an in ini ely la ge numbe o playe s. This esul allows me o show ha
as he numbe o playe s becomes la ge, he o al e o con e ges o he p ize’s alue
(o pe ec ly compe i i e ou come mo e gene ally) ega dless o he con es s uc u e.
Howe e , he speed o con e gence o his le el is di e en unde di e en disclosu e
policies. In simul aneous con es s, he a e o con e gence is linea , whe eas in sequen-
ial con es s i is exponen ial.
These esul s pain a di e en pic u e o highly compe i i e s a egic in e ac ions.
In simul aneous con es s, a high deg ee o compe i i eness equi es a la ge numbe o
playe s, all choosing a minuscule e o le el. Con as ingly, in a sequen ial con es , he
same o al e o equi es a much smalle numbe o playe s, each exe ing di e en e -
o le els. The i s playe chooses a much highe e o han anyone else, he second
one much highe han he i s , and so on. By any de ini ion, his is a highly concen-
a ed ma ke . Howe e , he ea ly mo e s canno capi alize on hei posi ion, as la e
mo e s would eac by inc easing hei e o s. The e o e, despi e he di e en e o
le els, hei payo s a e s ill close o ze o. These esul s hus p o ide an al e na i e oun-
da ion o he con es abili y heo y (Baumol (1982)). Ins ead o in oducing a sepa a e
class o inac i e playe s— he compe i i e inge—in his model, he compe i i e inge
a ises endogenously h ough he o de o mo es.
Li e a u e: The simul aneous e sion o he model has been s udied ex ensi ely, s a -
ing om Cou no (1838). The li e a u e on Tullock con es s was ini ia ed by Tullock
(1967,1974) and mo i a ed by en -seeking (K uege (1974)).1The mos gene al ea -
men o simul aneous con es s is p o ided by he li e a u e on agg ega i e games (Sel en
1See Ni zan (1994), Kon ad (2009), and Vojno i´
c(2015) o li e a u e e iews on con es s.
210 Toomas Hinnosaa Theo e ical Economics 19 (2024)
(1970), Acemoglu and Jensen (2013), Jensen (2018)). My model is an agg ega i e game
only in he simul aneous case.
The only sequen ial con es ha has been s udied ex ensi ely is he i s -mo e con-
es . I was in oduced by on S ackelbe g (1934), who s udied quan i y leade ship in
an oligopoly. Dixi (1987) showed ha he e is a i s -mo e ad an age in con es s. Rel-
a i ely li le is known abou (Tullock) con es s wi h mo e han wo pe iods. The only
pape p io o his ha s udied sequen ial Tullock con es s wi h mo e han wo pe iods
is Glaze and Hassin (2000), which cha ac e ized he equilib ium in he sequen ial h ee-
playe Tullock con es . Kahana and Kluno e (2018) is an independen and concu en
wo k ha uses a simila app oach o cha ac e ize he equilib ium in an impo an spe-
cial case o my model: an n-playe ully sequen ial Tullock con es . In con as o my
pape , hey do no s udy any o he ques ions ha a e he main ocus o my pape , such
as he op imal con es s, ea lie -mo e ad an age, and la ge con es s. Mo eo e , as I a -
gue in Sec ion 8, he cha ac e iza ion alone is no su icien o answe hese ques ions.
The only class o con es s whe e equilib ia a e ully cha ac e ized o sequen ial con es s
a e oligopolies wi h linea demand.2
Mo e is known abou la ge con es s. Pe ec compe i ion (Ma shall equilib ium) is
a s anda d assump ion in economics, and i is a baseline wi h which o unde s and i s
ounda ions. No shek (1980) showed ha Cou no equilib ium exis s in la ge ma ke s
and con e ges o he Ma shall equilib ium. Robson (1990) p o ided u he ounda ions
o Ma shall equilib ium by p o ing an analogous esul o la ge sequen ial oligopolis-
ic ma ke s. In his pape , I ake an al e na i e app oach. Unde s onge assump ions
abou payo s, I p o ide a ull cha ac e iza ion o equilib ia wi h any numbe o playe s
and any disclosu e s uc u e, including simul aneous and sequen ial con es s as oppo-
si e ex emes. This allows me no only o show ha he la ge con es limi is he Ma shall
equilib ium bu also o s udy he a es o con e gence unde any con es s uc u e.
The pape also con ibu es o he con es design li e a u e. P e ious pape s on con-
es design include Taylo (1995), Che and Gale (2003), Moldo anu and Sela (2001,2006),
and Olszewski and Siegel (2016), which ha e ocused on con es s wi h p i a e in o ma-
ion. Halac, Ka ik, and Liu (2017) s udied con es design in he p esence o in o ma-
ional ex e nali ies when playe s lea n abou he easibili y o he p ojec . In his pape ,
I s udy con es design on a di e en dimension: how o op imally disclose o he playe s’
e o s, when playe s mo e sequen ially, o minimize o maximize o al e o .3
Simila connec ions be ween disclosu es and subsequen ac ions ha e been ound
in o he se ings. Fo example, Fe sh man and Ni zan (1991), Va ian (1994), and Wi l
(1996) used a model o dynamic olun a y public goods p o ision o show ha i con-
ibu ions a e adjus ed a e obse ing ea lie con ibu ions, his may inc ease he ee-
iding p oblem. Adma i and Pe y (1991)andBona i and Hö ne (2011) showed simila
2Daughe y (1990) used such a model o show ha an oligopoly whe e playe s a e di ided be ween wo
pe iods is mo e concen a ed bu also close o compe i i e equilib ium han an oligopoly whe e all playe s
mo e a once. Hinnosaa (2021) p o ides a li e a u e e iew and shows ha he linea oligopoly model has
unique p ope ies ha ail when he demand is no linea .
3Recen ly, Ely, Geo giadis, Kho asani, and Rayo (2022) s udied eedback design in a con inuous- ime
model whe e he designe wan s o p olong pa icipa ion o as long as possible and con es an s do no
obse e hei successes.

Theo e ical Economics 19 (2024) Op imal sequen ial con es s 211
e ec s in dynamic eam p oduc ion p oblems. While he d i ing o ces in hese pape s
a e simila o he discou agemen e ec s udied he ein, none o hese wo ks add essed
highe -o de e ec s and hei implica ions o esul ing equilib ia.4
The pape also helps o explain empi ical indings. Fo example, he e is widesp ead
empi ical e idence o ea lie -mo e ad an age in consume goods ma ke s. Acco ding
o a su ey by Kalyana am, Robinson, and U ban (1995), he e is a nega i e ela ionship
be ween a b and’s en y ime and he b and’s ma ke sha e in many ma u e ma ke s,
including pha maceu ical p oduc s, in es men banks, semiconduc o s, and d illing
igs. Fo example, B onnenbe g, Dha , and Dubé (2009) s udied b ands o ypical con-
sume packaged goods and ound a signi ican ea ly en y ad an age. The ad an age is
s ong enough o d i e he ank o de o ma ke sha es in mos ci ies. Lemus and Ma -
shall (2021) used obse a ional da a and a lab expe imen o s udy he impac o public
leade boa ds in p edic ion con es s. They ound ha public leade boa ds encou aged
some playe s and discou aged o he s, bu he o e all e ec was posi i e, imp o ing he
p edic ion con es ’s quali y.
The es o he pape un olds as ollows. Sec ion 2in oduces he model. Sec ion 3
uses a h ee-playe example o illus a e why he s anda d backwa d induc ion is no
ac able and shows how he in e ed bes - esponse app oach sol es he ac abili y
p oblem. Sec ion 4p o ides he cha ac e iza ion esul . Sec ion 5discusses he sec-
ond main esul , connec ing in o ma ion and o al e o , and discusses i s implica ions.
Sec ion 6s udies ea lie -mo e ad an age and Sec ion 7analyzes la ge con es s. Sec-
ion 8shows how he analysis applies o a b oade class o models. Finally, Sec ion 9
concludes. All p oo s a e in Appendix A.
2. Model
The e a e niden ical playe s N={1, ,n}who a i e o he con es sequen ially and
make e o choices on a i al. A T−1 poin s in ime, he sum o e o s by p e ious
playe s is publicly disclosed. These disclosu es pa i ion playe s in o Tg oups, deno ed
by I=(I1,,IT). In pa icula , all playe s in I1a i e be o e he i s disclosu e and,
he e o e, ha e no in o ma ion abou o he playe s’ e o s. All playe s in I a i e be-
ween disclosu es −1and and, he e o e, ha e exac ly he same in o ma ion: hey
obse e he o al e o o playe s a i ing p io o disclosu e −1.5I e e o he ime
in e al in which playe s in he g oup I a i ed as pe iod . As all playe s a e iden ical,
he disclosu e ule o he con es is ully desc ibed by he ec o n=(n1,,nT),whe e
n =|I |is he numbe o playe s a i ing in pe iod .6
4Mo e b oadly, he e is a connec ion wi h he sequen ial in o ma ion design li e a u e. Fo example,
Do al and Ely (2020) and Mak is and Renou (2023) p o ide cha ac e iza ion esul s in sequen ial models
whe e in o ma ion design may in ol e signals abou playe s’ ac ions in addi ion o unknown ypes o s a es.
Li and No man (2021) s udy a sequen ial pe suasion model and ind ha playe s gene ally wan o mo e
only once.
5As he payo s depend on he o al e o o o he playe s and no hei indi idual e o s, obse ing he
sumo p e iousplaye s’e o sisequi alen oobse ing hei indi iduale o s.
6Equi alen ly, he model can be s a ed as ollows: nplaye s a e di ided ac oss T ime pe iods, ei he
exogenously o by he con es designe .
212 Toomas Hinnosaa Theo e ical Economics 19 (2024)
Figu e 1. A con es wi h 7 playe s and 3 disclosu es. Playe s 1 o 3 choose e o s x1,x2,andx3
independen ly; playe 4 obse es X1=x1+x2+x3,playe 5obse esX2=X1+x4,andplaye s
6and7obse eX3=X2+x5.
Each playe ichooses an indi idual e o xi≥0 a he ime o a i al. I deno e he
p o ile o e o choices by x=(x1,,xn), he o al e o in he con es by X=n
i=1xi,
and he cumula i e e o up (and including) o pe iod by X =
s=1i∈Isxi.Bycon-
s uc ion, he cumula i e e o be o e he con es is X0=0, and he cumula i e e o
a e pe iod Tis he o al e o exe ed du ing he con es , i.e., XT=X. Figu e 1illus-
a es he no a ion wi h an example o he ou -pe iod con es n=(3, 1, 1, 2):
Playe s compe e o a p ize o size one, he p obabili y o winning is p opo ional o
he le el o e o , and he ma ginal cos o e o is one. I he e o e assume he no mal-
ized Tullock payo s,wi h
ui(x)=xi
X−xi.(1)
I s udy pu e-s a egy subgame-pe ec equilib ia, a na u al equilib ium concep in
his se ing: he e is no p i a e in o ma ion, and ea lie a i als can be in e p e ed as
ha ing g ea e commi men powe . I show ha he e always exis s a unique equilib-
ium. Th oughou he pape , I main ain a ew assump ions ha simpli y he analysis.
Fi s , he e is no p i a e in o ma ion. Second, he a i al imes and he disclosu e ules
a e ixed and common knowledge. Thi d, each playe makes an e o choice jus once
upon a i al. Fou h, disclosu es make cumula i e e o s public.7In Sec ions 8and 9,
I discuss he ex en o which he esul s ely on each o hese assump ions and explain
how he esul s ex end o mo e gene al sequen ial games.
3. Example
The s anda d Tullock con es has niden ical playe s who make hei choices in isola ion.
Each playe ichooses e o xi o maximize payo (1). The op imal e o s ha e o sa is y
he i s -o de condi ion
1
X−xi
X2−1=0, (2)
7Speci ically, each playe obse es he sum o ea lie -mo e s’ e o s wi h ce ain y and uncondi ionally.
Mo e complex disclosu e ules would change he conclusions. Fo example, p obabilis ic disclosu es may
limi he ea lie -mo e s commi men powe (Bagwell (1995)) and condi ional disclosu es may subs an ially
expand he se o possible ou comes (Bizzo o, Hinnosaa , and Vigie (2023)).
Theo e ical Economics 19 (2024) Op imal sequen ial con es s 213
whe e X2is he o al e o squa ed. Combining he op imali y condi ions leads o a
o al equilib ium e o X∗=(n−1)/n and indi idual e o s x∗
i=(n−1)/n2.Theequi-
lib ium is unique, easy o compu e, and easy o gene alize in a ious di ec ions, which
may explain he widesp ead use o his model in a ious b anches o economics.
3.1 The p oblem wi h s anda d backwa d induc ion
Conside nex a h ee-playe e sion o he same con es , bu he playe s a i e sequen-
ially and hei e o s a e ins an ly publicly disclosed. Tha is, playe s 1, 2, and 3 make
hei choices x1,x2,andx3a e obse ing he e o s o p e ious playe s. I will i s y
o ind equilib ia using he s anda d backwa d-induc ion app oach.
Playe 3 obse es he o al e o o he p e ious wo playe s, X2=x1+x2<1and
maximizes he payo . The op imali y condi ion o playe 3 is
1
X2+x3−x3
(X2+x3)2−1=0. (3)
Sol ing i o x3gi es he bes - esponse unc ion x∗
3(X2)=√X2−X2.8Now, playe 2
obse es x1<1 and knows x∗
3(X2)and, he e o e, sol es he maximiza ion p oblem
max
x2≥0
x2
x1+x2+x∗
3(x1+x2)−x2=max
x2≥0
x2
√x1+x2−x2.
The op imali y condi ion o playe 2 is
1
√x1+x2−x2
2(x1+x2)3
2−1=0.
Fo each x1∈[0, 1), his equa ion de ines a unique bes - esponse,
x∗
2(x1)=1
12 −x1+827x3
1(27x1+1)+216x2
1+36x1+12
3+24x1+1
12827x3
1(27x1+1)+216x2
1+36x1+11
3
.(4)
Finally, playe 1’s p oblem is
max
x1≥0
x1
x1+x∗
2(x1)+x∗
3x1+x∗
2(x1)−x1,
whe e x∗
2(x1)and x∗
3(X2)a e de ined by equa ions (3)and(4). Al hough he p oblem is
no complex, i is no ac able. Mo eo e , he di ec app oach is no gene alizable o an
a bi a y numbe o playe s. In ac , he bes esponse unc ion does no ha e an explici
ep esen a ion o con es s wi h a la ge numbe o pe iods.
8In his example, I ocus only on in e io solu ions. I is s aigh o wa d o e i y ha co ne solu ions
canno occu in equilib ium, as hey equi e ha a leas one playe chooses an e o le el gi ing inducing
a nonposi i e payo , and he e is always a de ia ion wi h a s ic ly posi i e payo .
214 Toomas Hinnosaa Theo e ical Economics 19 (2024)
3.2 In e ed bes - esponse app oach
In his pape , I use a di e en app oach. Ins ead o cha ac e izing indi idual ( educed)
bes - esponses x∗
i(X −1), o he o al e o s induced by X −1, i.e., X∗(X −1), I cha ac-
e ize he in e se o X∗(X −1). Fo any le el o o al e o X, he in e ed bes - esponse
unc ion −1(X)speci ies he cumula i e e o X −1up o pe iod −1(i.e.,be o e he
mo e o playe s in pe iod ), ha is consis en wi h o al e o being X, gi en ha he
playe s in pe iods ,,Tbeha e op imally.
To see how he cha ac e iza ion wo ks, conside he h ee-playe sequen ial con es
again. In he las pe iod, playe 3 obse es X2and chooses x3. Equi alen ly, we can
hink o his p oblem as choosing he o al e o X≥X2by se ing x3=X−X2, i.e.,
max
X≥X2
X−X2
X−(X−X2).
Di e en ia ing he objec i e wi h espec o Xgi es us he op imali y condi ion
1
X−X−X2
X2−1=X2
X2−1=0,
which implies X2=X2. Tha is, i he o al e o in he con es is X, henbe o e
playe 3’s ac ion, he cumula i e e o had o be 2(X)=X2; o he wise, playe 3 would
no be beha ing op imally.
We can now hink o playe 2’s p oblem as choosing X≥X1=x1, which he can in-
duce by making su e ha he cumula i e e o up o his mo e is X2= 2(X), se ing
x2= 2(X)−X1. The e o e, his maximiza ion p oblem can be w i en as
max
X≥X1
2(X)−X1
X− 2(X)−X1.
Again, di e en ia ing wi h espec o X, we ge he op imali y condi ion

2(X)
X− 2(X)−X1
X2− 
2(X)=0. (5)
This is he key equa ion ha shows he ad an age o he in e ed bes - esponse ap-
p oach. Equa ion (5)isnonlinea inXand, he e o e, in x2, which causes he di icul y
o he s anda d backwa d-induc ion app oach. Sol ing his equa ion e e y pe iod o
he bes - esponse unc ion leads o complex exp essions, and he complexi y inc eases
wi h each s ep o he ecu sion. Howe e , (5) is linea in X1, making i easy o de i e he
in e ed bes - esponse unc ion
1(X)=X1= 2(X)− 
2(X)X(1−X)=X2(2X−1).
The condi ion X1= 1(X)agg ega es he wo necessa y condi ions o equilib ium in o
one, by cap u ing he bes esponses o playe s 2 and 3. I simply s a es ha i he o al
e o a he end o he con es is X, hen he cumula i e e o X1had o be 1(X)a e
playe 1. O he wise, ei he playe 2 o playe 3 is no beha ing op imally.
Theo e ical Economics 19 (2024) Op imal sequen ial con es s 221
The equilib ium payo o a playe iis in ui(x∗)=x∗
i(1/X∗−1), and since X∗is he
same o all he playe s, payo s a e p opo ional o e o s. The e o e, i su ices o show
ha he e o s o ea lie playe s a e s ic ly highe . Using Theo em 1and equa ion (9),
I can exp ess he di e ence be ween he equilib ium e o s o playe s iand j om con-
secu i e pe iods and +1as
x∗
i−x∗
j=
T−

k=1Skn −Skn +1gk+1X∗(11)
whe e n +1=(n +2,,nT)is he subcon es s a ing a e pe iod +1andn =
(n +1,n +1)is he subcon es s a ing a e pe iod . Clea ly, Sk(n )>S
k(n +1) o all k;
i.e., he e is mo e in o ma ion on all le els in a s ic ly longe con es . As gk+1(X∗)>0,
o each k he whole sum is s ic ly posi i e. The in ui ion o he esul is s aigh o -
wa d: playe s in ea lie pe iods a e obse ed by s ic ly mo e ollowe s han he playe s
om he la e pe iods. The e o e, in addi ion o he incen i es ha la e playe s ha e,
he ea lie playe s ha e an addi ional incen i e o exe mo e e o o discou age la e
playe s.
7. La ge con es s
Nume ic compa ison o simul aneous and sequen ial con es s highligh s ha he in o -
ma ion abou o he playe s’ e o s is a leas as impo an in de e mining he o al e o
as o he pa ame e s, such as he numbe o playe s. Fo example, he o al e o in he
simul aneous con es wi h 10 playe s is 0.9, whe eas he o al e o wi h ou sequen ial
playe s is 0.9082. A i h sequen ial playe inc eases he o al e o o 0.9587. A simul-
aneous con es wi h he same o al e o equi es 24 playe s. Figu e 2shows ha he
compa ison becomes e en mo e a o able o sequen ial con es s wi h la ge n.
The ollowing p oposi ion gi es he eason o his connec ion. As he numbe o
playe s becomes la ge, he o al e o con e ges o 1 no ma e he con es s uc u e,
bu he con e gence is di e en depending on he s uc u e. Fo la ge simul aneous
con es s, he con e gence is linea , wi h 1 −X∗≈1/n, while o la ge sequen ial con-
es s, he con e gence is exponen ial, wi h 1 −X∗≈1/2n.13 I is also wo h no ing ha ,
al hough he indi idual payo s con e ge o ze o, he indi idual e o s may no .
P oposi ion 2 (La ge Con es s). Fix T∈Nand a sequence o con es s (nn)∞
n=3,such ha
con es nnis n-playe con es wi h a mos Tpe iods. Le Xn=X∗(S(nn)) and o each
playe i, le xn
i he equilib ium e o in con es nn. Fo all ≤Tand all i∈In
,
lim
n→∞
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1−Xn−1
T

=11+nn

⎤
⎥
⎥
⎥
⎥
⎥
⎦=0and lim
n→∞
⎡
⎢
⎢
⎢
⎢
⎢
⎣
xn
i−1

s=11+nn
s
⎤
⎥
⎥
⎥
⎥
⎥
⎦=0. (12)
13P oposi ion 2is s a ed o a bi a y ixed T. The e o e, i is s aigh o wa d o apply i o he limi o
con es s whe e Ti sel becomes in ini ely la ge (e.g., la ge ully sequen ial con es s) by aking ano he limi
wi h espec o T.

222 Toomas Hinnosaa Theo e ical Economics 19 (2024)
Figu e 2. Numbe o playe s in a sequen ial con es ha leads o he same o al e o as a si-
mul aneous con es wi h nplaye s.
These esul s shed new ligh on he meaning o a highly compe i i e con es o ma -
ke . In a la ge simul aneous con es , each con es an chooses a minuscule e o le el.
Such a ma ke is clea ly no concen a ed. Fo example, wi h n=16,000 he s anda d
measu e o concen a ion, he He indahl–Hi schman Index is HHIsim ≈0.
In con as , a sequen ial con es equi es only a limi ed numbe o playe s o achie e
he same agg ega e esul s, and playe s beha e di e en ly. In a la ge sequen ial con-
es , he indi idual equilib ium e o s a e x∗≈(1/2, 1/4, ,1/2n). The ea lie mo e s
choose much la ge e o s and achie e la ge payo s han he ollowe s. Fo example,
wi h n=14 sequen ial playe s, he co esponding concen a ion index HHIseq ≈1/3,
which is ypically in e p e ed as a highly concen a ed ma ke . Howe e , in e ms o
ou comes, his ma ke is highly compe i i e: as o al e o is close o one, we ha e ull
dissipa ion o en s, and hus all playe s ea n equilib ium p o i s ha a e close o ze o.
This e ec is simila o con es abili y heo y (Baumol, Panza , and Willig (1988)),
whe e a small numbe o i ms canno capi alize on hei ma ke powe due o he p es-
ence o a compe i i e inge—a la ge numbe o po en ial compe i o s, who could ic-
ionlessly en e when a p o i oppo uni y a ises. In my model, he la e mo e s a e en-
dogenously aking he ole o he compe i i e inge. In equilib ium, hey decide o pu
in e y li le e o . Howe e , i he ea lie mo e s we e o y and exploi hei posi ion
by educing hei e o s, he la e mo e s would be he e o espond.
8. Gene aliza ion
In his sec ion, I discuss how o implemen he me hodology o a gene al class models.
I also p o ide su icien condi ions unde which he esul s abo e emain unchanged.
Theo e ical Economics 19 (2024) Op imal sequen ial con es s 223
Speci ically, I de ine a class o linea ly mul iplica i e payo unc ions and show ha i
i sa is ies P ope y 1,Theo em1 emains alid wi hou any modi ica ions. By adding
ano he su icien condi ion, P ope y 2, nea ly all o he esul s in he pape hold as
well. The di e ences be ween P ope y 1and P ope y 2also sugges ha Theo em 2
and mos o he esul s in he pape a e no di ec implica ions o Theo em 1.
Suppose ha each playe chooses an ac ion xi om a se Xiand i he p o ile o
ac ions is x=(x1,,xn), hen playe ige s a payo
Ui(x)=ui(xi,X). (13)
Take a playe i om he las pe iod T.Playe iobse es cumula i e e o XT−1be o e
pe iod Tand knows ha o he playe s in pe iod Ta e choosing e o s simul aneously
o him. The e o e, he sol es he maximiza ion p oblem
max
xi∈Xi
uixi,xi+XT−1+
j∈IT {i}
xj.
The s anda d bes - esponse unc ion would be x∗
i(XT−1).14 Bu suppose we can exp ess
he op imal e o xichoice as a unc ion o o al e o , φi(X). Then adding up indi id-
ual e o s in pe iod Tconsis en wi h o al e o Xgi es us a necessa y condi ion o
equilib ium,
XT−1=X−
i∈IT
φi(X).
I deno e he unc ion on he igh -hand side by T−1(X). I s in e se unc ion (assuming
i exis s), −1
T−1(XT−1)is he o al e o induced by cumula i e e o XT−1, i all playe s
in pe iod Tbeha e op imally.15
Suppose by induc ion ha he same a gumen holds s a ing om pe iod , i.e., i
cumula i e e o a e is X hen he o al e o induced is −1
(X ). Then playe iin
pe iod sol es he ollowing p oblem:
max
xi∈Xi
uixi, −1
(X ).
I again, we can exp ess he op imal xionly as a unc ion φi(X), hen adding up he
condi ions would gi e us a necessa y condi ion o equilib ium
X −1=X −
i∈I
xi= (X)−
i∈I
φi(X),
which I deno e by −1(X). Finally, in he beginning o he game cumula i e o al ac ion
is X0=0, which gi es us an equilib ium condi ion o he whole game.
14This unc ion is also called he educed bes - esponse unc ion as i only depends on he sum.
15When T=1, he game becomes a linea ly agg ega i e game, as in oduced in Sel en (1970), wi h a
known equilib ium condi ion X=n
i=1φi(X),whe en
i=1φi(X)is he agg ega e backwa d co espon-
dence. See Jensen (2018) o a li e a u e e iew. I T>1, he game is no agg ega i e, so he analysis p e-
sen ed he e is a dynamic gene aliza ion o linea ly agg ega i e games.
224 Toomas Hinnosaa Theo e ical Economics 19 (2024)
The e a e some gaps in his analysis ha need o be illed. I ha e al eady shown
ha wi h he Tullock con es payo s, ui(xi,X)=xi/X −xiand Xi=R+, his app oach
cha ac e izes he unique equilib ium. I is equally clea ha he app oach is no alid o
all payo unc ions, as in e io op imums may no exis o be unique. Nex , I in oduce a
mo e es ic ed class o payo unc ions and su icien condi ions whe e all esul s hold
and he analysis emains ac able.
Linea ly mul iplica i e payo s: Assume ha he payo unc ions a e iden ical and he
u ili y is linea ly mul iplica i e wi h espec o playe s’ own ac ions,
ui(xi,X)=xih(X),xi∈Xi=R+. (14)
Fo Tullock con es payo s, h(X)= /X −c,whe e ep esen s he p ize alue and cde-
no es he ma ginal cos o e o . This class o games also includes oligopolies wi h linea
cos s, whe e h(X)=P(X)−c,wi hxias he i m’s own quan i y, Xas he o al quan i y,
P(X)as he in e se demand unc ion, and cas he ma ginal cos . Addi ionally, his class
includes public goods games, in which xideno es p i a e consump ion and h(X) ep-
esen s he ma ginal bene i o p i a e consump ion, which dec eases wi h public good
con ibu ions and, he e o e, wi h o al p i a e consump ion.
I is na u al o assume in hese applica ions ha h(X)is s ic ly dec easing up o
some uppe bound X, a which i akes alue h(X)=0andabo ewhichh(X)≤0.
The e o e, e ec i ely he ac ion space is Xi=[0, X]. Wi hou loss o gene ali y, we can
change he scale o ac ions so ha X=1.
The i s -o de op imali y condi ion o playe s in pe iod Tis hen
h(X)+xih(X)=0⇐⇒ xi=g1(X),
whe e g1(X)=−h(X)/h(X). The e o e, we can w i e he in e ed bes - esponse unc-
ion as
T−1(X)=X−nTg1(X).
Simila ly, i he in e ed bes - esponse unc ions a pe iod is (X), which is in e -
ible in he ele an ange, he payo unc ion o playe iin pe iod is ui(xi, −1
(X )) =
xih( −1
(X )) and, he e o e, he i s -o de condi ion o playe s in pe iod is
h(X)+xih(X)1

(X)=0⇐⇒ xi=g1(X) 
(X). (15)
The e o e, −1(X)= (X)−n 
(X)g1(X). This shows ha we can use he cha ac e -
iza ion de i ed in he pape , wi h wo modi ica ions. Fi s , ins ead o speci ic exp es-
sion X(1−X),weha ea unc iong1(X)=−h(X)/h(X). And second, we need o
impose some condi ions on he unc ion h(X)so ha he condi ions o he exis ence
and uniqueness a e sa is ied.
In Appendix A, I de ine P ope y 1, which is a su icien condi ion o all unc ions
o be well beha ed so ha he cha ac e iza ion heo em (Theo em 1) holds wi hou any
modi ica ions. In ui i ely, P ope y 1pu s wo es ic ions on unc ions. Fi s , o
Theo e ical Economics 19 (2024) Op imal sequen ial con es s 225
su icien ly high X, hey a e s ic ly inc easing and, he e o e, in e ible in he ele an
ange. Second, a leas one o he unc ions is aking a nega i e alue o lowe alues
o X, which elimina es such Xas a candida e o equilib ium. P oposi ion 3in Ap-
pendix Ap o es ha Tullock payo s sa is y P ope y 1and below I discuss some o he
cases when i is sa is ied.
The e o e, unde P ope y 1, he equilib ium is s ill unique and can be compu ed as
he highes oo o 0(X)in [0, 1]. Mo eo e , he limi o la ge con es s (P oposi ion 2)
holds as well, wi h a pa icula adjus men in o mulas. Le α=−g
1(1)>0. Then he
o mulas in equa ion (12) would be adjus ed as
lim
n→∞
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1−Xn−1
T

=11+αnn

⎤
⎥
⎥
⎥
⎥
⎥
⎦=0and lim
n→∞
⎡
⎢
⎢
⎢
⎢
⎢
⎣
xn
i−α

s=11+αnn
s
⎤
⎥
⎥
⎥
⎥
⎥
⎦=0. (16)
Fo he in o ma ion heo em (Theo em 2) i s co olla ies (Co olla y 1), as well as he
ea lie -mo e ad an age esul (P oposi ion 1) I need an addi ional assump ion. Fi s ,
le us adjus gk unc ions by de ining hese as g1(X)=−h(X)/h(X)and gk+1(X)=
−g
k(X)g1(X) o all k. The addi ional assump ion, P ope y 2in Appendix A, s a es
essen ially ha each gk(X∗)>0 nea equilib ium. This assump ion can be in e p e ed
as ac ions being highe -o de s a egic subs i u es. P oposi ion 4p o es ha Tullock
payo s sa is y P ope y 2and below I discuss some unc ional o ms ha sa is y his
assump ion.
The only esul ha does no gene alize is Lemma 1 ha showed ha wi h Tullock
payo s, he weigh s gk(X∗)a e dec easing in k. I is easy o see ha his esul depends
on he unc ion h(X). Fo example, conside he case when h(X)=α
√1−X o all X∈
[0, 1]and 0 o he wise, whe e α>0 is a cons an . Then g1(X)=α(1−X),g2(X)=α2(1−
X),andsoon,gk(X)=αk(1−X).Whene e α>1, he weigh s a e inc easing in his
case.
The emaining ques ion is when a e p ope ies 1and 2sa is ied? Fo example, one
special class o unc ions whe e hese assump ion a e sa is ied, is he class o unc ions,
whe e g1(X)=−h(X)/h(X)is comple ely mono one, i.e., (−1)kdkg1(X)/dXk≥0 o
all k∈N.16 This includes many unc ions, including linea h(X), powe unc ion
h(X)=α
√1−X, bu also many o he na u al unc ions. Fo example, he ollowing unc-
ions a e all comple ely mono one: g(X)=α(1−Xm),g(X)=α(1−X)m, o allm∈N,
g(X)=α((X+γ)s−(1+γ)s) o all s<0, γ>0, g(X)=α[e− X −e− ] o all >0,
and g(X)=−αlog(X),allwi hanyα>0. Also, all sums and p oduc s o comple ely
mono one unc ions a e comple ely mono one.17
16I su ices ha g1(X)is only T- imes mono one, which is less es ic i e, bu pe haps ha de o e i y.
17In he wo king pape e sion (h ps://a xi .o g/pd /1802.04669.pd ), I gi e mo e examples: (1) An
oligopoly wi h loga i hmic demand, whe e he analysis can be di ec ly ex ended, e en wi h non-mono onic
g1(X); (2) An example whe e p ope ies 1and 2a e iola ed, and equilib ium may no exis o be unique;
226 Toomas Hinnosaa Theo e ical Economics 19 (2024)
No e ha linea ly mul iplica i e payo s combined wi h p ope ies 1and 2a e su i-
cien and con enien assump ions, bu hey a e no necessa y. These assump ions en-
su e ha he i s -o de condi ions a e linea in he cumula i e ac ion o p eceding play-
e s, and as a esul , he in e ed bes - esponse unc ions can be easily cha ac e ized.
This aises he ques ion: is he unc ional o m assump ion ele an only o ac abil-
i y o does i ha e economic implica ions? A simple con inui y a gumen demons a es
ha he esul s will no change wi h small pe u ba ions nea he o iginal model.18
9. Discussion
I showed ha each con es has a unique equilib ium. I is in pu e s a egies and sim-
ple o compu e. The main esul o he pape shows ha he o al equilib ium e o is
s ic ly inc easing in in o ma ion. This implies ha he op imal con es o maximiz-
ing o al e o is ully sequen ial, e.g., R&D con es s bene i om ull anspa ency. On
he o he hand, i he goal is o minimize he o al e o , such as en -seeking con es s,
he op imal con es is non anspa en , i.e., he simul aneous con es . Fu he , he e is a
s ic ea lie -mo e ad an age: playe s in ea lie pe iods exe s ic ly g ea e e o s and
ob ain s ic ly highe payo s. To al e o con e ges o ull dissipa ion linea ly wi h he
numbe o playe s in la ge simul aneous con es s bu exponen ially in la ge sequen ial
con es s.
The esul s in his pape hold much mo e gene ally han he model discussed he ein.
In addi ion o he gene aliza ion discussed in Sec ion 8, some assump ions abou he
iming o a i als can be elaxed. I assumed ha playe s exe e o s only a hei a i al
and ha hei e o s a e publicly obse able o playe s in he ollowing pe iods. Gi en
ha playe s bene i om he discou agemen e ec , hey would no hide o delay hei
ac ions. Thus, he ou comes would be unchanged i playe s could ake hidden ac ions o
ake ac ions o e mul iple pe iods. This was shown by Yildi im (2005) in he wo-playe
case.
The analysis can also be ex ended o he e ogeneous playe s. Howe e , he e is a new
complica ion: playe s may ind i op imal o s ay inac i e a di e en h esholds. This
means ha ea lie mo e s may some imes ind i op imal o de e en y by ollowe s
and he o de o playe s becomes an impo an de e minan o ou comes. Xu, Zhang,
and Zhang (2020) use he app oach in oduced he e o s udy he h ee-playe asymme -
ic sequen ial con es s.19
Hinnosaa (2023) ex ends he me hodology o ano he ype
o playe he e ogenei y, whe e he game is played on a ne wo k. Playe s only obse e
he choices o playe s hey a e linked o. This analysis shows ha he e is a connec-
ion be ween weigh ed measu es o in o ma ion and s anda d cen ali y measu es om
ne wo k heo y.
(3) An example whe e e o s may be di ec s a egic subs i u es in he s anda d sense bu s a egic comple-
men s due o indi ec e ec s. In his ins ance, he equilib ium wi h wo pe iods beha es as one would ex-
pec wi h s a egic subs i u es, while in oducing a hi d pe iod al e s he conclusions as a esul o indi ec
e ec s.
18In he wo king pape e sion, I show ha in Tullock con es s wi h quad a ic cos s, he analysis s ill
applies when he pa ame e mul iplying he quad a ic e m is su icien ly close o ze o.
19The wo-playe case has been s udied by Mo gan (2003) and Se ena (2017), who also conside ed en-
dogenous o de o mo es.

Theo e ical Economics 19 (2024) Op imal sequen ial con es s 227
Appendix A: P oo s
A.1 P oo o he cha ac e iza ion heo em (Theo em 1)
Be o e p o ing Theo em 1, i is use ul o de ine he ollowing p ope y.
P ope y 1 (In e ed Bes Responses a e Well Beha ed). Clea ly, T(X)=Xhas
unique oo XT=0. Fo all =0, ,T−1, he unc ion has he ollowing p ope ies:
(a) (X)=0 has a oo in [X +1,1
].Le X be he highes such oo .
(b) (X)<0 o allX∈[X +1,X ).
(c) 
(X)>0 o allX∈[X ,1
].
Mo eo e , X0∈(0, 1).
The p oo o Theo em 1has wo pa s. The i s pa is P oposi ion 3in Appendix A.2
ha shows ha unc ions sa is y P ope y 1. The p oo elies on keeping ack o he
oo s o unc ions. The second pa in Appendix A.3 es ablishes he heo em’s claims.
B ie ly, i shows ha beha io whe e each playe iin each pe iod beha es acco ding o
equa ion (8) and expec s ha o al e o induced by cumula i e e o X o be −1
(X),
is an equilib ium and in ac i is he only equilib ium. The p oo is di ided in o i e
lemmas:
1. Lemma 5shows ha in all his o ies whe e X −1<1, each playe in pe iod chooses
s ic ly posi i e e o , bu hese added e o s in pe iod a e small enough so ha
he cumula i e e o a e pe iod emains s ic ly below one, X <1. On he o he
hand, in his o ies whe e X −1≥1, he playe s in pe iod exe no e o . The e o e,
on he equilib ium pa h X <1 o all .
2. Lemma 6shows ha X = (X)is a necessa y condi ion o equilib ium. In pa -
icula , 0(X)=0 is a necessa y condi ion o equilib ium and, he e o e, X∗mus
be a oo o 0(X).
3. Lemma 7shows ha unde P ope y 1, he in e se unc ion −1
−1(X −1)is well-
de ined and s ic ly inc easing, −1
−1(0)=X −1and −1
−1(1)=1.
4. Lemma 8shows ha he bes - esponse unc ion o playe i∈I a e cumula-
i e e o X −1is x∗
i(X −1)=n−1
[ ( −1
−1(X −1)) −X −1] o all X −1<1and
x∗
i(X −1)=0 o allX −1≥0. On he equilib ium pa h, he indi idual e o s a e
x∗
i=n−1
[ (X∗)− −1(X∗)]. No e ha his s ep in he p oo implici ly also shows
ha s a egies o all playe s in pe iod a e iden ical.
5. Finally, Lemma 9 e i ies ha he unique candida e o equilib ium, i.e., x∗speci-
ied in he heo em, is indeed an equilib ium, which comple es he p oo .
The combina ion o hese esul s p o es Theo em 1.
228 Toomas Hinnosaa Theo e ical Economics 19 (2024)
A.2 P oo ha P ope y 1is sa is ied (P oposi ion 3)
P oposi ion 3. In e ed bes esponses 0,, Tde ined by equa ion (7) a e well be-
ha ed.
Be o e gi ing he p oo o P oposi ion 3, le me b ie ly desc ibe i s key idea. The
unc ion +1is a polynomial o deg ee =T− , so i can ha e a mos oo s. By
keeping ack o all he oo s, I show by induc ion ha all oo s a e eal and in [0, 1),
wi h he highes being X +1. The e o e, all −1 oo s o he de i a i e 
a e also eal
and in [0, X +1). E alua ing a X +1and 1, we ge
(X +1)= +1(X +1)
 
=0
−n +1 
+1(X +1)
 
>0
X +1(1−X +1)
 
>0
<0
(1)= +1(1)−n +1 
+1(1)1(1−1)
 
=0= +1(1)=···= T(1)=1>0.
This implies ha mus ha e a oo X ∈(X +1,1
). Mo eo e , since he highes oo
o i s de i a i e is again below X , i is s ic ly inc easing in [X ,1
]. Finally, I show ha
he second highes oo o is s ic ly below X +1,so ha (X)<0 o all[X +1,X ).
P o ing his equi es keeping ack o all he oo s.
P oo o P oposi ion 3.Fi s ,no e ha T(X)=Xis a polynomial o deg ee 1, and
each s ep o he ecu sion adds one deg ee, so (X)is a polynomial o deg ee T+1− ,
which I deno e by o b e i y. The ollowing wo echnical lemmas desc ibe he alues
o he polynomials a 1 and he numbe o oo s a 0.
Lemma 2. (1)=1 o all =0, ,T.
P oo . −1(1)= (1)−n 
(1)1(1−1)= (1)= T(1)=1.
Lemma 3. (0)=0 o all =0, ,T. Depending on n, he e could be ei he one o wo
oo s a ze o:
(a) I ns=1 o some s> , hen (X)hasexac ly wo oo sa ze o.
(b) O he wise, i.e., i ns=1 o all s> , hen (X)has exac ly one oo a ze o.
P oo .As (X)is a polynomial o deg ee =T+1− , i can be exp essed as
(X)=

s=0
c
sXs⇒ 
(X)=

s=1
c
ssXs−1,
whe e c
0,,c
a e he coe icien s. The e o e,
−1(X)=c
0+c
1(1−n )+

s=2c
s(1−sn )+n c
s−1(s−1)Xs+n c
T+1− (T+1− )XT+2− .
Theo e ical Economics 19 (2024) Op imal sequen ial con es s 229
As T(X)=X,weha e ha cT
0=0andsoc
0=0 o all . The e o e, each has a leas
one oo a 0. Nex , −1(X)has wo oo s a ze o i and only i c −1
1=c
1(1−n )=0. This
can happen only i ei he c
1=0(i.e., (X)has wo oo s a ze o) o n =1. As T(X)=X,
we ha e ha cT
1=1 and, he e o e, (X)does indeed ha e wo oo s a ze o i and only
i ns=1 o somes> .
Finally, −1(X)would ha e h ee oo s a ze o only i c −1
2=c −1
1=0=c −1
0.This
would equi e ha c −1
2=c
2(1−2n )+n c
1=c
2(1−2n )=0. Since 2n = 1, his can
happen only when c
2=0. Bu no e ha T−1(X)=nTX2−(1−nT)X,so ha cT−1
2=
nT=0. The e o e, (X)canno ha e mo e han wo oo s a ze o.
Lemma 4. The leading coe icien o is (T− )!T
s= +1ns>0.
P oo . Using he same no a ion as in Lemma 3, he leading coe icien o −1(X)is
c −1
+1= n c
= !T
s= ns.
Now I can p oceed wi h he p oo o P oposi ion 3i sel . The p oo uses ha ac
ha he is a polynomial o deg ee =T+1− and keeps ack o all o i s oo s. In
pa icula , i can be exp essed as
(X)=c

s=1
(X−Xs, ), (17)
whe e c >0 is he leading coe icien and X1, ,,X , a e he oo s. By Lemma 3,
ei he one o wo o hese oo s a e equal o ze o. I show by induc ion ha all o he oo s
a e dis inc and in (0, 1).
Le us conside he case o a single ze o oo i s , i.e., assume ha 0 =X1, <X
2, <
···<X
, <1. We can exp ess he de i a i e o as

(X)=c

i=1
s=i
(X−Xs, ).
The e o e, a oo Xj, , he polynomial 
(X) akes alue

(Xj, )=c 
s=j
(Xj, −Xs, ). (18)
In pa icula , a he highes oo , 
(X , )>0, and a he second highes 
(X −1, )<
0; he e o e, 
mus ha e a oo Y −1, ∈(X −1, ,X , ). By he same a gumen , he e
mus bea oo Ys, o 
be ween each o he wo adjacen dis inc oo s o .As 
is a
polynomial o deg ee −1, his a gumen implies ha all he oo s o 
a e dis inc and
such ha
X1, =0<Y
1, <X
2, <Y
2, <···<X
−1, <Y
−1, <X
, <1.
In pa icula , sgn 
(Xs, )=sgn (Ys, ) o all s∈{1, , −1}. Nex , no e ha (1)=
1>0 and, as he highes oo o 
is Y −1, <X
, , his implies 
(X , )>0, and so
−1(X , )= (X , )−n 
(X , )X , (1−X , )<0.
230 Toomas Hinnosaa Theo e ical Economics 19 (2024)
The e o e, −1mus ha e a oo X +1, −1∈(X , ,1
). Now, o each s∈{2, −1}
−1(Ys, )= (Ys, )and −1(Xs, )=−n 
(Xs, )Xs, (1−Xs, ).
Hence, sgn −1(Ys, )=sgn (Ys, )=sgn 
(Xs, )=− −1(Xs, ). Thismeans ha −1
mus ha e a oo Xs+1, −1∈(Xs, ,Ys, ). This a gumen de e mines −2 dis inc oo s in
(X2, ,Y −1, ). By Lemma 3, −1also has a leas one oo X1, −1=0.
We ha e he e o e ound 1+ −2−1= dis inc eal oo s o −1 ha is a polynomial
o deg ee +1. Thus, he inal oo X2, mus also be eal. By Lemma 3,i n =1, hen
he −1mus ha e wo oo s a ze o; so, X2, =0. Le us conside he emaining case
whe e n >1. By Lemma 3,X2, = 0. To de e mine i s loca ion, conside he unc ion
X
−1(X)= −1(X)/X.No e ha
X
(X)= (X)
X=c 
s>0
(X−Xs, )⇒ X
(0)=c 
s>0
(−Xs, )
and

(0)=c 
s>0
(−Xs, ).
The e o e,
X
−1(0)= X
(0)−n 
(0)(1−0)=c 
s>0
(−Xs, )[1−n ]= 
(0)[1−n ].
We assumed ha n >1; so, sgn X
−1(0)=−sgn 
(0). E alua ing he unc ion sgn X
−1a
Y1, gi es
sgn X
−1(Y1, )=sgn (Y1, )=sgn 
(X1, )=−sgn X
−1(0).
Hence, X
−1mus ha e a oo X2, −1∈(0, Y1, ).As −1(X)=X X
−1(X), i mus be a oo
o −1as well. We ha e he e o e loca ed all +1 oo s o −1, which a e all dis inc in
his case.
Le usnowge back o hecasewhe e had wo oo s a ze o. By he same a gu-
men as abo e, he e mus be a oo o 
be ween each posi i e oo o .As he e
a e −2 posi i e oo s, his de e mines −3 dis inc posi i e oo s o 
.I isalso
clea ha 
mus ha e exac ly one oo a ze o. Polynomial 
has −1 oo s, and
we ha e de e mined ha −2 o hem a e eal and dis inc . Thus, he emaining
oo mus be eal. To de e mine i s loca ion, using he abo e app oach, le X
(X)=

(X)/X.Thenas 
(X , )>0, we ha e X
(X , )>0. Simila ly, X
(X −1, )<0,
and so on. In pa icula , X
(X3, )<0i is e en, and X
(X3, )>0i is odd.
Now,
X
(0)=2c 
s>2
(−Xs, ),
which is s ic ly posi i e i is odd and s ic ly nega i e i is e en, so ha sgn X
(0)=
−sgn X
(X3, ).Hence, X
mus ha e a oo Y2, ∈(0, X3, ). Clea ly, his Y2, is
Theo e ical Economics 19 (2024) Op imal sequen ial con es s 237
2. Lemma 12 es ablishes a connec ion be ween he o al equilib ium e o X∗and
Zk−1:k. I shows ha i we ake he sequen ial n-playe con es n=(1, ,1
), hen
n−k(X)=gk(X)X/(1−X) o all k=1, ,n. The e o e, i we ake he ully se-
quen ial con es wi h nplaye s, we ge 0(X)=gn(X)X/(1−X),andso he o al
equilib ium e o X∗o his con es is exac ly equal o he second highes oo o
gn, i.e., Zn−1:n.
This p o es he “weak” pa o he p oposi ion, i.e., i nis ully sequen ial, hen
X∗=Zn−1:n, which is a oo o gnand, he e o e, gk(X∗)=0.
3. Lemma 13 shows di ec ly20 ha X∗is s ic ly inc easing in each n . The e o e, i he
con es is no sequen ial (n >1 o some ), hen he o al e o in his con es is
s ic ly highe han in he ully sequen ial T-playe con es . Thus, X∗>Z
T−1:Tand
gT(X∗)>0.
4. Finally, Lemma 11 also shows ha he adjacen gk’s a e in e laced; i.e., he second
highes oo s a e inc easing in k,so ha o allk<T,Zk−1:k<Z
T−1:T≤X∗and,
he e o e, gk(X∗)>0 o allk<T.
Lemma 11. Each gkhas he ollowing p ope ies:
(a) gk(1)=g
k(1)=−1.
(b) gkcan be exp essed as
gk(X)=−
k

j=0
(X−Zj:k), (21)
whe e 0=Z0:k<Z
1:k<···<Z
k:k=1.
(c) Zs:k+1∈(Zs−1:k,Zs:k) o all s=1, ,k.
P oo .Fi s ,no e ha g1(X)=g(X)=X(1−X)is a polynomial o deg ee 2. Each
s ep o he ecu sion gi es a polynomial o one deg ee highe ; i.e., gk(X)is a polyno-
mial o deg ee k+1, so g
k(X)is a polynomial o deg ee kand, he e o e, gk+1(X)=
−g
k(X)X(1−X)is a polynomial o deg ee k+2.
1. gk+1(1)=−g
k(1)g(1)=0, because g(1)=1(1−1)=0. The e o e, g
k(1)=
−g
k−1(1)g(1)−g(1)g
k−1(1)=g
k−1(1)=···=g
1(1)=g(1)=1−2·1=−1.
2. The claim clea ly holds o g1(X)=X(1−X)wi h Z0:1 =0<Z
1:1 =1. Suppose
i holds o k. Since all k+1 oo s o gka e eal and in [0, 1], by he Gauss–Lucas
heo em all k oo s o g
ka e in (0, 1).Thengk+1(X)=−g
k(X)X(1−X)clea ly has
oo s a 0 and 1 and k oo s in (0, 1). To see ha he oo s a e all dis inc , no e ha
g
k(X)=−
k

s=0
j=s
(X−Zj:k).
20No e he i s pa o Co olla y 1p o es he same claim, bu since P oposi ion 4es ablishes a su icien
condi ion o Theo em 2, and hence i s Co olla y 1, o a oid a ci cula a gumen I p o e i he e di ec ly.

238 Toomas Hinnosaa Theo e ical Economics 19 (2024)
The e o e, g
k(Zs:k)=−j=s(Zs:k−Zj:k), which is s ic ly nega i e o s=k, s ic ly
posi i e o s=k−1, and so on. The e o e, o each s=1, ,k, unc iong
k;hence,
gk+1also has a oo Zs:k+1=(Zs−1:k,Zs:k). This de e mines he kin e io oo s.
3. The p e ious a gumen also p o es he las claim.
Lemma 12. I n=(1, ,1
), hen n−k(X)=gk(X)X/(1−X) o all k=1, ,T.
P oo . Suppose ha n=(1, ,1
).Fi s , n−1(X)=X−X(1−X)=X2=g1(X)X/
(1−X). Now, suppose ha n−k(X)=gk(X)X/(1−X). Then since
dX
1−X
dX X(1−X)=1
1−X−−X
(1−X)2X(1−X)=X
1−X,
we ge ha
n−(k+1)(X)=gk(X)X
1−X−gk(X)
dX
1−X
dX X(1−X)−g
k(X)X
1−XX(1−X)
=gk+1(X)X
1−X.
Lemma 13. X∗is s ic ly inc easing in each n .
P oo . I i s show ha X∗is independen o pe mu a ions o n.Fixacon es nand a
pe iod >1. To sho en he no a ion, le φ (X)= 
(X)X(1−X):
−1(X)= (X)−n φ (X),

−1(X)= 
(X)−n φ
(X)=φ (X)
g(X)−n φ
(X),
−2(X)= (X)−[n −1+n ]φ (X)+n −1n φ
(X)X(1−X).
Swi ching n −1and n in ndoes no a ec −2and, he e o e, i also does no a ec 0.
This means ha any such swi ch lea es X∗una ec ed, which means ha X∗is indepen-
den o pe mu a ions o n.
To p o e ha X∗is s ic ly inc easing in each n , i he e o e su ices o p o e ha i
is s ic ly inc easing on n1.Take
n=(n1+1, n2,,nT).Then 1is unchanged and he
co esponding 
0a he o iginal equilib ium X∗is

0X∗= 1X∗−(n1+1) 
1X∗X∗1−X∗= 0X∗− 
1X∗X∗1−X∗<0,
because 0(X∗)=0and 1(X∗)>0byP ope y1.ByP ope y1,
0is s ic ly inc easing
be ween i s highes oo 
X∗and 1, hus 
X∗>X∗.
Theo e ical Economics 19 (2024) Op imal sequen ial con es s 239
A.6 P oo o dec easing weigh s lemma (Lemma 1)
This lemma allows o o de some con es s, which canno be anked acco ding o hei
in o ma ion measu es. Fo example, wo 10-playe con es s n=(5, 5)and 
n=(8, 1, 1)
ha e co esponding in o ma ion measu es S(n)=(10, 25)and S(
n)=(10, 17, 8).Con-
es nhas mo e second-o de in o ma ion, bu 
nhas one mo e disclosu e, and hus
mo e hi d-o de in o ma ion. Howe e , he sum o all in o ma ion measu es is 10 +
25 =10 +17 +8=35. Since he weigh s a e highe in lowe -o de in o ma ion, his
implies ha he o al e o is highe in he i s con es . Indeed, di ec applica ion The-
o em 1con i ms his, as X∗=(13 +√41)/20 ≈0.9702 >
X∗=(31 +√241)/48 ≈0.9693.
P oo o Lemma 1. By Lemma 12,gk(X)=

n−k(X)(1−X)/X,whe e

n−kis de ined
o a sequen ial 
n≥k-playe con es . Simila ly, gk−1(X)=
n+1−k(X)(1−X)/X.The e-
o e,
gk−1X∗−gkX∗=

n+1−kX∗−

n−kX∗1−X∗
X∗=


n+1−kX∗1−X∗2.
Now, ake 
n=T. Then by Lemma 13,X∗is weakly highe han he highes oo o 
0.
By P ope y 1, he highes oo o 
T+1−kis e en (weakly) lowe and 
T+1−kis s ic ly
inc easing abo e i s highes oo , so ha 


n+1−k(X∗)>0. This p o es ha gk−1(X∗)>
gk(X∗).
A.7 P oo s o implica ions o he in o ma ion heo em (Co olla y 1)
P oo o Co olla y 1. Take wo con es s nand
nand le Xand 
Xbe he co espond-
ing o al equilib ium e o s.
1. Suppose ha n<
n.ThenS(n)<S(
n)and, he e o e, X<
X.
2. I nis a pe mu a ion o 
n, henS(n)=S(
n)and, he e o e, X=
X.
3. I Iis a coa se pa i ion han

I, henS(n)<S(
n)and, he e o e, X<
X.
4. I  n = 
n =nand he e exis , such ha n n <
n 
n and ns=
ns o all
s= , , henbycons uc ionS1(n)=S1(
n)=nand Sk(n)<S
k(
n) o all k>1.
The e o e, X<
X.
5. Le n=(n).Then o any
n=n,S(n)<S(
n), so indeed Xis he unique minimum
o X∗o e all con es s. Simila ly, i 
n=(1, 1, ,1
), any o he con es has s ic ly
lowe measu es o in o ma ion and, he e o e, 
Xis he unique maximum o X∗
o e all con es s.
To es ablish he inal claim o he op imali y o equal di ision o playe s, le nbe
n-playe con es s whe e playe s a e dis ibu ed among a mos Tpe iods. Suppose
by con adic ion ha he co esponding o al equilib ium e o X∗is a maximum
o e all such con es s and ndoes no spli playe s as equally as possible. In pa ic-
ula , le k=n/T . Equal spli equi es ha each pe iod has ei he n ∈{k,k+1}
playe s. Since his is no he case, he e exis s a pe iod whe e n ≤k−1anda
240 Toomas Hinnosaa Theo e ical Economics 19 (2024)
pe iod swhe e ns≥k+1(o ,ssuch ha n ≤kand ns≥k+2, hen he p oo is
analogous).
We can now cons uc a new con es , 
n, whe e we ha e mo ed one playe om
pe iod s o pe iod .Thenasns−1≥k>n
,

n 
ns=(n +1)(ns−1)=n ns−n +ns−1>n
ns.
The e o e, he con es 
nis mo e homogeneous han nand so X<
Xby he p e i-
ous s ep. Thus, we ound a con adic ion wi h he assump ion ha Xis a maximal
o al e o among such con es s.
A.8 P oo o he ea lie -mo e ad an age (P oposi ion 1)
P oo o P oposi ion 1. The equilib ium payo o playe iis ui(x∗)=x∗
i(1/X∗−1),
so he payo s a e anked in he same o de as he indi idual e o s (in ac hey a e
p opo ional o indi idual e o s). The e o e, i su ices o p o e ha i i∈I and j∈
I +1, henx∗
i>x
∗
j. Using Theo em 1and equa ion (9), he di e ence in equilib ium
e o s can be exp essed as
x∗
i−x∗
j=
T−

k=1Skn −Skn +1gk+1X∗.
Now, no e ha S(n )≥S(n +1)as he e is less in o ma ion emaining in he game ha
s a s one pe iod la e . Mo eo e , S1(n )>S
1(n +1)as n includes playe j, whe eas n +1
does no . Finally, no e ha by P oposi ion 4,g2(X∗)>0 and, he e o e, x∗
i−x∗
j>0.
A.9 P oo o he la ge con es s limi (P oposi ion 2)
P oo o P oposi ion 2.ByTheo em1, each Xn<1. Meanwhile, by Theo em 2,
Xn≥(n−1)/n, which is he o al equilib ium e o o he simul aneous n-playe con es
(see Sec ion 3). The e o e, limn→∞ Xn=1.
The o al equilib ium e o o a censo ed con es nnis he highes oo o 0(X),
which can be exp essed by equa ion (10)as
Xn=
T

k=1
SknngkXn. (22)
Fo each k, unc iongk(X)is a wice con inuously di e en iable unc ion (a polyno-
mial), gk(1)=0, and g
k(1)=−g
k−1(1)g(1)−g
k−1(1)g(1)=g
k−1(1)=···=g
1(1)=−1,
as g1(X)=X(1−X). The e o e, o all k>1,
lim
X→1
gk(X)
X(1−X)=lim
X→1−g
k−1(X)X(1−X)
X(1−X)=−g
k−1(1)=1.
Taking limi s om bo h sides o equa ion (22) and using he esul ha limn→∞ Xn=1,
1=lim
n→∞Xn=lim
n→∞
T

k=1
SknngkXn
Xn1−XnXn1−Xn=lim
n→∞1−XnT

k=1
Sknn.
Theo e ical Economics 19 (2024) Op imal sequen ial con es s 241
To sho en he no a ion, le Sn=T
k=1Sk(nn). Rea anging he p e ious equa ion gi es
0=lim
n→∞1−1−XnSn=lim
n→∞Xn−1−1
SnSn. (23)
We can exp ess Sn=T
k=1Sk(nn)=T
=1(1+nk
)−1. As limn→∞ Sn=∞,equa ion(23)
implies ha
lim
n→∞Xn−1−1
Sn=lim
n→∞
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Xn−⎛
⎜
⎜
⎜
⎜
⎜
⎝
1−1
T

=11+nn

⎞
⎟
⎟
⎟
⎟
⎟
⎠
⎤
⎥
⎥
⎥
⎥
⎥
⎦=0.
Fo indi idual e o o playe i∈In
, we can use Theo em 1and equa ion (9) oge
xn
i=g1Xn+
T−

k=1
Skn gk+1Xn.
Taking he limi , again using he ac s ha Xn→1andgk+1(Xn)/[Xn(1−Xn)] →1,
lim
n→∞xn
i=lim
n→∞1−Xn%1+
T−

k=1
Skn &.
Now, no e ha 1 +T−
k=1Sk(n )=T
s= (1+nn
s). The e o e, using he esul om abo e,
we can exp ess he las equa ion as
0=lim
n→∞%xn
i−1−Xn'1+
T−

k=1
Skn (&=lim
n→∞
⎡
⎢
⎢
⎢
⎢
⎢
⎣
xn
i−1
T

s= 1+nn
s
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
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