Boh en, J. Aislinn
A icle
Pe sis ence in a dynamic mo al haza d game
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Boh en, J. Aislinn (2024) : Pe sis ence in a dynamic mo al haza d game,
Theo e ical Economics, ISSN 1555-7561, The Econome ic Socie y, New Ha en, CT, Vol. 19, Iss. 1, pp.
449-498,
h ps://doi.o g/10.3982/TE2680
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Theo e ical Economics 19 (2024), 449–498 1555-7561/20240449
Pe sis ence in a dynamic mo al haza d game
J. Aislinn Boh en
Depa men o Economics, Uni e si y o Pennsyl ania
This pape explo es how he pe sis ence o pas choices c ea es incen i es in a
con inuous ime s ochas ic game in ol ing a la ge playe (e.g., a i m) and a se-
quence o small playe s (e.g., cus ome s). The la ge playe aces mo al haza d and
he ac ions a e dis o ed by a B ownian mo ion. Pe sis ence e e s o how ac ions
impac a payo - ele an s a e a iable (e.g., p oduc quali y depends on pas in-
es men ). I cha ac e ize ac ions and payo s in Ma ko pe ec equilib ia (MPE)
o a ixed discoun a e, show ha he pe ec public equilib ium (PPE) payo
se is he con ex hull o he MPE payo se , and de i e su icien condi ions o
a MPE o be he unique PPE. Pe sis ence can se e as an e ec i e channel o in-
e empo al incen i es in a se ing whe e adi ional channels ail. Applica ions o
pe sis en p oduc quali y and policy a ge ing demons a e he impac o pe sis-
ence on equilib ium beha io .
Keywo ds. Con inuous ime games, s ochas ic games, mo al haza d.
JEL classi ica ion. C73, L1.
1. In oduc ion
This pape s udies how he pe sis ence o pas choices can be used o c ea e incen i es
in a con inuous ime s ochas ic game in which a la ge playe in e ac s wi h a sequence
o small playe s. Pe sis ence e e s o he impac ha ac ions ha e on a payo - ele an
s a e a iable, such as a wo ke ’s a ing, a i m’s p oduc quali y, o a go e nmen ’s key
economic a iables. I can cap u e exogenous ea u es o he en i onmen , such as how
pas in es men in luences cu en quali y o how pas policy choices map in o he cu -
en le el o an economic a iable. I can also cap u e endogenous design choices, such
as how a a ing sys em agg ega es pas e iews and ewa ds a wo ke based on he a -
ing. The la ge playe aces mo al haza d and he pas ac ions a e no pe ec ly obse ed
by consume s: hey a e dis o ed by a B ownian mo ion. Incen i es can depend on he
noisy signal o ac ion choices as well as on how pe sis ence in luences u u e payo s
h ough he impac ha ac ions ha e on he s a e. The goal o his pape is o de e mine
whe he and how pe sis ence s eng hens incen i es o o e come mo al haza d.
J. Aislinn Boh en: [email p o ec ed]
Ea lie e sions o his pape we e ci cula ed unde he i les “S ochas ic games in con inuous ime: Pe sis-
en ac ions in long- un ela ionships” and “Using pe sis ence o gene a e incen i es in a dynamic mo al
haza d p oblem.” I hank Simon Boa d, Ma Ellio , Je Ely, Ben Golub, Alex Imas, Ba Lipman, Da id
Mille , Geo ge Maila h, Ma kus Mobius, Paul Niehaus, And ew Pos lewai e, Yuliy Sanniko , Andy Sk zypacz,
Joel Sobel, Je oen Swinkels, Joel Wa son, Alex Woli zky, and especially S. Nageeb Ali o use ul commen s.
I also hank nume ous semina pa icipan s o help ul eedback.
©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/TE2680
450 J. Aislinn Boh en Theo e ical Economics 19 (2024)
The amewo k cap u es many economic se ings in which pas choices shape key
ea u es o cu en and u u e in e ac ions. Fo example, a wo ke ’s a ing on a pla o m
depends on he quali y o se ice she has p o ided o p e ious cus ome s. She may be
ewa ded o ea ning a good a ing and punished o poo pe o mance. This p o ides
an incen i e o he o ea n and main ain a good a ing. Simila ly, a i m’s abili y o make
a high quali y p oduc is a unc ion no only o i s e o oday, bu also i s pas in es -
men s in de eloping echnology and aining i s wo k o ce. Quali y oday is linked o a
i m’s u u e quali y, in ha cus ome s expe ience simila quali y ac oss ime due o he
pe sis ence o in es men . When cus ome s a e willing o pay a highe p ice o buy a
la ge quan i y o a high quali y good, pe sis ence p o ides an incen i e o he i m o
in es in de eloping a high quali y p oduc . Finally, a go e nmen ’s success in each-
ing he a ge le el o an economic a iable depends on bo h pas and cu en policy
choices. When pas policy choices impac he u u e alue o an economic a iable, he
go e nmen may be willing o unde ake mo e cos ly ac ions oday, since he bene i o
such ac ions con inue o acc ue in u u e pe iods.
I s udy pe ec public equilib ia (PPE) in his amewo k; ha is, equilib ia in which
s a egies depend only on public in o ma ion. I es ablish ha he PPE payo se is equal
o he con ex hull o he Ma ko pe ec equilib ium (MPE) payo se . In a MPE, equilib-
ium ac ions and payo s depend only on he payo - ele an componen s o he game—
in his case, he obse able s a e. Any pa hs o in o ma ion ha lead o he same cu en
s a e p esc ibe he same con inua ion play. In con as , a PPE can depend on pas in-
o ma ion in an a bi a y way. The in ui ion o his esul s ems om he ype o in-
cen i es ha a e possible in games wi h small playe s and B ownian in o ma ion. In
a s ochas ic game, dynamic incen i es can ei he be in o ma ional—signals a e used
o coo dina e u u e equilib ium play—o s uc u al—ac ions impac he s uc u e o
u u e in e ac ions h ough hei impac on he s a e, including bo h he s a e’s di ec
impac on u u e easible payo s and i s indi ec impac h ough i s e ec on u u e
equilib ium play. The e a e wo main o ms o in o ma ional incen i es: bu ning alue,
whe e incen i es a e c ea ed by he h ea o swi ching o an ine icien ac ion p o ile,
and ans e ing con inua ion payo s angen o he se o equilib ium payo s. I is no
possible o p o ide incen i es wi h ans e s when acing small playe s, and B ownian
in o ma ion is oo noisy o c ea e e ec i e incen i es ia alue-bu ning (Sanniko and
Sk zypacz (2010)). The e o e, any non i ial incen i es in games wi h small playe s and
B ownian in o ma ion mus be s uc u al. This is p ecisely he channel o incen i es in
a MPE, as in o ma ional channels a e p ecluded by de ini ion.1
In es ablishing his esul , I cha ac e ize equilib ium payo s and ac ions in MPE o
a ixed discoun a e. This cha ac e iza ion yields sha p insigh s. I shows ha whe he
pe sis ence allows he la ge playe o o e come mo al haza d depends on he ma ginal
1In ea lie ela ed wo k, Faingold and Sanniko (2011) es ablish a simila esul when small playe s ha e
incomple e in o ma ion abou he la ge playe ’s ype and he s a e is he belie ha he la ge playe is com-
mi ed o choosing a ce ain ac ion. S ochas ic games wi h mul iple la ge playe s, B ownian in o ma ion,
and a ailu e o iden i iabili y will likely ha e a simila equilib ium cha ac e iza ion o his pape , as such
games ace simila issues wi h in o ma ional incen i es (Ab eu, Milg om, and Pea ce (1991), Sanniko and
Sk zypacz (2007)).
Theo e ical Economics 19 (2024) Dynamic mo al haza d game 451
impac o i s ac ion on he s a e and how sensi i e he con inua ion payo is o changes
in he s a e. In con as o a olk heo em, i de e mines wha ype o equilib ia one ex-
pec s o eme ge and wha pa e n o beha io gene a es a gi en payo . I shows how
he dynamics o beha io depend on obse able ou comes (e.g., a es au an a ing o
an economic a iable) and how incen i es and payo s depend on key pa ame e s o he
model (e.g., he dep ecia ion a e o in es men ). The cha ac e iza ion o he con inua-
ion payo cap u es bo h he di ec and equilib ium channels o s uc u al incen i es.
Fo example, when consume s ha e obse ed a gi en le el o quali y in he ecen pas ,
hei willingness o pay o a p oduc depends on bo h he pe sis ence o his quali y—
he di ec channel—as well as hei belie abou how quali y in luences he i m’s cu -
en in es men choice— he equilib ium channel. The in e ac ion o hese wo chan-
nels can signi ican ly s eng hen o dampen incen i es, depending on he s uc u e o
he game.
The second main esul de e mines when a MPE eme ges as he unique PPE. This
esul elies on de e mining when he e is a unique MPE in he class o Ma ko equilib-
ia. When he e is a unique MPE, he esul desc ibed abo e es ablishes ha his will
also be he unique PPE. Uniqueness depends on incen i es as he s a e app oaches he
bounda y o he s a e space. I bounda y incen i es a e unique, e.g., i is possible o
sus ain a unique equilib ium ac ion p o ile and payo a he bounda y, hen om he
MPE cha ac e iza ion, incen i es mus also be unique on he in e io o he s a e space.
I p esen su icien condi ions o uniqueness in wo cases: (i) an unbounded s a e space
and (ii) a bounded s a e space. In case (i), hese condi ions ule ou complemen a i-
ies be ween he di ec and he equilib ium channels o incen i es nea he bounda y,
such as mul iple op imal ac ion p o iles due o coo dina ion mo i es. In case (ii), hese
condi ions ensu e ha incen i es collapse as he s a e app oaches he bounda y, which
ules ou he possibili y o sus aining mul iple equilib ium ac ion p o iles a he bound-
a y.
Se e al applica ions illus a e how pe sis ence can be used o c ea e e ec i e incen-
i es. The i s applica ion modi ies he canonical p oduc choice se ing o allow a i m’s
e o o ha e a pe sis en e ec on he quali y o i s p oduc . I show ha pe sis ence
p o ides e ec i e incen i es o he i m o in es in building a high quali y p oduc .
These incen i es a e p esen in he long un, in ha he i m con inues o choose a pos-
i i e le el o in es men as he ime pe iod g ows la ge. I also conside a a ia ion o
he p oduc choice game in which he ma ginal e u n o quali y is non-mono onic and
show ha his can lead o i ms specializing in low o high quali y. In he second ap-
plica ion, cons i uen s elec a boa d o implemen a policy ha a ge s an economic
a iable. The le el o he a iable depends on cu en and pas decisions by he boa d.
Fo example, he Fede al Rese e a ge s an in e es a e o a boa d o di ec o s se s a
g ow h a ge o a company. I show ha he boa d’s incen i e o unde ake cos ly in-
e en ion is s onges when he economic a iable is an in e media e dis ance om i s
a ge ; when i is a om i s a ge , he bene i o in e en ion is signi ican ly delayed,
while when i is close o i s a ge , he bene i o u he in e en ion is small. In he inal
applica ion, a go e nmen and a g oup o inno a o s in es in in ellec ual capi al, and
he e is a s a egic complemen a i y be ween hei in es men s. This complemen a i y
452 J. Aislinn Boh en Theo e ical Economics 19 (2024)
gi es ise o mul iple Ma ko equilib ia, including one in which nei he pa y in es s
and se e al ha sus ain a posi i e le el o in es men . The equilib ium cha ac e iza ion
in each applica ion can be used o add ess impo an design ques ions. Fo example,
a compa a i e s a ic on how a i m’s payo a ies wi h he pe sis ence o i s e o p o-
ides insigh in o he op imal du abili y o a p oduc ion echnology, while a compa a-
i e s a ic on how a wo ke ’s e o a ies wi h he pe sis ence o i s a ing is use ul o
designing a ing sys ems.
1.1 Li e a u e
Recen esul s on epea ed games be ween a long- un/la ge and sho - un/small play-
e s show ha he in e sec ion o noise in moni o ing and ins an aneous adjus men o
ac ions c ea es a genuine challenge in p o iding in e empo al incen i es (Fudenbe g
and Le ine (2007,2009), Faingold and Sanniko (2011)).2In he analogue o his pape
wi h no pe sis ence, he la ge playe canno ea n an equilib ium payo abo e he bes
s a ic Nash payo .3In con as , he equilib ium cha ac e iza ion in his pape demon-
s a es ha pe sis ence can lead o e ec i e in e empo al incen i es and enable he
la ge playe o o e come mo al haza d.
The li e a u e on epu a ion wi h beha io al ypes is ano he impo an and well
unde s ood mechanism o o e come mo al haza d in simila se ings (Fudenbe g and
Le ine (1989,1992), Faingold and Sanniko (2011), Faingold (2020)). I consume s be-
lie e ha he e is a chance ha he i m is commi ed o choosing high e o , hen he
i m will be able o cha ge a highe p ice o i s p oduc . Incomple e in o ma ion abou
he i m’s ype c ea es a o m o pe sis ence, as consume s’ belie s depend on pas e -
o choices. Howe e , ixing a s a egic i m’s pa ience, such epu a ion e ec s anish
in he ex an e p obabili y o beha io al ypes, and so he e ec i eness o pe sis ence ia
incomple e in o ma ion equi es a non i ial ac ion o beha io al ypes.4
The connec ion wi h he epu a ional li e a u e mo i a es se e al key insigh s. Fi s ,
when he i m is known o be s a egic, his pape shows ha o he o ms o pe sis ence
can o e come mo al haza d.5Second, in con as o he empo a y incen i es in epu a-
ion models (C ipps, Maila h, and Samuelson (2004), Faingold and Sanniko (2011)), he
incen i es in a s ochas ic game pe sis in he long un.6Finally, a a heo e ical le el, his
2Ab eu, Milg om, and Pea ce (1991) i s examined incen i es in epea ed games wi h impe ec mon-
i o ing and equen ac ions. They es ablished ha sho ening he pe iod be ween ac ions has a c ucial
impac on he abili y o s uc u e e ec i e incen i es.
3Sanniko and Sk zypacz (2007) show ha his is also he case in games be ween mul iple long- un play-
e s in which de ia ions be ween indi idual playe s a e indis inguishable.
4K eps, Milg om, Robe s, and Wilson (1982), K eps and Wilson (1982), and Milg om and Robe s (1982)
i s demons a ed ha epu a ion, in he o m o incomple e in o ma ion abou a playe ’s ype, has a d a-
ma ic e ec on equilib ium beha io . Maila h and Samuelson (2001) show ha epu a ional incen i es can
also come om a i m’s desi e o sepa a e i sel om an incompe en ype.
5Along hese lines, Dilmé (2019) shows ha adjus men cos s can help a i m o e come mo al haza d by
endogenously c ea ing pe sis ence.
6Long- un epu a ion e ec s a e also possible in models wi h beha io al ypes when consume s canno
obse e all pas signals (Ekmekci (2011)) o he ype o he i m is eplaced o e ime.
Theo e ical Economics 19 (2024) Dynamic mo al haza d game 453
pape explo es he gene al p ope ies o a s ochas ic game ha has powe ul in e em-
po al incen i es. The epu a ional game can be iewed as a speci ic ype o s ochas ic
game. Fo ins ance, i ins ead o in luencing he unce ain y abou whe he i is a beha -
io al ype, a s a egic i m makes a cos ly ini ial in es men in a new p oduc ion ech-
nology ha bene i s cus ome s oday and in he u u e, we obse e simila in e empo al
incen i es in he esul ing s ochas ic game.
This inal poin me i s a close compa ison wi h Faingold and Sanniko (2011), who
cha ac e ize he unique MPE in he s ochas ic game ha co esponds o a con inuous
ime epu a ion model. In hei pape , payo s and he e olu ion o he s a e ake a
speci ic o m due o Bayesian upda ing. My cha ac e iza ion builds on he echniques
in hei pape o unde s and mo e gene ally wha p ope ies o s ochas ic games a e
needed o uniqueness o MPE and nondegene a e in e empo al incen i es. I analyze
a gene al class o s ochas ic games ha places ew es ic ions on he p ocess go e ning
he e olu ion o he s a e and he s uc u e o payo s. The key echnical ad ancemen ,
ela i e o hei pape , is o he case o an unbounded s a e space and payo o he
la ge playe , as i equi es signi ican ly di e en echniques o comple e he analysis.
Beyond epu a ion models wi h beha io al ypes, a ich li e a u e analyzes dynamic
games wi h a s a e a iable in which e o is di ec ly linked o u u e payo s ia he s a e.
E icson and Pakes (1995) we e he i s o analyze hidden in es men and s ochas ic cap-
i al accumula ion ( he s a e) in a model ha is simila in spi i o he quali y example
p esen ed in Sec ion 2. They s udy i m and indus y dynamics, and es ablish equilib-
ium exis ence. Do aszelski and Sa e hwai e (2010)modi yE icson and Pakes (1995) o
gua an ee he exis ence o a pu e s a egy MPE, which is compu a ionally ac able. Nei-
he pape es ablishes uniqueness, bu ins ead ocus on he dynamics associa ed wi h a
pa icula MPE. Mo e b oadly, MPE is he wo kho se solu ion concep ac oss indus ial
o ganiza ion and poli ical economy. A comp ehensi e e iew o his li e a u e is beyond
he scope o his pape .
This pape also ela es o a li e a u e on s ochas ic games wi h an unobse able s a e.
In hese games, incen i es s em om he la ge playe ’s abili y o manipula e he public
belie abou he s a e h ough he e o choice. Cis e nas (2018) cha ac e izes necessa y
condi ions o he exis ence o Ma ko equilib ia in a con inuous ime s ochas ic game
wi h an unobse able s a e and su icien condi ions in wo mo e es ic i e classes o
games. Hidden s a es signi ican ly complica e he model, and i is no possible o es ab-
lish uniqueness esul s o a ull equilib ium cha ac e iza ion. Boa d and Meye - e -Vehn
(2013) s udy a se ing in which a i m’s hidden quali y depends on pas e o and con-
sume s lea n abou his quali y om noisy signals. My pape di e s in ocus in ha he e
is no ad e se selec ion, he e is s a egic in e ac ion be ween he la ge and small playe s,
and i allows o a iche class o s age game payo s.
Se e al olk heo ems exis o disc e e ime s ochas ic games wi h obse able s a es,
beginning wi h a pe ec moni o ing se ing in Du a (1995) and ex ending o impe -
ec moni o ing en i onmen s in Fudenbe g and Yamamo o (2011) and Hö ne , Takuo,
Sa o u, and Vieille (2011). My se ing di e s in ha he e is a single la ge playe and
in o ma ion ollows a di usion p ocess. I is al eady known ha hese wo changes sig-
ni ican ly al e incen i es in s anda d epea ed games (compa e he olk heo em in Fu-
denbe g, Le ine, and Maskin (1994) o he equilib ium degene acy in Fudenbe g and
454 J. Aislinn Boh en Theo e ical Economics 19 (2024)
Le ine (2007,2009) and Faingold and Sanniko (2011)). The in ui ion is simila o he
disc e e ime s ochas ic game olk heo ems compa ed o he MPE uniqueness esul in
his pape .7
The o ganiza ion o he pape p oceeds as ollows. Sec ion 2p esen s a p oduc
choice example o mo i a e he model. Sec ion 3se s up he model and cha ac e izes he
s uc u e o PPE. Sec ion 4p esen s he h ee main esul s: exis ence o a Ma ko equi-
lib ium, cha ac e iza ion o he PPE payo se , and uniqueness o a Ma ko equilib ium
in he class o all PPE. Sec ion 5p esen s s uc u al esul s on he shape o equilib ium
payo s. Sec ion 6explo es se e al applica ions. All p oo s a e p o ided in he Appendix.
2. Example 1: P oduc choice wi h pe sis en quali y
Conside a a ia ion o he canonical p oduc choice se ing in which a monopolis i m
p o ides a p oduc o consume s and he i m’s e o has a pe sis en e ec on he qual-
i y o he p oduc . A each ime , he i m chooses an unobse able e o le el a ∈[0, a],
whe e a>0. The quali y o he i m’s p oduc a ime depends on bo h cu en and
pas e o , q(a ,X )=(1−λ)a +λX , whe e pas e o in luences quali y h ough he
obse able s ock quali y
X =
0
e−θ( −s)(asds +dZs),
θ>0 de e mines he decay a e o pas e o , (Z ) ≥0is a s anda d B ownian mo ion,
and λ∈[0, 1]cap u es he ela i e impo ance o pas e o in de e mining cu en qual-
i y.8E o inc eases quali y bo h oday and in he u u e.
The e is a con inuum o iden ical consume s o uni mass. Consume s alue quali y:
when hey belie e ha he i m will choose e o le el ˜
a a ime , hey a e willing o pay
q(˜
a ,X ) o one uni o he p oduc . Each consume pu chases he p oduc o a p ice
equal o he willingness o pay when i is posi i e and o he wise does no pu chase.
The e o e, he i m ea ns a low e enue o b =q(˜
a ,X )when q(˜
a ,X )>0andb =0
o he wise. This exac o m o e enue is chosen o simplici y; he impo an ea u e is
ha he low e enue is inc easing in quali y and independen o he ue cu en e o
choice. E o has low cos a2
/2 and he i m discoun s a a e >0. The e o e, he i m’s
a e age discoun ed payo equals
∞
0
e− b −a2
/2d .
In he unique PPE wi h no pe sis ence, λ=0, he i m exe s ze o e o , quali y is
equal o ze o, and he i m ea ns ze o p o i ( his is a di ec applica ion o Theo em 3
7The pape also ela es o an olde li e a u e on s ochas ic games and he exis ence o Ma ko equilib ia
in disc e e ime, including Shapley (1953), Du a and Sunda am (1992), Nowak and Ragha an (1992), Du ie,
John Geanakoplos, and McLennan (1994).
8In a sligh abuse o no a ion, he Lebesgue in eg al and he s ochas ic in eg al a e placed unde he
same in eg al sign.
Theo e ical Economics 19 (2024) Dynamic mo al haza d game 455
om Faingold and Sanniko (2011)). In e empo al incen i es b eak down, despi e he
ac ha he i m would ea n highe p o i s i i could commi o highe e o .9
In his pape , I show ha pe sis en quali y incen i izes he i m o choose a posi i e
le el o e o and ea n posi i e p o i s. Theo ems 1 o 3es ablish ha he e is a unique
PPE, which is Ma ko in he s ock quali y X . The e o le el and p o i in his unique
equilib ium a e cha ac e ized as a unc ion o he impac o pas e o on cu en qual-
i y λ, he dep ecia ion a e o quali y θ, and he discoun a e .Fo anyλ>0, he i m
chooses a posi i e le el o e o and ea ns posi i e p o i s a posi i e and some (pos-
sibly all) nega i e le els o s ock quali y. Fu he , he i m has a long- un incen i e o
choose high e o . This con as s wi h models in which he incen i e o p oduce high
quali y is de i ed om consume s’ unce ain y o e he i m’s payo s and long- un e -
o con e ges o ze o (C ipps, Maila h, and Samuelson (2004), Faingold and Sanniko
(2011)).
Pe sis ence inc eases he i m’s payo s h ough wo complemen a y s uc u al
channels. Fi s , he i m’s e o inc eases he s ock quali y, which inc eases u u e e -
enue h ough i s impac on u u e p ices. This is he di ec e ec o pe sis ence, as
discussed in he In oduc ion. Second, pe sis ence c ea es a link wi h u u e payo s,
which allows he i m o c edibly choose a posi i e le el o e o oday, he eby inc eas-
ing he cu en p ice, and hence, e enue. This second channel a ises om he s a egic
in e ac ion be ween he i m and consume s: i is he equilib ium e ec discussed in
he In oduc ion. When quali y is high, he con inua ion alue is app oxima ely linea
and i is possible o quan i y he sha e o p o i a ising om each o hese channels. The
p esen alue o he di ec e ec minus he cos o e o is app oxima ely λ2/2( +θ)2,
which is highe when pas e o plays a la ge ole in de e mining cu en quali y (highe
λ), quali y dep ecia es a a lowe a e (lowe θ), o he i m is mo e pa ien (lowe ). The
p esen alue o he equilib ium e ec is app oxima ely (1−λ)λ/( +θ),whichisalso
highe when quali y dep ecia es a a lowe a e o he i m is mo e pa ien . In con as
o he di ec e ec , he equilib ium e ec is la ges o in e media e alues o λ.Thisis
because he incen i e o exe e o is inc easing in λwhile he impac o e o on he
cu en p ice is inc easing in 1 −λ.
This example will be used h oughou he pape o demons a e he esul s. The
p oduc choice amewo k lends i sel o o he a ia ions, se e al o which a e discussed
in Sec ion 6.1.
3. Model
3.1 Model se up
S a es and ac ions A la ge playe and a con inuum I≡[0, 1]o iden ical small play-
e s, indexed by i, play a con inuous ime s ochas ic game wi h impe ec moni o ing.
A each ins an o ime ∈[0, ∞), a publicly obse able s a e a iable X in nonemp y
closed in e al X⊂Rde e mines he ac ion se and easible low payo s. I Xis
9In con as o Ab eu, Milg om, and Pea ce (1991) and Sanniko and Sk zypacz (2007), his b eakdown
o incen i es akes place despi e he e being no ailu e o iden i iabili y.
456 J. Aislinn Boh en Theo e ical Economics 19 (2024)
bounded, deno e he uppe and lowe bounda y s a es by X≡supXand X≡in X, e-
spec i ely, and assume X0∈(X,X). La ge and small playe s simul aneously choose
ac ions a om Aand bi
om B(X ), espec i ely, whe e Ais a nonemp y compac sub-
se o a Euclidean space and B(X)is a nonemp y compac subse o a closed Euclidean
space Bwi h con inuous co espondence X→ B(X). Deno e he se o easible pai s
o small playe ac ions and s a es as E≡{(b,X)∈B×X|b∈B(X)}. Assume ha he
bounda y o he easible se o ac ions o small playe s g ows a mos linea ly wi h he
s a e; ha is, he e exis s a Kb,cb>0such ha o all(b,X)∈E,|b|≤Kb|X|+cb.10 Indi-
idual ac ions a e p i a ely obse ed. Playe s obse e he agg ega e dis ibu ion o small
playe s’ ac ions, b ∈B(X ), and do no obse e he la ge playe ’s ac ion.
Gi en ini ial s a e X0, he s a e e ol es s ochas ically acco ding o
dX =μ(a ,b ,X )d +σ(b ,X )dZ ,(1)
whe e (Z ) ≥0is a one-dimensional B ownian mo ion, and he d i and ola ili y a e
de e mined by Lipschi z con inuous unc ions μ:A×E→Rand σ:E→R,whicha e
linea ly ex ended o A×{(b,X)∈B ×X|supp b⊂B(X)}and {(b,X)∈B ×X|supp b⊂
B(X)}, espec i ely.11 The d i depends on he la ge playe ’s ac ion, he agg ega e ac-
ion o he small playe s, and he s a e. Vola ili y is independen o he la ge playe ’s
ac ion o main ain he assump ion ha i is no pe ec ly obse ed. I he s a e space
is bounded, hen o p e en he s a e om escaping i s bounda y and main ain im-
pe ec moni o ing a he bounda y, he ola ili y mus be ze o a he uppe and lowe
bounds, σ(b,X)=0 o allb∈B(X)and σ(b,X)=0 o allb∈B(X), and he d i mus
be weakly nega i e a he uppe bound, weakly posi i e a he lowe bound, and in-
dependen o (a,b)a bo h bounds, μ(a,b,X)=m≤0 o all(a,b)∈A×B(X)and
μ(a,b,X)=m≥0 o all(a,b)∈A×B(X). To ensu e ha he u u e pa h o he s a e is
s ochas ic, excep a bounda y s a es, assume ha i s ola ili y is posi i e a all in e io
s a es.
Assump ion 1 (Posi i e Vola ili y). When X=R,in Eσ(b,X)>0.WhenXis compac ,
he e exis s a C>0such ha σ(b,X)≥C(X−X)(X−X) o all (b,X)∈E.
This assump ion ules ou in e io abso bing s a es, whe e s a e Xis abso bing i
he d i and ola ili y a e equal o ze o, μ(a,b,X)=0andσ(b,X)=0 o all(a,b)∈
A×B(X).
The pa h o he s a e p o ides a public signal o he la ge playe ’s ac ion. The e a e no
addi ional public signals. This is wi hou loss o gene ali y, as addi ional public signals
ha e no e ec on he equilib ium cha ac e iza ion (see he discussion in Sec ion 3.3).
Le (F ) ≥0 ep esen he il a ion gene a ed by he public in o ma ion (X ) ≥0.Small
playe s obse e no in o ma ion abou he la ge playe ’s ac ion beyond wha is con ained
in (F ) ≥0.
10Iuse|·| o deno e he Euclidean no m o ec o s..
11Func ions μand σa e ex ended o dis ibu ions as B(X)μ(a,b,X)db(b)and B(X)σ(b,X)2db(b).
Theo e ical Economics 19 (2024) Dynamic mo al haza d game 463
4.1 Exis ence o Ma ko equilib ia
In a Ma ko equilib ium, he con inua ion alue and ac ions depend solely on he cu -
en alue o he s a e; hey a e independen o he pas pa h o he s a e. Since he pa h
o he s a e p o ides a signal o he la ge playe ’s ac ion, using i o punish o ewa d he
la ge playe could gi e ise o PPE in which di e en pa hs o he s a e speci y di e en
con inua ion payo s and equilib ium ac ions, e en when hese pa hs map o he same
cu en s a e. In a Ma ko equilib ium, his is no allowed.
Theo em 1es ablishes exis ence o a Ma ko equilib ium and cha ac e izes equilib-
ium beha io and payo s in Ma ko equilib ia. The con inua ion alue is cha ac e ized
as he solu ion(s) U:X→R o an o dina y di e en ial equa ion ha maps each s a e o
a payo . I he e a e mul iple solu ions, hen each solu ion cha ac e izes a Ma ko equi-
lib ium (see Sec ion 6.3 o an illus a ion o a se ing wi h mul iple Ma ko equilib ia).
Gi en a solu ion U, he co esponding Ma ko equilib ium ac ion p o ile is he sequen-
ially a ional ac ion p o ile a s a e Xand incen i e weigh U(X)/ . The la ge playe
has nondegene a e incen i es a any s a e wi h U(X)= 0.
Theo em 1. Assume Assump ions 1 o 3. Gi en ini ial s a e X0,i Uisasolu ion o he
op imali y equa ion
U(X)= g∗X,U(X)+U(X)μ∗X,U(X)+1
2U(X)σ∗X,U(X)2(7)
on X(on (X,X)i Xis compac ) and Uhas linea g ow h (is bounded i gis bounded),
hen Ucha ac e izes a Ma ko equilib ium wi h he ollowing payo s and ac ions:
(i) Equilib ium payo U(X0).
(ii) Con inua ion alues (W ) ≥0=(U(X )) ≥0.
(iii) Equilib ium ac ions (a ,b ) ≥0=(S∗(X ,U(X ))) ≥0,whe eS∗(X,U(X)) is he
unique solu ion o (6)a s a e Xand incen i e weigh U(X)/ .
The op imali y equa ion has a leas one wice con inuously di e en iable solu ion ha
lies in he ange o easible payo s o he la ge playe and has linea g ow h (is bounded
i gis bounded). Thus, he e exis s a leas one Ma ko equilib ium.
F om he op imali y equa ion, he con inua ion alue U(X)is equal o he sum o
he equilib ium low payo g∗(X,U(X)) and he expec ed change in he con inua ion
alue. This expec ed change has wo componen s: (i) he in e ac ion be ween he slope
o he con inua ion alue and he d i o he s a e, U(X)μ∗(X,U(X))/ , and (ii) he in-
e ac ion be ween he conca i y o he con inua ion alue and he ola ili y o he s a e,
U(X)σ∗(X,U(X))2/2 .
In ela ion o he discussion in Sec ion 3.3, a non i ial incen i e weigh is possi-
ble a some s a es wi hou he con inua ion alue escaping he payo se . Theo em 1
shows ha he ola ili y o he con inua ion alue in a Ma ko equilib ium is equal o i s
slope, β =U(X ). A any in e io s a e X∗ ha yields he maximum con inua ion alue
464 J. Aislinn Boh en Theo e ical Economics 19 (2024)
ac oss all s a es, U(X∗)=0. The e o e, when X =X∗, he ola ili y o he con inua ion
alue is ze o, β =0, which ensu es ha he con inua ion alue does no escape he
payo se . In hese pe iods, he la ge playe ac s myopically and ea ns he s a ic Nash
payo in s a e X∗. A o he s a es, he con inua ion alue can be sensi i e o changes in
he s a e, U(X)= 0, gene a ing non i ial incen i es.
Ou lineo p oo In a Ma ko equilib ium, con inua ion alues ake he o m o W =
U(X ) o some unc ion U. Assuming ha Uis wice con inuously di e en iable, by
I o’s o mula he con inua ion alue mus ollow he law o mo ion,
dU(X )=U(X )μa∗
,b∗
,X d +1
2U(X )σb∗
,X 2d +U(X )σb∗
,X dZ .
By Lemma 1, he con inua ion alue mus also ollow he law o mo ion in (5). Ma ching
he d i s o hese wo laws o mo ion yields he op imali y equa ion, while ma ching
he ola ili ies yields he equilib ium ola ili y o he con inua ion alue, β =U(X ).
Showing ha he op imali y equa ion has a leas one wice con inuously di e en iable
solu ion ha lies in he ange o easible payo s o he la ge playe es ablishes exis ence.
Faingold and Sanniko (2011) ollow simila s eps o de i e a Ma ko equilib ium in
a game o incomple e in o ma ion. Rela i e o hei de i a ion, he inno a i e pa o
my p oo lies in es ablishing exis ence o a solu ion o he op imali y equa ion when he
s a e space is unbounded, pa icula ly when gis also unbounded. I show by cons uc-
ion ha he e exis lowe and uppe solu ions o he op imali y equa ion, α:X→Rand
α:X→R, ha ha e linea g ow h. This is only possible when he maximum d i o he
s a e has linea g ow h a a e less han (Assump ion 2). The lowe and uppe solu ions
cha ac e ize bounds on he solu ion o he op imali y equa ion, α(X)≤U(X)≤α(X)
o all X. Nex I show ha he bound on he op imali y equa ion g ows linea ly wi h
espec o U(X)and, he e o e, he op imali y equa ion does no g ow oo quickly
( echnically speaking, i sa is ies a g ow h condi ion on any compac subse o he s a e
space). These condi ions es ablish ha he op imali y equa ion has a wice con inuously
di e en iable solu ion wi h linea g ow h. When gis bounded, he lowe and uppe so-
lu ions a e cons an , which es ablishes exis ence o a bounded solu ion.
The inal s ep is o show ha he con inua ion alue and ac ions cha ac e ized abo e
cons i u e a Ma ko equilib ium. Gi en a solu ion U(X)and an ac ion p o ile uniquely
speci ied a s a e X by (a∗
,b∗
)=S∗(X ,U(X )) (whe e uniqueness ollows om As-
sump ion 3), he s a e a iable e ol es uniquely acco ding o (1), he con inua ion alue
(U(X )) ≥0sa is ies he law o mo ion (5), and he ac ion p o ile sa is ies he condi ions
o sequen ial a ionali y (6). The e o e, (a∗
,b∗
,U(X )) cons i u e a PPE.
Fo a gi en solu ion U(X), he s a e e ol es uniquely and ac ions a e uniquely spec-
i ied as a unc ion o he s a e. The e o e, each solu ion o he op imali y equa ion cha -
ac e izes a unique Ma ko equilib ium. I he e a e mul iple solu ions, hen he e will be
mul iple Ma ko equilib ia.
Example 1 (P oduc Choice, con .). Gi en a(X,z)and b(X,z)cha ac e ized in Sec-
ion 3.2, any solu ion o
U(X)= bX,U(X)−
2aX,U(X)2+U(X)aX,U(X)−θX+1
2U(X)(8)
Theo e ical Economics 19 (2024) Dynamic mo al haza d game 465
wi h linea g ow h as X→∞and bounded as X→−∞cha ac e izes a Ma ko equilib-
ium wi h equilib ium ac ions a(X,U(X)) and b(X,U(X)).♦
4.2 The PPE payo se
Le ξ:X⇒Rdeno e he co espondence ha maps each s a e on o he co esponding
se o PPE payo s o he la ge playe , and le ϒ:X⇒Rdeno e he analogous co e-
spondence o he Ma ko equilib ium payo s cha ac e ized by he op imali y equa ion
in Theo em 1.Theo em2shows ha in any PPE, he la ge playe canno achie e a payo
abo e he highes o below he lowes Ma ko equilib ium payo in ϒ.
Theo em 2. Assume Assump ions 1 o 3. Then o any s a e X∈X(s a e X∈(X,X)i X
is compac ), he se o PPE payo s o he la ge playe a s a e Xis equal o he con ex hull
o he se o Ma ko equilib ium payo s a s a e X,ξ(X)=co(ϒ(X)).
The impossibili y o he la ge playe achie ing a PPE payo abo e he highes
Ma ko payo in ϒyields insigh in o he ype o incen i es gene a ed by pe sis ence.
As discussed in he In oduc ion and Sec ion 3.3, incen i es can be ei he in o ma ional
o s uc u al. When a Ma ko equilib ium yields he highes equilib ium payo , i p e-
cludes he exis ence o equilib ia ha achie e highe payo s using in o ma ional incen-
i es. The e o e, any non i ial incen i es a ising om pe sis ence a e s uc u al.
Ou lineo p oo The key a gumen in he p oo shows ha any PPE wi h an ini ial pay-
o abo e he highes Ma ko equilib ium payo in ϒwill e en ually yield a con inua ion
alue ha lies ou side he se o easible payo s o he la ge playe , which is a con a-
dic ion. Suppose ha a PPE wi h con inua ion alues (W ) ≥0yields a payo highe han
he maximum Ma ko equilib ium payo in ϒa s a e X0.Le D ≡W −U(X )be he
di e ence be ween he con inua ion alues in hese wo equilib ia a ime .Ishow ha
whene e D0>0, D will g ow a bi a ily la ge wi h posi i e p obabili y, independen
o X . By Lemma 1,|W (S)|is bounded wi h espec o X .Thus,D can only g ow
a bi a ily la ge when X g ows a bi a ily la ge, so i canno be ha D0>0.
This escape a gumen is simila o o he pape s in he li e a u e, in pa icula
Faingold and Sanniko (2011). Thei p oo elies on he compac ness o he s a e space
o show ha he ola ili y o D is bounded away om ze o and elies on he bounded-
ness o he low payo o each a con adic ion when D g ows a bi a ily la ge. The e-
o e, hei p oo s do no i ially ex end o an unbounded s a e space o an unbounded
low payo . The inno a i e pa s o his p oo a e o es ablish ha he ola ili y o D is
bounded away om ze o on an unbounded s a e space and o show ha when D g ows
a bi a ily la ge, i can jump ou side o he easible payo se (a con adic ion) p o ided
he s a e does no g ow oo quickly.
Equilib ium degene acy wi hou pe sis en ac ions I he s a e e ol es independen ly
o he la ge playe ’s ac ion, hen he e is no link be ween he cu en ac ion and he
con inua ion alue. I is no possible o gene a e e ec i e in e empo al incen i es and
he la ge playe ac s myopically. In he unique PPE, bo h playe s play he s a ic Nash
equilib ium ac ion p o ile S∗(X,0
)a all s a es X.
466 J. Aislinn Boh en Theo e ical Economics 19 (2024)
Co olla y 1. Assume Assump ions 1 o 3and suppose μis independen o a o all X.
Then in he unique PPE, (a ,b )=S∗(X ,0
) o all ≥0and he con inua ion alue is
cha ac e ized by he unique solu ion o he op imali y equa ion (7).
This is he s ochas ic game analogue o he equilib ium degene acy esul in e-
pea ed games wi h a long- un playe and sho - un/small playe s (Fudenbe g and
Le ine (2007,2009), Faingold and Sanniko (2011)).
4.3 Equilib ium uniqueness
This sec ion es ablishes su icien condi ions o he e o be a unique PPE, which is
Ma ko . The main s ep is o de e mine when he op imali y equa ion has a unique easi-
ble solu ion. When his is he case, Theo em 2es ablishes ha PPE payo s a e uniquely
speci ied as he payo s in his unique Ma ko equilib ium. The beha io o he op imal-
i y equa ion as he s a e app oaches i s bounda y plays a key ole in es ablishing when
i has a unique solu ion. Any wo easible solu ions ha sa is y he same bounda y con-
di ions canno di e on he in e io o he s a e space: hey mus be equi alen (see
Lemma 7in Appendix A.4). The e o e, es ablishing ha all easible solu ions sa is y he
same bounda y condi ions is necessa y and su icien o es ablish a unique solu ion. I
ou line a se o su icien condi ions o gua an ee his when X=R; he case o a compac
s a e space equi es no addi ional condi ions. The applica ion in Sec ion 6.3 illus a es
how mul iple Ma ko equilib ia can a ise when his condi ion ails.
4.3.1 Unbounded s a e space (X=R)Assump ion 4(below) ou lines a se o su icien
condi ions o a unique Ma ko equilib ium when X=R. The i s condi ion equi es
he la ge playe ’s equilib ium low payo and he equilib ium d i o be addi i ely sep-
a able in he s a e Xand incen i e weigh zas Xapp oaches ∞and −∞.This ulesou
complemen a i ies be ween he di ec and equilib ium channels o incen i es nea he
bounda y, which p e en s mul iple equilib ium incen i e weigh s—and hence, equilib-
ium ac ion p o iles—a a gi en s a e. I is used o es ablish ha he slope o he con in-
ua ion alue con e ges o he same limi in all Ma ko equilib ia. The second condi ion
ela es o he ola ili y: i is a echnical condi ion ha helps es ablish ha wo dis inc
solu ions o he op imali y equa ion canno ha e he same limi slope. The hi d condi-
ion applies o a g ow h model whe e he d i o he s a e app oaches in ini y as X→∞
(o app oaches nega i e in ini y as X→−∞); i ensu es ha he ola ili y does no also
g ow a bi a ily la ge. I is also used o pin down a unique bounda y con inua ion alue.
Assump ion 4.
(i) Addi i e Sepa abili y Nea Bounda y. The e exis s a δ>0and con inuously di e -
en iable unc ions g1,μ1:X→Rand g2,μ2:R→Rwi h μ1mono one such ha
o |X|>δ,g∗(X,z)=g1(X)+g2(z)and μ∗(X,z)=μ1(X)+μ2(z).
(ii) Vola ili y. The unc ion σ∗(X,z)2is Lipschi z con inuous.
Theo e ical Economics 19 (2024) Dynamic mo al haza d game 467
(iii) G ow h Case. When limX→∞ μ1(X)=∞, hen he e exis s an ε,δ>0such ha o
X>δand z∈R,|μ1(X)|/σ∗(X,z)2>ε, and simila ly when limX→−∞ μ1(X)=
−∞.23
Gi en Assump ion 4(i), selec g1(X)and g2(z)such ha g2(z)con ains any cons an
e m in g∗(X,z) o uniquely pin down each unc ion, and simila ly o μ1(X)and μ2(z).
When gis bounded, i is possible o es ablish uniqueness wi hou addi i e sepa abili y;
Assump ion 5 in Supplemen al Appendix D.3 (a ailable a h p://econ heo y.o g/supp/
2680/supplemen .pd ) p esen s an al e na i e condi ion.
Theo em 3es ablishes uniqueness and cha ac e izes he limi o he con inua ion
alue and i s slope as he s a e g ows la ge.
Theo em 3. Suppose X=Rand assume Assump ions 1 o 4. Fo each ini ial s a e X0∈
X, he e exis s a unique PPE ha is Ma ko and cha ac e ized by he unique solu ion Uo
(7)on Xwi h linea g ow h (bounded when gis bounded). The slope o he con inua ion
alue con e ges o a cons an ,
lim
X→xU(X)=zxwhe e zx≡lim
X→x g1(X)/ X −μ1(X),(9)
and he con inua ion alue con e ges o
lim
X→xU(X)−y(X)=g2(zx)+zxμ2(zx)/ (10)
o x∈{−∞,∞},whe ey(X)≡−φ(X)( g1(X)/φ(X)μ1(X))dX and φ(X)≡
exp(( /μ1(X))dX)when limX→xμ1(x)= 0,andy(X)≡g1(X)when limX→xμ1(X)=
0.Whengis bounded, his implies he con inua ion alue con e ges o he limi s a ic
Nash equilib ium payo and he slope o he con inua ion alue con e ges o ze o: o
x∈{−∞,∞},
lim
X→xU(X)−g∗(X,0
)=0and lim
X→xU(X)=0. (11)
Theo em 3es ablishes ha he slope o he con inua ion alue con e ges o a
unique limi slope, which is equal o he a io o he g ow h a e o he low payo
o he g ow h a e o he d i wi h espec o he s a e. Gi en his slope, he bound-
a y condi ion (10) highligh s he impac o s uc u al incen i es on he con inua ion
payo . Repea ed play o he s a ic Nash equilib ium p o ile yields a payo UNE ha
sa is ies limX→xUNE(X)−y(X)=g2(0)+zxμ2(0)/ . The e o e, om (10), he con-
inua ion alue app oaches he sum o his epea ed s a ic Nash payo and a con-
s an g2(zx)−g2(0)+zx(μ2(zx)−μ2(0))/ . This cons an de e mines he ex en o
which s uc u al incen i es pe sis a he bounda y o he s a e space. The i s e m,
g2(zx)−g2(0), cap u es he equilib ium e ec o pe sis ence. I is he po ion o he
equilib ium low payo ha a ises om u u e s a egic in e ac ion; i cap u es he e -
ec o he la ge playe ’s ac ion on he small playe s’ ac ions, ne o he cos o a.The
23Pa (ii) is unnecessa y when gis bounded. No e ha pa (iii) holds i ially when σ(b,X)is bounded.
468 J. Aislinn Boh en Theo e ical Economics 19 (2024)
second e m, zx(μ2(zx)−μ2(0))/ , cap u es he di ec e ec o pe sis ence on u u e
easible payo s, measu ed by how he con inua ion alue changes wi h espec o he
s a e and how he s a e changes wi h espec o he la ge playe ’s equilib ium ac ion el-
a i e o he s a ic Nash ac ion. I his cons an is posi i e, hen as he s a e becomes
la ge, s uc u al incen i es p o ide he la ge playe wi h a payo ha is s ic ly highe
han he payo om playing he s a ic Nash p o ile a each s a e.
When he asymp o ic slope zxis nonze o, i is possible o sus ain non i ial in e em-
po al incen i es as he s a e g ows la ge. This is an impo an and no el insigh o his
pape . I i is possible o sus ain non i ial incen i es a he bounda y o he s a e space,
hen incen i es a e pe manen in he sense ha hey do no dissipa e wi h ime, ega d-
less o he asymp o ic beha io o he s a e wi h espec o ime. In he case o a bounded
low payo , zx=0. The e o e, incen i es collapse a he bounda y and he con inua ion
alue con e ges o he limi o he s a ic Nash payo . Howe e , his does no p eclude
he exis ence o long- un incen i es: e en when incen i es collapse a he bounda y, he
s a e does no necessa ily con e ge o a bounda y s a e as →∞. The e o e, i can be
possible o sus ain non i ial incen i es in he long un.
The con inua ion o Example 1below illus a es how o e i y Assump ion 4and de-
i e he bounda y condi ions in Theo em 3when he low payo is unbounded, while
Sec ion 6.1 illus a es how o do so o a bounded low payo . Sec ion 6.3 shows ha
he e can be mul iple MPE in an applica ion in which Assump ion 4(speci ically, addi-
i e sepa abili y) ails.
Ou line o p oo I i s show ha all solu ions o he op imali y equa ion ha e he same
bounda y condi ions. Faingold and Sanniko (2011) also cha ac e ize bounda y con-
di ions as a s ep owa ds es ablishing ha he op imali y equa ion in hei pape has a
unique solu ion. Rela i e o hei esul , he inno a i e pa o my p oo is in es ablishing
bounda y condi ions o an unbounded low payo and s a e space, as I nex desc ibe.
Le ψ(X,z)≡g∗(X,z)+zμ∗(X,z)/ be he sum o he la ge playe ’s low payo and e-
u n on e o a he sequen ially a ional ac ion p o ile (a(X,z),b(X,z)),andle U(X)
be a solu ion o he op imali y equa ion. Suppose ha U(X)does no con e ge as X→
∞. Then o any slope zsuch ha he con inua ion alue has slope zin ini ely o en a
la ge X,U(X)will al e na e be ween being con ex and conca e a slope z.F om heop-
imali y equa ion, ψ(X,z)will lie abo e U(X)when i is conca e a slope zand will lie
below U(X)when i is con ex a slope z. The e o e, he oscilla ion o ψ(X,U(X)) is a
leas as la ge as he oscilla ion o U(X). This iola es he mono onici y o ψ,soi mus
be ha U(X)has a limi z∞∈R. Since U(X)has linea g ow h (by Theo em 1), his limi
mus be ini e. Mo eo e , i is equal o limX→∞ U(X)/X. Gi en addi i e sepa abili y, as
well as he Lipschi z con inui y and mono onici y o μ1and g1, he limi s o ψ(X,z)/X
and ψ(X,z)exis and a e equal as X→∞. Deno e hese limi s by ψ∞(z).Weuse hese
p ope ies and he op imali y equa ion o show ha limX→∞ σ∗(X,U(X))2U(X)/X =
0 and, he e o e, limX→∞ U(X)/X −ψ(X,U(X))/X =0. This es ablishes ha he
limi slope z∞is a ixed poin o ψ∞(z). The addi i ely sepa able assump ion on g∗
and μ∗issu icien oensu e ha ψ∞(z)has a unique ixed poin , which is equal o
z∞=limX→∞ g1(X)/( X −μ1(X)). This gua an ees ha all solu ions o he op imali y
equa ion ha e he same limi slope.
Theo e ical Economics 19 (2024) Dynamic mo al haza d game 469
Using he cha ac e iza ion o he limi slope, i can be shown ha any solu ion
U(X) o he op imali y equa ion sa is ies limX→∞ U(X)−U(X)μ1(X)/ −g1(X)=
g2(z∞)+z∞μ2(z∞)/ . Conside he linea i s -o de di e en ial equa ion (FODE)
y(X)−y(X)μ1(X)/ −g1(X)=0. Es ablishing ha any solu ion U(X)sa is ies
limX→∞ U(X)−y(X)=g2(z∞)+z∞μ2(z∞)/ o any linea g ow h solu ion y o his
FODE yields he bounda y condi ion o U(X)(i.e., (10)). The e o e, all solu ions o he
op imali y equa ion app oach he same alue and slope as he s a e g ows la ge o small.
Finally, I show ha any wo such solu ions Uand Vcanno di e on he in e io o
he s a e space. Simila o Faingold and Sanniko (2011), i he e exis s an Xsuch ha
U(X)−V(X)>0, he s uc u e o he op imali y equa ion p e en s hese solu ions om
sa is ying he same bounda y condi ions o a leas one bounda y.
Example 1 (P oduc Choice, con .). This example sa is ies Assump ion 4.F om he
cha ac e iza ion in Sec ion 3.2, he sequen ially a ional e o a(X,z)is independen
o Xand he consume s’ willingness o pay b(X,z)is addi i ely sepa able in (X,z).
The e o e, he low payo g∗(X,z)=b(X,z)−a(X,z)2/2and hed i μ∗(X,z)=
a(X,z)−θX a e addi i ely sepa able in (X,z). F om hese exp essions, g1(X)=λX
o X>0andg1(X)=0 o X<0, while μ1(X)=−θX o all X, which is mono one.
Finally, σ∗(X,z)2=1 i ially sa is ies Lipschi z con inui y, and he g ow h condi ion is
no ele an since limX→∞ μ1(X)=−∞and simila ly o X→−∞.
F om Theo em 3, he limi slopes a e z∞= λ/( +θ)and z−∞ =0. The e o e, equi-
lib ium e o app oaches a(X,z∞)=λ/( +θ)as Xg ows la ge, which is s ic ly posi-
i e. As discussed in Sec ion 2, his con as s wi h se ings in which e o does no ha e
a pe sis en e ec on quali y and long- un e o con e ges o ze o (C ipps, Maila h, and
Samuelson (2004), Faingold and Sanniko (2011)). F om (10), o la ge X he con inua-
ion alue app oxima es
U(X)≈ λ
+θX+(1−λ)λ
+θ+λ2
2( +θ)2,
whe e he i s e m is he payo om epea ed play o he s a ic Nash equilib ium p o-
ile, and he second and hi d e ms cap u e he impac o s uc u al incen i es on he
equilib ium payo : he equilib ium e ec o pe sis ence s emming om u u e s a e-
gic in e ac ion be ween he i m and consume s, and he di ec e ec o pe sis ence
on u u e payo s ia he s ock quali y, espec i ely.24 In con as , as Xapp oaches
−∞, equilib ium e o app oaches ze o and he con inua ion alue con e ges o ze o,
limX→−∞ U(X)=0. The e o e, a la ge nega i e alues o he s a e, incen i es col-
lapse. ♦
24Gi en he exp ession o a(X,z)abo e, g2(z)=(1−λ)a(X,z)−a(X,z)2/2 and μ2(z)=a(X,z), he
cons an on he igh hand side o (10)is(1−λ)λ/( +θ)+λ2/2( +θ)2. The payo om epea ed play
o he s a ic Nash equilib ium p o ile, y(X)= λX/( +θ), is calcula ed om he exp ession o y(X)in
Theo em 3, using he exp essions o g1(X)and μ1(X)abo e and φ(X)=exp(−( /θX)dX)=X− /θ.
470 J. Aislinn Boh en Theo e ical Economics 19 (2024)
4.3.2 Bounded s a e space (Xcompac ) When Xis compac , uniqueness ollows om
Assump ions 1 o 3. No addi ional condi ions a e needed as in Theo em 3, as Lipschi z
con inui y oge he wi h he condi ions on he d i and ola ili y ha p e en he s a e
om escaping i s bounda y (i.e., posi i e d i and ze o ola ili y a X, and analogously
o X) es ablish ha he la ge playe plays a unique ac ion a he bounda y and pin down
a unique bounda y con inua ion alue. Theo em 4es ablishes uniqueness when he
s a e space is compac , and cha ac e izes he limi o he con inua ion alue and he
la ge playe ’s incen i e cons ain .25
Theo em 4. Suppose Xis compac and assume Assump ions 1 o 3. Fo each ini ial
s a e X0∈X, he e exis s a unique PPE ha is Ma ko and cha ac e ized by he unique
bounded solu ion Uo (7)on (X,X). When he bounda y s a es a e abso bing, he con-
inua ion alue con e ges o he s a ic Nash equilib ium payo and in e empo al incen-
i es collapse a he bounda y,
lim
X→xU(X)−g∗(X,0
)=0and lim
X→xμ∗X,U(X)U(X)=0 (12)
o x∈{X,X}. When he bounda y s a es a e no abso bing,
lim
X→XU(X)=g∗(X,0
)+mu/ and lim
X→X
U(X)=g∗(X,0
)+mu/ (13)
gi en unique ini e limi slopes u≡limX→XU(X)and u≡limX→XU(X).
The con inua ion alue a a bounda y s a e depends on whe he he bounda y s a e
is abso bing o e lec ing. When he bounda y is abso bing, he s a e emains a he
bounda y once i is eached and, he e o e, he con inua ion alue con e ges o he
s a ic Nash payo . When he bounda y is e lec ing, he limi o he con inua ion
alue also depends on i s (unique) limi slope and he bounda y d i , which cap u es
how quickly he s a e mo es away om he bounda y and how he con inua ion alue
changes as he s a e changes. In ei he case, he impac o he long- un playe ’s ac ion
on he d i o he s a e con e ges o ze o a he bounda y. The e o e, incen i es col-
lapse and he equilib ium ac ion p o ile con e ges o he s a ic Nash ac ion p o ile. This
ules ou he possibili y o sus aining mul iple equilib ium ac ion p o iles a he bound-
a y, a key s ep in es ablishing uniqueness. An impo an di e ence om Theo em 3is
ha incen i es collapse e en i he slope o he con inua ion alue does no con e ge
o ze o. This s ems om he equi emen ha he bounda y d i is independen o he
25Theo ems 1,2, and 4also hold o an al e na i e e sion o Assump ion 3when he s a ic Nash pay-
o g∗(X,0
)is inc easing in X. Speci ically, assume ha he es ic ion o S∗ o X×[0, ∞)is nonemp y,
is single- alued, and e u ns ¯
b=δb o some b∈B(X),whe eδbis he Di ac measu e on ac ion b,S∗is
Lipschi z con inuous on e e y bounded subse o X×[0, ∞)when Xis compac and on X×[0, ∞)when
X=R, and when X=R, he eexis saδ>0such ha o all|X|>δand z∈[0, ∞), he a e o change o
g(S∗(X,z),X)+zμ(S∗(X,z),X)/ wi h espec o Xis mono one in Xand σ(S∗(X,z),X)is mono one in
Xand cons an in z. Change he de ini ion o S∗ o se S∗(X,z)=S∗(X,0
) o z<0. By P oposi ion 2, he
solu ion U(X)is inc easing, and he alues o z<0 a e i ele an . An analogous es ic ion o (−∞,0
]is
possible when g∗(X,0
)is dec easing in X.
Theo e ical Economics 19 (2024) Dynamic mo al haza d game 471
long- un playe ’s ac ion—in o de o main ain impe ec moni o ing when he ola ili y
is ze o—combined wi h con inui y as he d i app oaches i s bounda y. As in Theo-
em 3, when incen i es collapse a he bounda y, his does no p eclude he exis ence
o long- un incen i es, as he s a e does no necessa ily con e ge o a bounda y s a e as
→∞.Sec ion6.2 p o ides an illus a ion o Theo em 4.
5. P ope ies o equilib ium payo s
The op imali y equa ion yields ich insigh s in o how he co espondence o PPE payo s
is ied o he unde lying s uc u e o he game. P oposi ions 1and 2show ha he shape
o he s a ic Nash equilib ium payo g∗(X,0
)is a key de e minan o he shape o he
Ma ko equilib ium con inua ion alue. No e ha g∗(X,0
)is s aigh o wa d o de i e
om he p imi i es o he game.
P oposi ion 1 ela es he numbe and ype o ex ema o U(X) o he shape o
g∗(X,0
). Gi en a solu ion U(X) o he op imali y equa ion, de ine an in e al mini-
mum o Uon a closed p ope in e al I⊂Xas [Xa,Xb]⊂in Isuch ha U(X)=0
o all X∈[Xa,Xb],and he eexis sanε>0such ha U(Xa)<U
(X) o all X∈
(Xa−ε,Xa)∪(Xb,Xb+ε), wi h an analogous de ini ion o in e al maximum.26
P oposi ion 1. Assume Assump ions 1 o 3.Le I⊂Xdeno e a closed p ope in e al
o s a es and le U(X)deno e a linea g ow h o bounded (when gbounded) solu ion o
(7).
(i) I g∗(X,0
)is cons an on I, henU(X)has a mos one in e al ex emum on I.
(ii) I g∗(X,0
)is s ic ly mono one on I, henU(X)has a mos wo in e al ex ema
on Iand is no cons an on I.I g∗(X,0
)is s ic ly inc easing (dec easing) on I,
and U(X)has an in e al minimum [X1a,X1b]and maximum [X2a,X2b], hen
X1b<X
2a(X2b<X
1a).
(iii) I g∗(X,0
)has nin e al ex ema on I, henU(X)has a mos n+2in e al ex-
ema on I.
The in ui ion o P oposi ion 1s ems om he beha io o he con inua ion alue a
in e io ex ema. Gi en solu ion U(X), i he e is an ex emum a s a e X, henU(X)=
0 and he op imali y equa ion simpli ies o U(X)=g∗(X,0
)+U(X)σ∗(X,0
)2/2 .I
he ex emum is a minimum, U(X)≥0, and, he e o e, U(X)≥g∗(X,0
). Simila ly, a a
maximum, U(X)≤0, and, he e o e, U(X)≤g∗(X,0
). Hence, he oscilla ion o U(X)
is bounded by he oscilla ion o g∗(X,0
).
When he con inua ion alue con e ges o he s a ic Nash payo a bounda y s a es,
hen i is possible o cha ac e ize addi ional esul s on he shape o payo s ac oss he
en i e s a e space. P oposi ion 2 ela es he mono onici y o single-peakedness o U(X)
o he mono onici y o single-peakedness o g∗(X,0
).
26No e ha since Uis wice con inuously di e en iable, i U(X)=0 o allXin some open in e al
(Xa,Xb), henU(Xa)=U(Xb)=0 and, he e o e, U(X)=0onclosedin e al[Xa,Xb].In hecaseo
Xa=Xb, his de ini ion co esponds o a s ic ex emum poin .
472 J. Aislinn Boh en Theo e ical Economics 19 (2024)
P oposi ion 2. Assume Assump ions 1 o 3and gbounded. When X=R, assume As-
sump ion 4,andwhenXis compac , assume he bounda y s a es {X,X}a e abso bing.
Le U(X)deno e he unique bounded solu ion o (7).
(i) The e m g∗(X,0
)is cons an on Xi and only i U(X)is cons an on X.
(ii) I g∗(X,0
)is mono onically inc easing (dec easing) on X, henU(X)is mono on-
ically inc easing (dec easing) on X.
(iii) I g∗(X,0
)is single-peaked wi h a unique in e al maximum (minimum) and
g∗(X,0
)=g∗(X,0
)(o , in he case o X, unbounded, limX→∞ g∗(X,0
)=
limX→−∞ g∗(X,0
)), hen U(X)is single-peaked wi h a unique in e al maximum
(minimum).
(i ) I g∗(X,0
)has Nin e al ex ema on X, henU(X)has a mos Nin e al ex ema
on X.
Applying P oposi ions 1and 2 o speci ic applica ions will yield s uc u al empi ical
p edic ions abou how equilib ium beha io and payo s change wi h he s a e. This is
illus a ed in Sec ion 6.1 when he s a ic Nash payo is mono onic and in Sec ion 6.2
when he s a ic Nash payo is single-peaked.
P oposi ion 3es ablishes a bound on he PPE payo ac oss all s a es when he
con inua ion alue con e ges o he s a ic Nash payo a he bounda y s a es. Le
W≡supX∈XU(X)and W≡in X∈XU(X)be he leas uppe bound and g ea es lowe
bound o he la ge playe ’s PPE payo ac oss all s a es, and le XHand XLdeno e he
se s o s a es ha yield hese payo s (whe e, in a sligh abuse o no a ion, I say ∞∈XH
i limX→∞ U(X)=Wand simila ly o −∞ and he case o XL). The ollowing esul
shows ha he smalles s a ic Nash payo in XHbounds he PPE payo om abo e and,
simila ly, he la ges s a ic Nash payo in XLbounds he PPE payo om below.27
P oposi ion 3. Assume Assump ions 1 o 3and gbounded. When X=R, assume As-
sump ion 4,andwhenXis compac , assume he bounda y s a es {X,X}a e abso bing.
Then he PPE payo is bounded abo e (below) by he leas (g ea es ) s a ic Nash payo a
he s a es ha yield he highes (lowes ) PPE payo ,
sup
X∈XL
g∗(X,0
)≤W≤W≤in
X∈XH
g∗(X,0
),
whe e, in a sligh abuse o no a ion, i X∈{−∞,∞}, heng∗(X,0
)co esponds o
limx→Xg∗(x,0
).
These bounds ollow di ec ly om he op imali y equa ion. To see his, conside
he case in which he e is an in e io s a e XHsuch ha W=U(XH).ThenU(XH)=
27In gene al, i may be di icul o cha ac e ize XH om he p imi i es o he game, as XHdoes no
necessa ily co espond o he se o s a es ha maximizes he s a ic Nash payo . A weake bound ha can
be easily cha ac e ized is he highes s a ic Nash payo ac oss all s a es, W≤supXg∗(X,0
), and simila ly,
W≥in X∈Xg∗(X,0
).
Theo e ical Economics 19 (2024) Dynamic mo al haza d game 479
7. Conclusion
This pape shows ha pe sis ence p o ides an impo an channel o in e empo al in-
cen i es and de elops a ac able me hod o cha ac e ize Ma ko equilib ium beha io
and payo s. The ools de eloped in his pape will yield insigh s in o equilib ium be-
ha io in a b oad ange o se ings, om indus ial o ganiza ion o poli ical economy
o mac oeconomics. Once unc ional o ms a e speci ied o payo s and he e olu ion
o he s a e, i is s aigh o wa d o use Theo em 1 o cons uc Ma ko equilib ia. This
in u n can be used o de i e empi ically es able compa a i e s a ics and p edic ions
abou he dynamics o equilib ium beha io based on obse able ea u es o he en i-
onmen . Fu u e esea ch can use his amewo k o add ess design ques ions in spe-
ci ic applica ions, such as de e mining he op imal s uc u e o pe sis ence in a a ing
mechanism.
Fu he mo e, he equilib ium cha ac e iza ion can be used o s uc u al es ima ion.
Ma ko equilib ia ha e an in ui i e appeal in empi ical wo k due o hei simplici y and
dependence on payo - ele an a iables o s uc u e incen i es. Playe s do no need
o condi ion on pas beha io in a complex way, as ac ions and payo s a e ully de e -
mined by he cu en alue o he s a e. Es ablishing ha a Ma ko equilib ium exis s
and is unique p o ides a s ong jus i ica ion o ocusing on his equilib ium concep ,
while he equilib ium cha ac e iza ion yields exp essions o payo s and ac ions ha
can calib a ed and es ima ed.
Appendix A: P oo s
A.1 P oo o Lemma 1
I i s show ha (V (S)) ≥0is a ma ingale and (W (S)) ≥0is bounded wi h espec o
(X ) ≥0.
Claim 1. Unde Assump ion 2, o any public s a egy p o ile S=(a ,b ) ≥0, ini ial s a e
X0, and pa h o he s a e a iable (X ) ≥0 ha e ol es acco ding o (1)gi en S,V (S)is a
ma ingale and he e exis s a KW>0such ha |W (S)|≤KW(1+|X |) o all ≥0.
Suppose gis unbounded. By Assump ion 2, he eexis sak∈[0, )and c>0
such ha o all (a,b,X)∈A×E,i X≥0, hen μ(a,b,X)≤kX +c, and i X≤0,
hen μ(a,b,X)≥kX −c. Lipschi z con inuous unc ions ha e linea g ow h. The e-
o e, by Lipschi z con inui y o gand σ, he compac ness o A, and he assump ion
ha |b|≤Kb|X|+cb o all (b,X)∈E, he eexis saKg,Kσ,c>0such ha o all
(a,b,X)∈A×E,|g(a,b,X)|≤Kg(c
k+|X|)and |σ(b,X)|≤Kσ(1+|X|).
I i s de i e a bound on Eτ|g(a ,b ,X )|, he expec ed low payo a ime condi-
ional on a ailable in o ma ion a ime τ≤ . This bound will be independen o he
480 J. Aislinn Boh en Theo e ical Economics 19 (2024)
s a egy p o ile. De ine :X→Ras
(X)≡⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
Kgc
k−Xi X≤−1
−1
8KgX4+3
4KgX2+3
8Kg+Kg
c
ki X∈(−1, 1)
Kgc
k+Xi X≥1.
No e ha ∈C2, ≥0, | |≤Kg,and
(X)=⎧
⎨
⎩
0i |X|≥1
3
2Kg1−X2i |X|<1.
I o’s lemma holds o any C2 unc ion. Gi en a s a egy p o ile S=(a ,b ) ≥0, ini ial s a e
Xτ<∞, and pa h o he s a e a iable (X ) ≥τ ha e ol es acco ding o (1),
(X )= (Xτ)+
τ (Xs)μ(as,bs,Xs)+1
2 (Xs)σ(bs,Xs)2ds
+
τ
(Xs)σ(bs,Xs)dZs
≤ (Xτ)+
τKgk|Xs|+c+3KgK2
σds +KgKσ
τ1+|Xs|dZs
≤ (Xτ)+k
τ
(Xs)ds +3KgK2
σ( −τ)+KgKσ
τ1+|Xs|dZs
o all ≥τ, whe e he i s inequali y ollows om (X)μ(a,b,X)≤Kg(k|X|+c),
1
2 (X)σ(b,X)2≤3KgK2
σ,and (X)σ(b,X)z≤KgKσ(1+|X|)z o all z∈Rand o
all (a,b,X)∈A×E, and he second inequali y ollows om he de ini ion o .Thead-
di ion o he absolu e alue sign in (X)μ(a,b,X)≤Kg(k|X|+c) ollows om he sign
o , and he bound on 1
2 (X)σ(b,X)2 ollows om (X)σ(b,X)2=0i |X|≥1and
(X)σ(b,X)2=3
2Kg1−X2σ(b,X)2≤3
2Kg1−X2K2
σ1+|X|2≤6KgK2
σ
i |X|<1. Taking expec a ions and no ing ha (1+|Xs|)is squa e-in eg able on [τ, ],
so he expec a ion o he s ochas ic in eg al is ze o,
Eτ (X )≤ (Xτ)+3KgK2
σ( −τ)+k
τ
Eτ (Xs)ds
≤ (Xτ)+3KgK2
σ( −τ)ek( −τ),
whe e he las line ollows om G onwall’s inequali y. No e ha |g(a,b,X)|≤ (X) o
all (a,b,X)∈A×E. The e o e,
e− ( −τ)Eτg(a ,b ,X )≤e− ( −τ)Eτ (X )≤ (Xτ)+3KgK2
σ( −τ)e−( −k)( −τ).
Theo e ical Economics 19 (2024) Dynamic mo al haza d game 481
I nex show ha i X <∞, henW (S)<∞,
W (S)=E ∞
e− (s− )g(as,bs,Xs)ds
≤ ∞
e− (s− )E g(as,bs,Xs)ds
≤ ∞
(X )+3KgK2
σ(s− )e−( −k)(s− )ds
=
−k (X )+3 KgK2
σ
( −k)2,
which is ini e o any X <∞and k< . Also, gi en ha has linea g ow h, he e exis s
aKW>0such ha |W (S)|≤KW(1+|X |). By simila easoning, E|V (S)|<∞ o any
X0<∞since
EV (S)=EE ∞
0
e− sg(as,bs,Xs)ds≤E ∞
0
e− sg(as,bs,Xs)ds
is ini e o any X0<∞and k< .
Finally,
E V +k(S)=E +k
0
e− sg(as,bs,Xs)ds +e− ( +k)W +k(S)
=
0
e− sg(as,bs,Xs)ds
+E +k
e− sg(as,bs,Xs)ds
+e− ( +k)E +k ∞
+k
e− (s−( +k))g(as,bs,Xs)ds
=
0
e− sg(as,bs,Xs)ds +e− W (S)=V (S).
Taken oge he , his implies V (S)is a ma ingale and es ablishes Claim 1 o he case o
gunbounded. I gis bounded, hen i ially, W (S)<∞and E|V (S)|<∞ o all ≥0
and X0∈X, and only he inal s ep is needed o es ablish he claim.
I nex de i e he e olu ion o he con inua ion alue. This pa o he p oo ollows
om almos iden ical easoning o he p oo o P oposi ion 2 in Faingold and Sanniko
(2011). The de i a i e o V (S)wi h espec o is:
dV (S)= e− g(a ,b ,X )d − e− W (S)d +e− dW (S).
By he ma ingale ep esen a ion heo em (Ka a zas and Sh e e (1991)), he e exis s a
p og essi ely measu able p ocess (β ) ≥0such ha V can be ep esen ed as dV (S)=
482 J. Aislinn Boh en Theo e ical Economics 19 (2024)
e− β σ(b ,X )dZ . Combining hese wo exp essions o dV (S)yields he law o mo-
ion o he con inua ion alue,
dW (S)= W (S)−g(a ,b ,X )d + β σ(b ,X )dZ
= W (S)−g(a ,b ,X )d + β dX −μ(a ,b ,X )d ,
whe e β cap u es he sensi i i y o he con inua ion alue o he s a e a iable. As
shown abo e, any con inua ion alue has linea g ow h wi h espec o X and is
bounded when gis bounded.
Finally, I es ablish sequen ial a ionali y. This pa o he p oo ollows om almos
iden ical easoning o he p oo o P oposi ion 3 in Faingold and Sanniko (2011). Con-
side s a egy p o ile (a ,b ) ≥0played om pe iod τonwa d and al e na i e s a egy
(
a ,b ) ≥0played up o ime τ. Recall ha all alues o X a e possible unde bo h s a e-
gies, bu ha each s a egy induces a di e en measu e o e sample pa hs (X ) ≥0.A
ime τ, he s a e a iable is equal o Xτ.Ac ionaτwill induce dXτ=μ(aτ,bτ,Xτ)d +
σ(bτ,Xτ)dZτ, whe eas ac ion
aτwill induce dXτ=μ(
aτ,bτ,Xτ)d +σ(bτ,Xτ)dZτ.Le
Vτbe he expec ed a e age payo condi ional on in o ma ion a ime τwhen he la ge
playe ollows
aup o τand aa e wa d, and le Wτbe he con inua ion alue when he
la ge playe ollows s a egy (a ) ≥0s a ing a ime τ:
Vτ= τ
0
e− sg(
as,bs,Xs)ds +e− τWτ.
Conside changing τso ha he la ge playe plays s a egy (
a ,b ) o ano he ins an :
d
Vτis he change in a e age expec ed payo s when he la ge playe swi ches o (a ) ≥0
a τ+dτ ins ead o τ. When he la ge playe swi ches s a egies a ime τ,
d
Vτ= e− τg(
aτ,bτ,Xτ)−Wτdτ +e− τ dWτ
= e− τg(
aτ,bτ,Xτ)−g(aτ,bτ,Xτ)dτ + e− τβτdXτ−μ(aτ,bτ,Xτ)dτ
= e− τg(
aτ,bτ,Xτ)−g(aτ,bτ,Xτ)+βτμ(
aτ,bτ,Xτ)−βτμ(aτ,bτ,Xτ)dτ
+ e− τβτσ(bτ,Xτ)dZτ.
The e a e wo componen s o his s a egy change: how i a ec s he immedia e low
payo and how i a ec s he u u e s a e X , which impac s he con inua ion alue. The
p o ile (
a ,b ) ≥0yields he la ge playe a payo o
W0=E0[
V∞]=E0
V0+∞
0
d
V
=W0+E0 ∞
0
e− g(
a ,b ,X )+β μ(
a ,b ,X )−g(a ,b ,X )
−β μ(a ,b ,X )d .
I
g(a ,b ,X )+β μ(a ,b ,X )≥g(
a ,b ,X )+β μ(
a ,b ,X )
Theo e ical Economics 19 (2024) Dynamic mo al haza d game 483
holds o all ≥0, hen W0≥
W0and de ia ing o S=(
a ,b )is no a p o i able de ia ion.
A s a egy (a ) ≥0is sequen ially a ional o he la ge playe i , gi en (β ) ≥0, o all ,
a ∈a gmax
a∈Ag(a,b ,X )+β μ(a,b ,X ).
A.2 P oo o Theo em 1
In a Ma ko equilib ium, he con inua ion alue and equilib ium ac ions a e cha ac e -
ized as a unc ion o he s a e a iable as W =U(X ),a∗
=a(X ),andb∗
=b(X ).ByI o’s
o mula, i a Ma ko equilib ium wi h a wice con inuously di e en iable con inua ion
alue exis s, he con inua ion alue will e ol e acco ding o
dU(X )=U(X )dX +1
2U(X )σb∗
,X 2d
=U(X )μa∗
,b∗
,X d +1
2U(X )σb∗
,X 2d
+U(X )σb∗
,X dZ . (16)
Simila o he de i a ion o a Ma ko equilib ium in Faingold and Sanniko (2011),
ma ching he d i o (16) wi h he d i o he con inua ion alue cha ac e ized in (5)
yields he op imali y equa ion
U(X)=2 U(X)−ga(X),b(X),X
σb(X),X2−2μa(X),b(X),XU(X)
σb(X),X2, (17)
which is a second-o de nonhomogenous di e en ial equa ion, and ma ching he
ola ili ies cha ac e izes he p ocess go e ning incen i es, β =U(X ). Subs i u ing
his exp ession in o he condi ion o sequen ial a ionali y cha ac e ized in (6)yields
he Ma ko ian ac ion p o ile (a(X),b(X)) =S∗(X,U(X)) (by Assump ion 3,S∗is
single- alued.) Plugging his in o (17)yields(7).
I i s es ablish ha (7) has a leas one solu ion U∈C2 ha akes on alues in he
in e al o easible payo s o he la ge playe . In he case o an unbounded s a e space,
Theo em 5.6 om De Cos e and Habe s (2006) gi es su icien condi ions o he exis-
ence o a solu ion o a second-o de di e en ial equa ion de ined on R3.Icons uc
uppe and lowe solu ions o (17) a ac ion p o ile S∗(X,U(X)) o show ha hese con-
di ions a e sa is ied. This leads o he ollowing lemma, which is he inno a i e pa o
his p oo and is p o en in Supplemen al Appendix B.
Lemma 2. I X=R, hen(7)has a leas one solu ion U∈C2on X ha lies in he ange o
easible payo s o he la ge playe .
In he case o a bounded s a e space, I use an ex ension o a s anda d exis ence e-
sul om De Cos e and Habe s (2006), which was de eloped in Faingold and Sanniko
(2011). The ex ension is necessa y because (7) is unde ined a he bounda y o he s a e
space, {X,X}. This leads o he ollowing lemma, which is p o en in Supplemen al Ap-
pendix B.
484 J. Aislinn Boh en Theo e ical Economics 19 (2024)
Lemma 3. I Xis compac , hen (7)has a leas one solu ion U∈C2on (X,X) ha lies in
he ange o easible payo s o he la ge playe .
Finally, I cons uc a Ma ko equilib ium ha yields payo U(X0),whe eUis a
solu ion o (7). The unc ion X→ S∗(X,U(X)) is Lipschi z con inuous, as a e X→
μ∗(X,U(X)) and X→ σ∗(X,U(X)). The e o e, he s a e a iable s a s a X0and
e ol es acco ding o he unique s ong solu ion (X ) ≥0 o he s ochas ic di e en ial
equa ion
dX =μ∗X ,U(X )d +σ∗X ,U(X )dZ .
Mo eo e ,
dU(X )=U(X )μ∗X ,U(X )d +1
2U(X )σ∗X ,U(X )2d
+U(X )σ∗X ,U(X )dZ
= U(X )−g∗X ,U(X )d +U(X )σ∗X ,U(X )dZ
and, he e o e, he p ocess o con inua ion alues W =U(X )sa is ies (5)wi hp ocess
o incen i e weigh s β =U(X )/ . Finally, he s a egy p o ile (a∗
,b∗
) ≥0sa is ies (6)
gi en (β ) ≥0wi h β =U(X )/ . The e o e, (a∗
,b∗
) ≥0is a PPE yielding equilib ium
payo U(X0).
A.3 P oo o Theo em 2
Le Ube he linea g ow h (when gis unbounded) o bounded (when gis bounded)
solu ion o (7) ha yields he highes MPE payo a X0. Suppose he e exis s an ini ial
s a e X0∈Xand a PPE s a egy p o ile S=(a ,b ) ≥0 ha yields an equilib ium payo
W0>U(X0). In such a PPE, he s a e (X ) ≥0e ol es acco ding o (1)gi enS=(a ,b ) ≥0
and, by Lemma 1, he con inua ion alue e ol es acco ding o
dW (S)= W (S)−g(a ,b ,X )d + β dX −μ(a ,b ,X )d (18)
o some p ocess (β ) ≥0. By Assump ion 3, a unique ac ion p o ile sa is ies (6) a each
(X, β). The e o e, by Lemma 1, equilib ium ac ions sa is y (a ,b )=S∗(X , β ).By
I o’s o mula, he p ocess (U(X )) ≥0e ol es acco ding o
dU(X )=U(X )μ∗(X , β )d +1
2U(X )σ∗(X , β )2d
+U(X )σ∗(X , β )dZ . (19)
De ineap ocessD ≡W (S)−U(X )wi h ini ial condi ion D0=W0(S)−U(X0)>0.
Then D e ol es acco ding o dD =dW (S)−dU(X ). Plugging in Eqs. (18)and(19),
he p ocess has ola ili y (X ,β ),whe e (X,β)≡( β −U(X))σ∗(X, β),andhas
Theo e ical Economics 19 (2024) Dynamic mo al haza d game 485
d i D +d(X ,β ),whe e
d(X,β)≡ U(X)−g∗(X, β)−U(X)μ∗(X, β)−U(X)σ∗(X, β)2/2
= g∗X,U(X)−g∗(X, β)+U(X)μ∗X,U(X)−μ∗(X, β)
+U(X)σ∗X,U(X)2−σ∗(X, β)2/2,
and he second line ollows om subs i u ing he igh hand side o (7) o U(X).
Lemma 4. I (X,β)=0and σ(X, β)>0, hend(X,β)=0.
P oo . Suppose (X,β)=0 o some(X,β)and σ(X, β)>0. Then β =U(X).The
ac ion p o ile associa ed wi h S∗(X,U(X)) co esponds o he ac ions played in he
Ma ko equilib ium wi h con inua ion alue U(X)a s a e X. The e o e, d(X,β)=0.
Lemma 5. Fo e e y ε>0, he eexis saη>0such ha ei he d(X,β)>−εo
| (X,β)|>η.
P oo . Suppose he s a e space is unbounded, X=R. No e ha in his case, σ∗(X, β)
is bounded away om 0 by Assump ion 1, so, by Lemma 4,i (X,β)=0, hen d(X,β)=
0. Fi s show ha he e exis s an M>0 such ha his is ue o (X,β)∈a≡{X×R:
|β|>M}. Since Uis bounded by Lemma 9in hecaseo gunbounded o Lemma 26 in
Supplemen al Appendix D.2 in he case o gbounded (no e ha nei he lemma equi es
Assump ion 4), and since σ∗(X, β)is bounded away om 0, he e exis s an M>0and
η1>0such ha | (X,β)|>η
1 o all |β|>Mand X∈X, ega dless o d. Nex show ha
he e exis s an δ>0 such ha his is ue o (X,β)∈b≡{X×R:|β|≤M,|X|>δ
}.
Conside he se b⊂bwi h d(X,β)≤−ε. I mus be ha βis bounded away
om U(X)/ on b. Suppose no . Then ei he (i) he e exis s some (X,β)∈bwi h
β=U(X)/ , which implies (X,β)=0 and, he e o e, d(X,β)=0—a con adic ion—
o (ii) as Xbecomes la ge, he bounda y o he se bapp oaches β=U(X)/ . The la -
e implies ha o any δ1>0, he e exis s an (X,β)∈bwi h β −U(X)<δ
1. Choose
δ1so ha |g∗(X,U(X)) −g∗(X, β)|<ε/4 ,|U(X)||μ∗(X,U(X)) −μ∗(X, β)|<ε/4,
and |U(X)||σ∗(X,U(X))2−σ∗(X, β)2|=0, which is possible gi en ha g∗and μ∗
a e Lipschi z, Uis bounded, and σ∗is independen o z o la ge X.Then|d(X,β)|<
ε/4+ε/4+ε/4=3ε/4, which is a con adic ion. The e o e, he e exis s a η2such ha
| (X,β)|>η
2on b. Thenon hese b,i d(X,β)≤−ε, hen| (X,β)|>η
2.Fi-
nally show his is ue o (X,β)∈c≡{X×R:|β|≤Mand |X|≤δ}. Conside he se
c⊂cwhe e d(X,β)≤−ε.The unc iondis con inuous and cis compac , so cis
compac . The unc ion | |is con inuous and, he e o e, achie es a minimum η3on c.
I η3=0, hen d=0 by Lemma 4—a con adic ion. The e o e, η3>0and| (X,β)|>η
3
o all (X,β)∈c.Takeη≡min{η1,η2,η3}. Then when d(X,β)≤−ε,| (X,β)|>η.
The p oo o a bounded s a e space is analogous (see Supplemen al Appendix C).
Lemma 6. Gi en X0, any PPE payo W0is such ha W0≤U(X0).
486 J. Aislinn Boh en Theo e ical Economics 19 (2024)
P oo . Choose ε= D0/4 and suppose D ≥D0/2. Then, by Lemma 5, he eexis sa
η>0 such ha whene e he d i o D is less han D −ε> D
0/2− D0/4= D0/4>0,
| (X ,β )|>η. Thus, as long as D ≥D0/2>0, i has ei he posi i e d i o posi i e
ola ili y. This implies i g ows a bi a ily la ge wi h posi i e p obabili y, i espec i e o
X . This is a con adic ion, since in he case ha gis unbounded, by Lemma 1,D is he
di e ence o wo p ocesses ha a e bounded wi h espec o X , and in he case ha gis
bounded, D is he di e ence o wo bounded p ocesses. Thus, i canno be ha D0>0
andi mus be hecase ha W0≤U(X0).
Le ing Ube he linea g ow h (when gis unbounded) o bounded (when gis
bounded) solu ion o (7) ha yields he lowes MPE payo a X0, by analogous eason-
ing i is no possible o ha e D0=W0(S)−U(X0)<0, implying W0≥U(X0). The p oo
o Theo em 2immedia ely ollows om Lemma 6, he analogue o W0≥U(X0),and he
ac ha a any s a e X∈X, i is possible o he la ge playe o achie e any payo in he
con ex hull o he se o Ma ko equilib ium payo s a s a e Xby andomiza ion a ime
ze o.
P oo o Co olla y 1The exis ence o a Ma ko equilib ium ollows om Theo em 1.
When μis independen o a, he sequen ial a ionali y condi ion (6)inaMa ko equi-
lib ium collapses o maximizing he s a ic low payo , and he la ge playe plays he
unique s a ic Nash ac ion p o ile S∗(X,0
)in each s a e. The e o e, any solu ion o (7)
mus sa is y
U(X )=E ∞
e− sg∗(Xs,0
)d , (20)
whe e he measu e o e he s a e is independen o he solu ion Usince equilib ium
ac ions a e independen o U. Gi en ha he igh hand side o (20) is independen o
U,(7) mus ha e a unique solu ion and he e is a unique Ma ko equilib ium. By The-
o em 2, his is also he unique PPE. The solu ion o (7) e alua ed a s a e X analy ically
cha ac e izes (20).
A.4 P oo o Theo ems 3and 4
Ip o eTheo ems3and 4simul aneously. The p oo p oceeds in h ee s eps:
S ep 1. Any solu ion o he op imali y equa ion has he same bounda y condi ions.
S ep 2. I all solu ions ha e he same bounda y condi ions, hen he e is a unique linea
g ow h (bounded) solu ion.
S ep 3. When he e is a unique solu ion, hen he e is a unique PPE.
Le ψ(X,z)≡g∗(X,z)+zμ∗(X,z)/ be he alue o he la ge playe ’s incen i e con-
s ain a he sequen ially a ional ac ion p o ile o incen i e weigh z/ . All in e medi-
a e heo ems and lemmas main ain Assump ions 1 o 3and, as s a ed, Assump ion 4.As
a eminde , |·|deno es he Euclidean no m o ec o s. I i s p esen an in e media e
esul ha will be used in S eps 1 and 2.
Theo e ical Economics 19 (2024) Dynamic mo al haza d game 487
Lemma 7. Suppose Uand Va e bo h linea g ow h (bounded) solu ions o (7),wi h
U(X)<V(X) o some in e io s a e X∈X.ThenV−Udoes no ha e an in e io maxi-
mum and is mono one o la ge |X|.
P oo . Fi s suppose Xis compac . I ollows om iden ical easoning o Lemma C.7 in
Faingold and Sanniko (2011) ha i Uand Va e wo linea g ow h (bounded) solu ions
o (7)such ha U(X0)≤V(X0)and U(X0)≤V(X0), wi h a leas one s ic inequali y,
hen U(X)<V(X)and U(X)<V(X) o all X∈(X0,X).31 Simila ly i U(X0)≤V(X0)
and U(X0)≥V(X0), wi h a leas one s ic inequali y, hen U(X)<V(X)and U(X)>
V(X) o all X∈(X,X0).
Suppose Uand Va e bo h bounded solu ions o (7), wi h U(X)<V(X) o some
X∈(X,X). Suppose V−Uhas an in e io maximum a some X∗∈(X,X).Thenby
con inui y, his implies ha U(X∗)=V(X∗).I U(X∗)<V(X∗), hen by he abo e
s a emen , U(X)<V(X) o all X>X
∗and, he e o e, V(X)−U(X)is s ic ly inc eas-
ing o X>X
∗. This con adic s ha X∗is an in e io maximum. I U(X∗)>V(X∗),
hen by he abo e s a emen , U(X)>V(X)and U(X)>V(X) o all X>X
∗,and
U(X)>V(X)and U(X)<V(X) o all X<X
∗. The e o e, X∗is a global maximum.
This con adic s U(X)<V(X) o some X∈(X,X). The e o e, V−Udoes no ha e
an in e io maximum. Gi en his, V−Uhas a mos one in e io minimum. The e o e,
he e exis s a δ>0such ha V−Uis mono one o |X−X|<δand |X−X|<δ.The
p oo o he case o X=Ris analogous, eplacing Xand Xwi h ∞and −∞, espec-
i ely.
S ep 1: Bounda y condi ions Lemmas 8 o 19 as well as Lemmas 26 and 27 in Supple-
men al Appendix D.2 es ablish he ollowing bounda y condi ions o he case o X=R.
When gis unbounded, any solu ion Uo (7) wi h linea g ow h sa is ies limX→pU(X)−
yL(X)=g2(zp)+zpμ2(zp)/ ,limX→pU(X)=zp,andlimX→pσ(X,U(X))2U(X)=0
o p∈{−∞,∞},whe ezp≡ gp/( −μp)gi en μp≡limX→pμ∗(X,z)/X and gp≡
limX→pg∗(X,z)/X, which exis and a e ini e, and yL(x)≡− (x) g1(x)/ (x)μ1(x)dx
wi h in eg a ing ac o (x)≡exp( /μ1(x)dx)when limx→pμ1(x)= 0andyL(x)≡
g1(x)when limx→pμ1(x)=0. When gis bounded, his simpli ies o limX→pU(X)=gp,
whe e gp≡limX→pg∗(X,0
),andlimX→pU(X)=0. Supplemen al Appendix D.1 es-
ablishes analogous bounda y condi ions o he case o Xcompac , and Supplemen al
Appendix D.3 es ablishes he same bounda y condi ions o he case o X=Rand g
bounded unde an al e na i e o Assump ion 4.
De ine ψ(X,z)≡ψ(X,z)/X and U(X)≡U(X)/X.Le ψand ψdeno e he pa ial
de i a i es o ψand ψwi h espec o X.Le δ0>0 deno e he lowe bound abo e
which he la ge |X|p ope ies o Assump ions 3and 4hold. Se e al lemmas use he
p ope y ha g∗(X,z),μ∗(X,z),andσ∗(X,z)a e bounded in z, which ollows om he
compac ness o Aand B(X). The Lipschi z con inui y o g1,μ1,g2,andμ2is also used,
which ollows om he Lipschi z con inui y o g∗(X,z)and μ∗(X,z). The ollowing
se ies o lemmas a e s a ed o an unbounded low payo g; oapply hem oabounded
31Analogous o he de ini ion o φ1in hei esul , se X1≡in {X∈[X0,X):U(X)≥V(X)}and apply
he same easoning.
488 J. Aislinn Boh en Theo e ical Economics 19 (2024)
low payo , simply subs i u e “bounded solu ion o (7)” o “linea g ow h solu ion o
(7)” h oughou .
Lemma 8. Suppose X=R.Gi enp∈{−∞,∞},μp≡limX→pμ∗(X,z)/X and gp≡
limX→pg∗(X,z)/X exis and a e ini e. Mo eo e , limX→pψ(X,z)=limX→pψ(X,z)=
ψp(z) o all z∈R,whe eψp(z)≡gp+zμp/ .
P oo .Le p=∞and ix z∈R. Gi en Assump ion 4(i), ψ(X,z)=g
1(X)+zμ
1(X)/
o X>δ
0. By he Lipschi z con inui y o g1and μ1,g
1and μ
1a e bounded, and, he e-
o e, ψ(·,z)is bounded o any z∈R. By Assump ion 3,ψ(·,z)and g
1a e mono one
o la ge X( he la e ollows om he assump ion holding a z=0). The e o e, by he
mono one con e gence heo em, ψ∞(z)≡limX→∞ ψ(X,z)and g∞≡limX→∞ g
1(X)
exis and a e ini e. Gi en ha ψand g
1ha e well de ined limi s and ψ(X,z)=g
1(X)+
zμ
1(X)/ o la ge X,μ∞≡limX→∞ μ
1(X)exis s and is ini e. Mo eo e , ψ∞(z)=
g∞+zμ∞/ .Wheng1and μ1a e unbounded, hen by l’Hopi al’s ule, limX→∞ ψ(X,z)=
ψ∞(z),limX→∞ g1(X)/X =g∞,andlimX→∞ μ1(X)/X =μ∞.In hecasewhe eg1o
μ1is bounded, his immedia ely ollows om g∞=0o μ∞=0. Gi en ha g2(z)and
μ2(z)a e independen o X,limX→∞ g2(z)/X =0andlimX→∞ μ2(z)/X =0. This im-
plies limX→∞ g∗(X,z)/X =g∞and limX→∞ μ∗(X,z)/X =μ∞.No e ha μ∞< by As-
sump ion 2. The p oo o p=−∞is analogous.
Lemma 9. Suppose X=Rand Uis a solu ion o (7)wi h linea g ow h. Then o p∈
{−∞,∞}, he e exis s a ini e U
p∈Rsuch ha limX→pU(X)=limX→pU(X)=U
p.
P oo .Le p=∞ and le Ube a solu ion o (7) wi h linea g ow h. Suppose
limin X→∞ U(X)= lim supX→∞ U(X).Then o allδ>0, by he con inui y o U, he e
exis s a zand an inc easing sequence (Xn)n∈No al e na ing consecu i e Xsuch ha
X1>δ,U(Xn)=z,andU(Xn)≤0 o nodd, and U(Xn)=zand U(Xn)≥0 o n
e en, wi h one inequali y o U s ic . F om (7), his implies U(Xn)≤ψ(Xn,z) o n
odd and ψ(Xn,z)≤U(Xn) o ne en. Thus, he oscilla ion o ψ(X,z)is a leas as la ge
as he oscilla ion o U. Bu by Assump ion 3,ψ(X,z)is mono one o X>δ
0.The e-
o e,i mus be ha lim in X→∞ U(X)=lim supX→∞ U(X).Le U
∞deno e his limi .
Gi en Uhas linea g ow h, |U
∞|<∞and, by l’Hopi al’s ule, limX→∞ U(X)=U
∞.In
hecaseo gbounded, Ubounded implies U
∞=0andlimX→∞ U(X)=0. The p oo o
p=−∞is analogous.
Lemma 10. Suppose X=Rand Uisasolu iono (7)wi h linea g ow h. Then
limX→pψ(X,U(X)) =ψp(U
p) o p∈{−∞,∞},whe eU
p≡limX→pU(X).
P oo .Le p∈{−∞,∞}and le Ubeasolu iono (7) wi h linea g ow h. Gi en μ∗and
g∗a e Lipschi z con inuous and addi i ely sepa able in (X,z) o |X|>δ
0, he eexis sa
M1,M2,M3,c>0andδ>δ
0such ha o |X|>δ,
ψ(X,z1)−ψ(X,z2)≤M1|z1−z2|+M2|z1||z1−z2|+M3|z1−z2||X|+|z2|.
Theo e ical Economics 19 (2024) Dynamic mo al haza d game 495
P oo o P oposi ion 2.Le Ube he unique bounded solu ion o (7). A a s a e X
co esponding o an in e io ex emum on X,U(X)=0. F om (7), i Xis an in e io
minimum, g∗(X,0
)≤U(X), and i Xis an in e io maximum, U(X)≤g∗(X,0
).Le n
deno e he numbe o (s ic in e io ) in e al ex ema o Uon Xand le ngdeno e he
numbe o (s ic in e io ) in e al ex ema o g∗(X,0
)on X. Fi s conside Xcompac .
I em (i). Suppose g∗(·,0
)is cons an on X.Thenng=0and he eexis sac∈Rsuch
ha g∗(X,0
)=c o all X∈X. By pa (i ) (see below o p oo ), ng=0impliesn=0.
By he bounda y condi ions om Theo em 4,U(X)=cand U(X)=c, which implies
U(X)=U(X). Combined wi h n=0, his implies ha Uis cons an on X. To es ablish
he o he di ec ion, suppose g∗(·,0
)is no cons an on X. Then he e exis s a p ope
in e al I1⊂Xsuch ha g∗(·,0
)is s ic ly mono one on I1. Take a closed p ope subse
I2⊂I1. By P oposi ion 1(ii), Uis no cons an on I2. The e o e, Uis no cons an on X.
I em (ii). Suppose g∗(·,0
)is mono onically inc easing on Xand Uis no mono on-
ically inc easing. Then ng=0andU(X)<0 o someX∈X. By P oposi ion 2(i ) (see
below o p oo ), ng=0impliesn=0. The e o e, i mus be ha Uis mono onically de-
c easing on X, i.e., U(X)≤0 o allX∈X.Gi enU(X)<0 o someX∈X, his implies
ha U(X)<U(X). By he bounda y condi ions om Theo em 4,U(X)=g∗(X,0
)and
U(X)=g∗(X,0
), and by he mono onici y o g∗(·,0
),g∗(X,0
)≤g∗(X,0
). This implies
U(X)≤U(X), a con adic ion. The e o e, Uis mono onically inc easing. The p oo o
Umono onically dec easing is analogous.
I em (iii). Suppose g∗(X,0
)=g∗(X,0
)and suppose g∗(·,0
)is single-peaked wi h
a unique in e al ex emum, a maximum. Le Xg
1deno e he in e al o s a es co e-
sponding o his ex emum. By P oposi ion 2(i ), ng=1impliesn≤1. Gi en ha he e
is a unique in e al ex emum and i is a maximum, g∗(·,0
)is mono onically inc easing
on I1=[X,in Xg
1]and s ic ly so on some p ope in e al I
1⊂I1. The e o e, by P opo-
si ion 1(ii), Uis no cons an on I1. Simila ly, g∗(·,0
)is mono onically dec easing on
I2=[supXg
1,X]and s ic ly so on some p ope in e al I
2⊂I2,soUis no cons an on
I2. F om he bounda y condi ions, U(X)=g∗(X,0
)and U(X)=g∗(X,0
). The e o e,
U(X)=U(X). Since Uis no cons an and U(X)=U(X), by con inui y Umus ha e
a leas one in e al ex emum, n≥1. Gi en ha i was al eady es ablished ha n≤1, i
mus be ha n=1.
Suppose he unique in e al ex emum o Uis a minimum. Le X1deno e he in-
e al o s a es co esponding o his ex emum. Then g∗(x1,0
)≤U(x1) o all x1∈X1.
Gi en ha X1is a minimum and i is he unique in e al ex emum, Uis mono onically
dec easing on [X,in X1]and s ic ly so on some some p ope in e al I⊂[X,in X1].
This implies ha o all x1∈X1,U(x1)<U
(X)=g∗(X,0
). The e o e, o all x1∈X1,
g∗(x1,0
)≤U(x1)<g
∗(X,0
). Fu he , since Xg
1is he unique in e al ex emum o
g∗(·,0
)and a maximum, o all xg
1∈Xg
1,g∗(X,0
)=g∗(X,0
)<g
∗(xg
1,0
).Bu heng∗(·,0
)
mus ha e wo in e al ex ema, since g∗(x1,0
)<g
∗(X,0
)=g∗(X,0
) o x1∈X1and
g∗is con inuous. This is a con adic ion. The e o e, Uis single-peaked wi h a unique
in e al maximum. The p oo o Usingle-peaked wi h a minimum is analogous.
I em (i ). This ollows di ec ly om U(X)≥g∗(X,0
)a an in e io minimum o U,
U(X)≤g∗(X,0
)a an in e io maximum o U,U(X)=g∗(X,0
),U(X)=g∗(X,0
),and
he Lipschi z con inui y o g∗.
496 J. Aislinn Boh en Theo e ical Economics 19 (2024)
Fo he case o X=R, eplaceg∗(X,0
)and U(X)wi h limX→∞ g∗(X,0
)and
limX→∞ U(X), and analogously o X. These limi s exis by he p oo o Theo em 4.
P oo o P oposi ion 3.Le Ube he unique bounded solu ion o (7). Then Uis con-
inuous and bounded on a closed se . The e o e, ei he Ua ains a global maximum on
X, in which case W=U(XH) o some XH∈X,o in hecasewhe eXis unbounded,
i is also possible ha W=lim supX→XHU(X) o ei he XH=−∞o XH=∞.Sup-
pose Ua ains a global maximum a an in e io s a e XH∈in X.ThenU(XH)=0and
U(XH)≤0. F om (7), his implies
U(XH)=2 W−g∗(XH,0
)
σ∗(XH,0
)2≤0,
and, he e o e, W≤g∗(XH,0
).I Xis bounded and Ua ains a global maximum a
bounda y s a e XH=Xo XH=X, hen by Theo em 4,W=g∗(XH,0
). Simila ly, i X
is unbounded and limX→XHU(X)=W o ei he XH=−∞o XH=∞, hen by The-
o em 4,W=g∗(XH,0
). The e o e, W≤in XH∈XHg∗(XH,0
). The p oo o Wis analo-
gous.
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Co-edi o Thomas Ma io i handled his manusc ip .
Manusc ip ecei ed 2 No embe , 2016; inal e sion accep ed 29 Oc obe , 2019; a ailable on-
line 3 Ma ch, 2023.
