To infinity and beyond: A general framework for scaling economic theories

Goncza owski, Yannai; Komine s, Sco Duke; Sho e , Ran I.
A icle
To in ini y and beyond: A gene al amewo k o scaling
economic heo ies
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Goncza owski, Yannai; Komine s, Sco Duke; Sho e , Ran I. (2025) : To in ini y
and beyond: A gene al amewo k o scaling economic heo ies, Theo e ical Economics, ISSN
1555-7561, The Econome ic Socie y, New Ha en, CT, Vol. 20, Iss. 2, pp. 511-542,
h ps://doi.o g/10.3982/TE5878
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Theo e ical Economics 20 (2025), 511–542 1555-7561/20250511
To in ini y and beyond: A gene al amewo k o scaling
economic heo ies
Yannai A. Goncza owski
Depa men o Economics and Depa men o Compu e Science, Ha a d Uni e si y
Sco Duke Komine s
En ep eneu ial Managemen Uni , Ha a d Business School, Depa men o Economics and Cen e o
Ma hema ical Sciences and Applica ions, Ha a d Uni e si y, and a16z c yp o
Ran I. Sho e
Depa men o Economics, Penn S a e Uni e si y
Many economic models inco po a e ini eness assump ions ha , while in o-
duced o simplici y, play a eal ole in he analysis. We p o ide a p incipled ame-
wo k o scaling esul s om such models by emo ing hese ini eness assump-
ions. Ou su icien condi ions a e on he heo em s a emen only, and no on
i s p oo . This esul s in sho p oo s, and e en allows us o use he same a gu-
men o scale simila heo ems ha we e p o en using dis inc ly di e en ools.
Yannai A. Goncza owski: [email p o ec ed]
Sco Duke Komine s: [email p o ec ed]
Ran I. Sho e : [email p o ec ed]
An ea lie e sion o his pape , en i led “To In ini y and Beyond: Scaling Economic Theo ies ia Logical
Compac ness,” appea ed as a one-page abs ac in he P oceedings o he 21s ACM Con e ence on Economics
and Compu a ion. We hank Da id Ahn, Bob Ande son, Mo gane Aus e n, A chishman Chak abo yz,
Ch is Chambe s, Yunseo Choi, Hen y Cohn, Pio Dwo czak, And ew Ellis, Tamás Fleine , D ew Fuden-
be g, Wayne Gao, Je y G een, Joseph Halpe n, Ron Holzman, Ra i Jagadeesan, M. Ali Khan, Da id Laibson,
Rida La aki, Ba Ligh , Ellio Lipnowski, Ce Liu, Geo ge Maila h, Michael Mandle , Paul Milg om, Anku
Moi a, Yo am Moses, Juan Pe ey a, Ma ek Pycia, Deb aj Ray, John Rehbeck, Phil Reny, Joseph Roo , A iel
Rubins ein, Do Same , Ch is Shannon, Tomasz S zalecki, Se giy Ve s yuk, Shing-Tung Yau, Bill Zame, and
nume ous semina audiences o help ul commen s. Goncza owski was suppo ed in pa by he Adams
Fellowship P og am o he Is ael Academy o Sciences and Humani ies; his wo k was suppo ed in pa by
ISF g an s 1435/14, 317/17, and 1841/14 adminis e ed by he Is aeli Academy o Sciences; by he Uni ed
S a es–Is ael Bina ional Science Founda ion (BSF g an 2014389); and by he Eu opean Resea ch Council
(ERC) unde he Eu opean Union’s Ho izon 2020 esea ch and inno a ion p og amme (g an 740282), and
unde he Eu opean Union’s Se en h F amewo k P og amme (FP7/2007–2013) / ERC g an numbe 337122.
Komine s g a e ully acknowledges he suppo o he Na ional Science Founda ion (g an SES-1459912), as
well as he Ng Fund and he Ma hema ics in Economics Resea ch Fund o he Ha a d Cen e o Ma he-
ma ical Sciences and Applica ions. Sho e was suppo ed by a g an om he Uni ed S a es–Is ael Bina-
ional Science Founda ion (BSF g an s 2016015 and 2022417). Pa o his wo k was conduc ed du ing he
Simons Lau e Ma hema ical Sciences Ins i u e Fall 2023 p og am on he Ma hema ics and Compu e Sci-
ence o Ma ke and Mechanism Design, which was suppo ed by he Na ional Science Founda ion unde
G an DMS-1928930 and by he Al ed P. Sloan Founda ion unde g an G-2021-16778. Pa s o he wo k o
Goncza owski we e ca ied ou while a he Heb ew Uni e si y o Je usalem, a Tel A i Uni e si y, and a
Mic oso Resea ch.
©2025 The Au ho s. 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/TE5878
512 Goncza owski, Komine s, and Sho e Theo e ical Economics 20 (2025)
We demons a e he e sa ili y o ou app oach ia an a ay o examples om
e ealed-p e e ence heo y.
Keywo ds. Re ealed p e e ences, in ini e models.
JEL classi ica ion. C02, D00.
“Mo e is good... all is be e .”
—Fe engi Rule o Acquisi ion #242, S a T ek
1. In oduc ion
In economic heo y, we equen ly make ini eness assump ions o simplici y and/o
ac abili y—and hose assump ions can play a eal ole in he analysis. O cou se, he
eal wo ld is i sel ini e, so he e is in some sense no “loss” om assuming ini eness
in ou models. Bu ini eness assump ions ne e heless some imes lead o concep-
ual p oblems—i ou unde s anding o economic heo y hinges on ini eness, hen ou
models may no qui e ell he whole s o y.
Fo example, in decision heo y, e ealed p e e ence analysis seeks o unde s and
wha we can in e abou agen s om hei choice beha io . While a lis o obse ed
choices is always ini e, i we make pa ame ic assump ions such as homo he ici y, hen
each da a poin becomes in ini ely many da a poin s. E en wi hou such assump ions,
we would like o eason abou possible demand unc ions—de ined e e ywhe e— ha
a e consis en wi h he da a, and his equi es conjec u ing abou beha io o e an in-
ini e da ase . Fu he mo e, heo izing abou obse ing in ini e da ase s le s us sepa-
a e he limi a ions o in e ence abou agen s’ p e e ences ha a e jus imposed by da a
ini eness om hose ha a e inhe en e en wi h access o e e y possible obse a ion.
As ano he example, i a game- heo e ic inding is ue only when he se o agen s is
ini e, hen he e is an implici discon inui y, possibly elying on an edge e ec o a spe-
ci ic s a ing condi ion ha may no be obus o small ic ions o pe u ba ions.1Thus
ini e-ma ke esul s ha also hold in in ini e ma ke s a e in some sense mo e obus .
In his pape , we p esen a gene al amewo k o s eng hening esul s ha assume
ini eness, by scaling hem o in ini e se ings.2Ou app oach, by elying on esul s in
p oposi ional logic, implici ly le e ages opological p ope ies o he space o heo em
s a emen s a he han any ea u es o echniques om hei p oo s. As such, i allows
us o p o e esul s along he lines o “i a ce ain s a emen holds when assuming ini e-
ness ( ega dless o how one would p o e i ), hen—due me ely o he s uc u e o his
s a emen —i mus hold e en i he ini eness assump ion is d opped.”
Ou me hods allow us o elax a ious ini eness assump ions, such as da ase size
and ma ke size. In his pape , we ocus on applica ions o decision heo y, whe e he in-
ini y ha we ackle is he in ini y o da a. To demons a e he e sa ili y o ou app oach
1Fo an example o a di e en kind o discon inui y—be ween a ini e and a con inuum se ing—see he
wo k o Mi alles and Pycia (2017), showing ha a con inuum model may ule ou impo an phenomena
ha a e obse ed in he ini e models ha con e ge o i .
2As a side no e: economic heo y some imes also u ns o in ini e models when hei ini e analogues a e
ha d o analyze— o example, o smoo h ou in ege e ec s. Tha is no ou ocus he e.
Theo e ical Economics 20 (2025) To in ini y and beyond 513
ac oss dispa a e ields, we also demons a e an applica ion o game heo y, whe e he
in ini y ha we ackle is ha o he ma ke size.
In Sec ion 2, we s a e and p o e ou gene al scaling lemma ha we apply h oughou
he pape . Sec ion 3p esen s a “wa m up,” applying ou app oach o scale a undamen-
al esul o which he p oo o he ini e case is conside ably simple han ha o he
in ini e case. Speci ically, he esul ha we scale is ha any da ase sa is ying he s ong
axiom o e ealed p e e ences (SARP) is a ionalizable, o which he ini e case is sim-
ple and he in ini e case is usually p o en by appealing o Zo n’s lemma. In Sec ion 4,we
p oceed wi h using a p oo e y simila o he simple p oo om he wa m-up, o elax
a ini eness assump ion in a esul o which he in ini e case has no been p e iously
p o en. Speci ically, we p o e a no el in ini e-da a e sion o Masa lioglu, Nakajima,
and Ozbay’s (2012) cha ac e iza ion o limi ed-a en ion a ionalizabili y. The ini e-
case p oo s o his esul and o he wa m-up a e s a kly di e en . None heless, he
s a emen s o hese wo esul s a e simila , and his enables us o scale hem using es-
sen ially he same p oo . We u he mo e show ha since ou app oach only elies on
heo em s a emen s (and no on how hey a e p o en) i can be used o condi ionally
scale a ich amily o no -ye -p o en esul s (i.e., condi ional on he ini e case being
ue).
A i s glance, ou app oach migh seem limi ed o p o ing esul s ha a e disc e e
in na u e (see discussion below). None heless, in Sec ion 5we use ou app oach o
p o e esul s ega ding objec s ha a e nondisc e e (coming om a con inuum space).
Speci ically, he e we ep o e Reny’s (2015) in ini e-da a e sion o A ia ’s (1967) heo-
em, whe e u ili ies come om a con inuum space, as well as Caplin, Dean, and Leahy’s
(2017) in ini e-da a e sion o Caplin and Dean’s (2015) cha ac e iza ion o ha ing a
cos ly in o ma ion acquisi ion ep esen a ion, whe e p io s come om a con inuum
space. Again, ou scaling p oo s o bo h o hese heo ems a e nea ly iden ical. In Sec-
ion 6, we discuss limi a ions o ou app oach in he con ex o decision heo y.
In Sec ion 7, we conclude wi h an applica ion o a di e en ield and wi h a di e en
no ion o in ini y, ep o ing he exis ence o a Nash equilib ium in in ini e games on
g aphs. We hen discuss limi a ions o ou app oach mo e b oadly.
As al eady men ioned, some o he esul s ha we p o e in his pape a e no el o
his wo k. O he esul s ha we ( e)p o e ha e al eady been ob ained using o he , e y
di e en me hods, which allows us o compa e and con as p io p oo echniques wi h
ou s. As ou illus a i e applica ions demons a e, p oo s ha use ou amewo k ha e
se e al no able ea u es. Fi s , hey use one ool a he han ha ing o choose om a -
ious se ing-speci ic ools. Second, he p oo s uc u e is modula : ou condi ions o
scaling he ini e-case esul o he in ini e case depend only on he s a emen o he
ini e-case esul and a e comple ely agnos ic o he a gumen /me hods used o p o e
ha esul . Fu he mo e, he p oo s a e obus in ha e en hei dependence on he de-
ails o he model is qui e weak, and essen ially he same p oo can some imes be used
in qui e di e en models.
514 Goncza owski, Komine s, and Sho e Theo e ical Economics 20 (2025)
1.1 Technique
Ou gene al app oach, o mula ed ia Lemma 1in Sec ion 2, allows us o scale p oblems
p o ided ha hey ha e wha we call a well desc ip ion, and ha his well desc ip ion
sa is ies wha we call he ini e-subse p ope y. We de ine hese concep s p ecisely in
Sec ion 2.2, a e e iewing p elimina ies o p oposi ional logic in Sec ion 2.1.He e,we
p o ide an in o mal desc ip ion and an illus a i e applica ion (which we la e o mal-
ize): showing ha in ini e da ase s ha sa is y SARP a e a ionalizable.
Awell desc ip ion speci ies o each p oblem a (po en ially in ini e) se o indi id-
ually ini e logical s a emen s o e Boolean a iables, such ha he p oblem has a so-
lu ion i and only i he e is an assignmen o u h alues o hese Boolean a iables
unde which all hese s a emen s hold simul aneously.3Fo example,4in a e ealed-
p e e ences se ing, we can encode a a ionalizing p e e ence o de using a se o
Boolean a iables {ag b}(“ag ea e han b”), each being T u e i ab o he co e-
sponding aand b. We can hen exp ess all he equi ed p ope ies o a a ionalizing
o de (comple eness, ansi i i y, an isymme y, and consis ency wi h whiche e ou -
comes a e e ealed p e e ed o o he s) using logical s a emen s ph ased in e ms o
hese a iables (in ini ely many such s a emen s, bu each s a emen indi idually ini e).
An assignmen o u h alues o he a iables unde which all s a emen s hold simul-
aneously co esponds exac ly o a solu ion (a a ionalizing o de ), and ice e sa. In
pa icula , o each p oblem, ou se o logical s a emen s has such an assignmen i and
only i he p oblem has a solu ion and, he e o e, his is a well desc ip ion.
The p eceding well desc ip ion cap u es ini e and in ini e p oblems equally: he
only di e ence ha a ises is in he ca dinali ies o he se s o Boolean a iables and log-
ical s a emen s. When he p oblem is in ini a y (e.g., wi h an in ini e da ase o wi h
in ini ely many agen s), he associa ed se o logical s a emen s is in ini e as well. Ye ,
each o he logical s a emen s we cons uc is none heless indi idually ini e, ha is, i
con ains only ini ely many o he Boolean a iables.
Fix a well desc ip ion, and call he se o logical s a emen s associa ed wi h each
p oblem “ he desc ip ion o he p oblem.” The well desc ip ion sa is ies he ini e-subse
p ope y i e e y ini e subse o he desc ip ion o any p oblem belongs o he desc ip-
ion o a p oblem ha has a solu ion. In ou example, we iden i y such a p oblem ha
has a solu ion by es ic ing he o iginal p oblem o he da a poin s “men ioned” in he
gi en ini e subse . The gi en ini e subse is indeed pa o he well desc ip ion o he
es ic ed p oblem, and since his p oblem is ini e, i can be sol ed by known exis ence
esul s o ini e p oblems, so long as we e i y ha i “inhe i s” om he in ini e p ob-
lem any p ope ies equi ed by hese esul s (namely, o ou example, sa is ying SARP).
Lemma 1 hen gua an ees he exis ence o an app op ia e solu ion o he in ini e p ob-
lem.
We p o e Lemma 1using logical compac ness (see Sec ion 2.1), a cen al esul in he
heo y o p oposi ional logic. While he abo e example demons a es he applicabili y o
3The eade may hink o an assignmen o u h alues o he Boolean a iables as a “s a e o he wo ld.”
In he language o ma hema ical logic, such an assignmen unde which all s a emen s hold is called a
model o he s a emen s (no o be con used wi h he s anda d economic concep o a “model”).
4We u he elabo a e on his example in Sec ion 3.

Theo e ical Economics 20 (2025) To in ini y and beyond 515
ou app oach o exis ence esul s o inhe en ly disc e e objec s, we also show how o use
his app oach o scale economic esul s ha go beyond disc e e solu ions in o in ini e
se ings.
2. F amewo k
In his sec ion, we p o ide a b ie in oduc ion o p oposi ional logic (in Sec ion 2.1),5
and use i o s a e and p o e ou main echnical lemma (in Sec ion 2.2).
2.1 P oposi ional logic p elimina ies
In p oposi ional logic, we wo k wi h a se o Boolean a iables, and s udy he u h alues
o s a emen s—called o mulae—made up o hose a iables. We cons uc o mulae
by conjoining a iables wi h simple logical ope a o s such as o ,no ,andimplies.Va i-
ables a e abs ac , and do no ha e meaning on hei own—bu we can imbue hem wi h
“seman ic” meaning by in oducing o mulae ha e lec he s uc u e o economic (o
o he ) p oblems. Once gi en seman ic meaning, he u h o alsi y o s a emen s in ou
p oposi ional logic model imply he co esponding esul s in he associa ed economic
model.
We s a by o malizing he idea o (well- o med p oposi ional) o mulae. To de ine
he se o o mulae a ou disposal, we i s in oduce a basic ( ini e o in ini e) se o
(Boolean) a iables. In each sec ion o his pape , we in oduce a di e en se o a i-
ables buil a ound he economic se ing ha we model in ha sec ion.
Once we ha e in oduced a ( ini e o in ini e) se Vo a iables, we can de ine he
se o all well- o med o mulae induc i ely:
•‘φ’ is a well- o med o mula o e e y a iable φ∈V.
•‘¬φ’ is a well- o med o mula o e e y well- o med o mula φ.
•‘(φ∨ψ)’, ‘(φ∧ψ)’, ‘(φ→ψ)’, and ‘(φ↔ψ)’ a e well- o med o mulae o e e y wo
well- o med o mulae φand ψ.
Example 1. We could s a wi h a se o ou a iables V={P,Q,R,S}. Then each o he
ollowing is a well- o med o mula:
‘P’(1)‘(P∨Q)’(2)‘¬(P∧Q)’(3)‘((P∧R)→S)’(4)♦
We some imes abuse no a ion by omi ing pa en heses and w i ing, o exam-
ple, ‘φ∨ψ∨ξ’ when any a bi a y placemen o pa en heses in he o mula (e.g.,
‘((φ∨ψ)∨ξ)’o ‘
(φ∨(ψ∨ξ))’) will no make a di e ence. We some imes abuse no a-
ion e en u he by w i ing, o example, ‘10
i=1φi’ o mean ‘φ1∨φ2∨···∨φ10’(once
5Fo a mo e in-dep h look a p oposi ional logic p imi i es and a he compac ness heo em, see a ex -
book on ma hema ical logic (e.g., Ende on (2001), Goncza owksi and Nisan (2022)). P oposi ional logic
o mulae a e also used o s a ing Boolean sa is iabili y p oblems.
516 Goncza owski, Komine s, and Sho e Theo e ical Economics 20 (2025)
again, only when he p ecise placemen o omi ed pa en heses is o no consequence o
ou analysis).
We no e ha while well- o med o mulae can be a bi a ily long, each well- o med
o mula is always ini e in leng h. Thus, o example, a disjunc ion ‘φ1∨φ2∨···’o
in ini ely many o mulae is no a well- o med o mula. We he e o e ake special ca e
when we claim ha o mulae o he o m ‘φ∈φ’ a e well- o med, as his is ue only
i is ini e.
Amodel is a mapping om he se Vo all a iables o Boolean alues, ha is, each
a iable is mapped ei he o being T u e o o being False. This induces a u h alue o
e e y o mula ‘φ’whe eφ∈V. A model also induces a u h alue o all o he o mulae,
de ined induc i ely as ollows:
•‘¬φ’isT u e i and only i φis False;
•‘(φ∨ψ)’isT u e i and only i ei he o bo h o φand ψis T u e ;
•‘(φ∧ψ)’isT u e i and only i bo h φand ψa e T u e ;
•‘(φ→ψ)’isT u e i and only i ei he φis False o ψis T u e o bo h (i.e., ‘(φ→ψ)’is
False i and only i bo h φis T u e and ψis False); and
•‘(φ↔ψ)’isT u e i and only i φand ψa e ei he bo h T u e o bo h False.
Example 2. Gi en he concep o u h alues, we can ein e p e he o mulae (1)–(4)
as ollows:
‘P’“P[is T u e ],” (1)
‘(P∨Q)’“Po Q[is T u e ],” (2)
‘¬(P∧Q)’“no (Pand Q[a e bo h T u e ]),” (3)
‘((P∧R)→S)’“Pand R[bo h being T u e ],impliesS[being T u e ].” (4)
The o mula in (2) is T u e in a model i and only i ei he ‘P’o ‘Q’(o bo h)a eT u e in
ha model; he o mula in (3) is T u e in a model unless bo h ‘P’and‘Q’a eT u e in ha
model; and he o mula in (4) is T u e in a model unless bo h ‘P’and‘R’a eT u e in ha
model while ‘S’isFalse in ha model. ♦
We say ha a o mula is sa is ied by a model i i is T u e unde ha model. Fo ex-
ample, each o he o mulae (1), (2), and (4) is sa is ied by he model ha assigns alue
T u e o all a iables, howe e , he o mula (3) is no sa is ied by ha model. We say ha
a (possibly in ini e) se o o mulae is sa is ied by a model i e e y o mula in he se is
sa is ied by he model. Fo example, he se o he o mulae (1)–(4) is sa is ied by he
model ha assigns alue T u e o all a iables excep Q. We say ha a (possibly in ini e)
se o o mulae is sa is iable,o ha i has a model, i i is sa is ied by some model. Fo
example, he se con aining ‘P’and‘¬P’ is no sa is iable.
Theo e ical Economics 20 (2025) To in ini y and beyond 517
Clea ly, i a ( ini e o in ini e) se o o mulae is sa is iable, hen e e y subse o 
is also sa is iable (by he same model), and in pa icula e e y ini e subse o is sa is-
iable; he compac ness heo em o p oposi ional logic gi es a su p ising and non i ial
con e se o his s a emen .
Theo em 1 (Compac ness Theo em o P oposi ional Logic (Gödel,1930;Malce ,
1936)). A se o o mulae is sa is iable i (and only i ) e e y ini e subse ⊆is sa is-
iable.
2.2 A scaling lemma o economic heo ies
In his sec ion, we use p oposi ional logic o de i e a su icien condi ion o scalabili y
o an economic heo em o in ini e cases. This condi ion, o malized in Lemma 1,isa
he hea o all o ou p oo s.
Le Sbe a se o which we e e as he se o (po en ial) solu ions.Le Pbease
o which we e e as he se o (economic) p oblems, whose solu ions (i such exis ) a e
in S. Fo example, o he consume choice a ionaliza ion example om Sec ion 1.1,S
is he se o all consume p e e ences o e some se o objec s, and a p oblem P∈Pis o
a ionalize a speci ic da ase D.
De ine I:P×S→{T u e ,False}such ha I(P,S)is T u e i and only i Sis a solu ion
o P. No e ha gi enap oblemP, e en i we can easily de e mine o any gi en S
whe he Sis a solu ion o P(i.e., whe he I(P,S)is T u e ), i may no be clea jus om
examining P(and I) whe he o no i has any solu ion (i.e., whe he he e exis s S∈S
such ha I(P,S)is T u e ). Fo example, while i is easy o desc ibe when gi en p e e ences
a ionalize a gi en da ase , i is no immedia e om examining a da ase whe he he e
exis p e e ences ha a ionalize i . Simila ly, while i is easy o desc ibe when a gi en
s a egy p o ile cons i u es a Nash equilib ium in a gi en game, i is no immedia e om
examining a game whe he i admi s a Nash equilib ium. Rega dless o he economic
se ing, gi en a p oblem, ou goal will be o asce ain whe he a solu ion o i indeed
exis s.
We say ha he se Po p oblems is a se o well-desc ibable p oblems i o e e y
P∈P he e exis s a se Po well- o med o mulae such ha Phas a solu ion i and
only i Phas a model. We call a collec ion (P)P∈Po such se s a well desc ip ion o P.
Example 3. Conside a se Po p oblems whe e o each p oblem P=(X,D)∈P, he
se o solu ions o Pconsis s o all s ic p e e ences o e he uni e se X ha a ional-
ize he gi en choice da a D. To well desc ibe P, we may use he a iable o he o m
ag b om Sec ion 1.1 (whe e ag bhas he seman ic in e p e a ion “ais p e e ed o b”).
Speci ically, o each P∈P, one may ha e he se o o mulae P ha consis s o :
(i) o all dis inc aand bsuch ha in he gi en choice da a, ais chosen om a menu
ha con ains b, he o mula‘
ag b’, equi ing ha he p e e ences ( ha co e-
spond o any model o he o mulae) a ionalize he gi en choice da a D;
(ii) o all dis inc a,b∈X, he o mula‘
ag b∨bg a’, equi ing ha he p e e ences
be comple e;
518 Goncza owski, Komine s, and Sho e Theo e ical Economics 20 (2025)
(iii) o all dis inc a,b∈X, he o mula‘¬(ag b∧bg a)’, equi ing ha he p e e -
ences be an isymme ic;
(i ) o all dis inc a,b,c∈X, he o mula‘
(ag b∧bg c)→ag c’, equi ing ha he
p e e ences be ansi i e.
By cons uc ion, (P)P∈Pis a well desc ip ion o P.6(The p eceding o mulae co e-
spond exac ly wi h he o mulae (1)–(4) om Sec ion 2.1 upon aking P=ag b,Q=bg a,
R=bg c,andS=ag c.) ♦
Gi en a well desc ip ion o P, we say ha a p oblem P∈Psa is ies he ini e-subse
p ope y (wi h espec o he gi en well desc ip ion o P) i o e e y ini e subse ⊂P
he e exis s a p oblem P∈P ha has a solu ion and o which ⊆P.ByTheo em1,
we hen ha e he ollowing.
Lemma 1 (Scaling Lemma). Le Pbe a se o well-desc ibable p oblems and le (P)P∈P
be a well desc ip ion o P.Le P∈P.I Psa is ies he ini e-subse p ope y, hen Phas a
solu ion.
P oo .Le P∈Pbe a p oblem sa is ying he ini e-subse p ope y. By well desc iba-
bili y, i is enough o show ha Pis sa is iable. By Theo em 1, i is he e o e enough o
show ha e e y ini e ⊂Pis sa is iable. Le be such a ini e subse . Since Psa is-
ies he ini e-subse p ope y, he e exis s P∈P ha has a solu ion such ha ⊆P.
Since Phas a solu ion, by ou well-desc ibabili y assump ion, we ha e ha Pis sa -
is iable by some model. Since ⊆P, he same model also sa is ies ,andsois
sa is iable as equi ed.
As demons a ed in Sec ion 1.1, in many cases o in e es , he exis ence o a solu ion
o an app op ia e P o any can be es ablished by ini e-case heo ems (e.g., on a io-
nalizabili y o ini e da ase s o s able ma ching in ini e ma ke s). Thus, by Lemma 1,we
ob ain exis ence o solu ions o he in ini e case o such p oblems as well. In his pape ,
we demons a e he applicabili y o Lemma 1 o a wide ange o economic p oblems.
3. Wa m-up:Ra ional choice unc ions
We begin wi h a classic e ealed p e e ence se up. Le Xbe a (possibly in ini e) se o
goods. A menu is a ini e subse o X.Ada ase D⊆{(S,a)∈2X×X|a∈S}consis s o
he (unique) espec i e choices made by an agen in a (possibly in ini e) se o menus.
Apai (S,a)∈Dis in e p e ed o mean ha he agen selec ed a∈Swhen p esen ed
wi h he menu S. We say ha a da ase Dis a ionalized by a s ic p e e ence ela ion
(comple e, an isymme ic, and ansi i e) o e X, i o e e y (S,a)∈D, he agen ’s
choice ais he maximal elemen om Sacco ding o .Ada ase is a ionalizable i i is
a ionalized by some s ic p e e ence ela ion o e X.
6In ac , an e en s onge p ope y holds: he se o solu ions o Pis in one- o-one co espondence wi h
he se o models o P(see Sec ion 3 o mo e de ails on his co espondence). While such a one- o-one
co espondence holds in many o ou applica ions, his is no equi ed o ou a gumen s.
Theo e ical Economics 20 (2025) To in ini y and beyond 525
We i s claim ha e e ymodel ha sa is iesDco esponds o a solu ion o D.
Fix a model o D. Fo e e y ¯
x∈Rm
+and e e y n∈N,le n∈Vnbe he alue
such ha u ili yn
¯
x, nis T u e in he model (well-de ined by he i s and second o -
mula ypes abo e), and de ine u(¯
x)=limn→∞ n(well-de ined, e.g., by he hi d o -
mula ype abo e since nis a Cauchy sequence). The esul ing u ili y unc ion u
is a limi o nondec easing quasiconca e unc ions (by he ou h and i h o mula
ypes abo e) ha weakly a ionalize he da a (by he se en h o mula ype abo e).
Hence, ui sel is a nondec easing quasiconca e unc ion ha weakly a ionalizes
he da a. Fu he mo e, o e e y ¯
x,¯
y∈Rm
+such ha ¯
x¯
y, he eexis wo a io-
nal numbe ec o s “in be ween” hem, ha is, he e exis s k∈Nsuch ha ¯
x
¯
qk
1¯
qk
2¯
y. The e o e, we ha e ha u(¯
x)≤u(¯
qk
1)≤u(¯
qk
2)−2−k−1<u
(¯
qk
2)≤u(¯
y)
( he second inequali y s ems om ha inequali y holding o almos all unc ions
o which uis he limi , by he six h o mula ype abo e), so uis s ic ly inc easing
when all coo dina es s ic ly inc ease. Finally, since uweakly a ionalizes Dand is
also s ic ly inc easing when all coo dina es s ic ly inc ease, hen ualso a ional-
izes D.
Second, we claim ha i Dhas a solu ion, hen Dhas a model. Fix a solu ion u
o D,andle ¯
u(¯
x)1/4+(1/2π)·a c an(u(¯
x)) +k:¯
qk
2≤¯
x2−k−1 o e e y ¯
x∈Rm
+.As
his ans o ma ion o u ili ies is s ic ly mono one, he esul ing unc ion ¯
us ill a io-
nalizes he da a, and is quasiconca e, nondec easing, and s ic ly inc easing when all
coo dina es s ic ly inc ease. Fu he mo e, he sum o he i s wo summands is in
[0, 1/2], and so is he hi d summand, so he o e all sum is in [0, 1]. Finally, due o
he hi d summand, ¯
u(¯
qk
2)>¯
u(¯
qk
1)+2−k−1 o e e y k∈N. Using ¯
u, we can he e-
o e cons uc a model o D(by se ing each u ili yn
¯
x, o be T u e i and only i
=
¯
u(x)εn), and so Dhas a model. To sum up, (D)D∈Pis a well desc ip ion
o P.
Fini e-subse p ope y: Le D∈P.Le ⊂Dbe a ini e subse . Since is i-
ni e, he e a e only ini ely many o mulae o he abo e se en h ype ( he only o -
mula ype ha depends on he da ase ) in .Le D⊂Dbe he se o da apoin s
ha induce hese o mulae. By de ini ion, ⊆D. Fu he mo e, Dsa is ies GARP
since any subda ase o Dsa is ies GARP, and hence, by Theo em 7,Dis a ionaliz-
able. The e o e, Dsa is ies he ini e-subse p ope y. Thus, by Lemma 1,Dis a ional-
izable.
A na u al ques ion is why he same a gumen canno be used o scale Theo em 7
while main aining conca i y a he han quasiconca i y. The sho answe is ha —due
o he equi emen ha ube s ic ly mono one, and he inhe en need o make each
o mula ini e—ou p oo o Theo em 8 elies hea ily on he ac ha quasiconca i y,
unlike conca i y, is main ained unde weakly mono one ans o ma ions (such as he
mapping o u o ¯
u); we discuss his u he in Sec ion 6.
5.1 Addi ional applica ion o he same p oo : Ra ional ina en ion
Ou p oo o Theo em 8is qui e a bi mo e lexible han one migh imagine. In Ap-
pendix B, we use essen ially he same well desc ip ion o scale— om ini e o in ini e

526 Goncza owski, Komine s, and Sho e Theo e ical Economics 20 (2025)
da ase s— he seminal esul o Caplin and Dean (2015) in qui e a di e en a ionaliz-
abili y domain: a s a e-dependen s ochas ic choice da ase has a cos ly in o ma ion
acquisi ion ep esen a ion i and only i i sa is ies he No Imp o ing Ac ion Swi ches
(NIAS) and No Imp o ing A en ion Cycles (NIAC) condi ions. Caplin, Dean, and Leahy
(2017) ecen ly p o ed he in ini e e sion o his esul ia a no el p oo ha di e ges
om Caplin and Dean’s p oo o he ini e case.18 We ep o e his esul using essen ially
he same well desc ip ion as in ou p oo o Theo em 8, despi e he di e ences be ween
he wo se ings conside ed, and despi e he ac ha nei he he o iginal p oo s o he
ini e e sions no he o iginal p oo s o he in ini e e sions o any o hese qui e di e -
en heo ems sha e any common co e echnique. The main di e ence be ween he wo
well desc ip ions is ha his applica ion does no equi e s ic mono onici y. The e o e,
he six h o mula ype o he abo e well desc ip ion is no equi ed, and he p oo ha
a solu ion implies a model is simple as i does no equi e ca e ully “massaging” he
unc ion uin o ¯
uas abo e.
6. Rema ks and limi a ions
In he p eceding sec ions, we demons a ed he e sa ili y o ou app oach ac oss se -
e al e ealed-p e e ence se ings. Ou app oach can be used o scale many addi ional
ini e-da a esul s—like hose o Ca aneo, Ma, Masa lioglu, and Suleymano (2020)on
he exis ence o andom a en ion ep esen a ion, o Filiz-Ozbay and Masa lioglu (2023)
on p og essi e andom choice— o encompass in ini e da ase s.19 In addi ion o scaling
a wide a ay o ini e-da a esul s o encompass in ini e da ase s, ou app oach can also
be used o adap ini e-da a a ionaliza ion esul s o suppo pa ame ic es ic ions, as
in Hu, Li, Quah, and Tang (2021), since such es ic ions o en ansla e in o in ini ely
many cons ain s. Ou app oach, howe e , is no wi hou limi a ions. In his sec ion, we
p o ide some ema ks on limi a ions o ou p oo s wi hin e ealed p e e ence. A mo e
high-le el discussion o se ings in which ou app oach is no applicable is p o ided in
Sec ion 7.2.
A ia ’s heo em (Theo em 7) gua an ees ha ini e demand da ase s sa is ying
GARP can be a ionalized using a conca e u ili y unc ion. Bu he e a e well-known
examples o quasiconca e u ili y unc ions whose ull (in ini e) demand da ase (which
sa is ies GARP since i is de i ed om he choices o a u ili y unc ion) canno be a io-
nalized using a conca e u ili y unc ion. In Sec ion 5, we ep o ed he main esul o
Reny (2015) ha uni ied hese se ings—i.e., any demand da ase , ini e o in ini e, ha
sa is ies GARP can be a ionalized using a quasiconca e u ili y unc ion (Theo em 8).
By Lemma 1, he exis ence o a coun e example, oge he wi h he co ec ness o
A ia ’s heo em o ini e da ase s, implies ha he exis ence o a conca e a ionalizing
u ili y unc ion (as gua an eed by A ia ’s heo em) has no well desc ip ion ha sa is ies
he ini e-subse p ope y. This migh seem puzzling since a simple modi ica ion o he
ou h o mula ype in ou p oo (which imposes quasiconca i y) can be used o impose
18de Oli ei a, Den i, Mihm, and Ozbek (2017) p o ide a simila esul o in ini e da ase s o a di e en
kind.
19We hank Yusu can Masa lioglu o p oposing hese applica ions.
Theo e ical Economics 20 (2025) To in ini y and beyond 527
conca i y (as in ou simila scaling p oo in Appendix B), and so should seemingly esul
in a well desc ip ion as equi ed. The answe o his puzzle is ha his well desc ip ion
does no , in ac , sa is y he ini e-subse p ope y. Speci ically, he six h o mula ype
in ou p oo makes a s onge mono onici y equi emen han he mono onici y ha is
gua an eed by A ia ’s heo em and, he e o e, A ia ’s heo em canno be used o show
ha he ini e-subse p ope y equi ed by Lemma 1holds.
To add ess his issue, a na u al app oach would be o change he mono onici y e-
qui emen ha we use o equi e only s ic mono onici y, as gua an eed by A ia ’s he-
o em. Bu i is no possible o well-desc ibe s ic mono onici y wi h ou a iables (since
s ic inequali ies a e no p ese ed in he limi ). Ou way a ound his limi a ion was o
make a s onge equi emen ha is well desc ibable. Bu in o de o use A ia ’s heo em
o show ha he ini e-subse p ope y holds, we had o elax he conca i y equi emen
( ecall ha ou p oo applied mono onic ans o ma ions o he u ili y unc ion; while
hese ans o ma ions do no p ese e conca i y, hey do p ese e quasiconca i y). We
no e ha while his may appea o be an a e ac o using Lemma 1, he exis ence o he
abo emen ioned coun e example gua an ees ha no o he app oach could ci cum en
his issue. The adeo be ween s eng hening mono onici y and weakening conca i y,
so ha well desc ibabili y and he ini e-subse p ope y a e sa is ied, sheds some new
ligh on wha b eaks in he in ini e case, which a i s glance migh look like an issue
wi h conca i y, bu a a deepe look e eals i sel o be an issue wi h s ic mono onici y.
This a o ds some deg ee o in ui ion o “why” he conca i y assump ion in Theo em 7
mus be elaxed o quasiconca i y when scaling i o in ini e da ase s.
We no e ha when we use a e y simila p oo in Appendix B o scale a esul by
Caplin and Dean (2015), we do equi e conca i y a he han me ely quasiconca i y.
This is possible because in ha esul only weak ( a he han s ic ) mono onici y (in
in o ma ion) is equi ed, which can be well desc ibed wi hou being s eng hened. Con-
as ing hese wo p oo s p o ides ye ano he example o he powe o ou app oach o
angibly pinpoin why ce ain condi ions can be main ained when some heo ems a e
scaled bu no when o he s a e.
Meanwhile, Theo em 8illus a es some o he limi a ions o ou amewo k. A
p oposi ional logic o mula ion p ecludes he use o quan i ie s (e.g., “ he e exis s apos-
i i e gap by which he u ili y om ¯
q2is g ea e han he u ili y om ¯
q1”), and also p e-
cludes in ini ely long o mulae such as in ini e disjunc ions (e.g., “ he u ili y om ¯
q2is
g ea e han he u ili y om ¯
q1by a leas one o he ollowing in ini ely many posi i e
gaps”). This p ohibi s he well desc ip ion o ce ain p ope ies o in e es (e.g., s ic
mono onici y, unless s eng hened) wi hou he use o a iables ha e e o in ini ely
many objec s. Bu such a iables o en imes hinde he abili y o in oke ini e heo ems
o show ha he ini e-subse p ope y holds.
The equi emen o s ic mono onici y unde lies ano he well-known coun e ex-
ample. While a s ic p e e ence o de o e a ini e se o objec s can always be ep e-
sen ed by a u ili y unc ion, he same need no be ue when he se o objec s is un-
528 Goncza owski, Komine s, and Sho e Theo e ical Economics 20 (2025)
coun able.20 Acco dingly, any a emp o use ou s eng hened mono onici y equi e-
men o scale he ini e case is o cou se bound o ail when he se o objec s is un-
coun able. I is ins uc i e o conside how i would ail. Recall ha ou s eng hened
mono onici y equi es ixed posi i e gaps be ween a ious u ili y alues. In he case o
A ia ’s heo em, equi ing coun ably many such gaps su iced, and hence he equi ed
gap leng hs could be chosen so ha hei sum is ini e. By con as , scaling he exis-
ence o a u ili y ep esen a ion o s ic p e e ences o uncoun able se s would in ol e
equi ing uncoun ably many posi i e gaps. This means ha he sum o leng hs o e-
qui ed gaps would be in ini e, and so some objec s would no be associa ed wi h a ini e
u ili y.
7. Beyond e ealed p e e ences
Ou app oach is no in any way limi ed o e ealed p e e ences. In his sec ion, we illus-
a e i s applicabili y in ano he domain: noncoope a i e game heo y. We u he mo e
p o ide examples whe e ou amewo k is inapplicable.21
7.1 Nash equilib ia in games on in ini e g aphs
In his sec ion, we u n o he se ing o games on g aphs (see, e.g., Kea ns (2007), and he
e e ences he ein), which includes o e lapping gene a ions models, e en wi h in ini e
ime. We use Lemma 1 o show he exis ence o a Nash equilib ium in games on in i-
ni e g aphs. Ou esul he e is co e ed by Peleg (1969) (who di ec ly scales he seminal
exis ence esul o Nash (1951)), bu we gi e a new p oo ha uses he same p incipled
app oach ha we use h oughou his pape .
He e, we use Lemma 1 o scale he exis ence o a bi a ily good app oxima e Nash
equilib ia, and hen show ha he exis ence o such app oxima e equilib ia implies he
exis ence o an exac equilib ium. This wo-s ep p oo s a egy is chosen o con e-
nience: wi h addi ional a iables, i is easy o encode he second s ep o he p oo in o
he logical o mula ion jus like we did in Sec ion 5.22
In a game on a g aph, he e is a (po en ially in ini e) se o playe s I, each ha ing
a ini e se o pu e s a egies Si.Eachplaye i∈Iis linked o a ini e se o neighbo s
N(i)⊂Iwi h i∈N(i), and he u ili y only depends on he s a egies played by play-
e s in he se N(i).23 This se ing occu s, o example, in in ini e-ho izon o e lapping
gene a ions models, whe e a each poin in ime he e a e only ini ely many playe s
ali e, and a playe ’s u ili y depends only on he beha io o con empo a y playe s. Fo
each playe i,wedeno ebyi(Si) he se o mixed s a egies (i.e., dis ibu ions
o e pu e s a egies) o playe i.Amixed-s a egy p o ile (σi)i∈Iis a speci ica ion o a
20Fo example, lexicog aphic p e e ences o e R2o any s ic p e e ence o de o e 2R.
21An in-p epa a ion companion pape p esen s applica ions om ma ching heo y, some o which we e
in ou o iginal wo king pape (Goncza owski, Komine s, and Sho e ,2023); see also Choi (2024).
22The con e se does no hold o he analysis in ha sec ion, hough: The p oo he e hinges on a ull
in ini e sequence o app oxima ions being encoded by a single model.
23Reade s amilia wi h he wo k o Peleg (1969) will no e ha e en on g aphs, Peleg’s assump ions a e
weake han hose s a ed he e. Ou analysis can be gene alized o co e such weake assump ions.
Theo e ical Economics 20 (2025) To in ini y and beyond 529
mixed s a egy σi∈i o e e y playe i∈I. A mixed-s a egy p o ile (σi)i∈Iis a Nash
equilib ium i o e e y i∈Iand e e y possible de ia ing s a egy σ
i∈i,i holds ha
ui(σN(i))≥ui(σ
i,σN(i) {i}).
Games on ini e g aphs ha e ini ely many playe s and ini ely many s a egies pe
playe ; hence, he seminal analysis o Nash (1951) implies ha hey ha e Nash equilib ia.
Theo em 9 ( ollows om Nash (1951)). E e y game on a ini e g aph has a Nash equi-
lib ium.
Ou main esul o his sec ion is ha Nash equilib ia a e gua an eed o exis e en in
games on in ini e g aphs.
Theo em 10 ( ollows om Peleg (1969)). E e y game on a (possibly in ini e) g aph has a
Nash equilib ium.
As al eady no ed, we p o e Theo em 10 by i s using Lemma 1 o p o e he exis-
ence o a bi a ily good app oxima e Nash equilib ia, and hen showing ha he ex-
is ence o such app oxima e Nash equilib ia implies Theo em 10.Fo agi enε>0, a
mixed-s a egy p o ile (σi)i∈Iis an ε-Nash equilib ium i o e e y i∈Iand e e y possi-
ble de ia ing s a egy σ
i∈i,i holds ha ui(σN(i))≥ui(σ
i,σN(i) {i})−ε.
Lemma 2. Fo any ε>0, e e y (possibly in ini e) game on a g aph has an ε-Nash equi-
lib ium.
P oo .Le ε>0. Fo each playe i∈I, i will be con enien o conside he space o
p o iles o mixed s a egies o playe s in N(i)as a me ic space wi h he ∞me ic. No e
ha his me ic space is compac . As each playe ihas a con inuous u ili y unc ion
whose domain is his compac me ic space, playe s’ u ili y unc ions a e uni o mly con-
inuous by he Heine–Can o heo em. Thus, he e exis s ˆ
δi>0 ha assu es ha i wo
p o iles o mixed s a egies o playe s in N(i)a e less han ˆ
δiapa , hen he u ili ies hey
yield o idi e s by no mo e han ε/2.
Fo each playe i, choose δimin{ˆ
δj|j∈N(i)}>0. Recall ha ideno es he space
o playe i’s mixed s a egies, and le δi
i⊂ibe a ini e se o s a egies ha includes
all o i’s pu e s a egies, and includes o any mixed s a egy in ia s a egy ha is a
mos δiaway om i ; such a se exis s by he compac ness o i. We p o e he lemma by
p o ing ha he gi en game admi s an ε-Nash equilib ium in which each playe iplays
a s a egy in δi
i. We p o e his using Lemma 1.
De ini ion o P:Le Pbe all he subse s o I.Asolu ion o I∈Pis a s a egy p o ile
o I ha is a ε-Nash equilib ium in he induced game be ween all playe s in I(whe e
all o he “playe s” play any a bi a y s a egy), in which each playe i∈Iplays a s a egy
in δi
i.
Well desc ibabili y: We in oduce a a iable plays(i,σi) o e e y playe i∈Iand dis-
c e ized s a egy σi∈δi
i. In wha ollows, o each I∈Pwe de ine a se Io o mu-
lae o e hese a iables so ha models o Ia e in one- o-one co espondence wi h
530 Goncza owski, Komine s, and Sho e Theo e ical Economics 20 (2025)
he (no -ye -p o en- o-be-nonemp y) se o solu ions o I. The co espondence is ob-
ained by endowing he a iable plays(i,σi)wi h he seman ic in e p e a ion “iplays
he s a egy σi.” Tha is, i maps a model o I o he s a egy p o ile such ha o e e y
i∈I,weha e ha iplays he s a egy σii and only i he a iable plays(i,σi)is T u e in
ha model. Fo e e y playe iand e e y p o ile σN(i) {i}o mixed s a egies o N(i) {i},
we de ine he se o ε-bes esponses o i:
BRε
i(σN(i) {i})σiui(σi,σN(i) {i})≥max
σ
i∈iuiσ
i,σN(i) {i}−ε.
We de ine he se I o consis o he ollowing o mulae:
(i) o all i∈I, he ( ini e!) o mula ‘σ∈δi
i
plays(i,σ)’, equi ing ha iplays some
(disc e ized) s a egy ( his o mula is ini e because δi
iis);
(ii) o all i∈Iand all dis inc σi,σ
i∈δi
i, he o mula‘plays(i,σi)→¬plays(i,σ
i)’,
equi ing ha he s a egy ha playe iplays be unique;
(iii) o all i∈Iand all p o iles σ=(σj)j∈N(i) {i}∈×j∈N(i) {i}δi
io disc e ized mixed
s a egies o N(i) {i}, he ( ini e!) o mula
‘
j∈N(i) {i}
plays(j,σj)→
σi∈δi
i∩BRε
i(σ)
plays(i,σi)’,
equi ing o i o ε-bes espond o he s a egies played by he o he playe s.
By cons uc ion, (I)I∈Pis a well desc ip ion o P.
Fini e-subse p ope y: Le ⊂Ibe a ini e subse . Since is ini e, i “men ions”
( h ough a iables used) only ini ely many playe s; deno e he se o hese playe s by
I⊂I. By de ini ion, ⊆I. Conside he induced game on Iob ained by ha ing
each playe i∈I Imechanically play some ixed s a egy in δi
i.ByTheo em9, his
game has a Nash equilib ium. By choosing o each playe i∈Ia closes s a egy in δi
i
o he one she plays a his Nash equilib ium, each playe ’s u ili y changes by a mos ε/2
(by uni o m con inui y), and so does he u ili y a ainable by bes esponding. The e o e,
since we s a ed wi h a Nash equilib ium, i is assu ed ha each playe is now playing an
ε-bes esponse, so he esul ing s a egy p o ile is a solu ion o I. The e o e, Isa is ies
he ini e-subse p ope y. Thus, by Lemma 1, he e exis s an ε-Nash equilib ium in he
g and game (among all playe in I), as equi ed.
Now, we can use Lemma 2 o p o e Theo em 10 by way o a diagonaliza ion a gu-
men .
P oo o Theo em 10. Since eachplaye in he g aph has ini ely many neighbo s, e -
e y connec ed componen o he g aph consis s o a mos coun ably many playe s. As
i is enough o show he exis ence o a Nash equilib ium in each connec ed componen
sepa a ely (we use he axiom o choice he e), le us ocus on one connec ed componen .

Theo e ical Economics 20 (2025) To in ini y and beyond 531
By Lemma 2, he e exis s a sequence (σn)∞
n=1o 1
n-Nash equilib ia in he game on his
connec ed componen . Since each o he a -mos -coun ably-many coo dina es o each
elemen in his sequence lies in [0, 1], we can choose a subsequence (a “diagonal subse-
quence”) ha con e ges in all coo dina es; le σ∗deno e he limi o ha subsequence.
We claim ha σ∗is a Nash equilib ium. To see his, no e ha o e e y i∈Iand
σ
i∈i,weha e o hen h elemen s o he sequence ha
uiσn
N(i)≥uiσ
i,σn
N(i) {i}−1
n.
By he con inui y o ui, his means ha o e e y i∈Iand σ
i∈i,weha e
uiσ∗
N(i)≥uiσ
i,σ∗
N(i) {i},
so no playe has a p o i able de ia ion unde he p o ile σ∗.Hence,σ∗is indeed a Nash
equilib ium—and in pa icula , we see ha a Nash equilib ium exis s in he game, as
desi ed.
7.2 Nonapplica ions
When using ou amewo k, one aces an inhe en ension. On he one hand, each o -
mula in a well desc ip ion mus use ini ely many a iables. On he o he hand, o be
able o use ini e esul s o es ablish he ini e-subse p ope y, each a iable mus be
seman ically ela ed only o a ini e se o elemen s in he economic p oblem. A i s
glance, his seems o p eclude applica ions in which he desi ed solu ion has a pa am-
e e wi h an in ini e domain, since equi ing ha he pa ame e ake some alue would
equi e an in ini e disjunc ion. Indeed, i is simple o handle pa ame e s wi h ini e do-
mains (which, as we ha e seen, na u ally occu in many applica ions). Ne e heless, we
ha e success ully applied Lemma 1also o se ings wi h in ini e-domain pa ame e s,
such as u ili ies (Sec ion 5), cos s/p ices (Appendix B), o p obabili ies (Sec ion 7.1).
S ill, as he ollowing examples demons a e, in seemingly simila p oblems, his ap-
p oach could no possibly wo k, since he in ini e case has no solu ion.
Example 4 (Spli ing he Dolla ). A dolla mus be spli be ween a se Io agen s. A
solu ion is an e icien and en y- ee di ision. When |I|<∞, spli ing he dolla equally
is a solu ion. Bu he case I=Nhas no solu ion. ♦
Example 5 (Highe Numbe Wins). Two playe s can s a e a numbe in S⊆R. The playe
whose s a ed numbe is highe wins a p ize (which is sha ed in case o a ie). A solu ion
is a pu e-s a egy Nash equilib ium. When |S|<∞, a solu ion exis s (each playe s a es
maxS), bu he case S=Nhas no solu ion. ♦
Wha a e he limi a ions o ou app oach ha p e en i om co e ing he p eceding
wo examples? Ou app oach o well-desc ibing p oblems wi h in ini e-domain pa-
ame e s is o “encode” hese pa ame e s ia a sequence o alues, each om a ini e
532 Goncza owski, Komine s, and Sho e Theo e ical Economics 20 (2025)
domain. Speci ically, we ha e encoded each o he abo emen ioned pa ame e s using a
sequence o inc easingly ine disc e iza ions.
Two ea u es a e c i ical o he success o his app oach: Fi s , ha he deside a a
on he encoded pa ame e can be imposed by indi idually ini e o mulae on he dis-
c e iza ions. Fo example, in Sec ion 5, a u ili y unc ion weakly a ionalizes he da a i
and only i each o i s disc e iza ions weakly a ionalize he da a. This allowed us o ep-
esen a u ili y unc ion ha weakly a ionalizes he da a by a sequence o disc e iza ions
ha each weakly a ionalizes he da a. The second c i ical ea u e is ha no only each
pa ame e alue can be encoded by such a sequence o alues (disc e iza ions), bu also
each such sequence om any alid model encode a alid pa ame e alue (i.e., in he
pa ame e domain). In Sec ion 5, since he sequence o disc e ized u ili ies is poin wise
inc easing and bounded, i has a ini e limi . In ou scaling o he a ional ina en ion
esul o Caplin and Dean (2015) in Appendix B, since disc e ized cos s a e inc easing,
hey ha e a limi , which is in R≥0∪{∞}— he domain o cos s in ha p oblem. Fo each
o he abo e examples, he e is no encoding ha has bo h o hese ea u es.
In “spli ing he dolla ,” i , o example, we encode each playe ’s alloca ion using dis-
c e iza ions, hen he e is no se o indi idually ini e o mulae on he disc e iza ions
ha holds i and only i he limi di ision is e icien . Hence, he i s ea u e is missing.
And, any encoding ha has he i s ea u e would lose he second one.
In “highe numbe wins,” i , o example, we encode each playe ’s numbe using
disc e iza ions, hen he e is no se o indi idually ini e o mulae on he disc e iza ions
ha holds i and only i he limi is ini e. Hence, he second ea u e is missing. And, any
encoding ha has he second ea u e would lose he i s one.
8. Rela ed li e a u e
In decision heo y, bo h ini e- and in ini e-da a models a e impo an and common.24
Reny (2015) showed how o uni y hese wo app oaches in he se ing o A ia (1967).
In ou iew, ou main con ibu ion o his li e a u e is in gene alizing beyond any spe-
ci ic se ing by p o iding a way o sys ema ically uni y hese app oaches. In Sec ion 4,
we p o ed ha he esul o Masa lioglu, Nakajima, and Ozbay (2012) scales o in ini e
da ase s using essen ially he same p oo we used o ep o e he classic esul o Rich e
(1966)andHansson (1968) ha SARP su ices o a ionaliza ion by s ic p e e ences.
Like he o iginal heo em o Masa lioglu, Nakajima, and Ozbay, ou in ini e e sion o
ha heo em applies o ull da ase s. de Clippel and Rozen (2021) p o ide an analogous
heo em o ini e da ase s ha need no necessa ily be ull; ou p oo can be used o
simila ly scale hei heo em o in ini e da ase s.
To ou knowledge, we a e he i s o use p oposi ional logic as a gene al ool o
scaling esul s in economics. I is wo h men ioning wi hin his con ex , hough, he
wo k o Holzman (1984), who used logical compac ness o elax opological condi ions
o Fishbu n (1984).25
24Fo a ecen example o a ea men o ini e and in ini e da ase s, see Aguia , Hje s and, and Se ano
(2022).
25Logical compac ness is equen ly used o scale exis ence esul s in ma hema ics om ini e se ings
o in ini e ones (see, e.g., de B uijn and E dös (1951) and Halmos and Vaughan (1950)).
Theo e ical Economics 20 (2025) To in ini y and beyond 533
Ou app oach was s a ed using p oposi ional logic, bu , in ac , Lemma 1gene alizes
o well desc ip ions using i s -o de logic as well. P oposi ional logic is a special case
o i s -o de logic. Impo an ly, he o me does no use quan i ie s (i.e., ∀and ∃). We
chose o ocus on his special case o simpli y he exposi ion, since we we e no able o
iden i y any economic applica ion in which he added gene ali y would be bene icial.26
O he pape s ha e used i s -o de ( a he han p oposi ional) logic and nons an-
da d analysis o uni y, e ine, and scale esul s in economic heo y. Examples include
he wo k o Ande son (1978), B own and Khan (1980), Ande son (1991), Khan (1993),
Blume and Zame (1994), Halpe n (2009), and Halpe n and Moses (2016). Chambe s,
Echenique, and Shmaya (2014) used compac ness in i s -o de logic o o malize he
no ion o he empi ical con en o a model. Like us, Chambe s, Echenique, and Shmaya
(2014) s udy applica ions o e ealed p e e ence heo y (see also Chambe s, Echenique,
and Shmaya (2017)), howe e hey deal wi h di e en ques ions om us, and use di e -
en echniques.
Hellman and Le y (2019) use (s ill di e en ) ools om ma hema ical logic o p o e
concep ually ela ed, ye incompa able, esul s: while ou pape scales ce ain i-
ni e esul s o in ini e se ings, hei pape scales ce ain coun ably in ini e esul s o
uncoun ably in ini e se ings. Speci ically, hey gi e su icien condi ions o scale ce -
ain exis ence esul s ha a e known o hold whene e he e a e coun ably many possi-
ble s a es o he wo ld in o scena ios wi h uncoun ably many possible s a es o he wo ld.
Thei esul s a e incompa able o any o ou esul s, and e en o ou exis ence-in-la ge-
ma ke esul s, i s because hey always assume ha he numbe o agen s is ini e (an
in ini e numbe o agen s, e en wi h only wo possible ypes o each, would al eady
esul in an uncoun ably in ini e se o possible s a es o he wo ld o begin wi h), and
second, because hey equi e ha he heo ems ha hey scale be al eady known o hold
o he coun ably in ini e, a he han only he ini e case.
We ha e been asked abou he ela ion o a ious heo ems in opology. Lemma 1is
s a ed in e ms o logical p oposi ions and i s p oo elies on logical compac ness. Logi-
cal p oposi ions can be ansla ed in o closed se s in an applica ion-speci ic opological
(p oduc ) space, in which se ing logical compac ness ollows om Tychono ’s heo em
on opological compac ness. In o he wo ds, Lemma 1can be p o ed using Tychono ’s
heo em, and i s s a emen can be ansla ed o he language o opology. Howe e , in
ou iew, he esul ing lemma would be ha de o di ec ly o mula e and he condi ions
would be ha de o e i y. And while opological compac ness o he language o ne s a e
s onge and mo e gene al app oaches, in he domains we s udy, hey o en in oduce
echnical issues ha can ende a gumen s inco ec in sub le ways (e.g., ma chings may
con e ge o an objec ha is no a ma ching). We he e o e iew he me hodological pa
o ou con ibu ion as in oducing a uni ying app oach ha is simple and in ui i e o
26Well-desc ibing economic p oblems using he ull gene ali y o i s -o de logic is challenging. Fo
example, ixing se s o objec s (e.g., men and women) is no s aigh o wa d. In ac , by he (upwa d)
Löwenheim–Skolem heo em, i a i s -o de heo y has an in ini e model (a model wi h an in ini e do-
main), hen i has a model o any la ge ca dinali y, which implies ha i s -o de heo ies canno bound
he ca dinali y o hei in ini e models. Hence, cons an s would ha e o play an impo an ole in he well
desc ip ion.
534 Goncza owski, Komine s, and Sho e Theo e ical Economics 20 (2025)
wo k wi h, and ha does no equi e us o look o he “ igh ” opological space o apply
opological easoning di ec ly.27
9. Discussion
This pape p o ides a no el, p incipled app oach o scaling economic heo y esul s
om ini e models o in ini e ones. We iden i y a su icien condi ion o scaling a e-
sul : A esul can be scaled i i is well desc ibable wi h a desc ip ion sa is ying he ini e-
subse p ope y. The bulk o his pape is dedica ed o demons a ing ha many esul s
in e ealed-p e e ence heo y mee ou condi ion and, he e o e, hold e en wi h in ini e
da ase s. We also demons a e a game- heo e ic applica ion ha ocuses on a di e en
“ ype” o scaling o in ini y: allowing an in ini e ( a he han ini e) numbe o playe s.
Ou app oach is no wi hou limi a ions, and may ail whe e o he app oaches can
succeed. Tha said, we ha e cu a ed an a ay o applica ions showing ha i has me i
in decision heo y and beyond in p o ing no el esul s, as well as consolida ing and
sho ening p oo s o p e iously known esul s, in a way ha o en sheds new ligh on
hem.
We iew he main con ibu ion o his pape o be a me hodologicalone: a new, easy-
o-use, and e sa ile ool o he economic heo y oolbox. We hope ha eade s o his
pape will be able o u he le e age ou app oach.
Appendix A: P oo o Theo em 5
P oo o Theo em 5.Aswi hTheo em3, he “only i ” di ec ion is immedia e, so we
p o e he “i ” di ec ion using Lemma 1.
De ini ion o P:Fixing X,le Pbe he se o all pai s (X,D)such ha X⊆Xand
Dis a ull da ase o e X ha sa is ies WARP-LA. A solu ion o a pai (X,D)∈Pis a
pai comp ising a s ic p e e ence o de o e Xand an a en ion il e ha oge he
a ionalize D.
Well desc ibabili y: We in oduce a a iable ag b o e e y pai o dis inc a,b∈X
and a a iable a n(S,T) o each pai o ini e se s S,Tsuch ha ∅=T⊆S⊆X.Inwha
ollows, o each (X,D)∈Pwe de ine a se (X,D)o o mulae o e hese a iables so
ha models o (X,D)a e in one- o-one co espondence wi h he (no -ye -p o en- o-
be-nonemp y) se o solu ions o (X,D). The co espondence is ob ained by endowing
he a iable ag bwi h he seman ic in e p e a ion “ais p e e ed o b(when bo h a e
a en ion a ac ing),” and he a iable a n(S,T)wi h he seman ic in e p e a ion “Tis
he se o a en ion-a ac ing elemen s when he menu is S.” Tha is, i maps a model
o (X,D) o he p e e ence such ha o e e y dis inc a,b∈X,weha e ha abi
and only i he a iable ag bis T u e in ha model and o he a en ion il e such ha
o e e y S,Tsuch ha ∅=T⊆S⊆X,weha e ha (S)=Ti and only i he a iable
a n(S,T)is T u e in ha model. We de ine he se (X,D) o consis o he ollowing
o mulae:
27Once a p oo is de i ed using Lemma 1, i is o cou se possible o hen ansla e i o a opological
s a emen and a emp o achie e g ea e gene ali y, i /when such gene ali y is o in e es .
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