Lazza i, Na alia; Quah, John K.-H.; Shi ai, Koji
A icle
An o dinal app oach o he empi ical analysis o games
wi h mono one bes esponses
Quan i a i e Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Lazza i, Na alia; Quah, John K.-H.; Shi ai, Koji (2025) : An o dinal app oach o
he empi ical analysis o games wi h mono one bes esponses, Quan i a i e Economics, ISSN
1759-7331, The Econome ic Socie y, New Ha en, CT, Vol. 16, Iss. 1, pp. 235-266,
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Quan i a i e Economics 16 (2025), 235–266 1759-7331/20250235
An o dinal app oach o he empi ical analysis o games wi h
mono one bes esponses
Na alia Lazza i
Depa men o Economics, Uni e si y o Cali o nia a San a C uz
John K.-H. Quah
Depa men o Economics, Na ional Uni e si y o Singapo e
Koji Shi ai
School o Economics, Kwansei Gakuin Uni e si y
We de elop a nonpa ame ic and o dinal app oach o es ing pu e s a egy Nash
equilib ium play in games wi h mono one bes esponses, such as hose wi h
s a egic complemen s/subs i u es. The app oach makes minimal assump ions
on unobse ed he e ogenei y, equi es no pa ame ic assump ions on payo
unc ions, and no es ic ion on equilib ium selec ion om mul iple equilib ia.
The app oach can also be ex ended in o de o make in e ences and p edic ions.
Bo h model- es ing and in e ence can be implemen ed by a ac able compu-
a ion p ocedu e based on column gene a ion. To illus a e how ou app oach
wo ks, we include an applica ion o an IO en y game.
Keywo ds. Re ealed p e e ence, mono one compa a i e s a ics, single-c ossing
di e ences, supe modula games, e ealed mono onici y axiom.
JEL classi ica ion. C1, C6, C7, D4, L1.
Na alia Lazza i: [email p o ec ed]
John K.-H. Quah: [email p o ec ed]
Koji Shi ai: [email p o ec ed]
Fo help ul discussions and commen s, he au ho s a e g a e ul o S. Be y, J. Fox, K. Hi ano, T. Hoshino,
A.Kajii,Y.Ki amu a,B.Kline,E.K asnoku skaya,C.Manski,W.Newey,T.Sekiguchi,J.S oye,B.S ulo ici,
S. Takahashi, Y. Takahashi, and especially X. Tang. Va ious e sions o his p ojec ha e been p esen ed
o audiences a he ollowing e en s and we a e g a e ul o hei commen s: semina s a Uni e si y o A i-
zona, Johns Hopkins, Kyo o, Lou ain (CORE), New Yo k Uni e si y, Rice Uni e si y, Simon F ase Uni e si y,
he Na ional Uni e si y o Singapo e, No hwes e n Uni e si y, Uni e si y o Pa is (Dauphine), Queens-
land, Shanghai Uni e si y o Finance and Economics, Uni e si y o Sou he n Cali o nia, Singapo e Man-
agemen Uni e si y, S an o d, UC Da is, UC San Diego, he Canadian Economic Theo y Con e ence (Van-
cou e , 2017), he Con e ence on Econome ics o Incomple e Models (CeMMAP and No hwes e n, 2018),
13 h G ea e New Yo k Me opoli an A ea Econome ics Colloquium (P ince on Uni e si y, 2018), and he
Econome ic Socie y No h Ame ican Summe Mee ing (UC Da is, 2018). Koji Shi ai g a e ully acknowl-
edges inancial suppo om he Japan Socie y o P omo ion o Science (KAKENHI 19K00155) and he
hospi ali y o Johns Hopkins Uni e si y du ing his isi in he 2019–2020 academic yea .
©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 p://qeconomics.o g.h ps://doi.o g/10.3982/QE2192
236 Lazza i, Quah, and Shi ai Quan i a i e Economics 16 (2025)
1. In oduc ion
Economic analysis is o en conce ned wi h he e ec o an exogenous o s a egic a i-
able on an agen ’s decision: Would a consume buy mo e o good A i he p ice o good
B alls? Would a i m ollow i s i al when he la e aises i s p ice? Is someone mo e
likely o join a demons a ion i mo e people a e pa icipa ing? The heo y o mono one
compa a i e s a ics iden i ies he single-c ossing p ope y (see Milg om and Shannon
(1994)) as a su icien (and, in a speci ic sense, necessa y) condi ion o op imal choices
o be mono one wi h espec o opponen s’ s a egies and exogenous a iables. The
empi ically ele an ollow-up ques ion is he ollowing: Wha kind o obse ed choice
beha io a e necessa y and su icien o he eco e y o payo unc ions obeying he
single-c ossing p ope y? The con ibu ion o his pape is o answe his e ealed p e -
e ence ques ion and o show ha i o ms he basis o an econome ic analysis o games
wi h s a egic complemen s.
One ob ious and impo an a ea o applica ion o ou esul s is o he s udy o en y
games (as in B esnahan and Reiss (1990), Be y (1992), o Cilibe o and Tame (2009))
and o he games ha a ise in he empi ical IO li e a u e. In hese pape s, i ms’ en y
decisions a e modeled as games o comple e in o ma ion, whe e each i m’s decision on
whe he o no o en e a gi en ma ke is a bes esponse o he en y decisions aken by
o he i ms in ha ma ke . The payo unc ions a e assumed o depend on obse able
a iables in a speci ic pa ame ic o m while he unobse ed componen is addi i ely
sepa able. The unobse ed componen is he e ogenous ac oss ma ke s and belongs o
a known class o dis ibu ions. En y decisions by i ms ac oss many ma ke s a e ob-
se ed, om which one could hen es ima e i ms’ payo unc ions. A majo issue in his
wo k conce ns he e ec s o s a egic in e ac ion and ma ke cha ac e is ics in e ms o
i s di ec ion and size: How o en does he en y o ano he i m encou age o de e en-
y? To wha ex en does an exogenous a iable (such as ma ke size) encou age o de e
he en y o o he i ms?
Ou app oach has as i s s a ing poin a da a se o he same ype as he pape s ci ed
abo e. Wi h his da a se , we can es whe he i ms a e playing pu e s a egy Nash equi-
lib ia (PSNE), subjec o single-c ossing es ic ions on i s payo unc ions. Fo exam-
ple, we can es he hypo hesis ha a i m’s en y in o a ma ke is encou aged when he
ma ke is la ge and discou aged when ano he i m is also en e ing. Ou me hod wo ks
wi hou imposing any pa ame ic assump ions on payo unc ions, wi hou assuming
ha unobse ed he e ogenei y is addi i e o ha i s dis ibu ion belongs o a pa icula
amily, and wi hou assump ions on equilib ium selec ion. By speci ying a join dis i-
bu ion on he payo unc ions, we allow o co ela ion o o he o ms o dependence
among i ms’ payo unc ions, which is impo an in many se ings (see Chen, Ch is-
ensen, and Tame (2018)). To pass ou es means ha he hypo hesis ha he da a a e
explained as PSNE by i ms wi h payo unc ions sa is ying single-c ossing es ic ions
canno be e u ed.
A i s mos basic, ou app oach p o ides a way o esea che s o es he gene al
(nonpa ame ic) ea u es o a model, be o e he implemen a ion o a mo e es ic i e
pa ame ic model ha could be used o in e ence and p edic ion. In some cases, he
Quan i a i e Economics 16 (2025) An o dinal app oach o he empi ical analysis 237
con i ma ion o mono one ea u es, which a e pa o ou es could also acili a e es i-
ma ion p ocedu es.1Beyond his, since ou es eco e s he dis ibu ions on i ms’ pay-
o unc ions ha sa is y single-c ossing es ic ions and ag ee wi h he obse a ions, he
p ocedu e can also be ex ended o he pu poses o in e ence and p edic ion (when he
da a se passes he es ).
While we w i e o eco e ing “payo unc ions,” wha we a e eally eco e ing a e
a playe ’s p e e ence o e di e en ac ions, condi ional on co a ia es and he ac ions
o o he playe s; his is as i should be, because in an en i onmen whe e only PSNE
a e played, he in o ma ion eco e ed om he da a has o be jus o dinal. The speci ic
p e e ence p ope y we es (o when making in e ences, assume)— he single-c ossing
p ope y—is also an o dinal p ope y.
Ou econome ic app oach is simila o ha in Ki amu a and S oye (2018)(hence-
o h KS).2This pape es s a andom u ili y model o consume demand. In he i s
s ep, i is assumed ha he popula ion dis ibu ion o consume demand a a linea
budge se B,whichwedeno ebyP
(·|B), is known o a ini e collec ion o budge se s B.
Then one could o mula e necessa y and su icien condi ions unde which he s ochas-
ic demand sys em PKS ={P(·|B)}B∈Bis gene a ed by a popula ion o u ili y-maximizing
consume s, unde he condi ional independence assump ion; his assump ion equi es
he dis ibu ion o u ili y unc ions (which gene a es he dis ibu ion o demand) o be
he same a each budge se B∈B. The cha ac e iza ion o PKS in KS is acili a ed by he
well-known cha ac e iza ion o u ili y-maximizing demand beha io o a single con-
sume , known as he s ong axiom o e ealed p e e ence (SARP). The second s ep in he
KS app oach is o show how he cha ac e izing condi ions on PKS could be s a is ically
es ed o an ac ual da a se , wi h empi ical equencies es ima ed a each budge se
B∈B.
The key obse a ion in ou pape is ha a wo-s ep p ocedu e simila o ha im-
plemen ed in KS could also be used o analyzing speci ic classes o games. Suppose
ha he e is a la ge popula ion o g oups, wi h each g oup playing he same game. We
assume ha he popula ion dis ibu ion o e join ac ion p o iles a a gi en ec o o
co a ia e alues x,whichwedeno ebyP
(·|x), is known o a ini e se o co a ia e al-
1This in o ma ion could be used o build a mapping om speci ic momen s o he da a o he iden i ied
se o ele an pa ame e s. Fo ins ance, in wo-playe games he sign o he s a egic in e ac ion pa ame e s
allows us o iden i y ou comes ha could occu only as a unique equilib ium; i ollows ha he p obabil-
i ies o hese ou comes (condi ional on a ious obse able a iables) do no depend on any equilib ium
selec ion mechanism and can be nicely ela ed o payo ele an pa ame e s (see Tame (2003) and Kline
and Tame (2016)). Shape es ic ions can also educe he size o he iden i ied se o ele an pa ame e s
(see, e.g., Ma zkin (2007)) and allow o he mo e e icien use o small sample da a se s (see, e.g., Be es eanu
(2005,2007)) .
2Ou app oach is also close in spi i , hough no in speci ics, wi h he nonpa ame ic andom u ili y mod-
els in Tebaldi, To go i sky, and Yang (2023), Deb, Ki amu a, Quah, and S oye (2023), Apes eguia, Balles e ,
and Lu (2017), Hode lein and S oye (2014), Manski (2007), McFadden (2005), McFadden and Rich e (1991),
and Ma schak (1960). As a as we know, ou pape is he i s o exploi his nonpa ame ic app oach o
s udy games. No e ha Ki amu a and S oye’s empi ical app oach (and hence ou s) is based on linea p o-
g amming, which can be also ound in ea lie wo ks such as Hono é and Tame (2006) and Che nozhuko ,
Fe nández-Val, Hahn, and Newey (2013).
238 Lazza i, Quah, and Shi ai Quan i a i e Economics 16 (2025)
ues
X.3(The se
X akes he place o Bin he KS model.) We hen o mula e necessa y
and su icien condi ions unde which he se o choice dis ibu ions P={P(·|x)}x∈
Xis
consis en wi h a popula ion o g oups made up o agen s ha ing payo unc ions ha
sa is y single-c ossing condi ions and playing PSNE, unde he assump ion o condi-
ional independence (which, in his case, means ha he dis ibu ion o payo unc ion
p o ilesac ossg oupsis hesamea di e en x∈
X). The second s ep in ou app oach
shows how hese condi ions on Pcould be s a is ically es ed on an ac ual da a se , wi h
empi ical equencies o e ac ion p o iles a di e en co a ia e alues; o his second
s ep, we simply ollow he s a is ical p ocedu e in KS. As in KS, he sampling ame-
wo k equi es ha o each x∈
X, he ea eNxobse a ions o ac ion p o iles such ha
Nx/N →ρx∈(0, 1),whe eN=x∈
XNx→∞.
Simila o KS, he cha ac e iza ion o P={P(·|x)}x∈
X equi es ha we ind necessa y
and su icien condi ions unde which he join ac ions om a single g oup a di e en
co a ia e alues a e consis en wi h ou hypo hesis o PSNE play and payo unc ions
sa is ying single-c ossing condi ions (wi h espec o opponen s’ ac ions and co a i-
a es). Since, unlike KS, he e is no eady-made cha ac e iza ion o his class o games,
we need o de elop i ou sel es. We show ha his hypo hesis can be cha ac e ized by
a p ope y we call he e ealed mono onici y (RM) axiom. This axiom plays he ole o
SARP in he KS model.
When he da a se passes he es , ou app oach is in u n use ul o making in e -
ence and p edic ion in he spi i o Deb e al. (2023), which deals wi h a e sion o he
consume model. Fo example, we can es ima e he ac ion o playe s who a e e ec-
i ely nons a egic, in he sense ha hei ac ions depend only on co a ia e alues and
a e independen o wha o he playe s do. We can also bound he p opo ion o g oups
which (a a gi en co a ia e ec o ) has a pa icula equilib ium p o ile as a PSNE (along
he lines o he analysis in A adillas-Lopez (2011)); no e ha his po en ially di e s om
he obse ed ac ion o g oups playing ha ac ion p o ile, no jus because o sampling
a ia ion, bu also because a gi en ac ion p o ile could be a nonchosen PSNE when he e
a emul iplePSNE.
The p ocedu e in KS is ha d o implemen when he e is a la ge numbe o bud-
ge se s and Smeulde s, Che chye, and De Rock (2021) p opose a column gene a ion
me hod o deal wi h his di icul y. This me hod is also applicable in ou se ing and is
use ul in easing he compu a ional bu den o ou es when (e.g.)
Xis a big se . In ou
pape , we de elop a new esul on column gene a ion ha allows o his me hod o be
used, no jus o es ing bu also in e ence.
The es o he pape is o ganized as ollows. In Sec ion 2, we p o ide an ou line o
how ou p ocedu e wo ks in he con ex o an en y game and con as i wi h a pa a-
me ic app oach. Sec ion 3p esen s ou main esul s a he popula ion le el. We in-
oduce he e ealed mono onici y axiom and use i o cha ac e ize hose dis ibu ions
o e join ac ions ha a e consis en wi h ou hypo hesis; p ope ies o he unde lying
3Va ia ion o easible se s (as in KS) can be included in ou analysis o games (see Lazza i, Quah, and
Shi ai (2018)), bu we ha e a oided i , in o de no o bu den he eade wi h oo many model ea u es and
also because ou empi ical applica ion does no ha e such a ia ion. (See also Ca ajal (2004) o a ela ed
esul .)
Quan i a i e Economics 16 (2025) An o dinal app oach o he empi ical analysis 239
Table 1. P={P(·|x2=(0, 0)),P
(·|x2=(0, 1)),P
(·|x2=(1, 0))}.
x2=(0, 0)
Fi m 2
NE
Fi m 1 N3/12 3/12
E4/12 2/12
x2=(0, 1)
Fi m 2
NE
Fi m 1 N1/12 5/12
E3/12 3/12
x2=(1, 0)
Fi m 2
NE
Fi m 1 N2/12 4/12
E2/12 4/12
dis ibu ion o e payo unc ion p o iles can also be eco e ed. Sec ion 4explains how
he popula ion-le el analysis in Sec ion 3can be implemen ed on ini e sample da a.
In his sec ion, we also in oduce and ex end he column gene a ion me hod o Smeul-
de s, Che chye, and De Rock (2021). To illus a e ou app oach, we ca y ou an empi -
ical analysis o en y decisions made by ai lines; his is ound in Sec ion 5. The Supple-
men al Ma e ial (Lazza i, Quah, and Shi ai (2024)) con ains some addi ional heo e i-
cal/empi ical esul s as well as he omi ed de ails o he s a is ical p ocedu e.
2. Mo i a ing example
The e is a la ge empi ical li e a u e modeling oligopoly en y decisions. We shall use his
model o illus a e he basic ques ion we a e in e es ed in and he app oach we p opose
o add ess his ques ion. Fo simplici y, we ea he case o wo i ms. Le yi∈{N,E}
be he ac ion se o i m i,whe eEmeans ha he i m en e s he ma ke and N ha
i s ays ou and le xibe a eal- alued, ini e-dimensional ec o o exogenous p o i
shi e s (co a ia es) ha a ec i m i’s p o i and a e obse ed by he o he i m and he
esea che .
We assume ha he e is a la ge popula ion o ma ke s, wi h each ma ke consis ing
o a Fi m 1 and a Fi m 2 ha make hei en y decisions simul aneously. The designa ion
o a playe as Fi m 1 o Fi m 2 is made by he esea che and based on obse able cha ac-
e is ics; o example, in Kline and Tame (2016), one i m is he “Low-Cos Ca ie ” and
he o he i m is “O he Ai lines” (see Sec ion 5). The e is a ini e se o ealized p o i
shi e s, which we deno e by
X. Fo each (x1,x2)∈
X, we suppose ha he popula ion
dis ibu ion o join ac ion p o iles P(·|x1,x2)is known o he esea che . We deno e his
se o dis ibu ions by P={P(·|(x1,x2))}(x1,x2)∈
X.Table1gi es an example o Pwhe e
he e is only a ia ion in x2and i akes h ee possible ec o alues; o example, he
box on he le ells us ha P((E,N)|x2=(0, 0)) =4/12. We a e in e es ed in de elop-
ing a p ocedu e, which allows us o iden i y hose P ha a e compa ible wi h ou model
o i m en y. O cou se, in any empi ical analysis hese cha ac e izing condi ions on P
would ha e o be s a is ically es ed on an ac ual da a se wi h sampling a ia ion (as we
explain in de ail in Sec ion 4). Con ining ou discussion o Pa his s age allows us o
ocus on he mo e dis inc i e aspec s o ou analysis.
We now desc ibe he model, which (po en ially) gene a es P.Wedeno e hepay-
o /p o i o Fi ms 1 and 2 by 1(y1,y2,x1)and 2(y1,y2,x2), espec i ely. We pos ula e
ha en y decisions a e gene a ed as pu e s a egy Nash equilib ia (PSNE) o an en y
game be ween Fi ms 1 and 2. We allow o mul iple PSNE and impose no es ic ion on
240 Lazza i, Quah, and Shi ai Quan i a i e Economics 16 (2025)
how i ms selec among hese equilib ia. The e emains unobse ed ma ke he e ogene-
i y e en a e condi ioning on (x1,x2); his he e ogenei y is cap u ed by a join dis ibu-
ion on (1,2), which in u n leads o a dis ibu ion o e join ac ions P(·|x1,x2).We
assume ha he e is condi ional independence, in he sense ha he dis ibu ion o e
(1,2)does no a y wi h he ealized alue o (x1,x2).
Las ly, we pos ula e ha he i ms’ p o i unc ions sa is y single-c ossing es ic ions
(see Milg om and Shannon (1994)). In his con ex , i means ha Fi m 1’s en y in o he
ma ke is encou aged when he p o i shi e x1 akes highe alues and is discou aged
when Fi m 2 chooses o en e . Fo mally, we equi e
1E,y
2,x
1>
1N,y
2,x
1=⇒ 1E,y
2,x
1>
1N,y
2,x
1(1)
whene e x
1≥x
1and ei he y
2=y
2o y
2=Eand y
2=N. (A simila equi emen is
imposed on 2.) Fo example, in Cilibe o and Tame (2009),
1(y1,y2,x1)=α
1x1+δ11y2+ε1i y1=E,
0i y1=N,(2)
whe e 1E=1and1N=0. In his speci ica ion, he en y o Fi m 2 al e s he p o i o Fi m
1byδ1and unobse ed he e ogenei y in payo unc ions is cap u ed by ε1,whichen-
e s he p o i unc ion addi i ely. I is s aigh o wa d o check ha ou single-c ossing
es ic ions a e sa is ied i δ1<0andα1>0. No e, howe e , ha he con e se is no
ue, ha is, he e a e dis ibu ions o e payo unc ions sa is ying (1) ha canno be
ep esen ed in he addi i e o m gi en by (2), o any dis ibu ion on ε1.
We say ha Pis consis en wi h he single-c ossing model, o SC- a ionalizable,i
he e is a join dis ibu ion o payo unc ions (1,2) ha sa is y ou single-c ossing
condi ions (1) such ha he esul ing dis ibu ion o PSNE (gi en some equilib ium se-
lec ion ule) coincides wi h P(·|x1,x2) o each x∈
X. We would like o answe he ollow-
ing ques ion: Wha condi ions on Pcha ac e ize SC- a ionalizabili y? In o he wo ds,
when p esen ed wi h P, how could we check i i is SC- a ionalizable?
We i s obse e ha ou model does ha e s uc u al implica ions o P. Suppose he
obse able p o i shi e s weakly inc ease en y-by-en y om (x
1,x
2) o (x
1,x
2);4 hen,
a any pa icula ealiza ion 1o Fi m 1’s payo unc ion, i i p e e s o en e when
he o he i m en e s a (x
1,x
2), hen he single-c ossing condi ion gua an ees ha i
will con inue o p e e en y a (x
1,x
2). The same a gumen applies o Fi m 2, and so
we conclude ha i (E,E)is he Nash equilib ium a (x
1,x
2) o a gi en ealized p o i
unc ion p o ile (1,2), hen i will be he unique Nash equilib ium a (x
1,x
2) o his
ealized p o ile. Agg ega ing ac oss all p o iles, we es ablish ha
P(E,E)|x
1,x
2≥P(E,E)|x
1,x
2,
p o ided condi ional independence holds. This inequali y cons i u es a es ic ion on P
bu i is no he only es ic ion imposed by ou model. We now ske ch ou he p ocedu e
4Fo mally, (x
1,x
2)is weakly highe han (x
1,x
2)in he p oduc o de (see oo no e 8 o i s o mal de i-
ni ion).
Quan i a i e Economics 16 (2025) An o dinal app oach o he empi ical analysis 241
Table 2. Dis ibu ion o ypes a ionalizing he choice dis ibu ions in able 1.
x2=(0, 0)x2=(0, 1)x2=(1, 0)
Ac ion P o iles Ac ion P o iles Ac ion P o iles
Type Weigh N,NN,EE,NE,EN,NN,EE,NE,EN,NN,EE,NE,E
1 1/12 1/12 1/12 1/12
2 2/12 2/12 2/12 2/12
3 2/12 2/12 2/12 2/12
4 1/12 1/12 1/12 1/12
5 1/12 1/12 1/12 1/12
6 2/12 2/12 2/12 2/12
7 3/12 3/12 3/12 3/12
Sum 1 3/12 3/12 4/12 2/12 1/12 5/12 3/12 3/12 2/12 4/12 2/12 4/12
o sys ema ically checking whe he Pis SC- a ionalizable, using Pp esen ed in Table 1
as an example.
Gi en a pa icula ealiza ion (1,2), he i ms will choose an ac ion p o ile (ei he
(E,E),(E,N),(N,E),o (N,N)) a each ealiza ion o x2,andasx2 akes di e en al-
ues he ac ion p o ile o he wo i ms may change. We shall e e o he map om x2 o
he ac ion p o ile as a g oup ype. No ice ha e en hough i ms’ p o i unc ions may
be he e ogenous in in ini ely many ways, i s mani es a ion in beha io mus be ini e,
since he e a e only ini ely many possible ac ions and he ealized co a ia es (x1,x2)
ake alues in he ini e se
X.
To be p ecise, he e a e in o al 43=64 g oup ypes, bu no all a e consis en wi h
PSNE play and single-c ossing payo unc ions. Fo example, as we ha e al eady ex-
plained, a g oup ype whe e (E,E)is played a x2=(0, 0)and (N,N)a x2=(0, 1)is
no compa ible wi h single-c ossing. On he o he hand, i is qui e clea a g oup ype
whe e (N,E)is played a all h ee alues o x2can be jus i ied wi h single-c ossing p o i
unc ions.
Asce aining i Pcan be a ionalized in ol es a wo-s ep p ocedu e. Fi s , we mus
iden i y all single-c ossing g oup ypes, in he sense ha he ac ion p o ile (y1,y2)a each
alue o (x1,x2)could be gene a ed as PSNE om payo unc ions sa is ying (1). This
is do-able because we show in Sec ion 3 ha hese g oup ypes a e cha ac e ized by
an easy- o-check condi ion called he e ealed mono onici y axiom.Second,weha e
o check whe he he e a e weigh s on hese g oup ypes ha could accoun o he ob-
se ed dis ibu ion o ac ion p o iles; his in ol es sol ing a sys em o linea inequali ies.
We claim ha Pdepic ed in Table 1can be a ionalized. To unde s and why, we lis
in Table 2se en possible g oup ypes. One could check ha each o hese g oup ypes
is consis en wi h he single-c ossing p ope y. When hese ypes a e ep esen ed in he
242 Lazza i, Quah, and Shi ai Quan i a i e Economics 16 (2025)
popula ion wi h he weigh s indica ed in Table 2, hey gene a e he dis ibu ion o en-
y decisions obse ed in Table 1. (Compa e he en ies in Table 1wi h he las ow o
Table 2.)
Las ly, we poin ou ha while Pis SC- a ionalizable, i is no compa ible wi h a
model whe e p o i unc ions ha e he o m (2), so he la e speci ica ion does in ol e
a loss o gene ali y. Indeed, wi h his speci ica ion, Fi m 2’s p o i upon en y is
π2(E,y1,x21,x22,ε2)=α21x21 +α22x22 +δ211y1+ε2,(3)
whe e (α21,α22)>0andδ21 <0.5Whe he he boos o p o i s o an inc ease in x21 is
g ea e o smalle han ha ob ained om he same inc ease in x22 depends on whe he
α21 is bigge o smalle han α22 and is independen o he ealiza ion o ε2. So, i excludes
he case whe e he ealiza ion o ε2in luences he ela i e bene i o highe x21 e sus
highe x22. To see why his pa ame ic model canno explain he choice dis ibu ions in
Table 1, suppose ins ead ha i does. Then
P(E,E)|x1,(1, 0)−P(E,E)|x1,(0, 0)
=με1:π1(E,E,x1,ε1)≥0×{ε2:−δ21 ≥ε2≥−α21 −δ21},
whe e μis he p obabili y measu e on he space o (ε1,ε2); simila ly,
P(E,E)|x1,(0, 1)−P(E,E)|x1,(0, 0)
=με1:π1(E,E,x1,ε1)≥0×{ε2:−δ21 ≥ε2≥−α22 −δ21}.
Since he o me equals 2/12 while he la e equals 1/12, we conclude ha α22 <α
21.
Howe e ,
1
12 =P(N,N)|x1,(0, 0)−P(N,N)|x1,(1, 0)
=με1:π1(E,N,x1,ε1)≤0×{ε2:0≥ε2≥−α21}
and
2
12 =P(N,N)|x1,(0, 0)−P(N,N)|x1,(0, 1)
=με1:π1(E,N,x1,ε1)≤0×{ε2:0≥ε2≥−α22},
which ells us ha α22 >α
21. So, we ob ain a con adic ion.
In Supplemen a y Appendix A1, we p o ide a mo e elabo a e discussion o he con-
as be ween he obse able es ic ions imposed by a linea pa ame ic model and ou
(mo e gene al) nonpa ame ic model. In pa icula , using simula ions based on an ex-
ended e sion o he abo e example, we show ha he di e ence be ween he wo mod-
els is also picked up a he sample le el: he me hod o Kline and Tame (2016)(co ec ly)
inds ha he da a a e inconsis en wi h he linea model, whe eas ou me hod (also co -
ec ly) inds ha he da a a e consis en wi h he mo e gene al model.
5We a e g a e ul o Au eo De Paula o sugges ing ha we cons uc an example wi h his speci ic ea u e.
Quan i a i e Economics 16 (2025) An o dinal app oach o he empi ical analysis 249
SC- a ionalizable boils down o inding a posi i e solu ion o a se o equa ions linea in
he unknowns τB o all B ∈B.15
Rema k 4. I is pa o he de ini ion o SC- a ionalizabili y ha he dis ibu ion o =
(i)i∈Nis independen o x. Suppose we d op his condi ion bu s ill equi e all payo
unc ions o consis o single-c ossing unc ions; hen i is easy o see ha he payo
unc ions and equilib ium selec ion ules will induce a dis ibu ion o e g oup ypes in
Ba each x,whichwemaydeno eby(τB
x)B∈B, such ha he ollowing coun e pa o (7)
holds:
P(y|x)=
{B∈B:B(x)=y}
τB
x o all y∈Yand x∈
X.(8)
This condi ion is i ially ue in he sense ha one could always ind (τB
x)B∈Bsuch ha
i holds. Condi ional independence imposes he addi ional equi emen ha τB
x=τB
x
o any x,x ∈
X, and his condi ion in combina ion wi h (8)isob iouslyequi alen
o (7). One could imagine si ua ions whe e he modele has di e en iews o how he
dis ibu ion o (and hence he dis ibu ion o he associa ed g oup ypes) a ies wi h x,
which may be mo e pe missi e han o di e en om condi ional independence; hese
could be inco po a ed as u he condi ions on τB
x ha could be es ed in combina ion
wi h (8). Ob iously, such a es will emain a linea es i he added condi ions a e linea
in τB
x.
3.4 Reco e ing p ope ies o a a ionalizing dis ibu ion P
When Pis SC- a ionalizable, we a e also able o ex ac in o ma ion abou his a ional-
iza ion h ough he p ope ies o (τB)B∈B ha sol e (7). In pa icula , le SC∗beasubse
o single-c ossing payo unc ions (including all o i s s ic ly inc easing ans o ma-
ions) and le
B∗=B∈B: he eis∈SC∗ ha a ionalizes B.(9)
By a s aigh o wa d adap a ion o he p oo o Theo em 2(see he Supplemen al Ap-
pendix), we can show ha
max
B∈B∗
τB:τBB∈Bsol es (7)=max∈SC∗dP:P
a ionalizes P(10)
No ice ha he le -hand side o his equa ion is s aigh o wa d o compu e when B∗
and Ba e known, since i simply in ol es sol ing a linea p og am. Thus we can ind
he g ea es possible weigh on a gi en se o payo p o iles, o any dis ibu ion ha
15In some p oblems, i may no be compu a ionally easible o ind all he elemen s o B,bu in hose
cases, one could s ill es o SC- a ionalizabili y by p og essi ely enla ging he se o single-c ossing ypes
(see Sec ion 4.1).
250 Lazza i, Quah, and Shi ai Quan i a i e Economics 16 (2025)
a ionalizes P.16 We gi e wo cases whe e his exe cise is use ul, bo h o which a e em-
pi ically implemen ed in Sec ion 5. O he examples can be ound in he Supplemen a y
Appendix A3.
Applica ion 1. Bounds on he ole o s a egic in e ac ion
While ou model allows o he possibili y ha each playe eac s s a egically o
o he playe s in he game, i is concei able ha he condi ional choice dis ibu ions
could be explained mo e simply, wi hou appealing o s a egic e ec s o one o mo e
playe s in he game.
To be speci ic, suppose we wish o check whe he i is possible o ega d a subg oup
No he playe s as nons a egic. Le SC∗be he payo p o iles in SC such ha idoes
no depend on y−i o e e y i∈Nand le B∗be i s co esponding se o g oup ypes
(as de ined by (9)). The ypes in B∗can be cha ac e ized by a s ic e e sion o he RM
axiom: a g oup ype is in B∗i and only i i obeys he RM axiom and, o each i∈N,we
equi e ha y ∈B(x),y∈B(x),andx
i≥x
i=⇒ y
i≥y
i. Wi h his cha ac e iza ion, we
can cons uc B∗. I we ind ha
max
B∈B∗
τB:τBB∈Bsol es (7)=1,
we conclude (by (10)) ha Pcan be SC- a ionalized wi hou equi ing he playe s in N
o be s a egic; on he o he hand, i he uppe bound is s ic ly below 1, hen we mus
inco po a e s a egic in e ac ions among hese playe s o SC- a ionalize P.
Applica ion 2. P obabili y bounds o Nash equilib ium p o iles
Gi en a s a egy p o ile yand co a ia e x, we pose he ollowing ques ion: Among
all he possible SC- a ionaliza ions o P, wha is he g ea es ac ion o g oups, which
could ha e yas a pu e s a egy Nash equilib ium a x?He e,x∈Xmay o may no be
an elemen o
X,andwhenx/∈
X, he answe o his ques ion p o ides in o ma ion on
how he game would be played a an hi he o unobse ed co a ia e alue. Howe e , he
ques ion is in e es ing e en when x∈
X.
To see why, no ice ha he e is a dis inc ion be ween P(y|x), he obse ed ac ion
o g oups in he popula ion ha play ya x, and he ac ion o g oups o which yis a
Nash equilib ium. The o me is ypically smalle han he la e because g oups migh
ha e mul iple Nash equilib ia. Thus some g oups who play s a egy p o iles o he han y
may also ha e yas a Nash equilib ium.17 The dis inc ion be ween P(y|x)and he g ea es
possible weigh on hose g oups, which ha e yas a Nash equilib ium a x=xis ele an ,
16To ob ain min{∈SC∗dP:P
a ionalizes P}, we use he simila ly easy- o-p o e iden i y
min
B∈B0
τB:τBB∈Bsol es (7)=min∈SC∗dP:P
a ionalizes P,
whe e B0={B∈B: B can only be a ionalized by ∈SC∗}.
17In ou empi ical applica ion o an en y game wi h wo i ms, i (E,E)o (N,N)is played by a pai
o i ms, hen i has o be hei unique equilib ium, bu any pai ha plays (E,N)may also ha e (N,E)as
ano he (albei unselec ed) equilib ium. Thus i P(E,N|x)and P(N,E|x)a e he obse ed p obabili ies o
ac ion p o iles (E,N)and (N,E), espec i ely, hen he p obabili y ha (E,N)(simila ly, (N,E))isaNash
equilib ium p o ile a x=xis no g ea e han P(E,N|x)+P(N,E|x).
Quan i a i e Economics 16 (2025) An o dinal app oach o he empi ical analysis 251
because i he gap is small, hen we a e su e ha changing he equilib ium selec ion
scheme canno signi ican ly inc ease he equency wi h which yis played. This means
(e.g.) ha a policymake who wan s y o be played mo e o en mus al e payo s in some
way and i is no possible o simply con ince playe s o coo dina e on a di e en equilib-
ium. An ea lie analysis o ques ions o his ype can be ound in A adillas-Lopez (2011),
which ocuses on a di e en class o games.
To answe ou ques ion, le SC∗={∈SC :y∈NE(,x)}and le B∗be i s co e-
sponding se o g oup ypes. We can check whe he B belongs o B∗by using he RM
axiom. Indeed B ∈B∗i and only i he (possibly) mul i alued g oup ype Bde ined
as ollows obeys he RM-axiom: B(x)={B(x),y}and B(x)=B(x) o e e y x∈
X {x}.
The p opo ion o he popula ion which has yas a PSNE canno exceed max{B∈B∗τB:
(τB)B∈Bsol es (7)}and can equal his numbe .18
4. The s a is ical p ocedu e
This sec ion ou lines he s a is ical p ocedu e ha implemen s he esul s in he p e i-
ous sec ion, which a e based on popula ion dis ibu ions. The es o SC- a ionalizabili y
is explained in Sec ion 4.1 and elies on he s a is ical hypo hesis es ing p oposed by Ki-
amu a and S oye (2018). The e icien implemen a ion o his es when Bis la ge (and
canno be ully lis ed) uses he column gene a ion app oach p oposed in Smeulde s,
Che chye, and De Rock (2021). Sec ion 4.2 ou lines he p ocedu e (in essence p o ided
by Deb e al. (2023)) o ob ain con idence in e als o he weigh s on ce ain g oup
ypes; he e icien implemen a ion o his p ocedu e equi es a non i ial ex ension o
he column gene a ion me hod in Smeulde s, Che chye, and De Rock (2021)andwe
p o ide his in P oposi ion 3.
4.1 S a is ical hypo hesis es ing
We begin wi h a ma ix e o mula ion o he cha ac e iza ion gi en in Theo em 2.Each
gene alized g oup ype B :
X⇒Ycan be ep esen ed as a ec o b=(by,x)Y×
Xsuch ha
by,x=1i y∈B(x)and by,x=0 o he wise. Con e sely, o any b∈{0, 1}|Y×
X|co esponds
o a gene alized g oup ype, wi h a ec o b∈{0, 1}|Y×
X| ep esen ing a single- alued
g oup ype i and only i y∈Yby,x=1a e e yx∈
X. Simila ly, since Pconsis s o |
X|
dis ibu ions on Y, i can be cap u ed by he column ec o p∈[0, 1]|Y×
X|,whe e he
(y,x)- h en y o pis P(y|x)(and hence, y∈Ypy,x=1 o each x∈
X).
In wha ollows, we shall abuse no a ion and use B o deno e bo h he se o g oup
ypes obeying he RM axiom and also he ec o s co esponding o hose ypes. We de-
no e by B he ma ix whe e each column ep esen s a g oup ype in B.Theo em2s a es
ha Pis SC- a ionalizable i and only i he e is τ∈B, he se o dis ibu ions on B, ha
18Ou analysis he e gi es he mos op imis ic es ima e on he possibili y o swi ching he equilib ium
ac ion o y, in he sense ha i assumes ha e e y g oup ype, which can be a ionalized by an elemen in
SC∗, ac ually does ha e a payo unc ion p o ile in SC∗. We could also ind he mos conse a i e es ima e
o he p opo ion o he popula ion ha could swi ch o yby changing equilib ium selec ion ules; his is
explained in Supplemen a y Appendix A6.2.
252 Lazza i, Quah, and Shi ai Quan i a i e Economics 16 (2025)
sol es Bτ=p.(Bcould be hough o as elemen s o he s anda d (|B|−1)-simplex.)
We would like o es i he da a is consis en wi h he SC- a ionalizabili y o P.Equi a-
len ly, le ing PSC ={Bτ:τ∈B}(i.e., he se o SC- a ionalizable dis ibu ions in ec o
o m), ou null hypo hesis is
min
η∈PSC(p−η)·(p−η)=0. (11)
The da a se consis s o Nxobse a ions o he ac ion p o iles a each ealiza ion
o x∈
X. We assume ha Nx/N →ρx∈(0, 1)a each x∈
X,asN=x∈
XNx→∞.We
deno e he empi ical dis ibu ion o e ac ion p o iles by
Q=Q(·|x):x∈
X,
and we es ima e Pby his sample analog. As in he case wi h P, we can ep esen Qby a
column ec o q∈[0,1]|Y×X|whe e he (y,x)- h en y is equal o Q(y|x).
The es ing p ocedu e by Ki amu a and S oye (2018) depends on he simple, bu im-
po an obse a ion ha Bτ=pholds o some τ∈B, i and only i Bτ=pholds o
some τ≥0 (Theo em 3.1 in hei pape ). Thus, by le ing A={Bτ:τ≥0}, he null hy-
po hesis is equi alen o whe he pli es in his con ex cone, ha is,
min
η∈A(p−η)·(p−η)=0. (12)
Gi en his, ollowing Ki amu a and S oye (2018), we adop he es s a is ic
JN:=min
η∈AN(q−η)·(q−η)=min
τ∈R|B|
+
N(q−Bτ)·(q−Bτ). (13)
Calcula ing he c i ical alue. No e ha we canno simply adop a solu ion o he
p oblem (13) as he boo s ap es ima o o he empi ical choice dis ibu ion, due o he
possible discon inui y o he limi ing dis ibu ion o JN. Add essing his issue in ol es
in oducing a uning pa ame e and conside ing he co esponding igh ened p oblem.
We ollow he p ocedu e by Smeulde s, Che chye, and De Rock (2021), which is a modi-
ica ion o he one in Ki amu a and S oye (2018).
Choose B⊂Bso ha i con ains a basis o he space spanned by B, and de ine TκN=
{τ∈R|B|
+:τb≥κN/|B| o all b∈B},wi hκNbeing selec ed so ha κN↓0and√NκN↑
∞as N→∞. (See Supplemen a y Appendix A5, o he p ocedu e o cons uc ing B
and a de ailed jus i ica ion o ou p ocedu e.19) Le ing AκN={Bτ:τ∈TB
κN},weadop
η∗=a gmin
η∈AκN
N(q−η)·(q−η)=a gmin
τ∈TB
κN
N(q−Bτ)·(q−Bτ). (14)
19In he o iginal o mula ion by Ki amu a and S oye (2018), posi i e weigh s a e equi ed on all elemen s
in B, which is incon enien when applying he column gene a ion p ocedu e desc ibed la e in his subsec-
ion. The modi ica ion o ha app oach by Smeulde s, Che chye, and De Rock (2021) (which we a e using
he e) equi es posi i e weigh s only o he ypes in B.
Quan i a i e Economics 16 (2025) An o dinal app oach o he empi ical analysis 253
as he boo s ap es ima o o he empi ical choice equency. Compa ed o he p oblem
(13), he easible se in he minimiza ion p oblem is igh ened by he uning pa ame e ,
wi h posi i e weigh s equi ed o he elemen s in B. We hen gene a e a boo s ap sam-
ple q( )( o =1, 2, ,R) using s anda d nonpa ame ic boo s ap esampling om η∗
and ecen e his sample by se ing
q( ):=(q( )−q)+η∗.Wi h
q( ), we can calcula e he
boo s ap es s a is ic
J( )
N:=min
η∈AκN
N
q( )−η·
q( )−η=min
τ∈TB
κN
N
q( )−Bτ·
q( )−Bτ, (15)
and he empi ical dis ibu ion o J( )
Nallows us o ob ain he p- alue p=#{J( )
N>J
N}/R.
The null hypo hesis ha qis a sample om some p∈A(equi alen ly, p∈PSC)isno
ejec ed i he p- alue is g ea e han he c i ical alue.
Column Gene a ion. A majo hu dle in implemen ing he abo e es is ha he com-
pu a ion o JNand J( )
Nin ol es B, which is o en oo la ge o be lis ed in i s en i e y.
We cope wi h his p oblem by applying he column gene a ion p ocedu e in Smeulde s,
Che chye, and De Rock (2021). This p ocedu e in ol es i s es ing a mo e s ingen e -
sion o he model co esponding o a s ic subse B0o B, which is comple ely known.
Fo ins ance, we may choose he “s a e ” se B0 o be he se o cons an ypes,inwhich
e e y playe akes he same ac ion ega dless o opponen s’ ac ions and co a ia es; hese
g oup ypes ob iously obey he RM axiom. Then he se B0is p og essi ely enla ged by
including mo e g oup ypes om B, up o he poin whe e u he addi ions will no im-
p o e he model’s abili y o explain he da a.
To be p ecise, le B0be hema ixwhe e hecolumnsa eelemen so B0.Wecan
calcula e
JN,0 :=min
τ∈R|B0|
+
N(q−B0τ)·(q−B0τ). (16)
Ob iously, JN,0 ≥JNand we could check i i is possible o dec ease JN,0 by including
some b∈B. We say ha b∈Bimp o es B0i , when bis included in B0, henew alue
o JN,0 is s ic ly lowe . The ollowing esul , which ollows om he con ex p ojec ion
heo em, p o ides a necessa y and su icien condi ion o B0 o be imp o able.
P oposi ion 1. A se o g oup ypes B0is imp o ed by some b∈B,i and only i
max
b∈B(q−η0)·(b−η0)>0, (17)
whe e η0=B0τ0and τ0=a gminτ∈R|B0|
+
(q−B0τ)·(q−B0τ).
To sol e p oblem (17) wi hou ully enume a ing B, we mus ind a compu a ionally
e icien way o cha ac e ize B. Con enien ly o us, he RM axiom—and hence he se
B—can be cha ac e ized as solu ions o an in ege linea p og amming p oblem.
P oposi ion 2. We can cons uc a ma ix Cand a column ec o θ,bo h wi h nonneg-
a i e in ege en ies,such ha o any b∈{0, 1}|Y×
X|,we ha e b∈Bi and only i Cb≤θ.
254 Lazza i, Quah, and Shi ai Quan i a i e Economics 16 (2025)
The o mulae o Cand θa e ound in ou p oo o his p oposi ion in he Sup-
pleme nal Appendix. Combining his esul wi h P oposi ion 1,B0is imp o ed by some
b∈B, i and only i
max(q−η0)·(b−η0),subjec ob∈{0, 1}|Y×
X|and Cb≤θ, (18)
is s ic ly posi i e. I i is, we add his b o B0and hen epea he p ocess. In o he wo ds,
we ecalcula e JN,0 and η0based on he new B0, and y o ind ano he elemen in B
ha imp o es B0by checking i (18) has a s ic ly posi i e solu ion. Since Bis ini e, his
algo i hm mus e mina e, and a he end we can be su e we ha e ound B0such ha
JN,0 =JN.
The column gene a ion p ocedu e desc ibed abo e can be also applied o he com-
pu a ion o J( )
Nde ined by (15). Since he cons ain in p oblem (15) equi es posi i e
weigh s on B, his se needs o be con ained in he ini ial choice o B0.20
Summa y. Below is a s ep-by-s ep summa y o he es p ocedu e.21
I Ob ain he es s a is ic JNde ined in (13) as ollows:
(i) Based on B0, sol e he minimiza ion p oblem (16) oge JN,0 and η0.
(ii) Check he alue o (18). I i is s ic ly posi i e, hen upda e B0by adding a
solu ion o (18) and go o (i). Else,se JN=JN,0 and S op.
II Ob ain he igh ened es ima o η∗in (14) as ollows:
(i) Ob ain B⊂Busing he p ocedu e explained in Supplemen a y Appendix
A5.2.
(ii) Se Bas B0, and un he p ocedu e in S ep I eplacing R|B0|
+in p oblem (16)
wi h TB0
κN={τ∈R|B0|
+:τb≥κN/|B| o b∈B}. When i s ops, he esul ing η0
is η∗.
III Ob ain he boo s ap es s a is ics J( )
Nde ined in (15) o =1, 2, ,R:
(i) Ob ain he ecen e ed boo s ap sample
q( )=(q( )−q)+η∗.
(ii) Se Bas B0, and un he p ocedu e in S ep I eplacing qand R|B0|
+in p oblem
(16)wi h
q( )and TB0
κN, espec i ely. When i s ops, adop he esul ing JN,0 as
J( )
N.
IV Las ly, calcula e he p- alue p=#{J( )
N>J
N}/R.
20No e ha , e en i B0con ains B, and Bcon ains a linea basis o B, heconical hull o B0need no
coincide wi h he conical hull o B( hough he linea hull o B0o cou se coincides wi h he linea hull o
B). So, i is s ill possible o B0 o be imp o able.
21The Supplemen a y Appendix (Sec ion A5.2) p o ides a couple o sho cu s ha imp o es he compu-
a ion ime.
Quan i a i e Economics 16 (2025) An o dinal app oach o he empi ical analysis 255
4.2 In e ence on ypes
Suppose we ha e a da a se ha is consis en wi h SC- a ionalizabili y (in he sense ha
he null hypo hesis (12) is no ejec ed) and would now like o o m a con idence in-
e al on b∈B∗τb, he o al weigh o a subse o single-c ossing g oup ypes B∗(see
Sec ion 3.4). To do his, we ollow he p ocedu e in Deb e al. (2023). The p oblem o
de e mining whe he a gi en weigh o B∗ alls wi hin he con idence in e al can be
de e mined by es ing a sui ably modi ied e sion o he null hypo hesis (12), wi h PSC
eplaced by a di e en se o dis ibu ions. To be speci ic, suppose we would like o ind
he uppe bound o he con idence in e al. Fo each β∈(0, 1),wele
PSCβ;B∗=Bτ:τ∈Band
b∈B∗
τb≥β,
and es he null hypo hesis
min
η∈PSC(β;B∗)
(p−η)·(p−η)=0 (19)
a some signi icance le el ¯
p. We hen use bina y sea ch o ob ain he maximal alue
o βunde which he null hypo hesis is no ejec ed; he esul ing maximal alue o β
co esponds o he sup emum o he 100(1−¯
p)% con idence in e al o β.
Fo a gi en β, he es s a is ic is22
JN(β):=min
η∈PSC(β;B∗)
N(q−η)·(q−η)
=min
τ∈BN(q−Bτ)·(q−Bτ)subjec o
b∈B∗
τb≥β. (20)
As in he p eceding subsec ion, i may no be possible o ully enume a e Bo B∗,and
so a e sion o he column gene a ion p ocedu e ou lined he e is needed. This in u n
equi es an ex ension o P oposi ion 1, which we now explain.
Le B0⊂Bbe such ha B0∩B∗=∅, and le us calcula e
JN,0(β)=min
τ∈B0
N(q−B0τ)·(q−B0τ)s. .
b∈(B0∩B∗)
τb≥β, (21)
whe e B0is he s anda d (|B0|−1)-simplex. We say ha B0is imp o able gi en p oblem
(20),i JN,0(β)>J
N(β). The ollowing p oposi ion is he coun e pa o P oposi ion 1
and p o ides a necessa y and su icien condi ion o a gi en B0 o be imp o able.
P oposi ion 3. I he se B0⊂Bis imp o able gi en p oblem (20), hen he e is a pai o
ypes {b∗,b},wi h b∗∈B∗and b∈Bsuch ha
(q−η0)·βb∗+(1−β)b−η0>0, (22)
22As poin ed ou in Deb e al. (2023), unlike he case o he p eceding subsec ion (see (13)), τmus be
chosen om he simplex B, a he han he nonnega i e o han .
256 Lazza i, Quah, and Shi ai Quan i a i e Economics 16 (2025)
whe e η0=B0τ0wi h τ0being he dis ibu ion ha achie es JN,0(β).Con e sely,suppose
he e is b∗∈B∗and b∈Bsuch ha (22)holds; hen {b∗,b}imp o es B0gi en p oblem
(20).
We al eady know ( om P oposi ion 2) ha we can cons uc a ma ix Cand a column
ec o θso ha b∈Bi and only i Cb≤θ.Suppose ha ,inaddi ion,wecancons uc
ama ixC∗and a column ec o θ∗wi h in ege en ies so ha , o any b∈{0, 1}|Y×
X|,
we ha e b∗∈B∗⇐⇒ C∗b∗≤θ∗.Thenapai {b∗,b}obeying (22) exis s, i and only i he
p oblem
max(q−η0)·βb∗+(1−β)b−η0(23)
s. . b,b∗∈{0, 1}|Y×
X|and C∗O
OC
b∗
b≤θ∗
θ
has a posi i e op imal alue. No e ha e e y B∗in ou empi ical applica ion has a ma ix
cha ac e iza ion like he one desc ibed abo e (see Supplemen a y Appendix A4 o he
speci ic cons uc ion). I he e is a pai {b,b∗} ha imp o es B0, henweupda eB0by
including he pai in B0and ecalcula e JN,0(β). Since Bis ini e, his p ocess e mina es
and we ob ain JN,0(β)=JN(β).
To ind he alid c i ical alue, we need a sui able igh ening ha imposes s ic ly
posi i e weigh s on a ce ain subse o g oup ypes. The igh ening he e mus depend on
βand i s o mula ion is a he in ol ed, so we pos pone his discussion o Supplemen-
a y Appendix A5.3. Tha said, once we ha e cons uc ed a sui ably igh ened subse o
Bby some uning pa ame e κN, he es o he p ocedu e is simila o he one ou lined
in he p eceding subsec ion. Deno ing his subse by B
κN(β;B∗)and le ing
PSC
κNβ;B∗=Bτ:τ∈B
κNβ;B∗and
b∈B∗
τb≥β, (24)
he boo s ap es ima o and he ecen e ed boo s ap samples can be ob ained as in
(14)–(15), a e eplacing he se s AκNand TκNby PSC
κN(β;B∗)and B
κN(β;B∗), espec-
i ely. We can also implemen he e he column gene a ion p ocedu e we used be o e.
The de ails o he p ocedu e, including a s ep-by-s ep summa y, a e p o ided in Supple-
men a y Appendix A5.3.
5. Empi ical illus a ion
We apply ou esul s in he p eceding sec ions o an en y game using a da a se aken
om Kline and Tame (2016). The da a se con ains he en y decisions o ai lines in
7882 ma ke s, whe e a ma ke is de ined as a ip be ween wo ai po s i espec i e o
in e media e s ops. Ai line i ms a e di ided in o wo ca ego ies: LCC (low cos ca ie s)
and OA (o he ai lines).23 In Kline and Tame ’s analysis (and in ou s), he wo ca ego ies
23The da a we e collec ed om he second qua e o he 2010 Ai line O igin and Des ina ion Su ey
(DB1B). The low cos ca ie s a e Ai T an, Allegian Ai , F on ie , Je Blue, Midwes Ai , Sou hwes , Spi i ,
Sun Coun y, USA3000, and Vi gin Ame ica. A i m ha is no a low cos ca ie is, by de ini ion, an “o he
ai line”.
Quan i a i e Economics 16 (2025) An o dinal app oach o he empi ical analysis 257
a e ea ed as wo i ms. Thus, in each ma ke , he wo i ms, LCC and OA, can ei he
bo h en e a ma ke , bo h s ay ou , o one could en e wi h he o he s aying ou .
This da a se also con ains in o ma ion on wo co a ia es: ma ke p esence (MP) and
ma ke size (MS). Ma ke p esence is a ma ke - and ai line-speci ic a iable. Fo each
ai line and o each ai po , one coun s he numbe o ma ke s ha he ai line se es
om ha ai po and di ide i by he o al numbe o ma ke s se ed om ha ai po
by any ai line; he ma ke p esence a iable o a gi en ma ke and ai line is he a e -
age o hese a ios a endpoin s o ha ma ke / ip. The cons uc ion and inclusion o
his co a ia e is no no el and ollows Be y (1992). Since he ai lines a e agg ega ed in o
wo i ms, he ma ke p esence a iable is also agg ega ed: he ma ke p esence o LCC
( esp., OA) is he maximum among he ac ual ai lines in he LCC ca ego y ( esp., OA
ca ego y). The second co a ia e, ma ke size, is a ma ke -speci ic a iable (sha ed by all
ai lines in ha ma ke ) and is de ined as he popula ion a endpoin s o he co espond-
ing ip.
Fu he mo e, Kline and Tame (2016) disc e ize hese a iables, whe e each o hem
akes alue 1 i he a iable is highe han i s median alue and 0 o he wise. Thus, in
ou da a se , he e a e h ee bina y co a ia es, MPLCC ,MP
OA, and MS, and ma ke s a e
pa i ioned in o eigh g oups acco ding o ealiza ions o hem. Fo mally, X={0, 1}3,and
in his case, i also holds ha
X=X. No e ha MS simul aneously in luences he payo s
o bo h LCC and OA, and hence he co a ia es a ec ing LCC’s payo can be w i en as
xLCC =(MPLCC ,MS
),andsimila ly,xOA =(MPOA,MS
).
Obse a ions in he da a se can be used o calcula e he empi ical choice dis-
ibu ions ha we include in Table 3. I consis s o eigh blocks, wi h he ma ke s
in each block sha ing he same co a ia es. Fo example, he e a e 1271 ma ke s wi h
(MPLCC ,MP
OA,MS
)=(0, 0, 0), o which a ound 30% a e no se ed by ei he ai line
and abou 68% a e se ed only by ai lines in he OA ca ego y (an ac ion p o ile is w i en
as (yLCC ,yOA)∈{E,N}×{E,N}). The en ies in Table 3seem “ easonable,” in he sense
ha i appea s as hough a i m’s en y is encou aged whene e i s ma ke p esence is
la ge o he ma ke size is la ge, and i is de e ed by he en y o he o he i m. Fo ex-
ample, going om (0, 0, 0) o (1, 0, 0)(so he ma ke p esence o LCC has inc eased),
bo h Q(N,N)and Q(N,E) all, while Q(E,N)and Q(E,E)bo h inc ease.
Tes ing SC- a ionalizabili y. Ou hypo hesis is ha , in each ma ke , wo i ms
(LCC and OA) a e playing a pu e s a egy Nash equilib ium in a game o s a e-
gic subs i u es wi h mono one e ec s om co a ia es. The payo unc ion o LCC,
say, LCC(yLCC ,yOA,MP
LCC ,MS
), is equi ed o obey single-c ossing di e ences in
(yLCC;(−yOA,MP
LCC ,MS
)), and simila ly, he payo unc ion o OA, OA(yOA,yLCC ,
MPOA,MS
), is equi ed o obey single-c ossing di e ences in (yOA;(−yLCC ,MP
OA,
MS)). This ensu es ha a i m’s en y is discou aged by he opponen ’s en y and en-
hanced by an inc ease in own co a ia es. The da a se is supposed o a ise om a popu-
la ion o hose i ms, wi h unobse ed he e ogenei y gene a ing a dis ibu ion o ealiza-
ions o payo unc ions =(LCC ,OA),whichwedeno ebyP
, and an equilib ium
selec ion ule.
Employing he s a is ical es in Sec ion 4.1, we ind a p- alue o 0.138, and hence, he
hypo hesis ha he empi ical choice equencies a e explained by ou modeling es ic-
258 Lazza i, Quah, and Shi ai Quan i a i e Economics 16 (2025)
Table 3. Empi ical dis ibu ion ac oss each ealiza ion o co a ia es.
(MPLCC ,MP
OA,MS
)=(0, 0, 0)1271 ma ke s
Q(N,N)Q(N,E)Q(E,N)Q(E,E)
0.304 0.682 0.006 0.009
(MPLCC ,MP
OA,MS
)=(0, 1, 0)763 ma ke s
Q(N,N)Q(N,E)Q(E,N)Q(E,E)
0.190 0.785 0.003 0.022
(MPLCC ,MP
OA,MS
)=(1, 0, 0)1125 ma ke s
Q(N,N)Q(N,E)Q(E,N)Q(E,E)
0.194 0.367 0.253 0.186
(MPLCC ,MP
OA,MS
)=(1, 1, 0)782 ma ke s
Q(N,N)Q(N,E)Q(E,N)Q(E,E)
0.122 0.542 0.050 0.286
(MPLCC ,MP
OA,MS
)=(0, 0, 1)869 ma ke s
Q(N,N)Q(N,E)Q(E,N)Q(E,E)
0.159 0.823 0.001 0.017
(MPLCC ,MP
OA,MS
)=(0, 1, 1)1039 ma ke s
Q(N,N)Q(N,E)Q(E,N)Q(E,E)
0.078 0.889 0.000 0.033
(MPLCC ,MP
OA,MS
)=(1, 0, 1)677 ma ke s
Q(N,N)Q(N,E)Q(E,N)Q(E,E)
0.106 0.326 0.306 0.261
(MPLCC ,MP
OA,MS
)=(1, 1, 1)1356 ma ke s
Q(N,N)Q(N,E)Q(E,N)Q(E,E)
0.055 0.501 0.021 0.423
ions canno be e u ed a 5% (o 10%) signi icance le el. We choose he uning pa ame-
e κN=10−3logNx/Nx,whe eNx=minx∈
XNx, and he numbe o boo s ap samples
as R=2000.24 No e ha ha ing a p- alue s ic ly less han 1 means ha JNde ined in
(13) is s ic ly posi i e, ha is, he e is a s ic ly posi i e dis ance be ween ou empi i-
cal dis ibu ion and PSC, he se o (exac ly) SC- a ionalizable dis ibu ions. Using ou R
code on a desk op compu e wi h Apple M1 p ocesso and 16 GB RAM, he p- alue was
calcula ed in less han 3 minu es.
In his se ing, he se X=
Xhas exac ly eigh elemen s, and hence, he numbe o
possible g oup ypes is 48≈65,000. In his small en i onmen , i is in ac no di icul
o check he RM axiom o each o hese g oup ypes. Doing ha , we ind ha only 482
ypes sa is y single-c ossing (equi alen ly, sa is y he RM axiom). This gi es a sense o
he“empi icalbi e”o ou es : heda ase has obeexplainedbyusinga e ysmall
ac ion (less han 1%) o all possible g oup ypes.
Signi icance o s a egic in e ac ions. Ha ing es ablished ha he da a se is (s a is i-
cally) SC- a ionalizable, we can now go on o explo e i s p ope ies. In pa icula , we can
assess he ex en o which s a egic in e ac ions play a ole in explaining he da a, in he
sense discussed in Sec ion 3.4, by conside ing he subclasses o single-c ossing g oup
ypes ha co espond o: (i) he LCC i m ha ing a payo unc ion ha is independen
o he ac ions o OA; (ii) he OA i m ha ing a payo unc ion ha is independen o he
ac ions o LCC; and (iii) bo h i ms ha ing payo unc ions ha a e independen o he
o he i m’s ac ion. Applying he p ocedu e explained in Sec ion 4.2, we ind ha he
g ea es possible weigh s on hese h ee subclasses o single-c ossing g oup ypes a e (i)
0.923, (ii) 0.790, and (iii) 0.789 (wi hin 5% signi icance le el). Since hese weigh s a e all
s ic ly less han 1, we conclude ha any SC- a ionaliza ion o he da a equi es s a egic
24Recall ha Nxis he numbe o obse a ions wi h co a ia es x. We ollow Ki amu a and S oye (2018)in
ha ing κNp opo ional o log Nx/Nx.
Quan i a i e Economics 16 (2025) An o dinal app oach o he empi ical analysis 265
Jensen, Ma in K. (2010), “Agg ega i e games and bes - eply po en ials.” Economic The-
o y, 43, 45–66. [0245]
Ki amu a, Yuichi and Jö g S oye (2018), “Nonpa ame ic analysis o andom u ili y mod-
els.” Econome ica, 86, 1883–1909. [0237,0245,0246,0251,0252,0258]
Kline, B endan and Elie Tame (2016), “Bayesian in e ence in a class o pa ially iden-
i ied models.” Quan a i e Economics, 7, 329–366. [0237,0239,0242,0256,0257,0260,
0261]
Lazza i, Na alia, John K.-H. Quah, and Koji Shi ai (2018), “Nonpa ame ic analysis o
mono one choice.” Discussion Pape Se ies 184, School o Economics, Kwansei Gakuin
Uni e si y. [0238]
Lazza i, Na alia, John K.-H. Quah, and Koji Shi ai (2024), “Supplemen o ‘An o dinal ap-
p oach o he empi ical analysis o games wi h mono one bes esponses’.” Quan i a i e
Economics Supplemen al Ma e ial, 15, h ps://doi.o g/10.3982/QE2192.[0239]
Manski, Cha les F. (2007), “Pa ial iden i ica ion o coun e ac ual choice p obabili ies.”
In e na ional Economic Re iew, 48, 1393–1410. [0237]
Ma schak, Jacob (1960), Bina y Choice Cons ain s on Random U ili y Indica o s (Ken-
ne h A ow, ed.), S an o d Symposium on Ma hema ical Me hods in he Social Sciences.
S an o d Uni e si y P ess. [0237]
Ma zkin, Rosa L. (2007), Nonpa ame ic Iden i ica ion (James J. Heckman and Edwa d E.
Leame , eds.), Handbook o Econome ics, ol. 6b. Else ie Science. [0237]
McFadden, Daniel L. (2005), “Re ealed s ochas ic p e e ence: A syn hesis.” Economic
Theo y, 26, 245–264. [0237]
McFadden, Daniel L. and Ma cel K. Rich e (1991), “S ochas ic a ionali y and e-
ealed s ochas ic p e e ence.” In P e e ences, Unce ain y and Ra ionali y (John Chip-
man, Daniel L. McFadden, and Ma cel K. Rich e , eds.), 161–186, Wes iew P ess, Boul-
de . [0237]
Milg om, Paul and John Robe s (1990), “Ra ionalizabili y, lea ning, and equilib ium in
games wi h s a egic complemen a i ies.” Econome ica, 58, 1255–1277. [0244]
Milg om, Paul and Ch is Shannon (1994), “Mono one compa a i e s a ics.” Econome -
ica, 62, 157–180. [0236,0240,0243,0244]
Smeulde s, Ba , Lau ens Che chye, and B am De Rock (2021), “Nonpa ame ic analysis
o andom u ili y models: Compu a ional ools o s a is ical es ing.” Econome ica, 89,
437–455. [0238,0239,0251,0252,0253]
Tame , Elie (2003), “Incomple e simul aneous disc e e esponse model wi h mul iple
equilib ia.” Re iew o Economic S udies, 70, 147–165. [0237]
Tebaldi, Pie o, Alexande To go i sky, and Hanbin Yang (2023), “Nonpa ame ic es i-
ma es o demand in he Cali o nia heal h insu ance exchange.” Econome ica, 91, 107–
146. [0237]
266 Lazza i, Quah, and Shi ai Quan i a i e Economics 16 (2025)
Vi es, Xa ie (1990), “Nash equilib ia wi h s a egic complemen a i ies.” Jou nal o
Ma hema ical Economics, 19, 305–321. [0244]
Co-edi o S éphane Bonhomme handled his manusc ip .
Manusc ip ecei ed 29 June, 2022; inal e sion accep ed 20 Augus , 2024; a ailable online 25
Sep embe , 2024.
The eplica ion package o his pape is a ailable a h ps://doi.o g/10.5281/zenodo.13322699.
The Jou nal checked he da a and codes included in he package o hei abili y o ep oduce
he esul s in he pape and app o ed online appendices.
