On the limitations of data-based price discrimination

Xie, Hai ian; Zhu, Ying; Shishkin, Denis
A icle
On he limi a ions o da a-based p ice disc imina ion
Theo e ical Economics
P o ided in Coope a ion wi h:
The Econome ic Socie y
Sugges ed Ci a ion: Xie, Hai ian; Zhu, Ying; Shishkin, Denis (2025) : On he limi a ions o da a-based
p ice disc imina ion, Theo e ical Economics, ISSN 1555-7561, The Econome ic Socie y, New Ha en,
CT, Vol. 20, Iss. 1, pp. 303-351,
h ps://doi.o g/10.3982/TE5916
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/320287
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h ps://c ea i ecommons.o g/licenses/by-nc/4.0/
Theo e ical Economics 20 (2025), 303–351 1555-7561/20250303
On he limi a ions o da a-based p ice disc imina ion
Hai ian Xie
Depa men o Business S a is ics and Econome ics, Guanghua School o Managemen , Peking Uni e si y
Ying Zhu
Depa men o Economics, Uni e si y o Cali o nia San Diego
Denis Shishkin
Depa men o Economics, Uni e si y o Cali o nia San Diego
The classic hi d deg ee p ice disc imina ion (3PD) model equi es he knowledge
o he dis ibu ion o buye alua ions and he co a ia e o se he p ice condi-
ioned on he co a ia e. In e ms o gene a ing e enue, he classic esul shows
ha 3PD is a leas as good as uni o m p icing. Wha i he selle has o se a p ice
based only on a sample o obse a ions om he unde lying dis ibu ion? Is i s ill
ob ious ha he selle should engage in 3PD? This pape sheds ligh on hese un-
damen al ques ions. In pa icula , he compa ison o he e enue pe o mance
be ween 3PD and uni o m p icing is ambiguous o e all when p ices a e se based
on samples. This inding is in he na u e o s a is ical lea ning unde unce ain y:
a cu se o dimensionali y, bu also o he small sample complica ions.
Keywo ds. P ice disc imina ion, empi ical e enue maximiza ion, in o ma ion
heo y, p io -independen p icing, op imal a e o con e gence.
JEL classi ica ion. C14, C44, D42, D82.
1. In oduc ion
In he pas ew decades, he ad ances in he heo y o mechanism design ha e been
ollowed by a emendous in e es in i s p ac ical applica ions. A he same ime, classic
Hai ian Xie: [email p o ec ed]
Ying Zhu: [email p o ec ed]
Denis Shishkin: [email p o ec ed]
Hai ian Xie and Ying Zhu sha e he i s au ho ship and a e lis ed alphabe ically. This pape supe sedes a
p e iously ci cula ed d a by Xie and Zhu (h ps://a xi .o g/pd /2204.12723 1.pd ). All h ee au ho s a e
g a e ul o he cons uc i e commen s om wo anonymous e iewe s a Theo e ical Economics, and h ee
anonymous e iewe s and he me a e iewe a ACM Economics and Compu a ion. The au ho s would also
like o hank Di k Be gemann, Songzi Du, Fede ico Echenique, G aham Ellio , Yannai Goncza owski, Roge
Go don, Nima Haghpanah, Johannes Ho ne , Jona han Libgobe , Es andia Maasoumi, Maximilian Schae-
e , Joel Sobel, Ka l Schlag, La y Samuelson, Yixiao Sun, J. Miguel Villas-Boas, and Joel Wa son o aluable
commen s and discussions. Hai ian Xie is g a e ul o he UC San Diego Depa men o Economics whe e
his p ojec was de eloped du ing his doc o al s udies. Xie is suppo ed by he Fundamen al Resea ch
Funds o he Cen al Uni e si ies a Peking Uni e si y. Ying Zhu is g a e ul o he Socie y o Hellman Fel-
lows a Uni e si y o Cali o nia and he Cowles Founda ion a Yale Uni e si y, and also hanks pa icipan s
a he semina s.
©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/TE5916
304 Xie, Zhu, and Shishkin Theo e ical Economics 20 (2025)
heo e ical models ypically make s ong assump ions abou he designe ’s knowledge o
he en i onmen , which may lead he op imal mechanism o be sensi i e o he de ails
o he en i onmen (which is some imes e e ed o as he Wilson c i ique).1
Thi d deg ee p ice disc imina ion (3PD) equi es an obse able co a ia e alue asso-
cia ed wi h he buye alua ion. To se he p ice condi ioned on he co a ia e, he clas-
sic p icing model equi es he knowledge o he dis ibu ion o buye alua ions and he
co a ia e. In e ms o gene a ing e enue, he classic esul shows ha 3PD is a leas as
good as uni o m p icing. Wha i he selle has only pa ial in o ma ion abou hose dis-
ibu ions? Is i s ill ob ious ha he selle should engage in 3PD? On he one hand, se -
ing he op imal p ice o each obse ed alue o he co a ia e may no “ex apola e” well
o he unobse ed co a ia e alues, and yield a lowe expec ed e enue han a uni o m
p ice. Bu on he o he hand, oo li le disc imina ion unde u ilizes he in o ma ion
con ained in he co a ia e abou buye alua ions. This pape is conce ned wi h how
much in o ma ion he selle will need o make 3PD gene a e mo e e enue. Suppose
a uni demand buye wi h a p i a ely-known alua ion Yand a one-dimensional con-
inuous co a ia e Xd awn om a join dis ibu ion FY,X ha is unknown o he selle .
The con inuous co a ia e Xcan be a single index o sco e ha summa izes he ele an
cha ac e is ics o p icing and ma ke ing. Ha mann, Nai , and Na ayanan (2011)p o-
ide examples whe e ma ke ing i ms use a one-dimensional con inuous sco e unc ion
o cus ome cha ac e is ics, pas esponse his o ies, and ea u es o he zip code, and
casinos use a one-dimensional con inuous sco e e e ed o as he a e age daily win.
While ou selle is igno an o FY,X, he/she does ha e access o a andom sample
o i.i.d. {Yi,Xi}n
i=1d awn om FY,X. A na u al s a egy is o choose p ices ha op i-
mize agains he empi ical dis ibu ion o {Yi,Xi}n
i=1.TheK-ma ke s empi ical e enue
maximiza ion (ERM) di ides he co a ia e space in o Kequal-leng h segmen s, and he
op imal p ice based on he condi ional empi ical dis ibu ion o each segmen is cal-
cula ed. We show ha when K=(n1/4), heK-ma ke s ERM s a egy gene a es an
expec ed e enue con e ging o ha o he ue dis ibu ion 3PD op imum a he a e
O(n−1/2). The 1-ma ke ERM s a egy is simply he (uni o m) ERM s a egy, which we
show gene a es a e enue con e ging o ha o he ue-dis ibu ion uni o m op imum
a he a e O(n−2/3).TheK-ma ke s ERM is jus one possible s a egy and one may
wonde i a mo e sophis ica ed s a egy migh p o ide as e con e gence a es. In a
sense, he answe is no. We show ha hese a es a e asymp o ically unimp o able o
he wo s case dis ibu ions o (Y,X)subjec o some mild smoo hness condi ions. In
o he wo ds, o gua an ee a e enue de iciency o δuni o mly o e a class o dis ibu-
ions, he necessa y condi ion o he sample size is ha n=(δ−2)in he 3PD p oblem
and n=(δ−3/2)in he uni o m p icing p oblem.
Fo su icien ly small δ, heK-ma ke s ERM and he uni o m ERM s a egies a e op-
imal on he g ow h equi emen s o he sample size, espec i ely; ha is, n=(δ−2)
in he 3PD p oblem and n=(δ−3/2)in he uni o m p icing p oblem. To show his
op imali y esul , we es ablish a lowe bound o he e enue de iciency in any da a-
based p icing s a egy ela i e o he ue-dis ibu ion op imal s a egy in he wo s case
1In some cases, his leads o ex eme o un ealis ic esul s as in, o example, C éme and McLean (1988).
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
Theo e ical Economics 20 (2025) Limi a ions o da a-based p ice disc imina ion 305
(by conside ing he sup emum o e a class o join dis ibu ions, FY,X, subjec o some
mild smoo hness assump ions). In pa icula , da a-based uni o m p icing s a egies a e
algo i hms ha depend on {Yi}n
i=1only, and he ue-dis ibu ion op imal s a egy co -
esponds o he op imal uni o m p icing s a egy de i ed om FY. Simila ly, da a-based
3PD s a egies a e algo i hms ha depend on {Yi,Xi}n
i=1, and he ue-dis ibu ion op i-
mal s a egy co esponds o he op imal 3PD s a egy de i ed om FY,X. We show ha
he minimax e enue de iciency is (n−2/3)and (n−1/2)in he uni o m and 3PD cases,
espec i ely.
Ou esul s highligh he ollowing economic ade-o . When he selle has he ac-
cess o a sample o i.i.d. {Yi,Xi}n
i=1, she can choose he K-ma ke s ERM s a egy ha
exploi s bo h {Xi}n
i=1and {Yi}n
i=1, o he uni o m ERM s a egy ha igno es {Xi}n
i=1and
exploi s only {Yi}n
i=1. Inhe en ly, he o me is an algo i hm ying o lea n he FY,X-
op imal p icing unc ion p(·)while he la e is an algo i hm ying o lea n he FY-
op imal (cons an ) p icing unc ion. As a esul o he cu se om he ex a dimensional-
i y, he o me is mo e demanding in he sample size han he la e . Howe e , in e ms
o gene a ing e enue, he ue-dis ibu ion op imal 3PD s a egy is a leas as good as
he ue-dis ibu ion op imal uni o m p icing s a egy. This ade-o sugges s ha , e en
i Xcon ains use ul in o ma ion abou Y, heK-ma ke s ERM s a egy based on a an-
dom sample can be e enue in e io o he uni o m ERM s a egy when he sample size
nis no la ge enough, and ice e sa.
To e i y hese po en ial implica ions, we conduc se e al nume ical s udies. In pa -
icula , we calcula e he e enues o he K-ma ke s ERM and he uni o m ERM s a egies
based on a eal-wo ld da a se om eBay auc ions and wo simula ed da a se s. Ou nu-
me ical esul s illus a e he a o emen ioned ade-o . When he sample size is small,
he uni o m ERM s a egy can gene a e highe expec ed e enue han he K-ma ke s
ERM s a egy. As he sample size g ows, he K-ma ke s ERM s a egy ( he uni o m
ERM s a egy) ge s close o he ue-dis ibu ion op imal 3PD s a egy ( esp., he ue-
dis ibu ion op imal uni o m p icing s a egy). The slowe a e o con e gence in he
e enue om he K-ma ke s ERM s a egy (in con as o he as e a e o con e gence
in he e enue om he uni o m ERM s a egy) is domina ed by he bene i o p ice dis-
c imina ion (based on FY,X) o e uni o m p icing (based on FY). Consequen ly, he e -
enue o he K-ma ke s ERM s a egy o e akes ha o he uni o m ERM s a egy when
he sample size becomes su icien ly la ge and Xcon ains su icien in o ma ion abou
Y.
The key akeaways om his pape a e summa ized he e. Fi s , no sample-based
3PD s a egy is able o escape om he cu se o dimensionali y, shown by ou in o ma-
ion heo e ic lowe bounds. Second, absen unce ain y ega ding he unde lying p ob-
abili y laws, hi d-deg ee p ice disc imina ion is a leas as good as uni o m p icing in
gene a ing e enue. In con as , he compa ison o he e enue pe o mance be ween
he K-ma ke s ERM and he uni o m ERM s a egies is ambiguous o e all. This inding
is in he na u e o s a is ical lea ning unde unce ain y: a cu se o dimensionali y, bu
also o he small sample complica ions.2Empi ical e enue maximiza ion is no ee o
2Speci ically, he e exis s a dis ibu ion FYwhe e he e enue o he uni o m ERM s a egy is wo se wi h
wo obse a ions han wi h one; see Babaio , Goncza owski, Mansou , and Mo an (2018). We illus a e in
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
306 Xie, Zhu, and Shishkin Theo e ical Economics 20 (2025)
hese issues. Ul ima ely, his pape poses a challenging open ques ion o whe he he e
exis some n<¯
n<∞such ha o any n∈[n,¯
n]and dis ibu ion in he class de ined in
his pape , he K-ma ke s ERM s a egy ( o any K>1) is always e enue-in e io o he
uni o m ERM s a egy.
1.1 Rela ed li e a u e
Complexi y measu es and in o ma ion heo e ic lowe bounds In o ma ion heo e ic
lowe bounds and sample complexi y a e impo an no ions in machine lea ning. Bo h
aim o cha ac e ize lea nabili y, i.e., how easy i is o lea n an unknown objec o in e es
(in ou con ex , he ue-dis ibu ion op imal 3PD s a egy) om da a whe e he unce -
ain y a ises. Sample complexi y de i es he a e a which he sample size needs o g ow
o gua an ee a desi ed lea ning accu acy. In o ma ion heo e ic lowe bound de i es
a lowe bound as a unc ion o he sample size on he lea ning e o (in ou con ex ,
he e enue de iciency) in he wo s case. Sample complexi y and in o ma ion heo-
e ic lowe bounds a e in insically ied o he complexi y o size o he unde lying unc-
ion class o in e es . Vapnik–Che onenkis (VC) dimensions, sha e ing dimensions,
and me ic en opy (such as he ca dinali y o packing se s) a e popula measu es o
complexi y in machine lea ning. The e ha e been a numbe o inno a i e applica ions
o VC dimensions o sha e ing dimensions in economic heo y and algo i hmic eco-
nomics. Toge he wi h he P obably App oxima ely Co ec (PAC) amewo k, hey a e
used o s udy he complexi y o he classes o demand and u ili y unc ions (Beigman
and Voh a (2006), Balcan, Daniely, Meh a, U ne , and Vazi ani (2014)), k-demand buye ’s
alua ion (Zhang and Coni ze (2020)), heo ies o choices (Basu and Echenique (2020)),
p e e ence unc ions (Chambe s, Echenique, and Lambe (2021,2023)), as well as he
esul ing lea nabili y om da a. VC dimension is use ul o de i ing sample complex-
i y bounds conce ning disc e e unc ion se s and ini e-dimensional ec o spaces, and
sha e ing dimension is use ul o ce ain eal unc ions.
F om he heo y o machine lea ning, when a class has in ini e VC o sha e ing di-
mensions, his class is no PAC lea nable. Fo example, a collec ion o sinusoids ha e
subg aphs wi h in ini e VC dimension. The max-min expec ed u ili y model wi h a
leas h ee s a es o he wo ld has in ini e VC dimension (Basu and Echenique (2020)).
The class o demand unc ions has in ini e sha e ing dimension (Beigman and Voh a
(2006)). None heless, he no ion o “lea nabili y” can be gene alized using a di e en
ype o complexi y analysis ha gi es ise o ou in o ma ion heo e ic lowe bound in
he 3PD p oblem. This ype o analysis is buil upon he no ion o packing se s, along
wi h ools om in o ma ion heo y. In pa icula , packing se s a e use ul o s udying
classes wi h an in ini e numbe o elemen s (see Kolmogo o and Tikhomi o (1959)
and Wainw igh (2019)). This is he case o ou 3PD p oblem as we y o lea n an op i-
mal p icing unc ion o he co a ia e (an in ini ely-dimensional pa ame e ) and bound
he de iciency in he expec ed e enue, which conce ns he en i e p icing unc ion a all
co a ia e alues.
Sec ion 6 ha his seemingly coun e -in ui i e esul highligh s he di icul y o es ablishing gene al com-
pa a i e esul s wi h e y small sample size and sheds some ligh on he compa ison o he e enue pe o -
mance o he K-ma ke s ERM s a egy wi h K=1 s.K=2in hecaseo n=2.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License

Theo e ical Economics 20 (2025) Limi a ions o da a-based p ice disc imina ion 307
P io -independen mechanism design Mos o he classic monopoly p icing li e a u e
assumes a known dis ibu ion o alua ions (and co a ia es).3Mo e ecen ly, some pa-
pe s (e.g., hose su eyed in Ca oll (2019)) s udied “p io ”-independen mechanism de-
sign.4The main ocus o ha li e a u e is on de i ing a obus ly op imal mechanism in
he absence o bo h “p io ” and da a. In pa icula , Be gemann and Schlag (2008,2011)
de i e he minimax- eg e uni o m p icing s a egy in closed o m; ha is, he s a egy
ha gua an ees he smalles de iciency in e enue ela i e o he known dis ibu ion
case. Like Be gemann and Schlag (2008,2011), we s udy he e enue de iciencies, bu in
con as , we assume he a ailabili y o da a and ocus on he (ine i able) in o ma ion-
heo e ic limi a ions o any da a-based p icing s a egies and he achie abili y o he
limi a ion.
This pape is inspi ed by he li e a u e ha s udies app oxima ely op imal “p io ”-
independen mechanism design, in pa icula monopoly p icing wi h a single buye .5
This li e a u e assumes ha he selle has access o a andom sample o i.i.d. {Yi}n
i=1
d awn om FYand p oposes a ian s o he uni o m ERM s a egy o de i e he e -
enue gua an ee in ela ion o ha om he ue-dis ibu ion op imal uni o m p icing
s a egy. The e a e wo ypes o analyses in his li e a u e. The i s one ocuses on he
gua an ees o he speci ic case o n=1o n=2(Babaio e al. (2018), Allouah, Ba-
hamou, and Besbes (2023)). The second one (e.g., Huang, Mansou , and Roughga den
(2018)) es ablishes “sample complexi y bounds” such ha he uni o m ERM a ian s
achie e a (1−) ac ion gua an ee when he sample size g ows a a a e depending on
, and also de i es he a e a which he sample size needs o g ow (as a unc ion o ) o
any da a-based uni o m p icing s a egies o ob ain a gi en (1−) ac ion gua an ee.
Allouah, Bahamou, and Besbes (2022) in ol e bo h ypes o analyses.
In his pape , we ask he ela ed ques ion, how as he e enue de iciency decays as
a unc iono n, and p o ide an answe using in o ma ion- heo e ic lowe bounds (in-
dependen o algo i hms) and uppe bounds wi h espec o speci ic algo i hms in he
wo s case scena ios.6The main di e ence wi h he majo i y o he da a-based li e -
a u e is ha , we s udy hi d-deg ee p ice disc imina ion (3PD) wi h a con inuous co-
a ia e and compa e he e enue pe o mance o da a-based 3PD and uni o m p icing
s a egies.
To unde s and why he 3PD p oblem in ou con ex is mo e challenging han he
uni o m p icing p oblem, no e ha undamen ally he la e ies o lea n he cons an
3See also Segal (2003) o a s udy o op imal mul iuni auc ions whe e he selle has a p obabilis ic belie
abou he alua ion dis ibu ion o he i.i.d. buye s.
4He e, “p io ” dis ibu ion e e s o he selle ’s p io belie abou buye s’ alua ions and is o en aken o
be he ue dis ibu ion.
5The e is a less ela ed li e a u e ha s udies op imal auc ions; see, e.g., Cole and Roughga den (2014),
Dhangwa no ai, Roughga den, and Yan (2015), Fu, Immo lica, Lucie , and S ack (2015), Guo, Huang, and
Zhang (2019), Fu, Haghpanah, Ha line, and Kleinbe g (2021).
6A la ge li e a u e s udies da a-based auc ions by ocusing on gua an ees o e enue de iciencies (in-
s ead o ac ions), such as how he e enues om he da a-based s a egies con e ge in p obabili y o he
ue-dis ibu ion benchma k, e.g., Baliga and Voh a (2003), Goldbe g, Ha line, Ka lin, Saks, and W igh
(2006), Gonçal es and Fu ado (2024). This line o wo k does no conside he op imal a es o con e gence
o op imal sample size equi emen s.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
308 Xie, Zhu, and Shishkin Theo e ical Economics 20 (2025)
op imal p icing unc ion (a scala pa ame e ) while he o me ies o lea n an op i-
mal p icing unc ion o he co a ia e (an in ini ely-dimensional pa ame e ), whe e he
de iciency in he expec ed e enue conce ns he en i e p icing unc ion a all co a ia e
alues. Ou amewo k allows us o ackle se e al challenging aspec s o he 3PD p ob-
lem, which migh be di icul o analyze wi h he oolki in he exis ing p icing li e a u e.
We desc ibe one example below.
Somewha ela ed, De anu , Huang, and Psomas (2016) s udy sample complexi y o
op imal p icing wi h “side in o ma ion.” In hei “signals model” (Sec ions 5.1 and 5.3),
he e is a co a ia e (signal) X∈[0, 1], and he selle can condi ion he da a-based ese e
p ice on he co a ia e. Fo he single-buye case (which would co espond o ou 3PD
p oblem), hey de i e uppe and lowe sample complexi y bounds. Impo an ly, hey
assume ha he ue join dis ibu ion FY,Xhas he ollowing p ope y: la ge alues o
Xa e associa ed wi h la ge alues o Yin he sense o i s -o de s ochas ic dominance
o condi ional dis ibu ions. In con as , ou 3PD se up imposes no assump ions abou
he ela ionship be ween he co a ia e Xand he alua ion Y; meanwhile, ou p oposed
K-ma ke s ERM s a egy lea ns he ela ionship om he da a. Mo eo e , ou K-ma ke s
ERM s a egy a ains he op imal a e o con e gence in e enue de iciency (as desc ibed
be o e), while he uppe and lowe bounds in De anu , Huang, and Psomas (2016)ha e
di e en a es, and hence, he op imal sample size equi emen is unclea .
2. Se up
The selle is selling an i em o a buye . Le Y∈[0, 1]be he alua ion (i.e., willingness
o pay) o he buye , and X he co a ia e (such as a cha ac e is ic) associa ed wi h he
buye . The join dis ibu ion o (Y,X)is deno ed by FY,X. We assume ha Xis sup-
po ed on a bounded in e al, and wi hou loss o gene ali y, we ake he in e al o be
[0, 1].7
Gi en a co a ia e alue, he selle wan s o se a p ice acco ding o a mapping om
he co a ia e o a se o p ices. We use D o deno e he se o all p icing unc ions:
D≡p:[0, 1]→[0, 1], measu able.
Fo a gene ic p icing s a egy p∈D, he p ice depends on he co a ia e alue x.This
scheme alls in he ealm o hi d-deg ee p ice disc imina ion (3PD). Uni o m p icing
can be iewed as a special case whe e he p ice is he same o all co a ia e alues. We
use U o deno e he se o all uni o m p icing unc ions:
U≡{p∈D:pis a cons an unc ion}.
7The assump ion ha Y,X∈[0, 1]is made me ely o simplici y. Fi s o all, ou esul s in Sec ions 3and
4hold o gene al bounded suppo s. Second, he p ecise knowledge o he suppo bounda ies is unnec-
essa y because hey can be eadily es ima ed using ex emum o de s a is ics. The es ima o con e ges a
a supe consis en a e o n−1(see, e.g., Hi ano and Po e (2003)), signi ican ly as e han he con e gence
o e enue de iciency ha we show in Sec ion 3. The e o e, in ou analysis, he es ima ion e o esul ing
om he unknown suppo is negligible. We a e g a e ul o a e e ee o aising his discussion.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
Theo e ical Economics 20 (2025) Limi a ions o da a-based p ice disc imina ion 309
To ligh en he no a ion, we exp ess p∈Uas a scala a he han a unc ion o he uni-
o m p icing p oblem.
Le FY|Xbe he condi ional CDF and X he ma ginal densi y unc ion. Gi en a p ice
y∈[0, 1]and a co a ia e alue x∈[0, 1], he ea e1−FY|X(p|x)buye s whose alua ion
is abo e he p ice. The e enue gene a ed om hese buye s is
(y,x,FY,X)≡1−FY|X(y|x)y,(1)
and he expec ed e enue o a p icing unc ion pis
R(p,FY,X)≡1
0
p(x),x,FY,X X(x)dx.
In a ious places o he es o he pape , we will sligh ly abuse he no a ion and deno e
(p,x)≡ (p(x),x)when pis a p icing unc ion and also w i e (y,x)= (y,x,FY,X)
o b e i y when FY,Xis clea om he con ex . In he special case whe e he p icing
s a egy is uni o m (i.e., p∈U), he e enue only depends on he ma ginal dis ibu ion
FY:
R(p,FY,X)=pP(Y≥p)=p1−FY(p),p∈U.
The ue-dis ibu ion op imal 3PD s a egy p∗
Dis he one ha maximizes he e enue:
Rp∗
D,FY,X=sup
p∈D1
0
p(x),x,FY,X X(x)dx.
In a simila ashion, we deno e p∗
Uas he ue-dis ibu ion op imal uni o m p icing
s a egy such ha
Rp∗
U,FY=Rp∗
U,FY,X=sup
p∈U
p1−FY(p).
No e ha p∗
Ddepends on FY,Xand p∗
Udepends on FY.
In e ms o gene a ing e enue, he classic p icing heo y shows ha 3PD is a leas
as good as uni o m p icing when he join dis ibu ion FY,Xis known o he selle . In
his case, we can sol e analy ically o nume ically o he op imal p icing s a egies p∗
D
and p∗
U. Since Uis con ained in D,p∗
Dmus achie e a (weakly) be e e enue han p∗
U.
In ui i ely, when Yis co ela ed wi h X,p∗
Du ilizes he in o ma ion in X.
Now suppose ha he selle knows nei he FY,Xno FY, bu ins ead obse es a an-
dom sample o da a ≡{(Yi,Xi),1≤i≤n}d awn om FY,X,o da aY≡{Yi,1≤i≤n}
om FY, and wan s o cons uc a p icing s a egy based on he sample. The ollowing
assump ion is used h oughou his pape .
Assump ion 1. da a and da aYconsis o i.i.d. d aws om FY,Xand FY, espec i ely.
The ollowing assump ion is used o es ablish he esul s conce ning ou 3PD p ob-
lem. Ins ead o a single known join dis ibu ion FY,X, he e is a class Fo unknown
dis ibu ions, which a e deemed possible and ou da a-based p icing s a egies can be
e alua ed wi hin his class. The unc ions in Fsa is y se e al smoo hness and egula i y
condi ions s a ed below.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
310 Xie, Zhu, and Shishkin Theo e ical Economics 20 (2025)
Assump ion 2. Any dis ibu ion unc ion in he se Fsa is ies he ollowing condi ions:
(i) (Lipschi z con inui y) The e exis s C0∈(0, ∞)such ha , o any y,y,x∈[0, 1], he
condi ional densi y Y|Xsa is ies
 Y|X(y|x)− Y|Xy|x≤C0y−y.
(ii) (S ong conca i y) The e exis s C∗>0such ha he e enue unc ion (y,x)≡
y(1−FY|X(y|x)) is s ic ly conca e wi h he second-o de de i a i e
−2 Y|X(y|x)−y∂
∂y Y|X(y|x)≤−C∗,a.e. (2)
(iii) (In e io solu ion) Fo each x∈[0, 1], he op imal p ice is an in e io solu ion; ha
is, p∗
D(x;FY,X)∈(0, 1).
(i ) (Di e en iabili y) The condi ional dis ibu ion unc ion Y|X(y|x)is con inuously
di e en iable in (x,y)in a neighbo hood o he cu e {(x,p∗
D(x;FY,X)) :x∈
[0, 1]}.
( ) (Boundedness) The unc ions
2 Y|X(y|x)+y∂
∂y Y|X(y|x)and (3)

∂
∂xFY|X(y|x)+y∂
∂x Y|X(y|x)(4)
a e bounded om abo e by C∈(0, ∞)a.e.
( i) (Ma ginal densi y) The ma ginal densi y Xis bounded om abo e by C∈(0, ∞)
and bounded away om ze o; ha is, X≥C > 0.
Pa (i) equi es he condi ional densi y unc ion o be su icien ly smoo h. The pa -
ial de i a i e ∂
∂y Y|X(y|x)is well-de ined almos e e ywhe e because Y|Xis Lipschi z
con inuous, and hence, absolu ely con inuous. Pa (iii) ensu es ha he i s -o de con-
di ion holds o he op imal p ice. Pa (i ) ensu es ha he op imal p icing unc ion
p∗
D(x;FY,X)is su icien ly smoo h in x. Pa ( ) equi es he pa ial de i a i es o he
e enue o be bounded. Pa ( i) ensu es ha he co a ia e does no ake anishing o
domina ing alues.
Unde pa (ii), he op imal p ice is well-de ined. Pa (ii) is a s anda d assump ion
in he op imal auc ions/p icing li e a u e also known as egula i y (Mye son (1981)),
which is a so-called “s ong conca i y” condi ion om machine lea ning heo y. I is
well known ha any dis ibu ion Fwi h he mono one haza d a e sa is ies egula i y.
Analogously, he ollowing assump ion is used o es ablish he esul s o he uni o m
p icing p oblem, which conce ns a class FUo unknown ma ginal dis ibu ions ha a e
deemed possible.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
Theo e ical Economics 20 (2025) Limi a ions o da a-based p ice disc imina ion 317
he co a ia e (an in ini ely-dimensional pa ame e ) and he de iciency in he expec ed
e enue conce ns he en i e p icing unc ion a all co a ia e alues. The no ion o pack-
ing se s in Kolmogo o and Tikhomi o (1959) and he Gilbe –Va shamo bound om
coding heo y a e use ul ing edien s o p o ing Theo em 4. The mos in ica e pa o
he p oo in ol es ca e ully cons uc ing Mcondi ional densi ies (whe e Mg ows wi h
n) and bounding he sepa a ion be ween he op imal p ices associa ed wi h hese densi-
ies. The desi ed se o op imal p ices in ou p oo is a packing se whe e he sepa a ion
be ween elemen s is (n−1/4)wi h espec o he unweigh ed L2no m, and he ca di-
nali y o his se is (2n1/4).
4.2 Uni o m p icing
We ha e he ollowing heo em o uni o m p icing.
Theo em 5. Le Assump ion 1hold. Fo any FUsa is ying he condi ions in Assump-
ion 3wi h C∗∈(0, 2)in (2), he minimax di e ence in he e enues is bounded om
below as
RU
nFUn−2/3.
Theo em 5s a es ha he e is an ine i able de iciency, (n−2/3),in he e enue
om any da a-based uni o m p icing s a egy ela i e o he e enue om he ue-
dis ibu ion op imal uni o m p icing s a egy by aking he sup emum o e FU.
Recalling Co olla y 1on he con e gence a e O(n−2/3)o he 1-ma ke ERM s a -
egy, despi e i s simplici y, Theo em 5implies ha he e enue om his algo i hm
achie es he op imal a e o con e gence (as a unc ion o n) o he e enue om he
ue-dis ibu ion op imal uni o m p icing s a egy uni o mly o e FU.
4.3 Ske ches o he p oo s
To acili a e unde s anding, we s a wi h a p elimina y o he p oo o Theo em 3be o e
laying ou he p elimina ies o Theo ems 4and 5.
4.3.1 P elimina y o he p oo o Theo em 3Fo Theo em 3, we i s show ha he min-
imax di e ence in p ice a a gi en co a ia e alue x0is bounded om below as ollows:
in
ˇ
pD∈ˇ
D
sup
FY,X∈F
EFY,Xˇ
pD(x0;da a)−p∗
D(x0;FY,X)n−1/4,x0∈(0, 1).(6)
Using Taylo expansion ype o a gumen s and condi ion (2), we can ela e he e enue
di e ence o he minimax squa ed di e ence in p ice a x0:
RD
n(x0;F)in
ˇ
pD∈ˇ
D
sup
FY,X∈F
EFY,Xˇ
pD(x0;da a)−p∗
D(x0;FY,X)2
≥in
ˇ
pD∈ˇ
D
sup
FY,X∈FEFY,Xˇ
pD(x0;da a)−p∗
D(x0;FY,X)2
whe e he las line ollows om he Jensen’s inequali y.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License

318 Xie, Zhu, and Shishkin Theo e ical Economics 20 (2025)
The de i a ion o he lowe bound (6) can be educed o a bina y classi ica ion p ob-
lem. In a bina y classi ica ion p oblem, we ha e wo dis ibu ions F1
Y,X,F2
Y,X∈Fwhose
op imal p ices a e sepa a ed by some numbe 2ε; ha is,
p∗
Dx0;Fj
Y,X−p∗
Dx0;Fj
Y,X≥2ε,j,j∈{1, 2}.(7)
A bina y classi ica ion ule uses he da a o decide whe he he ue dis ibu ion is F1
Y,X
o F2
Y,X. To ela e he bina y classi ica ion p oblem o he p icing p oblem, no e ha ,
gi en any p icing unc ion ˇ
pD, we can use i o dis inguish be ween F1
Y,Xand F2
Y,Xin
he ollowing way. De ine he bina y classi ica ion ule
ψ(da a)=a gmin
j∈{1,2}p∗
Dx0;Fj
Y,X−ˇ
pD(x0;da a).
We claim ha when he unde lying dis ibu ion is Fj
Y,X he decision ule ψis co ec i
p∗
Dx0;Fj
Y,X−ˇ
pD(x0;da a)<ε.(8)
To see his, no e ha by he iangle inequali y, (7)and(8) gua an ee ha
p∗
Dx0;Fj
Y,X−ˇ
pD(x0;da a)
≥p∗
Dx0;Fj
Y,X−p∗
Dx0;Fj
Y,X−p∗
Dx0;Fj
Y,X−ˇ
pD(x0;da a)
>2ε−ε=ε,whe ej=j,j,j∈{1, 2}.
This implies ha
PFj
Y,Xψ(da a)=j≤PFj
Y,Xp∗
Dx0;Fj
Y,X−ˇ
pD(x0;da a)≥ε,j=1, 2.
The e o e, we can uppe bound he a e age p obabili y o mis akes in he bina y classi-
ica ion p oblem as
1
2PF1
Y,Xψ(da a)=1+1
2PF2
Y,Xψ(da a)=2
≤1
2PF1
Y,Xp∗
Dx0;F1
Y,X−ˇ
pD(x0;da a)≥ε
+1
2PF2
Y,Xp∗
Dx0;F2
Y,X−ˇ
pD(x0;da a)≥ε
≤sup
FY,X∈F
PFY,Xp∗
D(x0;FY,X)−ˇ
pD(x0;da a)≥ε.
By he Ma ko inequali y, we ha e
sup
FY,X∈F
Eˇ
pD(x0;da a)−p∗
D(x0;FY,X)
≥εsup
FY,X∈F
Pˇ
pD(x0;da a)−p∗
D(x0;FY,X)≥ε
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
Theo e ical Economics 20 (2025) Limi a ions o da a-based p ice disc imina ion 319
≥ε1
2PF1
Y,Xψ(da a)=1+1
2PF2
Y,Xψ(da a)=2.
Finally, we ake he in imum o e all p icing s a egies on he le -hand side (LHS), and
he in imum o e he induced se o bina y decisions on he igh -hand side (RHS). This
leads o
in
ˇ
pD∈ˇ
D
sup
FY,X∈F
Eˇ
pD(x0;da a)−p∗
D(x0;FY,X)
≥εin
ψ1
2PF1
Y,Xψ(da a)=1+1
2PF2
Y,Xψ(da a)=2.(9)
The RHS o he abo e inequali y consis s o wo pa s: (1) ε, ela ed o he sepa a ion
be ween wo op imal p ices, and (2) he a e age p obabili y o making a mis ake in dis-
inguishing he wo dis ibu ions. To ob ain a meaning ul bound, we wan o ind wo
dis ibu ions F1
Y,Xand F2
Y,X ha a e close o each o he (ha d o dis inguish) bu hei
op imal p ices a e su icien ly sepa a ed. We lea e he de ails o he cons uc ion o such
dis ibu ions o he p oo o Theo em 3gi eninAppendixB.
4.3.2 P elimina y o he p oo o Theo em 4Fo Theo em 4, we i s show ha he min-
imax (unweigh ed) L2-dis ance in p ice is bounded om below as ollows:
in
ˇ
pD∈ˇ
D
sup
FY,X∈F
E
ˇ
pD(da a)−p∗
D(FY,X)
2
2n−1/2
whe e

ˇ
pD(da a)−p∗
D(FY,X)
2
2=1
0ˇ
pD(x;da a)−p∗
D(x;FY,X)2dx.
Using Taylo expansion ype o a gumen s and condi ion (2), we can ela e he di e ence
in he expec ed e enues o he minimax (unweigh ed) L2-dis ance in p ice:
RD
n(F)in
ˇ
pD∈ˇ
D
sup
FY,X∈F
EFY,X
ˇ
pD(da a)−p∗
D(FY,X)
2
2
whe e he expec a ion EFY,Xis aken wi h espec o da a ∼FY,X.
The objec abo e conce ns he en i e p icing unc ion p∗
D(·;FY,X).Asa esul ,
bounding he RHS o he abo e inequali y is mo e complica ed han he p e ious one
(6). In pa icula , we conside a mul iple classi ica ion p oblem ha ies o dis inguish
among Mdis ibu ions, whe e Mis a unc ion o he sample size n. Simila as be o e,
we wan he op imal p ices o hese Mdis ibu ions o be su icien ly sepa a ed. Simila
de i a ions show ha he lowe bound o he e enue p oblem can be educed o ha o
a mul iple classi ica ion p oblem:
in
ˇ
pD∈ˇ
D
sup
FY,X∈F
EFY,X
ˇ
pD(da a)−p∗
D(FY,X)
2
2≥ε2in
ψ
1
M
M

j=1
PFj
Y,Xψ(da a)=j, (10)
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
320 Xie, Zhu, and Shishkin Theo e ical Economics 20 (2025)
whe e he in imum in ψis aken o e he se o all mul iple decisions (wi h Mchoices).
To p oceed, we apply he Fano’s inequali y om in o ma ion heo y (Co e and Thomas
(2005)). Fano’s inequali y gi es a lowe bound on he a e age p obabili y o mis akes:9
1
M
M

j=1
PFj
Y,Xψ(da a)=j≥1−
M

j,j=1
KLFj
Y,XFj
Y,X/M2+log2
logM, (11)
whe e KL(··)deno es he Kullback–Leible (KL) di e gence be ween wo dis ibu ions:
KL(F1F2)≡ 1(y,x)log 1(y,x)
2(y,x)dy dx.
To ob ain a sha p bound based on he mul iple classi ica ion p oblem, we wan o ind
a se o dis ibu ions (whe e he ca dinali y Mo hese isla geenough) ha a eclose
enough o each o he (small enough pai wise KL di e gence) bu hei op imal p ices
a e su icien ly sepa a ed. We lea e he de ailed p oo o Appendix B. Ou p oo is
based on a delica e cons uc ion o condi ional densi ies along wi h an applica ion o
he Gilbe –Va shamo lemma om coding heo y. Speci ically, we use he dis ibu ion
Y,X∼U[0, 1]wi h Xindependen o Yas he benchma k dis ibu ion and cons uc
i s pe u bed e sions wi h some co ela ion.
4.3.3 P elimina y o he p oo o Theo em 5Rela i e o he p oo s in he case o 3PD,
he p oo s o he p ice- and e enue-de iciency lowe bounds in uni o m p icing a e
simple . We i s show ha he minimax di e ence in p ice is bounded om below as
ollows:
in
ˇ
pU∈ˇ
U
sup
FY∈FU
EFYˇ
pU(da aY)−p∗
Un−1/3. (12)
As p e iously, we can ela e he e enue di e ence o he minimax squa ed di e ence in
p ice:
RU
nFUin
ˇ
pU∈ˇ
U
sup
FY∈FU
EFYˇ
pU(da aY)−p∗
U2
≥in
ˇ
pU∈ˇ
U
sup
FY∈FUEFYˇ
pU(da aY)−p∗
U2
whe e he las line ollows om he Jensen’s inequli y. The de i a ion o (12)only e-
qui es cons uc ing wo dis ibu ions, simila o he app oach discussed in Sec ion 4.3.1.
5. Nume ical e idence
Sec ions 3and 4es ablish ha he K-ma ke s ERM s a egy achie es he op imal a es
o con e gence in e enue uni o mly o e a class o dis ibu ions. In his sec ion, we
9We do no p esen he Fano’s inequali y in i s s anda d o m as in Co e and Thomas (2005). Ins ead, we
use a e sion om Wainw igh (2019) ha is mo e con enien o ou pu poses.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
Theo e ical Economics 20 (2025) Limi a ions o da a-based p ice disc imina ion 321
u n o speci ic dis ibu ions and s udy he e enue pe o mance o ou K-ma ke s ERM
s a egies in hese cases. We p esen nume ical e idence ha suppo s he implica ions
o ou heo e ical esul s. Speci ically, we calcula e he e enues o he p icing s a egies
p oposed in Sec ion 3using eal-wo ld and simula ed da a. We desc ibe he da a in
de ail below.
Da a Fo he empi ical s udy, we use an eBay auc ion da a se (Jank and Shmueli
(2010)). Because eBay uses a sealed-bid second-p ice auc ion o ma , he bid o each
pa icipan can se e as a p oxy o an indi idual alua ion o he objec . In pa icula ,
we use he da a on 194 7-day auc ions o he new Palm Pilo M515 PDAs.10 The da a has
3832 obse a ions a he bid le el, and each obse a ion includes an auc ion id, a bid
amoun , a bidde id, and a bidde a ing. Some bidde s appea in he da a se se e al
imes because ei he hey e ised hei bid du ing an auc ion o pa icipa ed in se e al
auc ions. To be consis en wi h ou assump ion o independen sampling, we analyze
he da a a he bidde le el and use he highes bid o each bidde ac oss all auc ions
she pa icipa ed in as he one ep esen ing he alua ion. This lea es 1203 obse a ions
om which we d aw samples o a ious sizes. Fo Yi, we use he bid (as desc ibed abo e)
o bidde ino malized o [0, 1].Fo Xiin he 3PD case, we use bidde i’s a ing on eBay,
which indica es he numbe o imes selle s le eedback a e a ansac ion wi h i.
Fo he simula ion s udy, we le he ma ginal dis ibu ion o Xbe uni o m on [0, 1]
and he CDF o Ycondi ional on X=xbe
FY|X(y|x)=yx+1. (13)
Implemen a ion Fo each ype o da a, we calcula e (a Mon e-Ca lo app oxima ion o )
he expec ed e enue gene a ed by he uni o m ERM and he K-ma ke s ERM s a egies
o a ious sample sizes as ollows. Fi s , ix nand K. Then d aw a sample {Yi,Xi}n
i=1
and, o each k=1, ,K,le
ma ke k≡{Yi:Xi∈Ik},ˆ
Fk( )≡|Yi∈ma ke k:Yi≤ |
|ma ke k|.
Then he empi ical op imal p ice in he k h ma ke is gi en by
ˆ
pD,k≡a gmax
y∈[0,1]
y1−ˆ
Fk(y)=a gmax
y∈ma ke k
y1−ˆ
Fk(y),
whe e he second equali y holds because ˆ
Fkis a s ep unc ion. No e ha he uni o m
ERM s a egy simply co esponds o he 1-ma ke ERM s a egy. When K>1anda
d awn sample esul s in emp y ma ke s ha con ain no obse a ions, we se he p ices
in hose ma ke s o one. Finally, we compu e he e enue de iciency o he uni o m
ERM and K-ma ke s ERM s a egies (unde Kn1/4).
10Jank and Shmueli (2010) also p o ide da a on Ca ie w is wa ches, Swa o ski beads, and Xbox game
consoles, bu each o hese da a se s may pool a ious con igu a ions o models o hese p oduc s ca e-
go ies. Thus, we choose he da a on he Palm Pilo M515 o minimize such a ia ions.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
322 Xie, Zhu, and Shishkin Theo e ical Economics 20 (2025)
Figu e 1. Re enue unde uni o m and K-ma ke s ERM s a egies.
Nume ical indings Figu e 1plo s he expec ed e enue gene a ed by he K-ma ke s
ERM s a egy o K∈{1, ,5
}as a unc ion o he sample size n(wi h K=1 co espond-
ing o he uni o m ERM s a egy). To acili a e he exposi ion, we use a loga i hmic scale
o he n-axis. Fo bo h ypes o da a, one can see ha o su icien ly small n, heK-
ma ke s e enue is dec easing in K.Asng ows, he pe o mance o highe Kimp o es
as e han ha o lowe K, and o su icien ly la ge n, heK-ma ke s e enue o e akes
ha wi h any lowe K. This inding can be explained by he bound (K/n)2/3+1/K2
in Theo em 1(ii), which implies ha highe K(mo e disc imina ion) app oxima es he
e enue gene a ed by he FY,X-op imum be e bu incu s a la ge “ a iance.” When he
sample size is small, a lowe Kcan indeed be mo e bene icial.
Figu e 1also sugges s ha , e en i Xcon ains use ul in o ma ion abou Y, he uni-
o m ERM s a egy may be e enue supe io o any K(>1)-ma ke s ERM s a egy when
nis su icien ly small. Recall om Theo em 1 ha he bound (K/n)2/3+1/K2is min-
imized a K=n1/4,whichgi esn−1/2, he op imal a e o con e gence o he e enue
gene a ed by he FY,X-op imal 3PD s a egy. This con e gence a e is slowe han n−2/3,
he op imal a e o con e gence o he e enue gene a ed by he FY-op imal uni o m
p icing s a egy (c . Co olla y 1). The slowe con e gence o he a e-op imal K-ma ke s
ERM s a egy can po en ially domina e he e enue gain om p ice disc imina ion o e
wi hou disc imina ion o small n. Figu e 2illus a es he di e ence in he con e -
gence a es o he uni o m ERM and he K-ma ke s ERM s a egies o hei espec i e
heo e ical benchma ks. In pa icula , we se K=1
5n1/4 o he simula ion s udy and
K=max{1, 2n1/4−7} o he empi ical s udy. As p edic ed by he a e n−1/2in Theo-
em 1and he a e n−2/3in Co olla y 1, he e enue om he uni o m ERM s a egy is
con e ging o he e enue om he FY-op imal uni o m p icing s a egy as e han he
K-ma ke s e enue o he e enue om he FY,X-op imal 3PD s a egy.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License

Theo e ical Economics 20 (2025) Limi a ions o da a-based p ice disc imina ion 323
Figu e 2. Da a-based e enue de iciency unde uni o m and K-ma ke s ERM s a egies (wi h
Kn1/4).
Figu e 3exhibi s he e enue unde he K-ma ke s ERM s a egy o K=1, ,5and
n=2, ,10
5,in hecasewhe eXand Ya e uni o m on [0, 1]and independen o each
o he . No su p isingly, he e is no bene i om p ice disc imina ion o e enue.
Figu e 3. Uni o m and K-ma ke s e enue o hecaseo Xand Yuni o m on [0, 1]and inde-
penden o each o he .
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
324 Xie, Zhu, and Shishkin Theo e ical Economics 20 (2025)
6. Discussions
Recall ha p∗
Dis he ue-dis ibu ion op imal 3PD s a egy and ˆ
pDis he K-ma ke s
ERM s a egy wi h K=(n1/4)gi ing he bes ade-o be ween he “ a iance” and ap-
p oxima ion e o as shown in Theo em 1;p∗
Uis he ue-dis ibu ion op imal uni o m
p icing s a egy and ˆ
pUis he uni o m ERM s a egy. We can decompose he di e ence
be ween he expec ed e enues gene a ed espec i ely om ˆ
pDand ˆ
pUas ollows:
ER(ˆ
pD)−ER(ˆ
pU)=−Rp∗
D−ER(ˆ
pD)
 
A1
+Rp∗
D−Rp∗
U
 
A2
+Rp∗
U−ER(ˆ
pU)
 
A3
.
The i s e m A1=(n−1/2)unde a wo s -case dis ibu ion FY,X∈F,and he hi d
e m A3=O(n−2/3)unde FY, he ma ginal o FY,X.Thesecond e mA2=(1)when
Xcon ains su icien in o ma ion abou he alua ion Y. Then a su icien condi ion
o ˆ
pD o be e enue supe io o ˆ
pUis ha n→∞. In heo y, his claim can be p o ed
wi h he uppe bounds in Sec ion 3and a di e en cons uc ion in he de i a ions o he
lowe bounds. Pa icula ly, his new cons uc ion would i s ind a densi y Y,Xsuch
ha he e enue gene a ed by he co esponding Y,X-op imal 3PD s a egy is well sep-
a a ed om he e enue gene a ed by he op imal uni o m p icing s a egy associa ed
wi h Y, and hen build a la ge enough class o pe u bed e sions o Y,X; inally, we
would bound he sepa a ion be ween he op imal p ices associa ed wi h hese densi-
ies, in a simila ashion as wha is done in Appendix B. In he pape , o make he analy-
sis ac able, we choose he dis ibu ion Y,X∼U[0, 1]wi h Xindependen o Yas he
benchma k dis ibu ion and cons uc i s pe u bed e sions wi h some co ela ion.
A challenging open ques ion is, can he condi ion on nbe weakened o some ini e
numbe and i so, when? To answe his ques ion, we would ha e o de i e he uni e sal
cons an s in ou bounds in meaning ul o ms. Un o una ely, due o he complexi y o
ou p oblem, his exe cise is in easible unde he exis ing echniques om ma hema ical
s a is ics, p obabili y heo y, and in o ma ion heo y.
Ou esul s sugges ha i is mo e bene icial o engage in sample-based uni o m
p icing when Xis independen o Y.11 The undamen al eason lies in he p oo s o
Theo ems 3and 4:unlessn=∞, no s a egies ha exploi {(Yi,Xi),1≤i≤n}a e able
o dis inguish wi h ce ain y he dis ibu ion Y,X∼U[0, 1]wi h Xindependen o Y
om i s pe u bed e sions wi h some co ela ion (see he de ailed cons uc ions in Ap-
pendix B). The cu se o dimensionali y om exploi ing he co a ia e Xmakes he con-
e gence o 3PD s a egies based on {(Yi,Xi),1≤i≤n}slowe han ha o he uni o m
p icing s a egies based on {Yi,1≤i≤n}.
Ou uppe and lowe bounds oge he sugges he ollowing possibili y: e en when
he co a ia e Xcon ains use ul in o ma ion abou he alua ion Y, heK-ma ke s ERM
s a egy can be e enue in e io o he uni o m ERM s a egy in ini e samples, due o
he cu se o dimensionali y and slowe con e gence o he K-ma ke s ERM s a egy o i s
ue-dis ibu ion op imal coun e pa (and hence, a mo e s ingen g ow h equi emen
o he sample size). Indeed, he nume ical e idence in Sec ion 5con i ms his possibili y.
Bu such an implica ion should be aken wi h cau ion in small samples.
11The in o ma ion o independence is unknown o he selle . She can s a is ically es o he indepen-
dence o Yand X om he da a bu any such es s would su e om Type I and Type II e o s.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
Theo e ical Economics 20 (2025) Limi a ions o da a-based p ice disc imina ion 325
Small sample complica ion Gi en he pa e n obse ed in ou nume ical s udies, one
migh conjec u e he ollowing: he e exis s some ¯
n>1 such ha when n< ¯
n, he uni-
o m ERM is always e enue-supe io o he K-ma ke s ERM (wi h K>1). In wha ol-
lows, we explain why his conjec u e may no hold.
Speci ically, in ou language, Babaio e al. (2018) cons uc a dis ibu ion FYsuch
ha he uni o m ERM e enue unde n=2 is s ic ly smalle han he uni o m ERM
e enue unde n=1. This seemingly coun e in ui i e esul highligh s he di icul y o
es ablishing gene al compa a i e esul s wi h e y small sample size. We now a gue ha
his cons uc ion also sheds some ligh on he compa ison o he e enue pe o mance
o he K-ma ke s ERM s a egy wi h K=1 s.K=2in hecaseo n=2.
To make his connec ion, we ake X o be uni o m on [0, 1]and independen o Y,
and assume ha in he case K=2, when one o he ma ke s is emp y, he p ice o his
ma ke is se a he same le el as o he o he ma ke . Then, i K=2 and bo h ma -
ke s a e nonemp y, he e enue in each ma ke equals he 1-ma ke ERM e enue unde
n=1. O he wise, i bo h obse a ions a e in he same ma ke , hen he e enue equals
he 1-ma ke ERM e enue unde n=2. The e o e, he expec ed 2-ma ke s ERM e -
enue wi h n=2 is he a e age o he 1-ma ke ERM e enue unde n=1andn=2, and
hence, s ic ly highe han he 1-ma ke ERM e enue wi h n=2 o a dis ibu ion FY
exhibi ing he p ope y discussed in Babaio e al. (2018).
Mo e o mally, le RK,ndeno e he expec ed e enue o he K-ma ke s ERM s a egy
wi h a sample o size n.Then
R2,2 =Eda an=2∼FY,XRˆ
pD(da a),FY,X
=1
2Eda an=2∼FY,X|I1=∅o I2=∅Rˆ
pD(da a),FY,X
+1
2Eda an=2∼FY,X|I1=∅and I2=∅Rˆ
pD(da a),FY,X
=1
2R1,2 +1
2R1,1.
The e o e, R1,1 >R
1,2 implies R2,2 >R
1,2.
Finally, we add he ca ea ha he cons uc ion in Babaio e al. (2018) is based on
an a omless app oxima ion o he censo ed equal- e enue dis ibu ion FY(y)=1−1/y,
y∈[1, ∞), which has a discon inuous densi y. Howe e , i is s aigh o wa d o e i y
ha he same p ope y holds o he equal- e enue dis ibu ion unca ed a any y>4,
which has a Lipschi z con inuous and di e en iable densi y. Mo eo e , he equal e -
enue dis ibu ion unca ed a yand ansla ed o he le by >1/y (so ha he suppo
is [1− ,y− ]) also has a Lipschi z con inuous and di e en iable densi y, he in e io
op imal p ice (in line wi h ou assump ions), and sa is ies he Babaio e al. (2018)p op-
e y, e.g., o y=4, =1/2.
An open p oblem To conclude, we would like o p opose a challenging open p oblem:
Do he e exis some 3 ≤n<¯
n<∞such ha o any n∈[n,¯
n]and dis ibu ion in F, he
K-ma ke s ERM s a egy ( o any K>1) is always e enue-in e io o he uni o m ERM
s a egy?
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
326 Xie, Zhu, and Shishkin Theo e ical Economics 20 (2025)
Appendix A: P oo s o uppe bounds
To acili a e he p esen a ion, we i s gi e he p oo o Co olla y 1.
P oo o Co olla y 1.Deno eκ≡in p∈[0,1]|R(p)|/2>0. By Taylo expansion, o
any p,
Rp∗
U−R(p)≥κp−p∗
U2.
Deno e ˆ
R(p)≡p(1−ˆ
F(p)). Combining he inequali y abo e wi h he basic inequali y
(i.e., ˆ
R(ˆ
pU)≥ˆ
R(p∗
U)), we ha e
κˆ
pU−p∗
U2≤Rp∗
U−R(ˆ
pU)≤Rp∗
U−ˆ
Rp∗
U−R(ˆ
pU)−ˆ
R(ˆ
pU). (14)
Fo δ∈(0, p∗
U], de ine
Gδ≡y→p1{y≥p}−p∗
U1y≥p∗:p∈p∗
U−δ,p∗
U+δ
and
Gδ(y)≡⎧
⎪
⎪
⎨
⎪
⎪
⎩
0, i y<p
∗
U−δ,
p∗
U,i p∗
U−δ≤y≤p∗
U+δ,
δ,i y>p
∗
U+δ.
Then Gδis an en elope unc ion o he class Gδ.TheL2(P)-no m o Gδis bounded by
GδL2(P)=p∗
U2PY∈p∗
U−δ,p∗
U+δ+δ2PY>p
∗
U+δ1/2≤C√δ.
As we a gue in he p oo o Lemma 6,Gδis a VC-subg aph class, so we ha e
Esup
g∈Gδ

1
n
n

i=1
g(Yi)−Eg(Yi)≤Cδ/n. (15)
We de i e he con e gence a e o ˆ
p−p∗ ia a peeling a gumen . Conside he ollowing
decomposi ion:
Pn1/3ˆ
pU−p∗
U>M=∞

j=M+1
P(n1/3ˆ
pU−p∗
U∈(j−1, j]).
Fo any j≥M+1, we ha e
ˆ
pU−p∗
U∈((j−1)n−1/3,jn−1/3]
=ˆ
pU−p∗
U>(j−1)n−1/3,ˆ
pU−p∗
U≤jn−1/3
⊂Rp∗
U−ˆ
Rp∗
U−R(ˆ
pU)−ˆ
R(ˆ
pU)≥κ(j−1)2n−2/3,ˆ
pU−p∗
U≤jn−1/3
⊂j,n≥κ(j−1)2n−2/3,
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
Theo e ical Economics 20 (2025) Limi a ions o da a-based p ice disc imina ion 333
The in eg al on he las line can be decomposed based on he Kma ke s:
E1
0ˆ
pD(x;da a)−p∗
D(x) X(x)dx
≤
K

k=1IkEˆ
pD(x;da a)−˜
pk+˜
pk−p∗
D(x) X(x)dx
=
K

k=1
E|ˆ
pk−˜
pk|/K +
K

k=1Ik˜
pk−p∗
D(x) X(x)dx
(K/n)1/3+1/K +exp−nc2
1
8K2+logKn−1/4,
whe e he las line ollows om he p oo o Theo em 1.
Fo pa (ii), since p∗
Uis a scala , he wel a e can be simpli ied o
Wp∗
U,FY=p∗
U
0
y Y(y)dy.
Then we ha e
EWˆ
pU(da aY),FY−Wp∗
U,FY=Eˆ
pU(da aY)
p∗
U
y Y(y)dy
≤sup
yy Y(y)Eˆ
pU(da aY)−p∗
U
n−1/3,
whe eweha eusedCo olla y1along wi h he ac ha y Y(y)is nonnega i e and
bounded o y∈[0, 1].
Appendix B: P oo s o lowe bounds
P oo o Theo em 3.Fo Theo em3, we use Lemma 4 o p o e he lowe bound. De-
ine
ωD()≡sup
F1,F2∈Fp∗
D(x0;F1)−p∗
D(x0;F2):H(F1F2)≤.
By Lemma 4,weha e
in
ˇ
pD∈ˇ
D
sup
FY,X∈F
EFY,Xˇ
pD(x0;da a)−p∗
D(x0)≥1
8ωD1/(2√n).
The e o e, we only need o ind a lowe bound o ωD. Based on he explana ion in
Sec ion 4.3.1, we wan o cons uc wo dis ibu ions ha a e ha d o dis inguish bu
hei op imal p ices a e well sepa a ed. We s a by de ining wo pe u ba ion unc ions.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License

334 Xie, Zhu, and Shishkin Theo e ical Economics 20 (2025)
Figu e 4. Pe u ba ion unc ions φYand φX.
Le φYbe de ined as
φY( )≡
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
+1, ∈[−1, 0],
− +1, ∈[0, 2],
−3, ∈[2, 3],
0, o he wise.
(27)
No ice ha φYis Lipschi z con inuous on R.Le φXbe de ined as
φX( )≡⎧
⎪
⎪
⎨
⎪
⎪
⎩
e−(4 −1)2/(1−(4 −1)2), ∈(0, 1/2),
−e−(4 −3)2/(1−(4 −3)2), ∈(1/2, 1),
0, o he wise.
No ice ha φXis in ini ely di e en iable on R. We plo he wo pe u ba ion unc ions
in Figu e 4.
Now we cons uc he wo dis ibu ions. Le δ∈(0, 1/4)be a small numbe ( ha
depends on n) o be speci ied la e . Le abe any numbe in he in e al (0, 4 −2C∗).
De ine he wo condi ional densi y unc ions o Ygi en Xas
1(y|x)≡1,
2(y|x)≡1+aδφYy−1/2
δφXx−x0
δ+1/4. (28)
We le he ma ginal dis ibu ion X(x)o Xbe he uni o m dis ibu ion on [0, 1].No e
ha 1(y|x), 2(y|x), 1(y,x)= 1(y|x) X(x),and 2(y,x)= 2(y|x) X(x)a e nonnega-
i e e e ywhe e, wi h in eg als o e hei espec i e en i e spaces all equaling o 1.
The i s ask is o e i y ha he wo dis ibu ions a e indeed in he class Fκ.Fo
C∗∈(0, 2), he i s dis ibu ion is in Fby Lemma 2and he ac ha Yis independen
o X. Gi en any x∈[0, 1], we can ea he whole e m aφX((x−x0)/δ +1/4)as he
coe icien bin Lemma 3. Then he esul s o Lemma 3applies since |φX|≤1. In pa -
icula , he e enue unc ion a xis wice-di e en iable a.e., he absolu e alue o he
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
Theo e ical Economics 20 (2025) Limi a ions o da a-based p ice disc imina ion 335
second-o de pa ial de i a i e wi h espec o yis bounded, and is also bounded om
below by C∗. The op imal p ice is an in e io solu ion and is in he in e io o a egion
on which he e enue unc ion is wice di e en iable. Las ly, he absolu e alue o he
pa ial de i a i e o 2(y|x)wi h espec o xis bounded. This ensu es ha he quan i y
|∂
∂x FY|X(y|x)+y∂
∂x Y|X(y|x)|is bounded.
Nex , we wan o de i e he Hellinge dis ance be ween he wo join densi ies
1(y,x)=1,
2(y,x)=1+aδφYy−1/2
δφXx−x0
δ+1/4.
Le ( )≡√1+ . I s second-o de de i a e is bounded when | |<1/2; ha is,
sup
| |<1/2( )<C.
We use H o deno e he Hellinge dis ance:
H( 1 2)2≡1
0 1(y)− 2(y)2dy.
The Hellinge dis ance can be bounded as
H2( 1 2)/2=1−1
01
0
aδφYy−1/2
δφXx−x0
δ+1/4dy dx
=1
01
0
(0)−aδφYy−1/2
δφXx−x0
δ+1/4dy dx
≤−a(0)δ1
01
0
φYy−1/2
δφXx−x0
δ+1/4dy dx
+a2Cδ21
01
0
φ2
Yy−1/2
δφ2
Xx−x0
δ+1/4dy dx,
whe e we ha e applied he second-o de Taylo expansion o ob ain he las inequali y.
By he change o a iables u=(y−1/2)/δ and =(x−x0)/δ +1/4, o su icien ly small
δ∈(0, 1/2],
1
01
0
φYy−1/2
δφXx−x0
δ+1/4dy dx =δ21
−1
φY(u)du1
0
φX( )d =0, (29)
and
1
01
0
φ2
Yy−1/2
δφ2
Xx−x0
δ+1/4dy dx =δ21
−1
φ2
Y(u)du1
0
φ2
X( )d ≤Cδ2.
The e o e, he Hellinge dis ance is bounded as
H2( 1 2)δ4.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
336 Xie, Zhu, and Shishkin Theo e ical Economics 20 (2025)
Now we ake δsuch ha δ41/n.No e ha (29)holdswhenδ>0 is small enough such
ha δ∈(0, 1/2],1/4−x0/δ ≤0, and (1−x0)/δ +1/4≥1; ha is, when x0n1/4≥cand
(1−x0)n1/4≥c o posi i e uni e sal cons an s cand c (independen o nand x0).
This ensu es ha H2( 1 2)1/n. Then om Lemma 4, we know ha
in
ˇ
pD∈ˇ
D
sup
FY,X∈F
EFY,Xˇ
pD(x0;da a)−p∗
D(x0)n−1/4,x0∈(0, 1).
Fo bounding he e enue, ecall ha he e enue achie ed a he p ice pand co a ia e
alue x0is (p,x0)=maxpp(1−FY|X(p|x0)). By Lemma 1,weha e
p∗
D(x0)− ˇ
pD(x0;da a)≥C∗
2p∗
D(x0)−ˇ
pD(x0;da a)2.
As a esul , we ha e
in
ˇ
pD∈ˇ
D
sup
FY,X∈F
E p∗
D,x0− ˇ
pD(da a),x0
≥in
ˇ
pD∈ˇ
D
sup
FY,X∈F
EC∗
2p∗
D(x0)−ˇ
pD(x0;da a)2
≥in
ˇ
pD∈ˇ
D
sup
FY,X∈F
C∗
2Ep∗
D(x0)−ˇ
pD(x0;da a)2n−1/2.
This p o es Theo em 3.
P oo o Theo em 4.Top o eTheo em4, we ollow he explana ion in Sec ion 4.3.2
and use he Fano’s inequali y o bound he p obabili y o mis akes in he mul iple clas-
si ica ion p oblem. Be o e sol ing he e enue p oblem, we i s s udy he lowe bound
o he L2-dis ance o p icing unc ions. Fo wo p icing unc ions p1and p2, we de ine
he (unweigh ed) L2-dis ance as
p1−p22≡1
0p1(x)−p2(x)2dx1/2
.
In pa (i), we de ined he pe u ba ion on he Xdimension a a ixed poin x0.Nowwe
wan o de ine a la ge se o pe u bed dis ibu ions. Each o hese dis ibu ions is pe -
u bed in a small in e al on he Xdimension. Le m≥8 be a la ge numbe (depending
on n) ha we speci y la e . Le α∈{0, 1}mbe a ec o o leng h m; ha is,
α≡(α1,,αm),whe eαj∈{0, 1},j=1, ,m.
We cons uc a se o condi ional densi y unc ions indexed by α:
α
Y|X(y|x)≡1+a
m
m

j=1
αjφYm(y−1/2)φXmx −(j−1).
The ma ginal dis ibu ion o Xis aken o be he uni o m dis ibu ion on [0, 1], i.e., X≡
1[0,1]. We deno e he join dis ibu ion by α
Y,X≡ α
Y|X X.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
Theo e ical Economics 20 (2025) Limi a ions o da a-based p ice disc imina ion 337
We b ie ly desc ibe his cons uc ion o he condi ional densi y. The uni in e al
[0, 1]is di ided equally in o msubin e als:
Ij≡(j−1)/m,j/m,j=1, ,m.
Fo x∈Ij,i αj=0, hen he condi ional densi y is 1. I αj=1, hen he condi ional
densi y
α
Y|X(y|x)≡1+a
mφYm(y−1/2)φXmx −(j−1),x∈Ij.
By ea ing 1/m as he scala δin pa (i), we can see ha , o mla ge enough, each α
Y,X
belongs o he se Fκ.
F om he se { α
Y,X:α∈{0, 1}m}, we wan o pick ou a la ge enough subse o dis-
ibu ions whose op imal p ice unc ions a e well sepa a ed. Fo his pu pose, we use
he Gilbe –Va shamo bound (Lemma 2.9, Chap e 2, Tsybako (2009)). The Gilbe –
Va shamo bound s a es ha o m≥8, he e exis s a subse A⊂{0, 1}mwi h ca dinali y
M≡|A|≥2m/8, and he pai wise escaled Hamming dis ance be ween elemen s in his
se is g ea e han 1/8. Tha is,
1
m
m

j=1
1αj=α
j≥1
8, o anyα,α∈A.
Applying he Gilbe –Va shamo bound, we can show ha o α,α∈A, he op imal
p icing unc ions o α
Y,Xand α
Y,Xa e well sepa a ed. Le pαbe he p icing unc ion
associa ed wi h α
Y,X; ha is,
pα(x)≡a gmax
p∈[0,1]
p1−Fα
Y|X(p|x),
whe e Fα
Y|X(y|x)is he co esponding condi ional cumula i e dis ibu ion unc ion.
No e ha α,α∈Adi e in a leas m/8 posi ions. This means ha α
Y|Xand α
Y|Xdi e
in m/8 in e als. Suppose ha Ijis such an in e al, whe e αj=0andα
j=1. We es ic
ou a en ion o a subse o his in e al:
˜
Ij≡1
6m+j−1
m,1
3m+j−1
m⊂Ij.
When x∈˜
Ij,weha e
mx −(j−1)∈[1/6, 1/3]=⇒ φXmx −(j−1)∈φX(0),φX(1/2). (30)
By Lemma 3(whe e b=aφX(mx −(j−1)) >0, δ=1/m), he choice a∈(0, 4 −2κ),and
he ac (30), i we ix x∈˜
Ij, henpα(x)=1/2 while
pα(x)≤1/2−c
mφXmx −(j−1)≤1/2−cφX(1/6)
m,x∈˜
Ij,
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
338 Xie, Zhu, and Shishkin Theo e ical Economics 20 (2025)
whe e c>0 is a uni e sal cons an ha does no depend on n.12 This implies ha
pα(x)−pα(x)1
m,x∈˜
Ij.
The e o e, on he in e al Ij, he sepa a ion be ween pαand pαis lowe bounded as
Ijpα(x)−pα(x)2dx ˜
Ij
1/m2dx =1
6m×1
m21/m3.
By he Gilbe –Va shamo bound, he e a e a leas m/8 such in e als. The e o e, we
can lowe bound he o al sepa a ion by
p1−p22m/8×1/m31/21/m.
Nex , we wan o compu e he KL di e gence be ween α
Y,Xand α
Y,X. No e ha he
e m φX(mx −(j−1)) is nonze o only when x∈Ij. The KL di e gence can he e o e be
ea ed as a sum o min eg als:
KL α
Y,X α
Y,X=1
01
0
α
Y,X(y,x)log α
Y,X
α
Y,X
dy dx =
m

j=1
Ej,
whe e
Ej≡Ij1
01+a
mαjφYm(y−1/2)φXmx −(j−1)
×log
1+a
mαjφYm(y−1/2)φXmx −(j−1)
1+a
mα
jφYm(y−1/2)φXmx −(j−1)dy dx.
No ice ha when αj=αj,Ej=0. The e o e, we only need o conside he j’s whe e
αj=α
j.Deno e1( )=−log(1+ )and 2( )=(1+ )log(1+ ).Thenwecanw i eEj
as
Ej=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
Ij1
0
1a
mφYm(y−1/2)φXmx −(j−1)dy dx,i αj=0, α
j=1,
Ij1
0
2a
mφYm(y−1/2)φXmx −(j−1)dy dx,i αj=1, α
j=0.
By he second-o de Taylo expansion a ze o, we ha e
1( )=− +1
21+ 2 2,
12Fo example, ccan be equal o a/8 acco ding o Lemma 3.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License

Theo e ical Economics 20 (2025) Limi a ions o da a-based p ice disc imina ion 339
o some be ween 0 and .When| |≤1/4,13 we ha e
1( )≤− +C 2,
o some uni e sal cons an C>0. Simila ly, we can show ha
2( )≤ +C 2.
Applying hese inequali ies o Ej,weha e
Ej≤±Ij1
0
a
mφYm(y−1/2)φXmx −(j−1)dy dx
+CIj1
0
a2
m2φ2
Ym(y−1/2)φ2
Xmx −(j−1)dy dx.
Simila o he de i a ion in pa (i), we know ha he i s e m on he RHS is ze o. Fo
he second e m, we can apply change o a iables u=m(y−1/2)and =mx −(j−1)
and ob ain ha
Ij1
0
φ2
Ym(y−1/2)φ2
Xmx −(j−1)dy dx
=1
m21
0
φ2
X( )d 3
−1
φ2
Y(u)du ≤C
m2
o some uni e sal cons an C>0. Pu ing he esul s esul s oge he , we know ha
Ej≤C
m4 o all j. Since he e a e min e als, we can bound he KL di e gence by
KL α
Y,X α
Y,X=
m

j=1
Ej1
m3.
This is he KL dis ance o a single obse a ion. Fo he en i e da a se wi h ni.i.d. ob-
se a ions, he KL di e gence is uppe bounded by Cn/m3.
Las ly, we can summa ize ou esul s in o he Fano inequali y p esen ed in Lemma 5.
We ha e
in
ˇ
pD∈ˇ
D
sup
FY,X∈F
E
ˇ
pD(da a)−p∗
D
2
2≥C1
m21−C2n/m3+log2
log2m/8
≥C1
m21−C2n/m3+log2
C3m.
By choosing m=c0n1/4 o a su icien ly la ge uni e sal cons an c0>0, we can make
he ac o (1−C2n/m3+log2
C3m)s ay abo e, say, 1/2. Then we ha e
in
ˇ
pD∈ˇ
D
sup
FY,X∈F
E
ˇ
pD(da a)−p∗
D
2
21
m2n−1/2.
13La e we show ha mis chosen o be c0n1/4whe e c0>0 is a uni e sal cons an . As a esul , | |≤1/4is
gua an eed as long as c0is su icien ly la ge.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
340 Xie, Zhu, and Shishkin Theo e ical Economics 20 (2025)
So a we ha e de i ed he lowe bound o he L2-dis ance o p icing. Mo ing on o
he e enue p oblem, ecall ha he e enue achie ed a he p ice pand co a ia e alue
xis (p,x)=maxpp(1−FY|X(p|x)). By Lemma 1,weha e
p∗
D,x− ˇ
pD(da a),x≥C∗
2p∗
D(x)−ˇ
pD(x;da a)2.
Since Xis bounded away om ze o, we ha e
in
ˇ
pD∈ˇ
D
sup
FY,X∈F
ERp∗
D−R(ˇ
pD)
=in
ˇ
pD∈ˇ
D
sup
FY,X∈F
E1
0 p∗
D,x− (ˇ
pD,x) X(x)dx
≥in
ˇ
pD∈ˇ
D
sup
FY,X∈F
EC∗
2in
x∈[0,1] X(x)1
0p∗
D(x)−ˇ
pD(x;da a)2dxn−1/2.
P oo o Theo em 5. We use Lemma 4 o p o e he lowe bound o Theo em 5.De-
ine
ωU()≡sup
F1,F2∈FUp∗
U(F1)−p∗
U(F2):H(F1F2)≤.
Then by Lemma 4,weha e
in
ˇ
pU∈ˇ
U
sup
FY∈FU
EFYˇ
pU(da aY)−p∗
U≥1
8ωU1/(2√n).
The e o e, we only need o ind a lowe bound o ωU. The p oo p oceeds in h ee s eps.
In he i s s ep, we cons uc wo dis ibu ions and compu e he sepa a ion be ween
hei op imal p ices. The second s ep bounds he Hellinge dis ance be ween hese wo
dis ibu ions. The hi d s ep summa izes.
S ep 1. We cons uc wo dis ibu ion unc ions. The i s dis ibu ion is he uni o m
dis ibu ion on he uni in e al [0, 1]. We deno e his densi y unc ion as
1(y)=1[0,1](y).
The dis ibu ion unc ion is F1(y)=yon he suppo [0, 1]. The e enue unc ion unde
his dis ibu ion is R1(p)=p(1−p). The op imal p ice is
p1=a gmax
p∈[0,1]
R1(p)=a gmax
p∈[0,1]
p−p2=1/2.
The second dis ibu ion unc ion is a small wis o he uni o m dis ibu ion. We use he
same pe u ba ion unc ion φYde ined in (27).
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
Theo e ical Economics 20 (2025) Limi a ions o da a-based p ice disc imina ion 341
Figu e 5. Densi y unc ions 1and 2.
We apply a small pe u ba ion o he uni o m densi y. Le δ>0beasmallnumbe
( ha depends on n) speci ied la e . Le a∈(0, 4 −2C∗). The o mula o he densi y 2is
gi en by
2(y)≡1+aδφYy−1/2
δ=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
1, i y∈[0, 1/2−δ),
ay +1−a
2+aδ,i y∈[1/2−δ,1/2),
−ay +1+a
2+aδ,i y∈[1/2, 1/2+2δ),
ay +1−a
2−3aδ,i y∈[1/2+2δ,1/2+3δ),
1, i y∈[1/2+3δ,1
].
We compa e he wo densi ies 1and 2in Figu e 5.
Deno e he op imal p ice unde 2by p2. By Lemma 3(ii), we ha e
|p2−p1|≥aδ/8
when δis su icien ly small.
S ep 2. We wan o bound he Hellinge dis ance H(F1F2). De ine he unc ion
( )=√1+ . I s second-o de de i a i e is bounded when | |<1/2; ha is,
sup
| |<1/2( )≤√2
2.
Since 1(y)=1, we ha e
H(F1F2)2/2=1−1
0
aδφYy−1/2
δdy
=1
0
(0)−aδφYy−1/2
δdy.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
342 Xie, Zhu, and Shishkin Theo e ical Economics 20 (2025)
By he second-o de Taylo expansion, we ha e
(0)−aδφYy−1/2
δ
≤−(0)aδφYy−1/2
δ+√2
4a2δ2φ2
Yy−1/2
δ.
By he cons uc ion o φY,weha e
1
0
φYy−1/2
δdy =0.
By hechangeo a iablesu=(y−1/2)/δ,weha e
1
0
φ2
Yy−1/2
δdy =δR
φ2
Y(u)du ≤4δ0
−1
(x+1)2dx =4
3δ.
Combining hese esul s oge he , we ob ain a bound on he Hellinge dis ance
H(F1F2)2≤2√2
3a2δ3.
S ep 3. By se ing δ=c
0(3/8√2)1/3a−2/3n−1/3 o c
0∈(0, 1), we can ensu e ha
H(F1F2)≤1/(2√n). P e iously, we assumed ha aδ ≤1/2 o he second-o de Taylo
expansion. This is ue i c
0is chosen o be su icien ly small. In his case, he sepa a ion
be ween p1and p2is lowe bounded as below:
|p1−p2|≥aδ/8=c
0
163
√21/3a
n1/3
.
By Lemma 4,weha e
in
ˇ
pU∈ˇ
U
sup
FY∈FU
Eˇ
pU(da aY)−p∗
U≥c
0
163
√21/3a
n1/3
.
Las ly,wewan olowe bound he e enue.ByLemma1,weha e
RU
nFU=in
ˇ
pU∈ˇ
U
sup
FY∈FU
ERˇ
pU(da aY),FY−Rp∗
U,FY
≥in
ˇ
pU∈ˇ
U
sup
FY∈FU
EC∗
2ˇ
pU(da aY)−p∗
U2
≥in
ˇ
pU∈ˇ
U
sup
FY∈FU
C∗
2Eˇ
pU(da aY)−p∗
U2
1
n2/3
.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
Theo e ical Economics 20 (2025) Limi a ions o da a-based p ice disc imina ion 349
Re e ences
Alexande , Kenne h S. (1987), “Ra es o g ow h and sample moduli o weigh ed em-
pi ical p ocesses indexed by se s.” P obabili y Theo y and Rela ed Fields, 75, 379–423.
[0313]
Allouah, Amine, Ach a Bahamou, and Oma Besbes (2022), “P icing wi h samples.” Op-
e a ions Resea ch, 70, 1088–1104. [0307]
Allouah, Amine, Ach a Bahamou, and Oma Besbes (2023), “Op imal p icing wi h a sin-
gle poin .” Managemen Science.[0307]
Babaio , Moshe, Yannai A. Goncza owski, Yishay Mansou , and Shay Mo an (2018), “A e
wo (samples) eally be e han one?” In P oceedings o he 2018 ACM Con e ence on
Economics and Compu a ion, 175. [0305,0307,0325]
Balcan, Ma ia-Flo ina, Ami Daniely, Ru a Meh a, Ru h U ne , and Vijay V. Vazi ani
(2014), “Lea ning economic pa ame e s om e ealed p e e ences.” In Web and In e ne
Economics: 10 h In e na ional Con e ence, WINE 2014, Beijing, China, Decembe 14–17,
2014. P oceedings 10, 338–353, Sp inge . [0306]
Baliga, Sandeep and Rakesh Voh a (2003), “Ma ke esea ch and ma ke design.” Ad-
ances in Theo e ical Economics,3.[0307]
Basu, Pa hik i and Fede ico Echenique (2020), “On he alsi iabili y and lea nabili y o
decision heo ies.” Theo e ical Economics, 15, 1279–1305. [0306]
Beigman, Eyal and Rakesh Voh a (2006), “Lea ning om e ealed p e e ence.” In P o-
ceedings o he 7 h ACM Con e ence on Elec onic Comme ce, 36–42. [0306]
Be gemann, Di k and Ka l H. Schlag (2008), “P icing wi hou p io s.” Jou nal o he Eu-
opean Economic Associa ion, 6, 560–569. [0307]
Be gemann, Di k and Ka l H. Schlag (2011), “Robus monopoly p icing.” Jou nal o Eco-
nomic Theo y, 146, 2527–2543. [0307]
Bousque , Oli ie (2003), “Concen a ion inequali ies o sub-addi i e unc ions using
he en opy me hod.” In S ochas ic Inequali ies and Applica ions, 213–247, Sp inge .
[0327]
Ca oll, Gab iel (2019), “Robus ness in mechanism design and con ac ing.” Annual Re-
iew o Economics, 11, 139–166. [0307]
Chambe s, Ch is ophe P., Fede ico Echenique, and Nicolas S. Lambe (2021), “Reco -
e ing p e e ences om ini e da a.” Econome ica, 89, 1633–1664. [0306]
Chambe s, Ch is ophe P., Fede ico Echenique, and Nicolas S. Lambe (2023), “Reco -
e ing u ili y.” a Xi p ep in a Xi :2301.11492.[0306]
Cole, Richa d and Tim Roughga den (2014), “The sample complexi y o e enue maxi-
miza ion.” In P oceedings o he Fo y-Six h Annual ACM Symposium on Theo y o Com-
pu ing, 243–252. [0307]
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License

350 Xie, Zhu, and Shishkin Theo e ical Economics 20 (2025)
Co e , Thomas M. and Joy A. Thomas (2005), Elemen s o In o ma ion Theo y.JohnWiley
& Sons, L d. [0320]
C éme , Jacques and Richa d P. McLean (1988), “Full ex ac ion o he su plus in
Bayesian and dominan s a egy auc ions.” Econome ica: Jou nal o he Econome ic
Socie y, 1247–1257. [0304]
De anu , Nikhil R., Zhiyi Huang, and Ch is os-Alexand os Psomas (2016), “The sample
complexi y o auc ions wi h side in o ma ion.” In P oceedings o he Fo y-Eigh h Annual
ACM Symposium on Theo y o Compu ing, 426–439. [0308]
Dhangwa no ai, Pee apong, Tim Roughga den, and Qiqi Yan (2015), “Re enue maxi-
miza ion wi h a single sample.” Games and Economic Beha io , 91, 318–333. [0307]
Folland, Ge ald B. (1999), Real Analysis: Mode n Techniques and Thei Applica ions,sec-
ond edi ion, olume 40. John Wiley & Sons, New Yo k, NY. [0343]
Fu, Hu, Nima Haghpanah, Jason Ha line, and Robe Kleinbe g (2021), “Full su plus
ex ac ion om samples.” Jou nal o Economic Theo y, 193, 105230. [0307]
Fu, Hu, Nicole Immo lica, B endan Lucie , and Philipp S ack (2015), “Randomiza ion
bea s second p ice as a p io -independen auc ion.” In P oceedings o he Six een h ACM
Con e ence on Economics and Compu a ion, 323. [0307]
Giné, E a is and Richa d Nickl (2015), Ma hema ical Founda ions o In ini e-
Dimensional S a is ical Models. Camb idge Se ies in S a is ical and P obabilis ic Ma he-
ma ics. Camb idge Uni e si y P ess. [0348]
Goldbe g, V. And ew, Jason D. Ha line, Anna R. Ka lin, Michael Saks, and And ew W igh
(2006), “Compe i i e auc ions.” Games and Economic Beha io , 55, 242–269. [0307]
Gonçal es, Dua e and B uno A. Fu ado (2024), “S a is ical mechanism design: Robus
p icing, es ima ion, and in e ence.” a Xi p ep in a Xi :2405.17178.[0307]
Guo, Chenghao, Zhiyi Huang, and Xinzhi Zhang (2019), “Se ling he sample complex-
i y o single-pa ame e e enue maximiza ion.” In P oceedings o he 51s Annual ACM
SIGACT Symposium on Theo y o Compu ing, 662–673. [0307]
Ha mann, Wesley, Ha ikesh S. Nai , and S idha Na ayanan (2011), “Iden i ying causal
ma ke ing mix e ec s using a eg ession discon inui y design.” Ma ke ing Science, 30,
1079–1097. [0304]
Hi ano, Keisuke and Jack R. Po e (2003), “Asymp o ic e iciency in pa ame ic s uc-
u al models wi h pa ame e -dependen suppo .” Econome ica, 71, 1307–1338. [0308]
Huang, Zhiyi, Yishay Mansou , and Tim Roughga den (2018), “Making he mos o you
samples.” SIAM Jou nal on Compu ing, 47, 651–674. [0307,0316]
Jank, Wol gang and Gali Shmueli (2010), Modeling Online Auc ions. John Wiley & Sons.
[0321]
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License
Theo e ical Economics 20 (2025) Limi a ions o da a-based p ice disc imina ion 351
Kolmogo o , And ei Nikolae ich, and Vladimi Mikhailo ich Tikhomi o (1959), “ε-
en opy and ε-capaci y o se s in unc ion spaces.” Uspekhi Ma ema icheskikh Nauk,
14, 3–86. [0306,0317]
Mye son, Roge B. (1981), “Op imal auc ion design.” Ma hema ics o Ope a ions Re-
sea ch, 6, 58–73. [0310]
Segal, Ilya (2003), “Op imal p icing mechanisms wi h unknown demand.” Ame ican Eco-
nomic Re iew, 93, 509–529. [0307]
Talag and, Michel (1996), “New concen a ion inequali ies in p oduc spaces.” In en-
iones Ma hema icae, 126, 505–563. [0327]
Tsybako , Alexand e B. (2009), In oduc ion o Nonpa ame ic Es ima ion, i s edi ion,
Sp inge Se ies in S a is ics. Sp inge , New Yo k, NY. [0337]
an de Gee , Sa a A. (2000), Empi ical P ocesses in M-Es ima ion, olume 6. Camb idge
Uni e si y P ess. [0313]
an de Vaa , Aad W. and Jon A. Wellne (1996), Weak Con e gence and Empi ical P o-
cesses. Sp inge , New Yo k, NY. [0313,0348]
Wainw igh , Ma in J. (2019), High-Dimensional S a is ics: A Non-Asymp o ic Viewpoin ,
olume 48. Camb idge Uni e si y P ess. [0306,0320,0347]
Zhang, Han ui and Vincen Coni ze (2020), “Lea ning he alua ions o a k-demand
agen .” In In e na ional Con e ence on Machine Lea ning, PMLR, 11066–11075. [0306]
Co-edi o Rakesh Voh a handled his manusc ip .
Manusc ip ecei ed 18 Oc obe , 2023; inal e sion accep ed 12 July, 2024; a ailable online 30
July, 2024.
15557561, 2025, 1, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.3982/TE5916 by ZBW Kiel - Hambu g (Ge man Na ional Lib a y o Economics), Wiley Online Lib a y on [04/07/2025]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License