Du mann, Julius; Obe lechne , Ma hias; Bichle , Ma in
A icle — Published Ve sion
Algo i hmic P icing and Algo i hmic Collusion
Business & In o ma ion Sys ems Enginee ing
Sugges ed Ci a ion: Du mann, Julius; Obe lechne , Ma hias; Bichle , Ma in (2025) : Algo i hmic
P icing and Algo i hmic Collusion, Business & In o ma ion Sys ems Enginee ing, ISSN 1867-0202,
Sp inge Fachmedien Wiesbaden, Wiesbaden, Vol. 67, Iss. 6, pp. 971-979,
h ps://doi.o g/10.1007/s12599-025-00965-z
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/333372
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CATCHWORD
Algo i hmic P icing and Algo i hmic Collusion
Ma in Bichle •Julius Du mann •Ma hias Obe lechne
Recei ed: 12 No embe 2024 / Accep ed: 22 Ap il 2025 / Published online: 29 Sep embe 2025
ÓThe Au ho (s) 2025
Keywo ds Algo i hmic collusion Online lea ning Game
heo y
1 In oduc ion
Wo ldwide, companies a e inc easingly using algo i hms
and a i icial in elligence (AI) in o de o powe hei
ope a ions, om p oduc de elopmen o manu ac u ing
and ma ke ing, and p oduc p icing is no excep ion. This
new ype o in o ma ion sys em based on lea ning algo-
i hms has d awn conside able a en ion in an e o o
unde s and i s socie al e ec s (Lysyako and Viswana han
2023; Lu and Zhang 2024;Fu
¨gene e al. 2021). An
impo an ques ion wi hin his b oade con ex is how
lea ning algo i hms a e used o p icing (B ackmann e al.
2024). Algo i hmic p icing is a p ac ice whe e so wa e
agen s au oma ically de e mine p ices o i ems o sale, in
o de o maximize he selle ’s p o i s. This p ac ice is
inc easingly common in online e ail ma ke s. Chen e al.
(2016) es ima ed ha , by 2015, algo i hms we e used o se
p ices o oughly one- hi d o he op 1,600 p oduc s on
Amazon. By 2018, he a e age p oduc p ice on Amazon
epo edly changed e e y en minu es ı
ˆn o de o adap o
ma ke condi ions.
1
Since hen, an indus y has de eloped
a ound au oma ed p icing so wa e.
The p ices de e mined by he di e en i ms depend on
each o he ’s ac ions, and he o e all en i onmen in which
hey ope a e is one o an oligopolis ic compe i ion. Eco-
nomic heo y has long aimed o p edic he ou come o
such compe i i e si ua ions. Models o oligopolis ic com-
pe i ion assume some knowledge abou he demand o
cus ome s, such as a eadily a ailable demand unc ion,
and hen hey de e mine a Nash equilib ium p ice. Fo
example, in he celeb a ed Be and compe i ion model,
companies p oducing iden ical (homogeneous) p oduc s
simul aneously choose hei p icing s a egies based on a
gi en demand unc ion ha akes he p ices o he com-
pe i o s in o accoun (Be and 1883). In his model, he
compe i o s play hei equilib ium s a egy om he s a .
Typically, new selle s ha e limi ed in o ma ion abou
compe i o s’ cos s o cus ome demand a di e en p ices,
and hey mus lea n o e ime which p ices op imize p o i .
Addi ionally, changes in demand and supply o e ime
equi e ecalcula ing he equilib ium s a egy, which
emphasizes he need o an adap i e and lea ning p icing
so wa e agen . F om he pe spec i e o an indi idual
selle , algo i hmic p icing aims o sol e an online lea ning
p oblem (Shale -Shwa z 2011), whe e he selle ’s ac ions
a e he p ices hey se , and he objec i e is o maximize
p o i . Bu wha is he ou come o ma ke s wi h such
lea ning agen s? Can we assume ha hey con e ge o an
equilib ium p ice?
Recen expe imen al esea ch showed ha lea ning
algo i hms can lead o p ices highe han he Nash equi-
lib ium in he s a ic oligopolis ic p icing game (Wal man
and Kaymak 2008; Cal ano e al. 2019; Abada and Lambin
Accep ed a e one e ision by Susanne S ah inge .
M. Bichle (&)J. Du mann M. Obe lechne
School o Compu a ion, In o ma ion, and Technology, Technical
Uni e si y o Munich, Munich, Ge many
e-mail: [email p o ec ed]
1
h ps://www.businessinside .com/amazon-p ice-changes-2018-8.
123
Bus In Sys Eng 67(6):971–979 (2025)
h ps://doi.o g/10.1007/s12599-025-00965-z
2023; Klein 2021; Abada e al. 2024b; B own and MacKay
2023). This phenomenon is e e ed o as algo i hmic
collusion. Explici collusion e e s o an i-compe i i e
conduc s ha a e main ained wi h explici ag eemen s.
Fi ms in e ac di ec ly and ag ee on he op imal le el o
p ice o ou pu (OECD 2017). In con as , ‘‘ aci collusion
e e s o o ms o an i-compe i i e co-o dina ion which can
be achie ed wi hou any need o an explici ag eemen ,
bu which compe i o s a e able o main ain by ecognizing
hei mu ual in e dependence. In a aci ly collusi e con ex ,
he non-compe i i e ou come is achie ed by each pa ici-
pan deciding i s own p o i -maximizing s a egy indepen-
den ly o i s compe i o s.’’ (OECD 2017)Algo i hmic
collusion is a o m o aci collusion in which lea ning
algo i hms p oduce sup a-compe i i e ou comes ha di e
om he Nash equilib ium o he s a ic game- heo e ical
model o compe i ion wi hou being p og ammed o do so.
Simila de ini ions a e p o ided by den Boe (2023) and
Abada e al. (2024a).
The e a e also cases sugges ing ha algo i hmic collu-
sion happens in eal-wo ld ma ke s. Fo example, Assad
e al. (2024) showed ha ma gins inc eased by 28% in
local duopoly e ail gasoline ma ke s in Ge many a e bo h
i ms adop ed algo i hmic p icing so wa e, while he e was
no p ice change in local monopolies. The pape inds
p icing algo i hms can lea n aci ly collusi e p icing
s a egies which a e legal in mos ju isdic ions wi hou
explici communica ion. An in es iga ion in he UK
e ealed ha online pos e e aile s we e using simple
p icing algo i hms o coo dina e hei p ices on Amazon as
pa o a ho izon al ca el.
2
The phenomenon has aised
conce ns among egula o s as i may educe consume
wel a e. In ecen yea s, se e al compe i ion au ho i ies,
including F ance, Ge many, Denma k, Japan, No way, and
Sweden, ha e published policy pape s conside ing he
ela ionship be ween algo i hms and compe i ion (OECD
2023).
Howe e , he magni ude o he h ea posed by algo-
i hmic collusion in o he ma ke s om au onomous, sel -
lea ning algo i hms is s ill dispu ed in academic li e a u e.
Ul ima ely, algo i hmic collusion ouches on a deep and
non- i ial ques ion, ha o lea ning in games. Unde
which condi ions do lea ning algo i hms con e ge o an
equilib ium in epea ed play, and when is his no he case?
When do we see algo i hmic collusion, ine icien p ice
cycling, o e en chao ic p ice dynamics as a esul o
au oma ed p icing decisions? Cu en ly, he e is no com-
p ehensi e heo y ha would p o ide answe s o hese
ques ions. Up o now, he e a e only e y ew a icles in
Business & In o ma ion Sys ems Enginee ing ou le s on
algo i hmic collusion (Kang e al. 2022; Douglas e al.
2024; Deng e al. 2024; Bichle e al. 2023). The opic is
cen al o he wel a e o elec onic ma ke s oday and i
d aws on lea ning algo i hms om compu e science as
well as game- heo e ical models as hey ha e been de el-
oped in economics.
The aim o his ca chwo d a icle is o discuss algo-
i hmic collusion and make he basic p oblem and he
ela ed ques ions accessible o a b oade communi y. This
will enable us o iden i y a enues o u u e esea ch in
Business & In o ma ion Sys ems Enginee ing.
2 Oligopoly P icing
P icing in online e ail ma ke s is an en i onmen ha can
be modeled as a Be and compe i ion (Be and 1883). In
his model, i ms compe e by se ing p ices o a homo-
geneous p oduc . The demand o he p oduc depends on
he p ices se by all i ms. The i ms’ objec i e is o
maximize hei p o i , which is he e enue om selling he
p oduc minus he cos o p oducing i . The e enue
depends on he p ice se by he i m and he demand o he
p oduc . The demand is a unc ion o he p ices se by all
i ms. The cos o p oducing he p oduc is ypically
assumed o be linea in he quan i y p oduced.
Mo e o mally, he Be and compe i ion can be
desc ibed as a no mal- o m game, whe e nplaye s (o
i ms) make decisions a he same ime. Each playe i=1,
2,..., n has a se o possible ac ions ai [Ai o choose om –
like di e en p ices hey could se o hei p oduc s. Each
playe ’s goal is o maximize hei payo hey ge based on
e e yones ac ions, which is gi en by hei indi idual
payo unc ion ui:A199An ?R. No mal- o m
games a e used o model en i onmen s whe e mul iple
agen s in e ac and in luence each o he s’ ou comes. O he
examples include, o ins ance, he Cou no oligopoly,
pla o m compe i ion, auc ions, and con es s. Mos o he
li e a u e on algo i hmic collusion is based on he classic
Be and compe i ion model.
In a Be and compe i ion, he i ms’ ac ion ai [Ai is
he p ice o a good i wan s o p oduce. All.
i ms p oduce he same good, and hey all ake ac ion
simul aneously. Fi ms compe e o demand wi h hei p i-
ces and a ec each o he s’ e enues. The demand di(ai,
a-i) depends on all ac ions, hei own ac ions ai and he
o he s’ ac ions a-i, and is dec easing in hei own p ice.
Assuming ha he cos unc ions a e linea in he demand,
he payo (o u ili y) unc ion o i m ican be desc ibed
by.
uiðai;aiÞ¼diðaÞðaieiÞð1Þ
2
h ps://www.go .uk/cma-cases/online-sales-o -disc e iona y-consu
me -p oduc s.
123
972 M. Bichle e al.: Algo i hmic P icing and Algo i hmic Collusion, Bus In Sys Eng 67(6):971–979 (2025)
This model allows o a ious assump ions abou con-
sume demand (e.g., all-o -no hing o logi ), and he
equilib ium p ices in hese scena ios ha e been analyzed.
In he s anda d case, wi h all-o -no hing demand, he i m
wi h he lowes p ice ge s all he demand. I mul iple i ms
o e he lowes p ice, he demand is sha ed equally. This
leads o a demand unc ion gi en by
diðai;aiÞ¼
D
nmin
ð1aiÞi i 2a g minj2Naj;
0 else
(
whe e D[0 is he maximum o al demand, Nis he se o
ma ke pa icipan s, and nmin: =|a gmin j[Naj| is he
numbe o i ms wi h he lowes p ice.
Ano he demand model used in a widely ci e a icle by
Cal ano e al. (2020), is he mul inomial o logi demand.
In his se ing, he demand is spli be ween he nagen s/-
goods and some ou side good (indexed wi h 0) acco ding o
diðai;aiÞ¼
exp aiai
l
exp a0
l
þPn
j¼1exp ajaj
l
The pa ame e s ai[0 cap u e di e en p oduc quali y
indices and l[0 models a p oduc di e en ia ion be ween
he goods, i.e., i l?0, he goods a e pe ec subs i u es.
In a s anda d Be and model wi h homogeneous p od-
uc s and symme ic i ms, p ices equal ma ginal cos s in
equilib ium. Howe e , his changes wi h asymme ies o
p oduc di e en ia ion. Wi h asymme ic cos s be ween
i ms, he Nash equilib ium ypically in ol es he low-cos
i m p icing a o jus below he ma ginal cos o he high-
cos i m. Wi h p oduc di e en ia ion (as in logi demand),
equilib ium p ices a e ypically abo e ma ginal cos s due
o i ms ha ing some ma ke powe . These equilib ium
p ices a e below a monopoly p ice because he compe i ion
d i es down p o i s.
No e ha he s a ic model o Be and compe i ion has
been ex ended o epea ed games. A epea ed game consis s
o a numbe o epe i ions o some base game (called a
s age game). In such dynamic se ings, a p ice abo e he
Nash equilib ium o he s age game can be an equilib ium
o his epea ed game depending on he s a egy used by
each i m and o he exogenous ac o s (Maskin and Ti ole
1988). Ac ually, he Folk Theo em in epea ed game heo y
shows ha i playe s a e su icien ly pa ien , a wide ange
o payo s can be sus ained as Nash equilib ia, including
ou comes ha a e be e o each playe han hei min–
max (o punishmen ) payo (Maschle e al. 2020). The
li e a u e on algo i hmic collusion akes he Nash equilib-
ium o he s age game o such epea ed games as a
baseline o de e mine algo i hmic collusion (Abada e al.
2024a).
Much o he discussion a ound algo i hmic collusion
elies on simula ions o lea ning algo i hms in epea ed
Be and compe i ion models wi h ixed selle s and speci ic
demand unc ions. Al hough eal-wo ld scena ios may be
mo e complex, his abs ac ion allows o an analysis o he
model and he compu a ion o i s s a ic Nash equilib ium.
This knowledge enables compa a i e s a ics based on cos
and demand knowledge, which is ypically una ailable in
empi ical s udies. I algo i hmic collusion does no a ise in
a epea ed Be and p icing game, i may be e en less
likely in mo e complex scena ios wi h luc ua ing demand
and supply. I i does a ise, i is a s ong indica o ha he
phenomenon is no limi ed o he speci ic model bu may
also occu in mo e complex se ings.
3 Lea ning Agen s in Agen ic Ma ke s
In wha ollows, we in oduce impo an online lea ning
algo i hms, he eme ging li e a u e on algo i hmic collu-
sion, and he key esul s om he li e a u e on lea ning in
games ha a e ela ed o i . These li e a u e s eams a e
impo an o he s udy o algo i hmic collusion and in e -
ela ed.
3.1 Online Lea ning
A majo challenge in de eloping algo i hmic p icing agen s
is deciding whe he o ocus on sho - e m p o i (by
exploi ing a known high-yield p ice) o on explo ing
al e na i e p ices ha may lead o be e long- e m ou -
comes. Online lea ning algo i hms a e designed o balance
his explo a ion–exploi a ion ade-o and can handle la ge
p oduc po olios e ec i ely (Bubeck 2011). On online
pla o ms, hese algo i hms ope a e wi h bandi eedback,
meaning ha a e se ing a p ice, a selle obse es he
p o i associa ed wi h ha speci ic p ice. The mul i-a med
bandi model is especially ele an o algo i hmic p icing, a
connec ion ecognized ea ly on. Bandi algo i hms we e
p oposed o p icing as a back as Ro hschild (1974), well
be o e digi al ma ke places eme ged. Today, mul i-a med
bandi algo i hms o p icing a e widely s udied in acade-
mia (T o o e al. 2015; den Boe 2015; Baue and Jannach
2018;
Muelle e al. 2019; El eedy e al. 2021; Taywade e al.
2023;Qu2024; Kasa and Rajan 2021), and p ac i ione s
also p o ide nume ous esou ces on implemen ing hese
algo i hms.
3
3
h ps:// owa dsda ascience.com/dynamic-p icing-wi h-mul i-
a med-bandi -lea ning-by-doing,h ps://www.g iddynamics.com/
blog/dynamic-p icing-algo i hms.
123
M. Bichle e al.: Algo i hmic P icing and Algo i hmic Collusion, Bus In Sys Eng 67(6):971–979 (2025) 973
Online op imiza ion and lea ning algo i hms a e p ime
candida es o p icing algo i hms on online pla o ms
(Muelle e al. 2019; El eedy e al. 2021; Taywade e al.
2023;Qu2024). Online op imiza ion is conce ned wi h
making sequen ial decisions in an unknown en i onmen
wi h he goal o op imizing a pe o mance me ic o e
ime. Le us b ie ly in oduce he basic model o online
lea ning in he con ex o algo i hmic p icing in i s mos
abs ac o m (see Algo i hm 1). A each s age =1,2,...,
he agen chooses an ac ion a [A om some ac ion se
acco ding o some s a egy s [Sand ge s a payo u (a ).
In algo i hmic p icing, his ac ion would be he p ice o an
agen . The agen would le e age he in o ma ion abou he
u ili y, i.e., eedback, she ge s in o de o upda e his ac ions
o p ices.
Algo i hm 1 Online Lea ning
Algo i hms a e widely analyzed in wo models, he
ad e sa ial model and he s ochas ic model. In bo h mod-
els, he objec i e goal is o maximize he expec ed ewa d
by selec ing he bes ac ion(s). In he s ochas ic se ing, we
assume ha he ewa ds a e d awn independen ly and
iden ically om some unde lying dis ibu ion ha does no
change o e ime. In he ad e sa ial model, he inpu can
be chosen by an ad e sa y ha can eac o he agen ’s pas
decisions and i s algo i hm. The e a e di e en ypes o
eedback a ailable o he agen s in online op imiza ion
algo i hms such as bandi o g adien eedback.
Real-wo ld implemen a ions o lea ning p icing agen s
a e bes modeled wi h bandi eedback. This means ha he
eedback consis s o a poin wise e alua ion o he payo
unc ion a he chosen ac ion: =ˆu (a ). Fo example,
a e se ing a p ice in an oligopoly compe i ion, an agen
obse es a speci ic p o i in he nex pe iod. They o en
ha e no o only incomple e in o ma ion abou he demand
model o he s a egies used by all o he selle s. This is
pa icula ly ue on la ge online e ail pla o ms whe e
he e a e many subs i u es o a good.
In he s ochas ic o ad e sa ial model, one can analyze
he pe o mance o an algo i hm. The key cha ace is ic in
his li e a u e is eg e , i.e. he di e ence be ween he
cumula i e payo i he algo i hm played he bes ixed
p ice in hindsigh and he cumula i e payo o he lea ning
algo i hm. Fo some online lea ning algo i hms he eg e
anishes o e ime. These algo i hms a e also e e ed o as
no eg e algo i hms. As an example, Exponen ial Weigh s
(o he Exp3 a ian ) is a well-known online lea ning
algo i hm ha uses bandi eedback and upda es he
weigh s associa ed wi h each ac ion based on he cumula-
i e ewa d obse ed o ha ac ion. The algo i hm hen
chooses ac ions wi h p obabili ies p opo ional o hei
weigh s. Exp3 is a no eg e algo i hm in he ad e sa ial
model.
The e a e also academic a icles using (deep) ein-
o cemen lea ning algo i hms o algo i hmic p icing
(Rana and Oli ei a 2014; Kas ius and Schlosse 2022;
Deng e al. 2024). Di e en om he online lea ning
algo i hms discussed so a , ein o cemen lea ning algo-
i hms allow he agen o ake in o accoun he s a e o a
sys em. This s a e could be his o ical p ices, he day o he
week, o o he a iables ha a e po en ially ele an . Wi h
an inc easing s a e and ac ion space, lea ning equi es a lo
o aining ounds and doesn’ scale well. In e es ingly, he
mos p e alen ein o cemen lea ning algo i hm in he
algo i hmic collusion li e a u e is Q-lea ning. Howe e , in
ela ed a icles, he s a e space is limi ed o he cu en
p ice, le ing he algo i hm esemble an online lea ning
algo i hm. Ye , Lambin (2024) shows ha algo i hmic
collusion can a ise wi h Q-lea ning e en in a s a eless
e sion o Q-lea ning.
3.2 Algo i hmic Collusion
While he li e a u e on online lea ning gi es us me hods o
ind op imal s a egies in s ochas ic o e en ad e sa ial
se ings, i does no cap u e he in e ac ion o mul iple
agen s using such algo i hms (see Fig. 1). The li e a u e on
algo i hmic collusion a emp s o ill his gap and has
a ac ed conside able a en ion om he esea ch com-
muni y and policymake s.
In he classical online lea ning se ing, a single agen
selec s ac ions and obse es s ochas ic (o ad e sa ial)
payo s. By con as , algo i hmic collusion s udies he
ou comes when mul iple agen s in e ac using online
Fig. 1 Lea ning agen s in di e en con ex s
123
974 M. Bichle e al.: Algo i hmic P icing and Algo i hmic Collusion, Bus In Sys Eng 67(6):971–979 (2025)
lea ning algo i hms. Unde s anding he esul s o hese
mul i-agen lea ning p ocesses equi es conside a ion o
he algo i hm, bu also o he s uc u e and p ope ies o he
unde lying game, which is analyzed in equilib ium
lea ning.
Mos o his li e a u e analyzes speci ic algo i hms such
as Q-lea ning o speci ic model a ia ions, i.e., Be and
oligopolies wi h s anda d all-o -no hing demand, linea , o
logi demand. Cal ano e al. (2020) analyzed a Be and
compe i ion wi h logi demand and cons an ma ginal cos .
They ound ha when all agen s employ Q-lea ning, he
compe i ion o hese agen s can lead o sup a-compe i i e
p ices highe han he Nash equilib ium. A ela ed
sequen ial mo e p icing duopoly en i onmen wi h linea
demand (ins ead o he simul aneous mo e Be and model
in Cal ano e al. (2020)) was analyzed by Klein (2021),
who also ound collusion wi h Q-lea ning agen s. Aske
e al. (2022) de ec ed in hei expe imen s on Be and
compe i ion wi h s anda d (all-o -no hing) demand ha
collusion depends on speci ics o he Q-lea ning algo i hm
(e.g., synch onous s. asynch onous upda ing).
In con as , Abada e al. (2024b) analyzed Q-lea ning in
Be and oligopolies and showed ha Q-lea ning algo-
i hms wi h su icien ly la ge e-g eedy explo a ion exhibi
no collusion. den Boe e al. (2022) p o ided a de ailed
analysis o he inne wo kings o Q-lea ning and a gue ha
Q-lea ning would no lead o collusion easily. In addi ion,
Eschenbaum e al. (2022) c i icized he claim ha algo-
i hms can be ained o line o success ully collude online
in di e en ma ke en i onmen s. The au ho s ound ha
collusion b eaks down when collusi e ein o cemen
lea ning policies a e ex apola ed om a aining en i-
onmen o he ma ke . While mos o his expe imen al
li e a u e on algo i hmic collusion is based on Q-lea ning,
he e is li le e idence ha his algo i hm is pa icula ly
impo an o widesp ead o algo i hmic p icing. Mo e
ecen ly, Hansen e al. (2021) analyzed he p ice le els ha
a ise in a duopoly se ing whe e agen s based on he UCB
(‘‘Uppe Con idence Bound’’) algo i hm de e mine p ices.
They an a se ies o expe imen s whe e a a ian o sym-
me ic UCB algo i hms in e ac simul aneously in a Be -
and economy compe i ion wi h linea demand unc ions.
The agen s obse ed a pe u bed es ima e o hei e enues
which a e a esul o hei p ices and he co esponding
demand. Hansen e al. (2021) ound ha some imes agen s
explo e p ices in a co ela ed manne , gi ing ise o sup a-
compe i i e ou comes.
The li e a u e on algo i hmic collusion in In o ma ion
Sys ems is s ill sca ce. Kang e al. (2022) and Douglas
e al. (2024) analyze he possibili y o collusion in a
epea ed P isonne ’s Dilemma a. (Deng e al. 2024) discuss
deep ein o cemen lea ning in a epea ed Be and
compe i ion, while Bichle e al. (2024) analyze he phe-
nomenon in he con ex o display ad e ising auc ions.
Online lea ning algo i hms add ess p oblems whe e
agen s op imize agains an unknown and independen
s ochas ic p ocess. Howe e , game- heo e ical p oblems
such as he Be and compe i ion di e because each
playe ’s ac ions impac he objec i es o o he s. In games,
he Nash equilib ium (NE) ep esen s a si ua ion whe e no
agen has an incen i e o unila e ally de ia e. No e ha his
is di e en om he ad e sa ial se ing in op imiza ion
discussed ea lie , because each agen aims o maximize his
payo . I agen s a e no in equilib ium, hen indi idual
agen s ha e an incen i e o de ia e om he cu en
s a egy p o ile, and he ou come is no s able. Indepen-
den ly, one migh ask i he equilib ium eached maximizes
wel a e.
3.3 Lea ning in Games
Al hough, he e m algo i hmic collusion is ela i ely new,
he opic is ela ed o a long s anding s eam o li e a u e in
game heo y. Ac ually, he ques ion i lea ning agen s
con e ge o a Nash equilib ium in epea ed play is as old as
he concep o he Nash equilib ium i sel (B own 1951).
Ac ually, e en Cou no ’s s udy o duopoly compe i ion ia
quan i y (Cou no 1838) al eady in oduced a pa icula
lea ning p ocess. Howe e , when lea ning algo i hms
con e ge o he Nash equilib ium and which p ope ies
hey need o possess, is s ill no ully esol ed (Young
2010; Cesa-Bianchi and Lugosi 2006).
Resea ch on lea ning in games (Young 2010; Fos e and
Voh a 1997) has shown ha no all games can be lea ned
(Ha and Mas-Colell 2006; Milionis e al. 2022): lea ning
dynamics may cycle, di e ge, o be chao ic (Me ikopoulos
e al. 2018; Bailey and Piliou as 2018). While he e is no
comp ehensi e cha ac e iza ion o games ha a e ‘‘lea n-
able’’, he e a e some impo an esul s ega ding lea ne s.
A classical esul is ha he class o no- eg e lea ning
algo i hms con e ges o he so-called coa se co ela ed
equilib ium (CCE) o a game Fudenbe g and Le ine
(1999). CCEs a e supe classes o Nash equilib ia. How-
e e , CCEs can also con ain domina ed s a egies and he
se o CCEs in a game can be e y la ge. Fo he analysis o
algo i hmic collusion, we wan o unde s and whe he
lea ning algo i hms.con e ge o a Nash equilib ium.
Less is known abou condi ions o games in which
lea ning algo i hms con e ge o a Nash equilib ium.
Monde e and Shapley (1996) in oduced he class o po-
en ial games, and hey showed ha Cou no oligopolies
wi h linea p ice o cos unc ions a e po en ial games.
Po en ial games a e gua an eed o ha e a leas one pu e
Nash equilib ium. Impo an ly, i was shown ha se e al
bandi algo i hms con e ge o a Nash equilib ium in
123
M. Bichle e al.: Algo i hmic P icing and Algo i hmic Collusion, Bus In Sys Eng 67(6):971–979 (2025) 975
po en ial games (Palaiopanos e al. 2017; Cohen e al.
2017). Howe e , while Cou no oligopoly models a e
po en ial games, his p ope y a ely holds in o he eco-
nomic games.
Ano he cen al condi ion o which posi i e esul s a e
known is ha o he (payo ) mono onici y o a game.
Games ha admi a s ic ly conca e po en ial a e s ic ly
mono one (Me ikopoulos and Zhou 2019). Games ha
sa is y his condi ion ha e a unique Nash equilib ium. I is
known ha simple algo i hms such as p ojec ed g adien
ascen con e ge o an equilib ium o s ic ly mono one
games (Dong e al. 2018). Ye , he class o games ha a e
s ic ly mono one is a he es ic ed.
O e all, he class o games o which lea ning algo-
i hms con e ge o a Nash equilib ium is no well unde -
s ood. Mo e speci ically, no much is known abou
p ope ies o Be and compe i ion models ha would
gua an ee con e gence o a Nash equilib ium. No e ha
each demand model assumed in he Be and compe i ion
model leads o a di e en game and hus migh ha e di -
e en con e gence p ope ies.
4 Implica ions o BISE Resea ch
A lo o he esea ch in BISE and mo e b oadly in he
economic sciences aims o unde s and human beha io in
ce ain ma ke in e ac ions. Elec onic ma ke s ha e been a
cen al esea ch opic in BISE o many yea s (Malone
e al. 1987; Schmid 2000; Bichle e al. 2010). Mo e and
mo e ma ke s a e au oma ed wi h lea ning agen s, and we
need o unde s and i hese ma ke s a e in equilib ium and
i hey lead o e icien ou comes. These ques ions a e no
new, bu he p esence implemen a ion o lea ning algo-
i hms is.
P icing agen s on e ail pla o ms such as Amazon a e an
example, and so a e display ad auc ions. The ques ion o
how hese agen s in e ac and wha ou comes hey p oduce
is o g ea in e es o esea che s and policymake s. Does
he use o lea ning algo i hms in p icing lead o e icien
equilib ium ou comes o does i jeopa dize consume
wel a e by leading o algo i hmic collusion? Algo i hmic
collusion challenges adi ional heo ies ha assume i ms
canno sus ain collusi e a angemen s wi hou explici
coo dina ion. The ques ion has implica ions on policy,
compe i ion law, and he de elopmen o esponsible
algo i hmic p ac ices in p icing. In wha ollows, we dis-
cuss a a ie y o esea ch ques ions ha he BISE com-
muni y is well-equipped o add ess. We p o ide a sho
o e iew in Fig. 2.
Algo i hms A key ques ion in he s udy o algo i hmic
collusion is de e mining which ypes o algo i hms a e
mo e p one o collusi e beha io . This inqui y in ol es
examining speci ic p ope ies o p icing algo i hms ha
may in luence hei endency o collude. Addi ionally, he
assump ions wi hin game- heo e ical models can a ec
how di e en algo i hms con e ge o equilib ium: some
model se ups may acili a e collusi e ou comes o ce ain
algo i hms, while o he s may no . Al hough he e a e ini ial
indings on Q-lea ning and some bandi algo i hms wi hin
speci ic Be and compe i ion models, a comp ehensi e
unde s anding o hese dynamics cons i u es a wide open
esea ch ques ion.
The ype o eedback an algo i hm ecei es signi ican ly
impac s i s po en ial o collusion. Algo i hms wi h bandi
eedback, ecei ing only in o ma ion on he ou comes o
hei own ac ions, a e likely o beha e di e en ly han
hose which can access mo e in o ma ion abou he en i-
onmen o compe i o s’ ac ions. S a e-based in o ma ion,
like his o ical p ices o obse ed demand pa e ns, can
u he enhance an algo i hm’s capaci y o p edic op imal
p ices, po en ially leading o aci collusion. Resea ch is
needed examining how di e en le els o in o ma ion
in luence he eme gence o collusi e s a egies.
De ec ion De ec ing algo i hmic collusion p esen s a
majo challenge o egula o s because p ice pa e ns
esul ing om collusion can closely esemble hose om
equilib ium s a egies, especially in dynamic ma ke s.
S a is ical me hods migh help o iden i y p icing pa e ns
ha may indica e collusion (Bonjou e al. 2022). Ano he
a ea o ocus is he de elopmen o algo i hms ained o
ecognize sub le signals o collusion, such as p ice
ma ching o e alia o y p ice adjus men s (Xu e al. 2024).
E ec i e de ec ion me hods could play a key ole in
helping egula o s moni o and add ess collusi e p icing
p ac ices on digi al pla o ms.
Regula ions & Moni o ing E en i a compe i ion
au ho i y we e o iden i y a po en ial case o aci collusion,
he cu en s a e o he law could make such p ac ice
i ep oachable in he absence o explici communica ion o
con ac among he companies using such au onomous
algo i hms. Exis ing compe i ion laws o en ocus on
explici , human-d i en collusion a he han implici algo-
i hmic coope a ion. New egula o y amewo ks may be
necessa y o add ess algo i hmic beha io s ha lead o
collusi e ou comes, e en wi hou di ec communica ion
be ween i ms. The OECD and o he egula o s a e awa e
o sho comings o he exis ing legisla ion, and se e al
calls o ac ion ha e been made o policy changes o
add ess his po en ial en o cemen gap (OECD 2023). Fo
example, he Eu opean Commission’s e ised guidelines
on ho izon al coope a ion ag eemen s, adop ed in 2023,
s ipula e ha an explici ag eemen among compe i o s o
use he same p icing algo i hm is conside ed an in inge-
men o a icle 101 o he T ea y on he Func ioning o he
Eu opean Union. No e ha his ac ion add esses a di e en
123
976 M. Bichle e al.: Algo i hmic P icing and Algo i hmic Collusion, Bus In Sys Eng 67(6):971–979 (2025)
o m o collusion whe e se e al i ms use he same hi d-
pa y p icing so wa e o de e mine hei p ices. This may
esul in a hub-and-spoke si ua ion ha can acili a e
in o ma ion exchanges in he con ex o an ag eemen o
conce ed p ac ice (OECD 2023). This is di e en om he
ype o algo i hmic collusion we discuss in his pape ,
whe e i ms use hei own p icing algo i hms and lea n o
collude wi hou explici communica ion. We need o
unde s and al e na i e means o de ec ing and p e en ing
algo i hmic collusion such as anspa ency equi emen s,
es ic ions on ce ain ypes o algo i hms, o eal- ime
moni o ing ools.
Accoun abili y Algo i hmic accoun abili y can play an
impo an ole in his con ex (Ho nebe and Laume 2023).
The concep e e s o he esponsibili y o o ganiza ions
and indi iduals o ensu e ha algo i hms ope a e ai ly,
anspa en ly, and e hically. In pa icula , anspa ency in
algo i hmic design could be an impo an ool in mini-
mizing he isk o collusion. T anspa ency assumes cen e
s age in he P e en ing Algo i hmic Collusion Ac o 2024,
4
a bill ha has been in oduced in he US Sena e. T ans-
pa ency is also an impo an aspec o he Eu opean
Union’s Digi al Ma ke s Ac (DMA)
5
and he Digi al
Se ices Ac (DSA),
6
al hough hey do no ye add ess
algo i hmic collusion. By es ablishing design guidelines
ha discou age collusi e s a egies o by equi ing algo-
i hms o be anspa en in hei decision-making p ocesses,
i ms may be able o mi iga e unin ended collusi e
beha io . Resea ch migh explo e how di e en le els o
anspa ency and design cons ain s a ec algo i hmic
beha io and whe he g ea e openness among algo i hms
would educe o inad e en ly inc ease he likelihood o
collusion. Ul ima ely, he esea ch in his ield migh lead
o ules ha can be implemen ed in egula o y audi s and
compliance es s. Fi ms migh be asked o disclose hei
use o p icing algo i hms and ensu e ha hese ools a e
designed o comply wi h an i us laws. Such measu es aim
o c ea e an en i onmen whe e algo i hmic beha io s a e
subjec o sc u iny, he eby discou aging collusi e ou -
comes (Beneke and Macken od 2021). I is c ucial o
unde s and he egula o y measu es needed o e ec i ely
minimize he isk o algo i hmic collusion.
Beyond oligopoly compe i ion While he ocus o
esea ch on algo i hmic collusion is on adi ional olig-
poloy models, he e is no eason o belie e ha he phe-
nomenon can only a ise he e. (Bichle e al. 2024)
analyzed display ad e ising auc ions which a e known o
be au oma ed ia lea ning agen s. P icing o pla o ms on
wo-sided ma ke s has d awn subs an ial a en ion in he
BISE li e a u e and he impac o algo i hmic p icing
desc ibes a na u al ex ension o his esea ch (Dou and Wu
2021; Cons an inides e al. 2018; Pa ke e al. 2016).
5 Conclusions
A cen al ques ion in he economic sciences has long been
unde which condi ions e icien ou comes can be achie ed
wi h ma ke mechanisms. The wel a e heo ems p o ide
condi ions o a compe i i e equilib ium o exis ha is
Pa e o e icien (Va ian 2014). O e decades, he game-
heo e ical li e a u e iden i ied condi ions unde which
e icien ou comes can be expec ed in equilib ium. Game
heo y highligh s he ole o incen i es and s a egic
in e ac ion in ma ke s, and he Nash equilib ium assumes
cen e s age. In game heo y, agen s a e assumed o ha e
enough in o ma ion and ha hey can de i e Nash equi-
lib ium s a egies ha hey use om he s a .
On eal-wo ld ma ke s, agen s o en don’ ha e he
equi ed in o ma ion o de i e an equilib ium, and e en i
hey had, he equilib ium p oblem is compu a ionally ha d
in gene al (Daskalakis e al. 2009). Impo an ly, agen s
Fig. 2 O e iew o BISE
esea ch oppo uni ies
4
h ps://www.cong ess.go /bill/118 h-cong ess/sena e-bill/3686/
ex .
5
h ps://digi al-ma ke s-ac .ec.eu opa.eu/.
6
h ps://commission.eu opa.eu/s a egy-and-policy/p io i ies-2019-
2024/eu ope- i -digi al-age/digi al-se ices-ac _en.
123
M. Bichle e al.: Algo i hmic P icing and Algo i hmic Collusion, Bus In Sys Eng 67(6):971–979 (2025) 977
don’ ha e he in o ma ion necessa y abou hei compe i-
o s cos s o alues o de i e an equilib ium. This is why on
au oma ed ma ke s, agen s ely on lea ning algo i hms,
explo ing a ious p ices, and exploi ing he in o ma ion
gained o e ime. The e is limi ed unde s anding o he
ci cums ances unde which such epea ed in e ac ions
among such lea ning agen s yield e icien ou comes.
Howe e , his unde s anding is impo an o unde s and he
ou comes o algo i hmic p icing in online e ail ma ke s.
Gaining insigh s in o hese algo i hmic ma ke s ep esen s
an impo an and undamen al challenge.
The BISE communi y has long analyzed elec onic
ma ke s. The e en o lea ning algo i hms in his ma ke is
mo e han a de ail. I is undamen al ha we unde s and
how such lea ning agen s impac he ou come o ma ke s
and whe he such algo i hmic ma ke s can be expec ed o
be e icien . This equi es echnical unde s anding o
lea ning algo i hms as well as he economic p inciples o
ma ke ins i u ions. As such, he Business and In o ma ion
Sys ems Enginee ing communi y is well equipped o
add ess his challenging and ye ha dly unde s ood
phenomenon.
Funding Open Access unding enabled and o ganized by P ojek
DEAL.
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