Aliaga-Mi anda, Augus o e al.
A icle
Analysis o he compe i ion o he Sou h-Eas e n Railway o
Pe u h ough a ime able auc ion
Games
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Sugges ed Ci a ion: Aliaga-Mi anda, Augus o e al. (2025) : Analysis o he compe i ion o he Sou h-
Eas e n Railway o Pe u h ough a ime able auc ion, Games, ISSN 2073-4336, MDPI, Basel, Vol. 16,
Iss. 2, pp. 1-19,
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Recei ed: 15 Janua y 2025
Re ised: 24 Ma ch 2025
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Published: 7 Ap il 2025
Ci a ion: Aliaga-Mi anda, A., Flo es-
Vilcapoma, L. R., Raqui-Rami ez, C. E.,
Claudio-Pé ez, J. L., Yanase-Rojas, Y.,
& Espinoza-Yangali, J. P. (2025).
Analysis o he Compe i ion o he
Sou h-Eas e n Railway o Pe u
Th ough a Time able Auc ion. Games,
16(2), 16. h ps://doi.o g/10.3390/
g16020016
Copy igh : © 2025 by he au ho s.
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(h ps://c ea i ecommons.o g/
licenses/by/4.0/).
A icle
Analysis o he Compe i ion o he Sou h-Eas e n Railway o
Pe u Th ough a Time able Auc ion
Augus o Aliaga-Mi anda 1, Luis Rica do Flo es-Vilcapoma 2,* , Ch is ian E ain Raqui-Rami ez 1,
José Luis Claudio-Pé ez 1, Yadi a Yanase-Rojas 2and Jo any Pompilio Espinoza-Yangali 3
1School o Business, Uni e sidad Nacional In e cul u al de la Sel a Cen al Juan San os A ahualpa,
La Me ced 12856, Pe u; [email p o ec ed] (A.A.-M.); [email p o ec ed] (C.E.R.-R.);
[email p o ec ed] (J.L.C.-P.)
2School o Business, Uni e sidad Tecnológica del Pe ú, Lima 15046, Pe u; [email p o ec ed]
3School o Business, Uni e sidad Con inen al, Junín 12000, Pe u; [email p o ec ed]
*Co espondence: [email p o ec ed]
Abs ac : Ou esea ch analyzes he design o an auc ion model o ailway anspo a ion
on he Sou h-Eas Railway o Pe u, managed by Fe oca il T ansandino S.A. (Fe ansa) and
ope a ed by Pe uRail. Ini ially, he egula o y amewo k aimed o p omo e compe i ion
in ailway anspo a ion h ough ime able auc ions and in as uc u e access. Howe e ,
he concession has esul ed in a e ically in eg a ed s uc u e ha a o s Pe uRail, which
aces minimal di ec compe i ion, con ols high-demand ime slo s, and hinde s he en y o
o he ope a o s due o s a egic and s uc u al access ba ie s. To add ess hese dis o ions,
we p opose e o ming he auc ion mechanism o neu alize hese ad an ages and enhance
compe i ion. In his e ised amewo k, he ack usage ee will se e as he compe i i e
ac o , wi h he highes bid abo e a minimum base a e secu ing he alloca ion. Addi ionally,
we p opose he implemen a ion o asymme ic a i s o compensa e o he highe cos s
aced by ope a o s wi h ewe economies o scale, echnological op imiza ions o acili a e
equi able access o ime slo s, and s ic e o e sigh mechanisms o ensu e anspa ency in
ime able alloca ion. These measu es aim o balance he ma ke and sa egua d compe i ion
h ough a mo e equi able and e icien auc ion design.
Keywo ds: auc ion heo y; game heo y; ailway ma ke ; ailway ime able
1. In oduc ion
The ques ion ega ding he consequences o implemen ing auc ions in Pe u’s ailway
se ice is c ucial, as he Sou h-Eas
1
Railway has demons a ed signi ican limi a ions,
pa icula ly in achie ing equi able access o ailway in as uc u e. This lack o compe-
i ion no only inc eases ees and limi s se ice op ions o use s bu also impac s public
pe cep ion and he e ec i eness o p i a iza ion policies in he coun y. Un il he 1980s,
he ailway sec o in Pe u was exclusi ely con olled and ope a ed by a s a e en i y wi hin
a highly egula ed sys em wi h no p i a e pa icipa ion. This model aligned wi h he
na u al monopoly s uc u e cha ac e is ic o he ailway indus y, whe e economies o scale
a o ope a ion by a single p o ide o minimize cos s and maximize ope a ional e iciency.
Howe e , as mode niza ion policies p og essed and mo e compe i i e ma ke models we e
p omo ed, he p i a iza ion and concession o s a egic sec o s we e encou aged o imp o e
se ice quali y and achie e g ea e economic e iciency. This ansi ion aimed o adap he
ailway sys em o a limi ed compe i ion amewo k, whe e p i a e ope a o s would ha e
incen i es o enhance he quali y o se ices o e ed o use s.
Games 2025,16, 16 h ps://doi.o g/10.3390/g16020016
Games 2025,16, 16 2 o 19
Since 1999, concessions o he managemen o ailway in as uc u e ha e been
implemen ed, allowing p i a e companies o pa icipa e in he sec o . In he case o he
Sou h-Eas Railway, in as uc u e managemen was en us ed o Fe oca il T ansandino
S.A. (Fe ansa), while ailway anspo a ion ope a ions we e assigned o Pe uRail S.A.
(Pe uRail). This concession scheme has been subjec o con o e sy, as con a y o he
in ended goals o p i a iza ion policies, Pe uRail con inues o ope a e in a ma ke wi h
limi ed di ec compe i ion
2
. This si ua ion has a ec ed public pe cep ion due o he high
a es and lack o se ice op ions. The absence o di ec compe i o s and he e ically
in eg a ed s uc u e o Fe ansa and Pe uRail ha e esul ed in a scena io whe e he p o i s
o p i a iza ion, in e ms o ees and se ice di e si y, emain limi ed.
Since he incep ion o he ailway concession, he s a e-owned ac ion and olling
s ock assigned o Fe ansa o i s ope a ion has been leased exclusi ely o Pe uRail. This
ag eemen , which o ms pa o he concession’s asse s, has been enewed on ou occa-
sions, wi h he mos ecen ex ension alid un il 2021. As a esul , Pe uRail’s dominan
posi ion in he ma ke has been u he consolida ed, signi ican ly limi ing oppo uni ies o
compe i ion and c ea ing a scena io whe e access o essen ial in as uc u e is exclusi ely
dependen on his single ope a o . This si ua ion p esen s conside able challenges o
po en ial new en an s, who mus con end wi h high ini ial cos s and signi ican s uc u al
ba ie s o compe e in a ma ke whe e s a e esou ces con inue o bene i a single company.
The exclusi e leasing o s a e-owned ac ion and olling s ock o Pe uRail has e ec ed
subs an ial ba ie s o compe i ion, p ima ily by es ic ing new ope a o s’ access and ais-
ing ma ke en y cos s. This, in u n, hampe s he ailway ma ke ’s dynamism. Howe e ,
hese obs acles could diminish in he long e m due o he na u al obsolescence o he
leased equipmen , which will ine i ably need o be eplaced. In oducing a anspa en
and inclusi e access mechanism could encou age new pa icipan s and os e a ai e
dis ibu ion o ailway in as uc u e.
Unlike he leasing o ac ion equipmen , access ees o ailway in as uc u e a e
egula ed unde he e ms o he concession con ac and a e adjus ed annually h ough an
indexa ion mechanism. This p ocess is o e seen by he Supe iso y Agency o In es men
in Public T anspo In as uc u e (OSITRAN), which ensu es compliance wi h he es ab-
lished egula o y amewo k. Al hough he concession allows ailway ope a o s o access
he in as uc u e ei he h ough di ec nego ia ion o auc ion, only he o me has been
employed o da e. The lack o incen i es, combined wi h an ine ec i e egula o y design
o auc ions, may in ensi y compe i ion- ela ed challenges, as he cu en sys em does no
gua an ee neu ali y o non-disc imina ion. The p e ailing auc ion condi ions, along wi h
he exis ing s uc u al and s a egic ba ie s, unde mine ma ke pe o mance by educing
oppo uni ies o new ope a o s and limi ing he o e all e iciency o he ailway ne wo k.
Ou analysis e eals ha he design o he ime able auc ion is inhe en ly an icompe i-
i e. The p ima y ope a o , Pe uRail, can o e high bids wi hou ully in e nalizing he
associa ed cos s, he eby educing he likelihood o success o o he po en ial ope a o s.
Fu he mo e, s uc u al ba ie s, such as he e ical in eg a ion be ween he in as uc u e
manage and he anspo se ice p o ide , ein o ce Pe uRail’s dominan posi ion. The
ope a ional ad an ages held by his company allow i o manipula e he alloca ion o ime
slo s, unde mining he compe i i eness o independen ope a o s and limi ing he po en ial
o se ice di e si ica ion in non-auc ioned ime slo s.
The auc ion e o m eme ges as he mos iable al e na i e o enego ia ing he olling
s ock lease con ac due o i s g ea e legal easibili y, as e implemen a ion, and impac on
compe i ion. Modi ying he concession con ac would equi e nego ia ions wi h Fe ansa
and Pe uRail, po en ially leading o li iga ion o e acqui ed igh s and signi ican ly delay-
ing i s execu ion. In con as , auc ion e o m could be implemen ed h ough egula o y
Games 2025,16, 16 3 o 19
adjus men s wi hou he need o complex con ac ual enego ia ions. Mo eo e , i s imple-
men a ion would allow o he co ec ion o dis o ions in he sho e m, whe eas opening
he leasing p ocess would equi e an ope a ional and inancial es uc u ing o Fe ansa,
ex ending he imeline. In e ms o compe i ion, while g an ing o he ope a o s access o
olling s ock would educe ce ain en y ba ie s, s uc u al obs acles such as access o
key in o ma ion and schedule dis ibu ion would pe sis , which could be mo e e ec i ely
add essed h ough a well-designed auc ion e o m.
In esponse o he cu en challenges, a e o m o he ime able auc ion sys em is
p oposed, inco po a ing measu es o le el he playing ield h ough a base ee. As com-
plemen a y condi ions, we sugges implemen ing asymme ic ees pe wagon-kilome e
a eled by he winning ailway ope a o du ing he auc ioned ime slo o o se he
addi ional cos s aced by smalle ope a o s, as well as adop ing echnological solu ions
o acili a e equi able access o ime slo s. Addi ionally, s ic e egula o y o e sigh is
ecommended o ensu e neu ali y and p e en he concen a ion o consecu i e ime slo s
in he hands o a single ope a o .
Ou esea ch p o ides a no el app oach by highligh ing he impo ance o adap ing
auc ion designs o con ex s o ma ke dominance, such as he Pe u ian case. I also
emphasizes he need o egula o y policies o e ol e in esponse o ma ke dynamics
and p oposes p ac ical solu ions ha could be applied o o he in as uc u e sec o s. The
combina ion o di e en ia ed a i s, access echnology, and con inuous o e sigh o e s an
adap i e egula o y model ha could ans o m he compe i i eness and e iciency o he
Sou h-Eas Railway in Pe u and simila con ex s.
2. Li e a u e Re iew
In he ups eam ma ke , he ailway indus y exhibi s dis inc i e cha ac e is ics ha
con igu e i as a na u al monopoly due o he economies o scale, scope, and densi y
equi ed o in as uc u e p o ision. Baumol e al. (1983) and B ock (1983) highligh ed he
sou ces o subaddi i i y in ailway ope a ions, iden i ying speci ic a eas whe e compe i ion
can be in oduced wi hou comp omising he o e all e iciency o he sys em. Fo eman-
Peck (1987) conduc ed a his o ical analysis o 19 h-cen u y ailway policies, e alua ing how
na u al monopolies in luenced he de elopmen o anspo a ion and consume choices.
In he downs eam ma ke , ocused on eigh and passenge anspo a ion se ices,
Bi zan (2003) emphasized he impo ance o in oducing compe i ion in ailway ne wo ks o
mi iga e monopolis ic powe and enhance e iciency. Howe e , such in e en ion equi es
obus egula o y o e sigh o ensu e equi able access and p e en an icompe i i e p ac ices.
The high concen a ion in he ele an ma ke g an s he ela ed company o Fe ansa
a dominan posi ion, making i di icul o new compe i o s o en e . Pei a e al. (2022)
conduc ed a sys ema ic e iew on he use o ailway anspo a ion in ou ism, highligh ing
how monopolies on speci ic ou es can impac ou ism de elopmen , accessibili y, and
passenge cos s.
The main ba ie s o en y in ailway anspo a ion include s uc u al ba ie s, such
as he high sunk cos s associa ed wi h acqui ing ac ion and olling s ock compa ed o
leasing, and s a egic ba ie s, such as he alloca ion o uncompe i i e schedules ha a ec
eigh ope a ions and educe demand. To add ess he la e , a ious au ho s ha e p oposed
i s -p ice auc ion models as a solu ion o enhance compe i i eness and op imize esou ces.
Se e al s udies, including hose by Joskow and Ti ole (2005), Bu ka (1995), and E inge
(2008), examined auc ion models o capaci y alloca ion, demons a ing how hese ools
can impac e iciency and p o i abili y in he sec o .
Klempe e (1999) analyzed auc ions in egula ed sec o s, highligh ing hei abili y o
alloca e esou ces o agen s who alue hem he mos . Ya ow (2003) examined auc ions
Games 2025,16, 16 4 o 19
wi h ese e p ices in ene gy ne wo ks, while McDaniel (2003) ocused on in o ma ional
asymme y and co ela ed alues in gas ne wo ks, demons a ing how i e a i e auc ions
enhance e iciency in capaci y alloca ion.
Simila ly, an Ko en (2012) employed a combina ion o i s -p ice and second-p ice
auc ions, along wi h oehold auc ion models, o demons a e ha when a gene a o owns
he in e connec o , i s bidding agg essi eness inc eases i s p o i s and auc ion e enues.
Howe e , his comes a he expense o e iciency and he ea nings o independen gene a o s.
In he ailway con ex , Ha od (2013) and Kuo and Mille -Hooks (2015) de eloped
models ha explo e capaci y alloca ion h ough i e a i e and combina o ial auc ions,
add essing challenges associa ed wi h e iciency and high demand in complex ne wo ks.
Pe ennes (2014) emphasized he need o cen alized planning, conside ing he echnical
limi a ions o la ge ailway sys ems.
O he s udies, such as hose by S edbe g e al. (2017), B oman e al. (2022), and
S ojadino i´c e al. (2019), examined he implemen a ion o auc ions and simula ions o
manage capaci y and p omo e compe i ion in ailway ma ke s. These wo ks highligh how
hyb id auc ions and open access can inc ease social wel a e and educe ees, al hough hey
en ail challenges such as he need o empo a y subsidies.
While his s udy ocuses on ailway compe i ion and d aws on auc ion models om
he ene gy sec o , he unde lying issue is b oade and closely ela ed o he li e a u e
on public p ocu emen auc ions. In pa icula , ex ensi e documen a ion highligh s how
he pa icipa ion o bidde s linked o he auc ionee can impac compe i ion in egula ed
ma ke s and in as uc u e concessions (Busu & Busu,2021).
The design o auc ions in e ically in eg a ed en i onmen s p esen s signi ican
challenges o e iciency and compe i ion, simila o hose obse ed in public p ocu emen
p ocesses and public–p i a e pa ne ships (Gao,2018). The li e a u e has shown ha
asymme ies in cos s uc u es and access o s a egic in o ma ion can c ea e ad an ages
o ce ain ope a o s, unde mining ai ness and os e ing a less compe i i e en i onmen
(Hanák e al.,2020). In his ega d, inco po a ing e e ences o public p ocu emen auc ion
models would s eng hen he heo e ical amewo k o his s udy, p o iding a b oade
pe spec i e o assess he dis o ions a ising om he cu en design o ailway auc ions.
The applica ion o auc ion heo y o egula ed ma ke s has been widely s udied in he
li e a u e. Joskow and Ti ole (2005) analyze how asymme ies in ma ke powe can dis o
compe i ion in capaci y alloca ion mechanisms, demons a ing ha s a egic in e ac ions
be ween bidde s linked o he auc ionee can educe e iciency. Simila ly, Busu and Busu
(2021) highligh he isks o bid igging in public p ocu emen auc ions, emphasizing he
ole o egula o y o e sigh in p e en ing collusion and ensu ing ai ness. Thei indings
a e pa icula ly ele an o he Pe u ian ailway case, whe e he link be ween Fe ansa
and Pe uRail c ea es an en i onmen suscep ible o s a egic bidding beha io s ha hinde
compe i ion. Gao (2018) u he explo es sus ainable winne de e mina ion mechanisms
in public–p i a e pa ne ship auc ions, p oposing me hodologies ha could be adap ed
o ailway capaci y alloca ion. The insigh s om hese s udies ein o ce he impo ance o
designing an auc ion mechanism ha neu alizes ma ke dis o ions and ensu es a le el
playing ield o all pa icipan s.
Finally, Adle e al. (2021), Ali and Eliasson (2022), and T i uno i´c e al. (2024), explo ed
he use o game heo y models and asymme ic auc ions o op imize capaci y alloca ion and
os e compe i ion in Eu opean ailway ma ke s. The indings e eal ha , while hese s a e-
gies enhance consume wel a e and encou age he en y o new pa icipan s, hey also pose
challenges ela ed o economic sus ainabili y and anspa ency in esou ce managemen .
Games 2025,16, 16 5 o 19
3. Me hodology
We p esen a de ailed analysis o he ime able auc ion design and i s e ec s on he
compe i i e dynamics o he passenge anspo a ion ma ke in he Sou h-Eas Railway in
Pe u. Following he p oposal o Adle e al. (2021), he use o game heo y-based models
is essen ial o de eloping auc ion mechanisms ha os e compe i ion, e en in highly
concen a ed ma ke s such as he ailway sec o . The analysis has been cohe en ly di ided
in o wo pa s. The i s desc ibes he cu en design o he ime able auc ion. The second
ou lines he main dis o ions a ising om he cu en auc ion design.
3.1. Desc ip ion o he Time able Auc ion Design
Acco ding o he concession con ac , he bidde o e ing he highes ack usage ee
will gain access o he in as uc u e o ope a e du ing he auc ioned ime slo . In e u n,
hey mus pay he o e ed ee pe wagon-kilome e mo ed un il he expi a ion o he access
con ac e m.
Likewise, i upon he expi a ion o he a o emen ioned con ac e m he e is once
again a concu ence o schedule eques s, he a ilia ed ope a o e ains he igh o ini ia e
an auc ion unde any ci cums ances, p o ided ha he eques ed schedule is no al eady
alloca ed as he esul o an ongoing auc ion p ocess wi h a alid e m.
The cu en cha ac e is ics o he auc ion e lec a i s -p ice model in ol ing ela ed
bidde s and independen bidde s. In his amewo k, he cos s associa ed wi h Pe uRail’s
ack usage ee a e no ully in e nalized. Al hough he e is pa ial in e naliza ion o hese
cos s wi hin he economic g oup, 37.25% o Fe ansa’s g oss e enues a e ans e ed o he
s a e. Howe e , he non-in e nalized po ion o cos s can c ea e signi ican asymme ies
compa ed o an auc ion model in ol ing independen companies.
In his ega d, an Ko en (2012) iden i ies a se o dis o ions in i s -p ice auc ions
wi h ela ed bidde s o he auc ionee , which can lead o s a egic beha io by he ela ed
company, e en when he cos in e naliza ion by he economic g oup is only pa ial.3
3.2. Dis o ions in he Auc ion Design
Based on he p e iously desc ibed auc ion mechanism, we ha e de eloped an analy -
ical model o iden i y he dis o ions gene a ed by he auc ion sys em es ablished in he
Concession Con ac . This model aims o p o ide a de ailed iew o how he mechanism’s
design a ec s alloca ions and economic ou comes o he ac o s in ol ed. The de ailed
de elopmen o he ma hema ical model and i s undamen al assump ions can be ound in
Appendix A.
In his analysis, we ha e modeled he paymen lows co esponding o ou key agen s
wi hin he auc ion mechanism in a wo-bidde scena io as ollows: one ela ed bidde and
one un ela ed bidde . The agen s conside ed a e he auc ionee , Fe ansa (A); he ela ed
bidde , Pe uRail (R); he un ela ed bidde , Inka Rail (U); and he s a e (G). This app oach
allows o an in-dep h analysis o he in e ac ion be ween he agen s and he po en ial
economic and egula o y implica ions.
I is impo an o highligh ha , o p ope ly iden i y he incen i es o bo h he ela ed
company and he auc ionee (R and A), hey mus be analyzed join ly. In his ega d, he
s a egy o he ela ed company in he auc ion will be aimed a maximizing he expec ed
p o i s o he Economic G oup (EG), which a e calcula ed as he sum o he expec ed p o i s
o bo h ac o s, namely he ela ed company and he auc ionee .
Addi ionally, o main ain consis ency wi h he la es auc ion announcemen conduc ed
by Fe ansa, i is assumed ha he auc ion pe ains o a newly iden i ied ime slo . The
p o i s o each agen will depend on he auc ion ou comes. Table 1summa izes he
paymen s co esponding o each scena io in he ime able auc ion.
Games 2025,16, 16 6 o 19
Table 1. Payo ma ix by agen .
U R A G
R wins 0 ΠR−CRbR(1−θ)CRbR−CAθCRbR
U wins ΠU−CUbU0(1−θ)CUbU−CAθCUbU
In he able,
ΠR
and
ΠU
ep esen he p esen alue o he p o i s o he ela ed and
un ela ed bidde s, espec i ely, when winning he auc ion. These alues a e s anda dized
wi hin a ange om 0 o 1 and do no include he paymen co esponding o he ack usage
ee. On he o he hand,
CRbR
and
CUbU
ep esen he p esen alues o he cos s associa ed
wi h he ack usage ee paymen , dis inguishing be ween he ela ed bidde
(bR)
and he
un ela ed bidde
(bU)
. Addi ionally,
CA
e lec s he ope a ing and main enance cos s bo ne
by he auc ionee , while
θ
ep esen s he pe cen age o emune a ion paid o he s a e by
he auc ionee o ack usage.
The pa ame e
θ
, which ep esen s he deg ee o cos -sha ing wi hin he economic
g oup, plays a c ucial ole in de e mining auc ion ou comes. I s de ini ion and egula ion
mus be clea ly communica ed o all pa icipan s be o e he auc ion p ocess begins o
ensu e anspa ency and p edic abili y.
Po en ial egula o y app oaches: Ex-An e ixed egula ion—OSITRAN se s a p ede-
e mined alue o
θ
based on ma ke s udies, ensu ing a s able bidding en i onmen .
Dynamic adjus men mechanism—
θ
is adjus ed o e ime based on auc ion ou comes,
main aining a a ge le el o ma ke compe i i eness. Sel - egula ing bidding s uc u e—a
θ
-indexed eba e sys em e u ns excess ees o disad an aged bidde s, p e en ing sys emic
exclusion wi hou di ec egula o y in e en ion.
We assume a uni o m dis ibu ion o he p i a e in o ma ion a iables
Πi
, de ined
wi hin he in e al
[0, 1]
. This choice allows o he modeling o a ma ke whe e in o ma ion
is dis ibu ed equally among pa icipan s, elimina ing po en ial in o ma ional asymme ies.
Unde his assump ion, he densi y emains cons an a
(Πi)=
1 h oughou he en i e
in e al, ensu ing a consis en and anspa en analysis o bidding s a egies.
Based on he p esen ed payo ma ix, he expec ed p o i s o he Economic G oup
(πEG)and he un ela ed company (πU)ha e been calcula ed as ollows:
πEG =P (R wins)E{[ΠR−θCRbR−CA]|bR>bU}+P (U wins)E{[(1−θ)CUbU−CA]|bR<bU}(1)
πU=P (U wins)E{[ΠU−CUbU]|bR<bU}(2)
whe e
P (R wins)
is he p obabili y ha he ela ed bidde wins he auc ion, and
P (U wins)
is he p obabili y ha he un ela ed bidde wins he auc ion.
The exp ession
E{·|bR>bU}
ep esen s he expec ed p o i s o each bidde when
he ela ed company wins he auc ion, ha is, when he bid o he ela ed bidde
(bR)
is
g ea e han he bid o he un ela ed bidde
(bU)
. In his ega d, no e ha when he ela ed
company wins he auc ion, he un ela ed company does no ob ain any p o i s.
On he o he hand, he exp essions
E{·|bR<bU}
ep esen he expec ed p o i s o each
bidde when he un ela ed company wins he auc ion, ha is, when he bid o he ela ed
bidde
(bR)
is lowe han he bid o he un ela ed bidde
(bU)
. In his ega d, no e ha
when he un ela ed company wins he auc ion, he ela ed company ecei es a p opo ion
o he ack usage ee.
To simpli y he esul s, i is assumed ha he numbe o wagon-kilome e s mo ed is
equal o 1 o bo h bidde s.
4
Thus, he expec ed p o i s o each bidde a e gi en by he
ollowing equa ions:
πEG =P (R wins)E{[ΠR−θCRbR−CA]|bR>bU}+P (U wins)E{[(1−θ)bU−CA]|bR<bU}(3)
Games 2025,16, 16 7 o 19
πU=P (U wins)E{[ΠU−bU]|bR<bU}(4)
Following he analysis de eloped by an Ko en (2012), he bids submi ed by he
ela ed and un ela ed bidde s (
bR
and
bU
) can be exp essed as unc ions o
ΠR
and
ΠU
,
espec i ely. Thus, he ela ed company will win he auc ion as long as bR>bU(ΠU).
Le he in e se bidding unc ion o he ela ed company be
U(·)=b−1
U(·)
. Replacing
his equali y in he condi ion o he ela ed company o win he ime slo , i ollows ha
o he ela ed company o win he auc ion, i mus hold ha
ΠU>U(bU)
. Thus, (3) can
be ew i en as ollows:
πEG =ZU(bR)
0[ΠR−θbR]dz +Z1
U(bR)[(1−θ)bU(z)]dz −CA(5)
Sol ing he i s in eg al, subs i u ing
ΠU≡U(bR)
, and in eg a ing he second in eg al
by pa s, he ollowing exp ession is ob ained, whe e bmax is he maximum bid as ollows:
πEG =U(bR)[ΠR−θbR]−CA+(1−θ)bmax −bRU(bR)−Zbmax
bR
U(q)dq(6)
To de e mine he i s -o de condi ion in he op imiza ion p ocess o he ela ed bidde ,
(6) is di e en ia ed wi h espec o
bR
, and hen
ΠR≡R(bR)
is subs i u ed, whe e
R(·)=
b−1
R(·). The op imali y condi ion o he ela ed company will be gi en by he ollowing:
[R(bR)−bR]U′(bR)=(1−θ)[U(bR)] (7)
The op imiza ion p ocess can be simila ly applied s a ing om (4) o he un ela ed
company. Thus, he op imali y condi ion o he un ela ed company will be gi en by
he ollowing:
[U(bU)−bR]R′(bU)=R(bU)(8)
4. Resul s
The esul s ob ained show ha he cu en s uc u e o auc ions, cha ac e ized by he
pa icipa ion o ela ed and un ela ed bidde s, p esen s signi ican dis o ions. Addi ionally,
we explain he po en ial an icompe i i e e ec s esul ing om he iden i ied dis o ions.
4.1. Compu a ional Resul s
Consequen ly, i is possible o analyze (7) and (8) based on he nume ical solu ions
de eloped by an Ko en (2012) o cases whe e θlies be ween 0 and 1.
Acco ding o he analysis p esen ed, he ack usage ee o e ed by he ela ed bidde
is consis en ly highe han ha o he un ela ed company, as shown in Figu e 1. This
is explained by dis o ions iden i ied in he auc ion model, as desc ibed by an Ko en
(2012), whe e he ela ed bidde , le e aging c oss-subsidies wi hin he economic g oup,
can bid mo e agg essi ely wi hou ully in e nalizing he cos s. This ad an age signi -
ican ly educes he likelihood o success o he un ela ed bidde , c ea ing an une en
compe i i e en i onmen .
In esponse o his disad an age, he un ela ed bidde is incen i ized o inc ease i s
bid, indi ec ly aising he expec ed e enue o he economic g oup while educing i s own.
The s a egic complemen a i y be ween he bids o bo h bidde s allows he economic g oup
o le e age his dynamic o maximize i s p o i s, ein o cing he incen i es o he ela ed
bidde o main ain an agg essi e bidding s ance.
This beha io in ensi ies as he s a e emune a ion a e dec eases, as a lowe a e
educes he in e naliza ion o cos s associa ed wi h he ees wi hin he economic g oup. As
a esul , i s -p ice auc ions end o p oduce ine icien ou comes i he emune a ion a e is
Games 2025,16, 16 8 o 19
below 100%, posing a signi ican challenge in e ms o ai ness and economic e iciency in
he auc ion model design.
Figu e 1. Bidding unc ions o bidde s in i s -p ice auc ions.
The sou ce o he iden i ied dis o ions is linked o he abili y o he ela ed company
o inc ease he auc ionee ’s expec ed p o i s h ough highe bids. In he scena io whe e
R
wins, A’s e enue di ec ly depends on
bR
. Con e sely, in he scena io whe e
U
wins, A’s
e enue indi ec ly depends on
bR
h ough i s s a egic complemen ,
bU
. The e o e, i he
auc ionee ’s e enue we e no in luenced by he bid o i s ela ed company, he dis o ions
obse ed in he ime able auc ion o he Sou h-Eas Railway would disappea .
On he o he hand, he cha ac e is ics o
ΠR
and
ΠU
a e non- i ial in he p ocess o
selec ing he op imal bid by he companies. These pa ame e s may a y depending on he
speci ic cha ac e is ics o each bidde , as any di e ences in cos s o expec ed e enues—
a ising om g ea e ma ke powe du ing he ime slo , economies o scale, o mo e
a o able access o speci ic asse s—impac he p o i s each bidde associa es wi h he
auc ioned ime slo .
To elimina e his dis o ion, a s a egy is p oposed o decouple he ela ed ope a o ’s
bids om he auc ionee ’s e enues. This p oposal in ol es main aining he ack usage
ee as he p ima y compe i i e ac o , wi h he winne being he bidde o e ing he highes
ee, while also es ablishing a minimum “base ee”. Unde his measu e, Fe ansa would
cha ge he winning ee pe wagon-kilome e and ans e o he s a e he di e ence be ween
he winning ee and he base ee o each wagon-kilome e a eled du ing he auc ioned
ime slo .
Thus, by applying his condi ion, he auc ionee ’s p o i s would become independen
o he bids om ela ed ope a o s, elimina ing incen i es o in la e bids and p omo ing
ai e compe i ion as ollows:
j wins (1−θ)Cjbmin −CA o j =R,U(9)
The auc ionee ’s e enue does no depend on
bR
in any scena io, as
bmin
is an exoge-
nous a iable.
Games 2025,16, 16 15 o 19
each s ep o he p ocess, including he assump ions made, he equa ions used, and he
implemen a ion p ocedu e.
Appendix A.1. Fundamen al Assump ions
To simpli y he analysis and ocus on he key dis o ions, he ollowing assump ions
we e made:
•
Dis ibu ion o p i a e in o ma ion: P i a e signals we e assumed o ollow a uni-
o m dis ibu ion wi hin he in e al
[0, 1]
. This choice e lec s he absence o sys em-
a ic di e ences in in o ma ion be ween ma ke pa icipan s.
•
Cos s uc u e: Fixed main enance cos s and a iable cos s associa ed wi h he dis ance
a eled (measu ed in wagon-kilome e s) we e conside ed. Ta i s we e based on
o icial a es speci ied in he cu en concession con ac .
•
Ope a o s’ s a egic beha io : Ope a o s we e modeled as a ional agen s compe ing
impe ec ly. The ela ed ope a o pa ially in e nalizes in as uc u e e enues, which
allows hem o adop mo e agg essi e bidding s a egies.
Appendix A.2. Ma hema ical Model
Ope a o s’ s a egic beha io was modeled using he ollowing equa ions:
• The p o i unc ion o he ela ed ope a o is exp essed as ollows:
πEG =E[ΠR−θCRbR−CA]+E[(1−θ)bU−CA](A1)
• The p o i unc ion o he independen ope a o is exp essed as ollows:
πU=E[ΠU−CUbU](A2)
These equa ions allow he analysis o how a ying he le el o s a e pa icipa ion
(θ)
in luences bidding s a egies.
The bidding s a egies o ela ed and un ela ed ope a o s s em om hei espec i e
p o i maximiza ion condi ions, shaped by di e ences in cos s uc u es and e enue
ans e s. Pe uRail pa ially in e nalizes ack access cos s, eco e ing a po ion h ough
Fe ansa, which gi es i g ea e lexibili y in i s bids. In con as , Inca Rail ully ex e nalizes
hese cos s, meaning ha any inc ease in i s bid di ec ly educes i s ne e enue, placing i
a a compe i i e disad an age in he auc ion p ocess.
To explici ly cap u e hese di e ences, he op imal bidding s a egy o each agen can
be ew i en as ollows:
• Fo he ela ed ope a o (Pe uRail)
bR=a g max
b[ΠR−θCRb−CA](A3)
• Fo he un ela ed ope a o (Inca Rail)
bU=a g max
b[ΠU−CUb](A4)
I he model we e pe ec ly compe i i e, hese cons ain s would gene a e di e en
bid unc ions, e lec ing he cos asymme y. Howe e , in he cu en s uc u e, he in e nal
e enue dynamics pa ially equalize he unc ions, explaining why he obse ed bidding
s a egies appea mo e simila han expec ed.
Games 2025,16, 16 16 o 19
Appendix A.3. Compu a ional Implemen a ion
The simula ion was implemen ed using Ma lab, a widely used ool o nume ical
analysis. The Runge–Ku a me hod o ou h o de was selec ed due o i s eliabili y and
accu acy when sol ing di e en ial equa ions.
Simula ion S eps:
1. Pa ame e de ini ion:
The ollowing ini ial pa ame e s we e se :
• P i a e in o ma ion uni o mly dis ibu ed wi hin [0, 1].
• Fixed and a iable cos s based on o icial concession da a.
• S a e pa icipa ion le el (θ) a ying be ween 0 and 1.
2. Gene a ion o p i a e signals:
We gene a ed 1000 andom obse a ions om a uni o m dis ibu ion, ep esen ing
each ope a o ’s subjec i e alua ion o he auc ioned ime slo .
3. Calcula ion o bids:
The bids we e calcula ed using he ollowing unc ions:
• Fo he ela ed ope a o : bR(Π)=bmin +(bmax −bmin)Π.
• Fo he independen ope a o : bU(Π)=bmin +(bmax −bmin)(1−Π).
4.
Sol ing he Equa ions: The bidding dynamics we e modeled using he ollowing
Runge–Ku a scheme:
yn+1=yn+h
6(k1+2k2+2k3+k4)(A5)
whe e
k1= ( n,yn)
,
k2= n+h
2,yn+h
2k1
,
k3= n+h
2,yn+h
2k2
and
k4=
( n+h,yn+hk3).
5. I e a ion and Da a Collec ion:
The calcula ions we e epea ed o all 1000 simula ed p i a e signals. Resul s we e
eco ded o he ope a o s’ bids, p o i s, and compa a i e ou comes unde di e en
egula o y scena ios.
Appendix A.4. Valida ion and Discussion
To alida e he eliabili y o he model, he simula ions we e epea ed wi h di e en
sample sizes. In all cases, he esul s emained s able. Fu he mo e, he simula ion ou comes
we e compa ed agains p edic ions de i ed om he analy ical model o linea bidding
unc ions, yielding a de ia ion o less han 2%.
Las ly, he model was used o explo e how a ying he le el o s a e pa icipa ion
(θ)
a ec s ma ke compe i ion. The analysis e ealed ha he p oposed e o m, which in ol es
ans e ing auc ion su pluses o he s a e, signi ican ly imp o es compe i i e condi ions
by elimina ing incen i es o he in as uc u e manage o a o i s ela ed ope a o .
Appendix B. Nume ical Example
Figu e A1 p esen s he simula ion o expec ed p o i s o he ope a o ela ed o he
ailway in as uc u e manage (panel a) and he independen ope a o (panel b) unde
di e en le els o s a e pa icipa ion
(θ=0.20, 0.50, 0.80)
. In bo h cases, expec ed p o i s
dec ease linea ly as p i a e in o ma ion
(Πi)
inc eases, indica ing ha a highe alua ion o
he auc ioned ime slo by he ope a o s ansla es in o lowe expec ed p o i s. Addi ionally,
he di e ences be ween he cu es o di e en alues o
θ
a e minimal, sugges ing ha
he ans e o e enues o he s a e does no ha e a signi ican impac on he ope a o s’
p o i s uc u e.
Games 2025,16, 16 17 o 19
(a) (b)
Figu e A1. Simula ion o expec ed p o i s wi h di e en le els o s a e pa icipa ion: (a) P o i s o he
ela ed ope a o . (b) P o i s o he independen ope a o .
To u he examine he impac o
θ
on ma ke compe i i eness, we conduc a sensi i i y
analysis by a ying
θ
be ween 0 and 1 and e alua ing i s e ec on he p obabili y o success
o he un ela ed bidde .
1. Simula ion se up:
•θ=
0.0: No in e nal cos -sha ing; Pe uRail bids as i i we e an independen
ope a o .
•θ=
0.5: Pa ial in e naliza ion, simula ing a mode a e ad an age o he ela ed
ope a o .
•θ=1.0: Full in e naliza ion, ep esen ing a comple ely dis o ed auc ion.
2. Resul s:
•
As
θ
inc eases, he likelihood o Pe uRail winning he auc ion ises non-linea ly,
con i ming ha cos -sha ing incen i izes mo e agg essi e bids.
•
Fo
θ>
0.5, he p obabili y o Inca Rail secu ing a ime slo d ops below 30%,
demons a ing a s ong an icompe i i e e ec .
•
A
θ
-adjus ed subsidy o co ec ion ac o could mi iga e hese dis o ions wi hou
imposing s ic egula o y in e en ions.
These indings sugges ha
θ
should be egula ed wi hin a compe i i e ange (0.2–0.5)
o ensu e ha new en an s ha e a iable pa hway o winning auc ions.
This esul implies ha he modi ica ion o he paymen scheme does no subs an ially
al e he compe i i e dynamics o he ailway ma ke . Bo h he ela ed and independen
ope a o s expe ience a simila educ ion in p o i s as p i a e in o ma ion inc eases, indica -
ing ha s a e pa icipa ion in he a i does no c ea e a signi ican compe i i e ad an age
o disad an age. Howe e , he pe sis ence o his downwa d end sugges s ha ope a o s’
s a egic incen i es may s ill a o he dominan company, making i necessa y o assess
whe he addi ional egula o y e o ms could s eng hen compe i ion and imp o e ai ness
in he alloca ion o ime slo s.
Games 2025,16, 16 18 o 19
No es
1
The Sou he n Railway ocuses on eigh anspo a ion along an ex ensi e ou e ha connec s majo ci ies and po s in sou he n
Pe u, while he Sou h-Eas Railway is p ima ily dedica ed o passenge anspo a ion o Machu Picchu, ea u ing a sho e ou e
and dis inc echnical cha ac e is ics.
2
Cu en ly, wo companies ope a e in he ailway anspo a ion ma ke : Inca Rail S.A.C., he esul o he 2013 me ge be ween
Inca Rail and Andean Railways Co p. S.A., and Pe uRail. Bo h companies compe e by p o iding se ices in his sec o , p ima ily
ca e ing o ou is and eigh ou es in he coun y.
3
an Ko en (2012) p oposes an economic model o ansmission capaci y auc ions among elec ici y gene a o s, whose esul s can
be gene alized and applied o he ailway indus y.
4
The conclusions o his s udy emain alid e en i he numbe o wagon-kilome e s mo ed by each bidde di e s om 1. The
assump ion o an iden ical numbe was a simpli ied app oach o be e iden i y dis o ions unde symme ical condi ions.
Howe e , in p ac ice, in as uc u e usage can a y signi ican ly due o di e ences in he ope a ional and s a egic cha ac e is ics
o each ope a o .
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au ho (s) and con ibu o (s) and no o MDPI and/o he edi o (s). MDPI and/o he edi o (s) disclaim esponsibili y o any inju y o
people o p ope y esul ing om any ideas, me hods, ins uc ions o p oduc s e e ed o in he con en .
