Jua ez, Ruben; Wu, Michael
A icle
Rou ing-p oo ness in conges ion-p one ne wo ks
Games
P o ided in Coope a ion wi h:
MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Jua ez, Ruben; Wu, Michael (2019) : Rou ing-p oo ness in conges ion-p one
ne wo ks, Games, ISSN 2073-4336, MDPI, Basel, Vol. 10, Iss. 2, pp. 1-18,
h ps://doi.o g/10.3390/g10020017
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h ps://hdl.handle.ne /10419/219240
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games
A icle
Rou ing-P oo ness in Conges ion-P one Ne wo ks
Ruben Jua ez * and Michael Wu
Depa men o Economics, Uni e si y o Hawaii, 2424 Maile Way, Saunde s Hall 542, Honolulu, HI 96822, USA;
[email p o ec ed]
*Co espondence: [email p o ec ed]
Recei ed: 31 Decembe 2018; Accep ed: 25 Ma ch 2019; Published: 3 Ap il 2019


Abs ac :
We conside he p oblem o sha ing he cos o connec ing a la ge numbe o a omless
agen s in a ne wo k. The cen alized agency elici s he a ge nodes ha agen s wan o connec ,
and cha ges agen s based on hei demands. We look o a cos -sha ing mechanism ha sa is ies
h ee desi able p ope ies: e iciency which cha ges agen s based on he minimum o al cos o
connec ing hem in a ne wo k, s and-alone co e s abili y which equi es cha ging agen s no mo e
han he cos o connec ing by hemsel es di ec ly, and limi ou ing-p oo ness which p e en s agen s
om p o i able epo ing as se e al agen s connec ing om A o C o B ins ead o A o B. We show
ha hese h ee p ope ies a e no always compa ible o any se o cos unc ions and demands.
Howe e , when hese p ope ies a e compa ible, a new egali a ian mechanism is shown o sa is y
hem. When he p ope ies a e no compa ible, we ind a ule ha mee s s and-alone co e s abili y,
limi ou ing-p oo ness and minimizes he budge de ici .
Keywo ds: cos sha ing; co e s abili y; ou ing p oo ness
1. In oduc ion
Conges ion is a o m o nega i e ex e nali y o he ne wo ks o any size. These ne wo ks can
include bu a e no limi ed o oad ne wo ks, in e ne ne wo ks, elecommunica ion ne wo ks as
well as u ili y powe g id ne wo ks and o he ne wo ks [
1
–
6
]. The eme gence o conges ion can be
a ibu ed o he ins ance in which a la ge numbe o sel ish agen s, each wishing o ou e a ic
ia one’s p e e ed links (o edges), wi hou aking in o accoun he ac ions o o he agen s in he
ne wo k [
7
]. Such sel ish ou ing beha io gene a es disc epancy be ween he socially op imal a ic
pa e n (minimizing o e all delay o cos ) and he sel ish- ou ing beha io al ou come (also known
as he non-coope a i e Nash equilib ium). [
8
,
9
] sugges ha he non-coope a i e Nash equilib ium
need no in gene al op imize he social wel a e (minimizing o e all delay o cos in his con ex ).
Indeed, he non-coope a i e Nash equilib ium could gene a e a loss subs an ially highe han he
social op imum. This implies ha he disc epancy could cause some unwan ed di ec o indi ec
consequences such as ne wo k conges ion, a ic pa alysis as well as social wel a e loss.
An ea lie wo k by Roughga den [
8
] in es iga es and quan i ies he e iciency loss o he ne wo k
pe o mance due o he sel -in e es ed agen s. The deg ada ion is calcula ed by he wo s -possible a io
be ween he Nash equilib ium and he social op imal a ic and is called he p ice o ana chy (POA) [
8
].
The close he POA is o one, he less e iciency loss i gene a es be ween he Nash equilib ium and
social op imum. Roughga den shows ha i he cos (la ency) unc ions on he links in he ne wo k a e
linea , hen he POA is equal o
4
3≈
1.333; i he cos unc ions a e cubic, hen he POA is app oxima ely
1.896; and i he cos unc ions a e polynomials o deg ee
≤p
whe e
p>
0, hen POA is a asymp o ically
Games 2019,10, 17; doi:10.3390/g10020017 www.mdpi.com/jou nal/games
Games 2019,10, 17 2 o 18
equal o
θp
ln p
as
p→∞1
. Finally, i he cos unc ions a e un es ic ed o any allowable class o
unc ions (e.g., linea , quad a ic, cubic, polynomials, e c.), hen he POA is unbounded. These esul s
indica e ha he e iciency loss can be immensely la ge and he consequences gene a ed by he sel ish
ou ing beha io can be ex emely se e e in he ne wo k.
In o de o a oid such consequences, i is he e o e impo an o manage he conges ion o
a ne wo k. In his pape , we conside a ne wo k wi h a se o nodes in which e e y node in e connec s
wi h a leas one o he node by a link. The e is a a iable conges ion cos associa ed wi h each link
ha depends on he uni s o demands sen by agen s. The cos may ep esen any main enance cos
o accessing ee o he links. Di e en links may ha e di e en conges ion cos s. The e a e a la ge
numbe o sel -in e es ed agen s who wish o ou e hei a ic om an a bi a y node o ano he node
using he links in he ne wo k. We assume he sel -in e es ed agen s a e in e es ed in ou ing only ia
links ha will incu he minimal cos o delay. In addi ion, we also include he possibili y o a single
agen pe o ming a special ype o s a egic maneu e (o simila ly, ou ing maneu e ). The concep o
s a egic maneu e was i s in oduced by Dominique and Moulin [5], and ollowed by Moulin [10],
who s udy mechanisms whe e coali ion o agen s in a la ge ne wo k can u i ely c ea e se e al alias
and mis epo hemsel es as se e al agen s. Fo mally, ha means an agen may epo and ou e he
a ic o he a ge node as wo o mo e agen s in hope o lowe ing he connec ion cos . To illus a e
such scena io, we suppose he e is an agen wi h ini ial node
i
and a ge node
j
. Then, she may epo
he sel as wo agen s i s connec ing om node
i↔k
, hen om node
k↔j
i she inds such s a egic
maneu e is less cos ly han connec ing di ec ly om node i↔j.
Unlike he decen alized ne wo ks such as oad ne wo ks o in e ne ne wo ks, he ne wo ks
being conside ed in his pape a e ope a ed in a cen alized manne . We imagine he e is a cen al
au ho i y ha possesses he powe o elici he demand cha ac e is ics om all agen s in he ne wo ks
and is capable o compu ing and implemen ing he op imal a ic pa e n a any ime. Meanwhile,
we imagine he e is a cen alized agency ha will help sa is y he needs demanded by he agen s
a minimal o al delay o cos . Since agen s a e sel -in e es ed, hey need no ha e he incen i e o
e eal hei ue ou ing schemes. The cen alized agency is he e o e ulne able o any mis epo s
by indi idual uni s, o by a coali ion o subg oup o uni s. Such s a egic maneu e beha io will
po en ially dis o he socially op imal a ic pa e n (minimizing o e all cos o delay) in he sense
ha links wi h ela i ely lowe cos a e mo e likely o be conges ed, which may in u ns incu ing
highe o al cos o delay.
The e o e, in o de o p e en e ec i ely such s a egic maneu e s, he eby imp o ing he
pe o mance o a ne wo k, he e is a need o design cos -sha ing mechanisms o cha ge on each agen
in he ne wo k so ha he ac ion o s a egic maneu e will make no di e ence in e ms o lowe ing
connec ion cos s. Moulin calls any cos -sha ing mechanisms o cos -sha ing ules ou ing-p oo
i hey a e in ulne able o such s a egic maneu e s [
10
]. Technically, i is no as di icul o ind
a ou ing-p oo mechanism, i ou ing-p oo ness is he only cons ain ha we a e aiming o sa is y.
Fo ins ance, he simple a e age cos mechanism (AC) will do
2
. By spli ing he o al cos equally
among all agen s in he ne wo k, each agen is o pay a sha e o he a e age cos . As a esul , any agen
who in ends o epo as wo o mo e agen s will simply ha e o pay double o mo e o he a e age cos .
Ne e heless, his mechanism may no be cha ging e e y agen o coali ion ai ly i he a e age cos
exceeds he cos o a subne wo k mee ing all connec ion needs o he g oup (s and-alone objec ions).
This ai ness- ela ed cons ain is known as s and-alone co e s abili y and is he p ima y design
cons ain in he minimum cos spanning ee p oblem [10] and a a ie y o o he p oblems.
1
See [
8
] o mo e POAs o di e en classes o unc ions such as he quad a ic, M/M/1 delay unc ions and M/G/1
delay unc ions.
2The a e age cos mechanism has been used success ully in a a ie y o ne wo k p oblems, e.g., [2,11].
Games 2019,10, 17 3 o 18
Ou wo k, he e o e, has he objec i e o design o p opose a cos -sha ing mechanism sa is ying
he ollowing h ee p ope ies:
•E iciency: he minimum o al cos o connec ing all agen s o hei a ge nodes;
•
S and-alone co e s abili y: no g oups o agen s pay mo e han he cos o a subne wo k mee ing
all connec ion needs o he g oup;
•
Rou ing-p oo ness: no agen can lowe i s cos by epo ing as se e al agen s along an al e na i e
pa h connec ing he a ge nodes.
1.1. Inno a ion o he Pape and O e iew o he Resul s
While e iciency and s and-alone co e s abili y a e s anda d p ope ies in he li e a u e (see ela ed
li e a u e sec ion), he main inno a ion o his pape comes om s udying he h ee p ope ies oge he
in a la ge economy wi h conges ion. P e ious wo k by Moulin [
10
] s udied hese p ope ies in
a model wi hou conges ion cos , whe e he cos associa ed wi h each link is ixed and he cos s a e
independen o he uni s o a ic sen by agen s. This assump ion, while con enien , is no ealis ic
in a a ie y o ne wo ks such as a ic, elecommunica ion and o he la ge and complex ne wo ks
(e.g., he in e ne ). Ou pape ex ends he model in oduced by Moulin [
10
] o he case wi h a iable
conges ion cos s ha depend on he uni s o a ic being connec ed in ou ne wo k.
The adop ion
o a iable conges ion cos s subs an ially inc eases he di icul y o compu ing he e iciency o a
gene al ne wo k. In pa icula , compu ing he social op imum (e iciency) o a gene al ne wo k is
Non-de e minis ic Polynomial- ime-ha d (NP-ha d) [
12
], unless he cos unc ions sa is y ce ain
condi ions (e.g., linea i y o all cos unc ions a e iden ical in he ne wo k [
12
]). Thus, ou wo k is
sha ply di e en om [
10
] whe e he minimum cos (e iciency) can easily be compu ed by applying a
simple g eedy algo i hm [13].
The second inno a ion o he pape comes om s udying he case whe e he e a e a la ge numbe
o sel -in e es ed agen s in he ne wo k such ha each o hem only con ols a negligible ac ion
o he o e all a ic (i.e., agen s ha e ze o mass). These ypes o assump ions a e c i ical o he
in e ne and o he la ge an complex ne wo ks, whe e e en i an agen e ou es in he ne wo k, hen
because he amoun o a ic each agen con ols is in ini esimally small ela i e o he o e all a ic,
he cos s cha ged o agen s should no be oo sensi i e and changing signi ican ly. This assump ion is
also ele an o e ou ing in ai line ne wo ks, whe e agen s wishing o ly om A o B, can ins ead
buy wo one-way icke s, om A o C and C o B, i p o en cheape han a di ec icke om A o
B. One agen manipula ing he ai line sys em will ha e no e ec on p ices and a subs an ially la ge
amoun o popula ion is needed o changes in p ices o ake e ec . Fu he mo e, in p ac ice, i is
ela i ely ha d o coo dina e a la ge g oup o agen s ha change p ices in a ne wo k. In o de o
add ess his ea u e, we in oduce a new axiom, he limi ou ing-p oo ness, ha adap s Moulin’s
concep o ou ing-p oo ness o he case whe e agen s a e ex emely small ela i e o he o e all a ic.
We s udy he limi a ion o such an axiom in a conges ion-p one ne wo k. In pa icula , we p o ide he
necessa y and su icien condi ions o he exis ence o ules ha sa is y limi ou ing-p oo ness along
wi h e iciency and s and-alone co e s abili y (Theo em 1).
A hi d inno a ion o he pape comes om ac ually in oducing a mechanism ha sa is ies hese
h ee p ope ies when hey a e compa ible. Ou mechanism builds on pe haps he mos s udied
mechanism in he cos -sha ing li e a u e, he uni o m ule [
14
,
15
], and adap i o ou ne wo k se ing
wi h conges ion in a way ha Limi ou ing-p oo ness is sa is ied. In o de o compu e he ou come o
he mechanism, we ac ually p o ide an algo i hm o compu e he p ices ha agen s pay, and show
ha such p ices coincide wi h he cos -sha es o ou new mechanism (P oposi ion 1).
1.2. Rela ed Li e a u e
The s and-alone co e s abili y and e iciency a e wo adi ional p ope ies in he ne wo k
li e a u e [
4
,
10
,
16
,
17
]; while ou ing-p oo ness is a ela i ely mo e ecen idea [
5
,
10
,
18
]. The p edecesso
Games 2019,10, 17 4 o 18
o ou ing-p oo ness was i s coined he axiom o no ansi o igina ed by Han ie and Moulin in [5].
In [
5
], Han ie and Moulin sugges wo cos -alloca ion me hods: p i a e-cos and ex e nal cos me hods
ha sa is y he axiom o addi i i y, sus ainabili y and no ansi . These me hods wo k only o he case
wi hou conges ion.
The closes pape o ou wo k is Moulin [
10
], who s udies a conges ion- ee ne wo k cos -sha ing
p oblem and whe e he no ansi axiom is enamed as ou ing-p oo ness. The pape cons uc s
wo co e s able and ou ing-p oo cos -sha ing ules: weigh ed Shapley alue (WSH) and weigh ed
spanning ule (WSP) oge he sa is ying he s and-alone co e s abili y and ou ing-p oo ness. The i s
amily o he ules equi es only ha he connec ing cos s o be ze o o one, whe eas he second amily
allows o a bi a y cos s bu equi es a spanning a ic. As such, nei he o hese ules wo k in ou
gene al conges ion-p one se ing.
Ano he ela ed pape is a eal-wo ld applica ion o ou ing-p oo ness by Dong, Guo and
Wang [
18
]. In he e, a highway oll p icing me hod is p oposed o assign he oll cha ges o di e en
ypes o agen s in a linea ne wo k. The p oposed me hod sa is ies he s and-alone es o ehicles,
cos eco e y ( o oad sec ions wi h no c oss-subsidizing) and a e sion o ou ing-p oo ness
in he sense ha no agen s can educe he oll cha ges by exi ing and e-en e ing he highway
consecu i ely [
18
]. In con as wi h he pape s abo e, he model p esen ed in [
18
] is a conges ion-p one
ne wo k a he han conges ion- ee, bu he p oposed me hod only wo ks in a linea ne wo k, whe eas
ou ne wo ks would ypically ha e cycles, and ha is whe e he s eng h o ou ou ing-p oo ness
p ope y comes om.
Rela ed wo k also comes om he cos -sha ing li e a u e on he implemen a ion o e icien
cos s. Jua ez and Kuma [
2
] discuss he app op ia eness on he implemen a ion o a ious e icien
cos -sha ing mechanisms when agen s choose hei pa hs s a egically as i hey a e in a ou ing game.
A compa ison o di e en classes o mechanisms such as he a e age cos mechanism, p opo ional
o s and-alone mechanism, (weigh ed) egali a ian mechanism and he Shapley mechanism is done.
I is impo an o no e ha he pape does no deal wi h ou ing-p oo ness issues, and as such, all he
mechanisms s udied he e ail o mee ou es .
O he implemen a ion wo k include Hougaa d and T ede [
19
,
20
], who cha ac e ize cos -minimizing
ne wo ks a he equilib ium o games by changing he announcemen ule. The sequen ial sha ing
o cos s and p o i s is s udied axioma ically in Jua ez e al. [
11
]. Kuma [
21
] also in es iga es secu e
implemen a ion in cos -sha ing models wi hou conges ion, while
Hougaa d e al. [22]
s udy he
sha ing o p o i s in ees and Moulin e al. [
23
] s udy he di ision o he cos s/p o i s o non- edundan
i ems. In con as wi h his li e a u e, ou wo k is he i s o add ess om an axioma ic iew he
di ision o cos s in conges ion-p one ne wo ks by a omless agen s.
2. The Model
A ne wo k cos -sha ing p oblem (o simply a p oblem) is a iple
P=hN
,
C
,
θi
, whe e
N= (V
,
L)
is a ne wo k wi h ini e se o nodes V={n1, ..., n }and a ini e se o undi ec ed links L={l1, ..., ls}
connec ing such nodes.
C= (C1(·)
, ...,
Cs(·))
is a ec o o cos unc ions associa ed o each link,
whe e he cos unc ion
Ci:R+→R+
associa ed o link
li
is s ic ly inc easing and s ic ly con ex.
The ec o
θ= (θ1
, ...,
θs)∈Rs
+
is he a ic demand associa ed o each link, whe e
θi
ep esen ing he
uni s o a ic demanded on link
li
(i.e.,
θi
ep esen s he mass o he agen s in e es ed in connec ed
he wo nodes o link li). We deno e by P he se o all ne wo k cos -sha ing p oblems.
Fo an a bi a y ne wo k cos -sha ing p oblem
P
, he link
li
will be simply deno ed by
i
i he e is
no con usion. Gi en he cos unc ion
Ci
associa ed o link
i
, we deno e by
ACi(x) = Ci(x)
x
he a e age

Games 2019,10, 17 5 o 18
cos associa ed o such link. Le
Pi={p1
, ...,
p }
be he se o all easible pa hs wi hou cycles ha
connec link iin he ne wo k.3
De ini ion 1.
(a) Gi en a pa h
k
and
α>
0, we deno e by
α·k= ( 1
, ...,
s)∈Rs
such ha
i=α
i
i∈k
and i=0i i 6∈ k.
(b) Gi en any link
i∈L
and a demand
θi
o ha link, a connec ion ec o
(wi
1
, ...,
wi
s)
gene a es demand
θi
i
he e exis s non-nega i e weigh s {αk}k∈Pisuch ha ∑k∈Piαk=θiand (wi
1, ..., wi
s) = ∑k∈Piαk·k.
(c) Gi en demands
(θ1
, ...,
θs)
, a connec ion ec o
(w1
, ..,
ws)
is easible i o e e y link
i∈ {
1, ...,
s}
he e
exis s a connec ion ec o (wi
1, ..., wi
s) ha gene a es demand θisuch ha (w1, ..., ws) = ∑s
i=1(wi
1, ..., wi
s).
While agen s demanding link
i
a e in e es ed in connec ing he nodes o such link, his connec ion
can be achie ed di ec ly ia link
i
o indi ec ly ia al e na i e pa hs connec ing link
i
. In pa icula ,
he demands o a ic
θi
on link
i
can be spli in o mul iple al e na i e pa hs whe e each pa icula
pa h indi ec ly connec s link
i
(pa (b) in de ini ion abo e). A connec ion ec o
(w1
, ...,
ws)
is said o
be easible i i is he sum o all hese pa hs.
De ini ion 2.
The o al cos (o simply a cos ) o a easible connec ion ec o
(w1
, ...,
ws)
is
C(w1
, ...,
ws) =
∑s
i=1Ci(wi).
Wi h each elemen
wi
ep esen s all possible amoun s o a ic on link
i
, he o al cos o connec ing
all agen s wi h connec ion ec o
(w1
, ...,
ws)
is he sum o each cos unc ion
Ci(·)
e alua ed a
wi
o
all i=1, ..., s.
Recall ha we a e in a cen alized se ing whe e he planne has he abili y o selec he pa hs
o some agen s demanding a gi en link. A na u al objec i e o he planne is o selec a pa h ha
minimizes he cos . This is inco po a ed in he ollowing de ini ion.
De ini ion 3.
Gi en a se o demands
(θ1
, ...,
θs)
, a easible connec ion ec o
w∗
1, ..., w∗
s
is e icien i o any
o he easible connec ion ec o (
w1, ...,
ws), we ha e C(w∗
1, ..., w∗
s)≤C(
w1, ...,
ws).
We say ha a easible connec ion ec o o a gi en se o demands is e icien i he e exis s
a easible connec ion ec o
(w∗
1
, ...,
w∗
s)
such ha he o al cos e alua ed a
(w∗
1
, ...,
w∗
s)
is always less
han o equal o he o al cos e alua ed any o he connec ion ec o s
(
w1
, ...,
ws)
. Such connec ion
ec o will gene a e he leas cos (also e e ed as e icien cos ) o se ing all he agen s in he ne wo k.
De ini ion 4
(Mechanism and e icien mechanism)
.
(a) A cos -sha ing mechanism (also called ule o
solu ion) is a con inuous unc ion
ξ:P→Rs
+
ha alloca es a se o cos sha es o he use o e e y link a
e e y ne wo k cos -sha ing p oblem. (b) A cos -sha ing mechanism is e icien a p oblem
P
i i di ides he
e icien cos among agen s:
∑
i∈L
ξi(P)·θi=C∗(P)
whe e C∗(P)is he e icien cos a a p oblem P.
We ocus on a mechanism ha alloca es paymen s based on he demands o single links. Tha is,
he planne canno dis inguish he names o agen s who demand he same link. This is a s anda d
assump ion in he cos -sha ing li e a u e. An e icien cos -sha ing mechanism is a cen alized
mechanism ha mee s he demands e icien ly and edis ibu e his cos based only on he demand o
3
A pa h is a sequence o consecu i e nodes in he ne wo k. We say ha a pa h has no cycles i he nodes do no epea . We say
ha a pa h connec s link
i
i he pa h s a s in one o he nodes o link
i
and ends in he o he . When he e is no con usion,
we also e e o a pa h by i s links ins ead o i s nodes.
Games 2019,10, 17 6 o 18
e e y link. Thus, he sum o he cha ges assigned by he mechanism, when mul iplied by he demands
o each link, is equal o he e icien cos a a gi en p oblem.
De ini ion 5
(S and-alone co e s abili y)
.
Conside a ne wo k cos -sha ing p oblem
P=hN
,
C
,
θi
.
A cos -sha ing mechanism
ξ
sa is ies he s and-alone co e s abili y a
P
i o any link
i
, we ha e ha
ξi(P)θi≤Ci(θi).
De ini ion 5 ensu es he absence o s and-alone objec ions om agen s caused by o e cha ging
om he implemen ed mechanism. This cons ain plays an impo an ole in e ms o ai ness o
p e en agen s om being cha ged mo e han he cos o he links hey ac ually demanded. Mo eo e ,
i also helps o p e en agen s who demand inexpensi e links om subsidizing hose who demand
expensi e links in a ne wo k.
De ini ion 6
(Limi ou ing-p oo ness)
.
Conside a ne wo k cos -sha ing p oblem
P=hN
,
C
,
θi
.
A cos -sha ing mechanism is limi ou ing-p oo (LRP) a
P
i o each link
i
, and each pa h
pk∈ Pi
ha
connec s link i we ha e ha :
ξi(P)≤∑
j∈pk
ξj(P). (1)
Limi ou ing-p oo ness p e en s agen s om ou ing hei demand along al e na i e pa hs ha
may also mee hei demand. In pa icula , a a p oblem
P
, by Equa ion (1), an agen demanding link
i
pays
ξi(P)
. I such an agen decides o demand an al e na i e pa h
pk
ha also connec s
i
(by posing
as sepa a e agen s along pa h
pk
) his cha ge will be
∑
j∈pk
ξj(P)
, which is no smalle han
ξi(P)
unde
limi ou ing-p oo (LRP).
3. E iciency, S and-Alone Co e S abili y and LRP
The ade-o be ween s and-alone co e s abili y and incen i e compa ibili y (he e, in e p e ed
as ou ing-p oo ness) o e icien mechanisms has al eady been iden i ied in he li e a u e. Indeed,
in a ela ed ne wo k model whe e he connec ion demands o agen s need o be me , Jua ez and
Kuma [
2
] s udy wo mechanisms: he a e age-cos mechanism (AC) ha di ides he cos o he
e icien ne wo k equally among all he agen s and he p opo ional o s and-alone mechanism (PR)
ha di ides he cos o he e icien ne wo k in p opo ion o he cos o hei demands.
4
In ou se ing,
he AC mechanism mee s LRP, as any agen pays he same cos -sha e ega dless o he link demanded,
hus an agen spli ing hei demand in wo o mo e links will pay wice o mo e hei cos -sha e.
Un o una ely, i is easy o see ha AC does no mee he s and-alone co e s abili y, as i migh happen
ha agen s demanding a e y inexpensi e link c oss-subsidize agen s demanding e y expensi e links.
PR is an exac opposi e si ua ion, mee ing s and-alone s abili y bu ailing o mee LRP. The e o e,
nei he o hem mee ou h ee objec i es. O he adi ional mechanisms, like Aumann–Shapley,
also ail o mee ou demanding LRP. The e o e, in add essing he compa ibili y be ween s and-alone
co e s abili y, e iciency and limi ou ing-p oo ness, new mechanisms need o be de eloped.
One o he mos s udied ules in he cos -sha ing li e a u e is he uni o m ule [
14
,
15
,
24
], which
is a a ia ion o AC ha espec s and-alone co e s abili y. This ule can be i ially adap ed o ou
ne wo k se ing by conside ing he cos o each link as he demand o he agen s.
5
Bo h he uni o m
ule and AC di ide he o al cos o he ne wo k ac oss all agen s, howe e , hei simila i y begins o
di e ge when he cha ges each he h eshold o he s and-alone cos o a link (p esumably he smalles
s and-alone cos i s ). In ha case, AC will dis ega d he s and-alone cos and con inue inc easing
4Fo b e i y, we do no epea he o mal de ini ion o hese mechanisms al eady discussed in Jua ez and Kuma [2].
5See [2,25] o he applica ion o he Uni o m ules o ne wo k p oblems.
Games 2019,10, 17 7 o 18
cha ges o agen s on each link
i
equally wi hou bound. In con as , he uni o m ule will s op assigning
cha ges o agen s whose links ha e eached hei s and-alone cos s. The emaining cos will hen be
co e ed by he links in which he s and-alone cos s a e ye o be eached. No e ha his cha ac e is ic
will de ini ely p ese e he p ope y o s and-alone co e s abili y. Un o una ely, he uni o m ule does
no mee LRP. The nex sec ion p o ides a a ia ion o he uni o m ule ha mee s LRP, s and-alone
co e s abili y and e iciency, whene e possible. We call i he egali a ian mechanism.
3.1. A New Egali a ian Mechanism
Gi en a ne wo k cos -sha ing p oblem
P=hN
,
C
,
θi
, ou ul ima e objec i e is o ind
a cos -sha ing mechanism such ha he se o cos sha es alloca ed by he mechanism will be e icien ,
limi ou ing-p oo and co e s able. In o de o achie e ha , we o mula e he ne wo k cos -sha ing
p oblem along wi h he cons ain s in o a se o sys em o equa ions.
s
∑
i=1
λiθi=EFF(θ1, ..., θs)(2)
λiθi≤Ci(θi) o all i=1, ..., s(3)
λi≤∑
l∈pk
λl o all pk∈ Piand o all i=1, ..., s(4)
λi≥0 o all i=1, ..., s. (5)
We deno e
λi
he paymen cha ged o each agen on link
i
. O he a iables a e de ined exac ly
he same way as he model in Sec ion 2.
EFF(θ1
, ...,
θs)
ep esen s he e icien cos o he ne wo k
compu ed as a unc ion o
(θ1
, ...,
θs)
. We exp ess e iciency in his o m because compu ing he
e iciency (social op imali y) o a ne wo k is gene ally a NP-ha d p oblem.6
Equa ion (2) equi es ha he sum o he cha ges, when mul iplied by he demands o e all links
i=
1, ...,
s
is exac ly equal o he e iciency o he ne wo k. In pa icula , his cons ain ep esen s he
e iciency p ope y.
Equa ion (3) desc ibes he s and-alone co e s abili y o all agen s demanding link
i
. Essen ially,
his cons ain says ha he amoun o assigned cha ges o each agen on link
i
should ne e exceed
he cos o connec ing he a ge pai s by hemsel es ia link i.
Equa ion (4) ep esen s he limi ou ing-p oo ness cons ain o a gene al ne wo k. Recall ha
o any link
i
,
Pi
is he se o all easible al e na i e pa hs
{p1
, ...,
p }
ha connec link
i
. The limi
ou ing-p oo ness cons ain hen implies ha he cha ge
λi
on link
i
mus be less han o equal o he
sum o λl o l∈pkon all possible pa hs pk∈ {p1, ..., p }7.
Equa ion (5) is an ob ious one, which says ha he cos -sha es assigned o agen s mus be g ea e
han o equal o ze o. The sys ems (2)–(5) migh ha e mul iple solu ions o hey migh no ha e any,
is i will be shown in he ollowing sec ions.
De ini ion 7
(Egali a ian solu ion a
λ∗
)
.
Conside a pa ame e
λ∗>
0. The egali a ian solu ion a
λ∗
,
deno ed by
(EGTλ∗)
, assigns a he ne wo k cos -sha ing p oblem
hN
,
C
,
θi
he se o cos -sha es
(λ∗
1
,
. . .
,
λ∗
s)
such ha he cos sha e o link i =1, ..., s is de e mined by
λ∗
i=min
pk∈Pi(λ∗,ACi(θi),"∑
l∈pk
λ∗
l#). (6)
6
Moulin [
4
] sugges s ha i no simple algo i hm can cha ac e ize he cos minimiza ion ou comes, hen we will ha e no
choice bu o d op he cos minimiza ion equi emen and u n o he exogenously gi en subop imali ies.
7
No e ha i a simple iangula ne wo k is analyzed he e, hen he ou ing-p oo ness cons ain is jus a se o
iangle inequali ies.
Games 2019,10, 17 8 o 18
This solu ion achie es e enue equal o Tλ∗=s
∑
i=1
λ∗
iθi.
No e ha
EGTλ∗
is a well de ined cos -sha ing solu ion, as he e exis s a unique se o alues ha
sa is y Equa ion (6). They can be easily compu ed by s a ing o small alues, say a
λ∗=
0, a which
poin he solu ion e u ns
λi=λ∗
o all
i=
1, ...,
s
. As we inc ease he alue o
λ∗
, he s and-alone
cons ain (
ACi(θi)
) o LRP cons ain s (
∑
l∈pk
λ∗
l
) bind. The o mal compu a ion o his ule is desc ibed
in he nex sec ion.
By cons uc ion, egali a ian mechanism (EG),
EGTλ∗
mee s s and-alone co e s abili y and LRP,
bu i may no be e icien . Indeed, he e enue gene a ed by EG
Tλ∗
equals o
Tλ∗=∑s
i=1λ∗
iθi
which
may be smalle o la ge han
EFF(θ1
, ...,
θs)
. Clea ly,
Tλ∗
is non-dec easing in
λ∗
, and i may be ha
Tλ∗
is ac ually cons an o a la ge alue o
λ∗
. Indeed, conside wo alues
λ∗
and
λ∗
such ha
λ∗>λ∗>ACi(θi)
o all
i
. Unde
EGTλ∗
, he paymen
λ∗
i
o he agen s along link
i
is no la ge
han
ACi(θi)
. Thus, any inc eases o
λ∗>λ∗
will also hi he same s and-alone cons ain s, and he
cos -sha es o he agen s emain unchanged.
F om he analysis, he maximum e enue gene a ed unde EG
Tλ∗
occu s when
λ∗=
max {AC1(θ1), ..., ACs(θs)}
. We deno e by
Tmax
λ∗
such a maximum e enue. No e ha i he e enue
Tmax
λ∗>EFF(θ1
, ...,
θs)
, hen by he con inui y o
Tλ∗(λ∗)
, we can always ind one
λ∗∗
such ha
Tλ∗∗ =EFF(θ1
, ...,
θs)
. This
λ∗∗
will gene a e an alloca ion EG
Tλ∗∗ 8
and i will be called he egali a ian
solu ion (
EGTλ∗
). No e also ha when his egali a ian solu ion exis s, i will be e icien , co e s able
and limi ou ing-p oo .
The compu a ion o EG abo e is simila o he compu a ion o he uni o m ule, excep o he
addi ional LRP cons ain s,
"∑
l∈pk
λ∗
l#
, in Equa ion (6). Such a a ia ion subs an ially changes how EG
assigns cos sha es o he agen in di e en links. In con as wi h he uni o m ule, he EG mechanism
p oposed in his pape migh no exis s o an a bi a y ne wo k cos -sha ing p oblem
P
. The exis ence
o EG depends on whe he he las s and-alone cos being eached can co e he loss incu ed by he
eplacemen o he limi ou ing-p oo ness cons ain s. We see his in he nex sec ions.
3.2. A Nume ical Example o EG
In his sec ion, we will p o ide a s ep-by-s ep nume ical example illus a ing how EG alloca es
cha ges o agen s. Conside he ne wo k in Figu e 1and a demand
θ= (
2, 6, 3
)
. Tha is, a o al o
11 uni s o a ic we e demanded: wo uni s a e demanded on link 1; six uni s we e demanded on link
2; and h ee uni s we e demanded on link 3.
1
2 3
(θ2)2
2
(θ3)23
2(θ1)2
1
Figu e 1. A simple ne wo k wi h h ee nodes.
8
I i happens ha
Tλ∗∗
is he maximum e enue ha EG can achie e, hen
Tλ∗∗ =Tmax
λ∗∗
and he alloca ion EG
Tλ∗∗
coincides
wi h EGTmax
λ∗∗ .
Games 2019,10, 17 15 o 18
Finally, ou wo k has only shown he exis ence o a mechanism mee ing e iciency, s and-alone
co e s abili y and limi ou ing-p oo ness, bu has no p o ided a ull cha ac e iza ion o a mechanism.
Mo e wo k is needed o de elop a cha ac e iza ion using axioms in he spi i o limi ou ing-p oo ness.
Au ho Con ibu ions:
R.J. and M.W. concep ualized he pape , ound he esul s and w o e hem. They
con ibu ed equally o he s udy.
Funding: This esea ch ecei ed no ex e nal unding.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
Appendix A. P oo s
Appendix A.1. P oo o P oposi ion 1
P oo .
Le
(P1
, ...,
Ps)
be he p ices gi en by he RP algo i hm and le
(λ∗
1
, ...,
λ∗
s)
be he EG
Tmax
λ∗
solu ion.
Fix a p oblem
P
and le
Tmax
λ∗
be he maximal e enue a ge achie ed by EG and he RP algo i hm,
we will show ha
(P1
, ...,
Ps) = (λ∗
1
, ...,
λ∗
s)
. Wi hou loss o gene ali y, suppose ha he a e age cos
unc ions sa is y AC1(θ1)<· · · <ACs(θs).
Sol ing by RP algo i hm: we assign empo a y p ices
Ti=ACi
on each link
i
. A each poin
in ime, he smalles empo a y p ice is eplaced by a pe manen p ice, i.e.,
Ti=ACi(θi) = Pi
un il
any semi-cycle is o med. Le
j
be he las link ha comple es he semi-cycle. We eplace
Tj
by
T0
j=min {ACj
|{z}
Tj
,∑l∈pkPl}.
Case 1: i T0
j=min nACj,∑l∈pkPlo=ACj, hen we ha e:
Subcase 1: i T0
j=ACjis he nex smalles ACj, hen ACjbecomes Pj.
Subcase 2: i
T0
j=ACj
is no he nex smalles
ACj
, hen we con inue eplacing o he empo a y
p ices
Tm
,
m/∈ {i∪j}
wi h
Pm
,
m/∈ {i∪j}
un il ei he
T0
j
becomes he smalles
ACj
o ano he
semi-cycle o ms.
No e ha i T0
j=ACjis he smalles p ice, hen T0
j=ACjbecomes Pj.
Case 2: i T0
j=min nACj,∑l∈pkPlo=∑l∈pkPl, hen we ha e:
Subcase 1: i T0
j=∑l∈pkPlis he nex smalles p ice, hen T0
j=∑l∈pkPlbecomes Pj.
Subcase 2: i
T0
j=∑l∈pkPl
is no he nex smalles p ice, hen we con inue eplacing o he
empo a y p ices
Tm
,
m/∈ {i∪j}
wi h
Pm
,
m/∈ {i∪j}
un il
T0
j=∑l∈pkPl
becomes he smalles p ice
o ano he semi-cycle o ms.
Toge he , his implies ha Ti=ACi=Piand T0
j=ACj=Pjo T0
j=∑l∈pkPl.
(Rema k: I ano he semi-cycle o ms be o e
T0
j
becomes he smalles p ice a some poin in ime,
hen we can ollow he same p oo as desc ibed abo e.)
Now, we sol e
P
by EG. F om Equa ion (6), we know ha
λ∗
i=min nλ∗,ACi(θi),h∑l∈pkλ∗
lio
.
As
λ∗→∞
,
AC1(θ1)
is eached, ollowed by
AC2(θ2)
, ... un il a semi-cycle is o med. Then,
λ∗
i=
ACi(θi)
o all
i
be o e a semi-cycle is eached. Suppose link
j
is he las link ha comple es he
semi-cycle. Then we ha e,
Case 1: I min nλ∗,ACj(θj),∑l∈pkλ∗
lo=ACj(θj), hen λ∗
j=ACj(θj).
Case 2: I min nλ∗,ACj(θj),∑l∈pkλ∗
lo=∑l∈pkλ∗
l, hen λ∗
j=∑l∈pkλ∗
l.
Hence, we ha e ha
Pi=ACi(θi) = λ∗
i
be o e any semi-cycle o ms. Since EG is cons uc ed as
he same way as he RP algo i hm, i.e.,
T0
j=min nACj,∑l∈pkPlo⇔min nλ∗,ACj(θj),∑l∈pkλ∗
lo=λ∗
j
.
Hence,
T0
j=ACj(θj) = Pj⇔λ∗
j=ACj(θj)
; on he o he hand,
T0
j=∑l∈pkPl=Pj⇔λ∗
j=∑l∈pkλ∗
l
.
Thus, Tmax
λ∗=P1θ1+· · · +Psθs=λ∗
1θ1+· · · +λ∗
sθssince Pi=λ∗
i o all i=1, ..., s.

Games 2019,10, 17 16 o 18
Appendix A.2. P oo o Theo em 1
P oo .
Pa (a):
⇒
) We i s p o e he su iciency pa , i.e., suppose a s and-alone co e
s able, e icien and limi ou ing-p oo solu ion exis s, hen we ha e
EFF(θ1,...,θs)−∑i6=j∗Piθi
θ∗
j≤
minpk∈Pj∗nACj∗,∑l∈pkPlo.
S ep 1: o p o e he su icien pa , we need o show he ollowing i s . Le
(λ∗
1
, ...,
λ∗
s)
be some p ices
on a limi ou ing-p oo and co e s able alloca ion. Then o any i6=j∗, we ha e λ∗
i≤Pi.
No e ha he empo a y p ice
Ti=ACi≥λ∗
i
by he s and-alone p ope y. We see ha a any
poin in ime, he RP algo i hm will eplace a pe manen p ice on link
i
. Fi s , no e ha p io o he
o ma ion o a semi-cycle,
Ti=ACi
and hen i becomes a pe manen p ice
≥λ∗
i
. Once a semi-cycle is
o med, we eplace
Ti
wi h a new empo a y p ice
T0
i=min {ACi
|{z}
Ti
,∑l∈pkPl}
and
Pl≥λ∗
l
o all
l
in
he pa h pk. Now, we need o show ha T0
i≥λ∗
i o all i.
No e ha
min nACi,∑l∈pkPlo≥min nλ∗
i,∑l∈pkλ∗
lo
since
Ti=ACi≥λ∗
i
and
Pl≥λ∗
l
.
Since T0
i=min nACi,∑l∈pkPlo, we ha e:
Case 1: I T0
i=min nACi,∑l∈pkPlo=ACi⇒T0
i=ACi≥λ∗
i.
Case 2: I
T0
i=min nACi,∑l∈pkPlo=∑l∈pkPl⇒T0
i=∑l∈pkPl≥∑l∈pkλ∗
l≥λ∗
i
. whe e he
las inequali y holds by limi ou ing-p oo ness. Toge he case 1 and case 2 imply
T0
i≥λ∗
i
. Con inue
checking in his ashion o all i6=j∗, we ha e ha Pi≥T0
i≥λ∗
i⇒Pi≥λ∗
i, as equi ed.
S ep 2: suppose a solu ion
(λ∗
1
, ...,
λ∗
s)
sa is ies limi ou ing-p oo ness, s and-alone co e s abili y and
e iciency, hen by s ep 1, we ha e ∑i6=j∗λ∗
iθi≤∑i6=j∗Piθi.
⇒λ∗
j∗=EFF(θ1,...,θs)−∑i6=j∗λ∗
iθi
θj∗≥EFF(θ1,...,θs)−∑i6=j∗Piθi
θj∗.
Since
Pi≥λ∗
i
o all
i6=j∗⇒min n∑l∈pkPlo≥min n∑l∈pkλ∗
lo≥λ∗
j∗
whe e he las inequali y
holds by limi ou ing-p oo ness. By he s and-alone co e s abili y,
ACj∗≥λ∗
j∗
. Toge he , his implies
minpk∈Pj∗nACj∗,∑l∈pkPlo≥EFF(θ1,...,θs)−∑i6=j∗Piθi
θj∗=λ∗
j∗, as equi ed.
⇐
) Now, we p oceed o p o e he necessi y pa . No e ha by he cons uc ion o he RP
algo i hm, he alloca ion
(P1
, ...,
Ps)
is co e s able and limi ou ing-p oo . By P oposi ion 1, we know
ha he p ices gi en by he RP algo i hm will coincide wi h EG
Tmax
λ∗
, ha is,
∑s
i=1λ∗
i(λ∗)θi=Tmax
λ∗=
∑s
i=1Piθi
. Thus, he only hing le o show is he e iciency. I we can show ha
EFF(θ1
, ..,
θs)≤Tmax
λ∗
,
hen egali a ian solu ion will be a solu ion ha sa is ies e iciency, s and-alone co e s abili y and limi
ou ing-p oo ness. By assump ion,
EFF(θ1,...,θs)−∑i6=j∗Piθi
θj∗≤minpk∈P∗
jnACj∗,∑l∈pkPlo
. By de ini ion,
T0
j∗=minpk∈P∗
j{ACj∗
|{z}
Tj∗
,∑l∈pkPl}=Pj∗. Hence, we ha e,
EFF(θ1, ..., θs)−∑i6=j∗Piθi
θj∗≤min
pk∈P∗
j(ACj∗,∑
l∈pk
Pl)=Pj∗
⇔EFF(θ1, ..., θs)−∑
i6=j∗
Piθi≤Pj∗θj∗
⇔EFF(θ1, ..., θs)≤Pj∗θj∗+∑
i6=j∗
Piθi
=
s
∑
i=1
Piθi
=Tmax
λ∗
as desi ed.
Games 2019,10, 17 17 o 18
Pa (b):
⇒)
Suppose he egali a ian solu ion EG
Tλ∗∗ = (λ∗
1
, ...,
λ∗
s)
exis s, hen i is e icien ,
co e s able and limi ou ing-p oo . By P oposi ion 1, we know ha he maximal e enue gi en by
he RP algo i hm is always an uppe bound o which EG can achie e. Mo eo e , he RP algo i hm
espec s s and-alone co e s abili y and limi ou ing-p oo ness. Thus, i he egali a ian exis s, hen he e
mus exis a p ice ec o
P= (P1
, ...,
Ps)
ha is also e icien , co e s able and limi ou ing-p oo . Hence,
by Theo em 1a, condi ion (i) o condi ion (ii) is sa is ied.
⇐)
Fi s ly, no e ha condi ion (i) clea ly implies condi ion (ii). Secondly, i condi ion (ii) is
sa is ied, hen we ha e ha
EFF(θ1
, ...,
θs)≤∑s
i=1Piθi=Tmax
λ∗
by Theo em 1. Hence, by p oposi ion 1
and con inui y, he e exis s
λ∗∗
such ha
Tλ∗∗ =EFF(θ1
, ...,
θs)
when we slowly dec ease he e enue
a ge om
Tmax
λ∗
o
EFF(θ1
, ...,
θs)
. This
λ∗∗
will gene a e exac ly he egali a ian solu ion EG
Tλ∗∗ =
(λ∗
1, ..., λ∗
s).
Appendix A.3. P oo o Co olla y 1
P oo .
Suppose wi hou loss o gene ali y ha agen s demand link
i
, EG s a s cha ging
λ∗>
0 o all
agen s. A any poin in ime, he s and-alone cons ain o e e y link will be eached.
Wi hou loss o gene ali y, suppose he s and-alone cons ain o link
i
is eached i s , i.e.,
λ∗=
ACi(θi) = λ∗
i. By assump ion,
λ∗
i=ACi(θi)≤min
pk∈Pi(∑
l∈pk
ACl(θl))⇒Ci(θi)
θi
≤∑
l∈pk
ACl(θl)∀pk∈ Pi
⇒λ∗
i≤∑
l∈p1
ACl(θl), ..., λ∗
i≤∑
l∈p
ACl(θl)
This implies ha agen s on link
i
a e cha ged a he lowes cos among all easible al e na i e
pa hs
pk
ha connec link
i
, so agen s will no ha e an incen i e o e ou e. Limi ou ing-p oo ness
is i ially sa is ied by he assump ion. No e ha
λ∗
i
sa is ies he s and-alone co e s abili y since
λ∗
i=ACi(θi). I also sa is ies e iciency since agen s on link ia e being cha ged a he lowes cos .
Now, i emains o check he cases λ∗
1, ...λ∗
i−1,λ∗
i+1, ...λ∗
s−1. No e ha he assump ion
ACi(θi)≤min
pk∈Pi(∑
l∈pk
ACl(θl))
holds o all
j=
1, ...,
s
, hence by he same easoning, we can deduce ha
λ∗
1
, ...,
λ∗
i−1
,
λ∗
i+1
, ...,
λ∗
s−1
a e also co e s able, limi ou ing-p oo and e icien . Now, wi hou loss o gene ali y, suppose he
s and-alone cos o link sis eached las , hen by e iciency,
λ∗
s=
EFF(θ1, ..., θs)−∑
i6=s
Ci(θi)
θs
The iangle inequali y cons ain assumed implies ha ,
λ∗
s=
EFF(θ1, ..., θs)−∑
i6=s
Ci(θi)
θs
≤ACs(θs)≤min
pk∈Pi(∑
l∈pk
ACl(θl))
Thus, each agen who demands link
s
is paying a
λ∗
s
, which essen ially is he lowes cos among
all o he easible pa hs. Hence,
λ∗
s
is also ou ing-p oo , co e s able and e icien . The e o e, EG has
a solu ion
EGTλ∗∗ = (λ∗
1
, ...,
λ∗
s)
such ha e e y
λ∗
i
o all
i=
1, ...,
s
is limi ou ing-p oo , co e s able
and e icien .
Games 2019,10, 17 18 o 18
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