Competition and efficiency in congested markets

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COMPETITION

AND

EFFICIENCY

IN

CONGESTED

MARKETS

.

Daron

Acemoglu

Asuman

E.

Ozdaglar

Working

Paper

05-06

February

1 7,

2005

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'>3SACHUSETTS INSTITUTE1

OF TECHNOLOGY

MAR

1 2005

Competition

and

Efficiency

in

Congested

Markets*

Daron

Acemoglu

Department

of

Economics

Massachusetts

Institute

of

Technology

Asuman

E.

Ozdaglar

Department

ofElectrical

Engineering

and

Computer

Science

Massachusetts

Institute

of

Technology

February

17,

2005

Abstract

We

study theefficiencyofoligopolyequilibria incongested markets.

The

moti-vatingexamplesaretheallocation ofnetworkflows inacommunication networkor

oftraJfi-Cina transportation network.

We

showthat increasingcompetition

among

oligopolistscanreduce efficiency, measuredasthe differencebetweenusers'

will-ingness topayanddelaycosts.

We

characterizeatight

bound

of 5/6onefficiency inpure strategyequilibria. This

bound

istighteven

when

the

number

ofroutes and oligopolists is arbitrarily large.

We

also study the efficiency properties of mixedstrategyequilibria.

'We

thank Xin Huang,RameshJohari,EricMaskin,NicolasStierMoses,JeanTirole,JohnTsitsiklis,

IvanWerning,

Muhamet

Yildizandparticipants atthe

INFORMS

conference, Denver 2004foruseful

1

Introduction

We

analj'ze pricecompetitioninthepresenceofcongestioncosts. Considerthe following environment: oneunit oftraffic can useone of/ alternative routes.

More

traffic ona particular route causes delays, exerting a negative (congestion) externahty

on

existing

traffic.^ Congestioncostsarecaptured

by

a route-specificnon-decreasingconvexlatency

function, li(•). Profit-maximizingoligopohstssetprices (tolls) fortravel

on

each route denoted

by

pi.

We

analyze

subgame

perfect

Nash

equilibria ofthis environment,

where

foreachprice vector,p,all trafficchoosesthepaththathas

minimum

(tollplus delay)

cost,

_h+Pi,

and

oligopohstschooseprices to

maximize

profits.

The

environment

we

analyze is ofpractical importance for a

number

ofsettings.

These

include transportation

and communication

networks,

where

additional use of a route (path) generates greatercongestionforallusers,

and

marketsinwhichthere are "snob" effects,so thatgoods

consumed by

fewer other

consumers

are

more

valuable(see for example, [38]).

The

key feature of these environments is the negative congestion externalitythat users exert

on

others. This externalityhas

been

well-recognized since the

work by

Pigou[27] ineconomics,

by

Samuelson

[33],

Wardrop

[41],

Beckmann

etal. [4] intransportation networks,

and by

Orda

et. al. [23], Koriliset. al. [18], Kelly [17],

Low

[20]in

communication

networks.

More

recently,therehas

been

agrowingliterature that focuses

on

quantification ofefficiency loss(referred to asthe price ofanarchy)that results

from

externaUties

and

strategicbehaviorindifferentclasses ofproblems; selfish

routing (Koutsoupias

and

Papadimitriou [19],

Roughgarden and

Tardos [31], Correa,

Schulz,

and

Stier-Moses [8], Perakis [26],

and Friedman

[12]), resource allocation by

market

mechanisms

(Johari

and

Tsitsiklis [16], Sanghavi

and Hajek

[32]),

and

network

design (Anshelevich et.al. [2]). Nevertheless, the game-theoretic interactions between

(multiple) serviceproviders

and

users, ortheeffectsofcompetition

among

theproviders

on

the efficiency losshas not

been

considered. This is an important areafor analysis sincein

most

applications, (competing) profit-maximizingentitieschargepricesfor use.

Moreover,

we

will

show

that the nature of the analysis changes significantly in the presenceof pricecompetition.

We

provideageneral

framework

forthe analysis of pricecompetition

among

providers

in a congested (and potentiaUycapacitated) network, studyexistence ofpurestrategy

and

mixed

strategy equihbria,

and

characterize

and

quantify theefficiencyproperties of

equilibria.

There

are foursetsof

major

resultsfrom our analysis.

First,

though

the equilibrium of traffic assignment without prices can be highly

inefficient (e.g., [27], [31], [8]), price-setting

by

a monopolist internalizesthe negative

externality

and

achievesefficiency.

Second,increasingcompetitioncanincreaseinefficiency. Infact,changingthe

market

structure from

monopoly

to

duopoly

almost always increases inefficiency. Thisresult

contrastswith

most

existing resultsintheeconomicsliterature

where

greatercompetition tendstoimprovethe allocation of resources(e.g. see Tirole [36]).

The

intuitionforthis resultisdrivenbythepresenceofcongestion

and

isillustrated

by

the

example

we

discuss below.^

^

_An

_{externalityariseswhenthe actionsofthe playerinagame}affectsthe payoff of otherplayers.

Third,

we

provide a tight

bound on

the extent of inefficiency in the presence of

pricing, which apphes irrespective of the

number

of routes, /.

We

show

that social

surplus (defined as the difference betweenusers'willingness to

pay and

the delay cost) in any purestrategyoligopolyequilibriumisalwaysgreater than 5/6ofthe

maximum

social surplus. Simple examples reach this 5/6 bound. Interestingly, this

bound

is

obtained even

when

the

number

of routes, /,is arbitrarilylarge.

Fourth, purestrategy equilibria

may

failto exist. Thisisnotsurprising inview of

thefactthat

what

we

havehereisaversion ofaBertrand-Edgeworth

game

where

pure strategy equilibriado notexist inthepresenceofconvexcostsofproductionor

capac-ity constraints (e.g.,

Edgeworth

[11], Shubik [35], Benassy [6], Vives [40]). However,

in our oligopoly environment

we

show

that

when

latency functions are linear, apure strategyequihbrium always exists, essentiallybecause

some

amount

ofcongestion ex-ternahties

remove

the payoff discontinuities inherentin theBertrand-Edgeworth game. Non-existence

becomes an

issuein our environment

when

latency functions are highly convex. Inthiscase,

we

provethat

mixed

strategy equilibriaalwaysexist.

We

also

show

that

mixed

strategy equilibriacanlead to arbitrarilyinefficientworst-caserealizations; inparticular, socialsurpluscan

become

arbitrarilysmallrelativetothe

maximum

social surplus.

The

following

example

illustrates

some

of theseresults.

Example

1 Figure 1

shows

asituation similar totheonefirstanalyzedby Pigou [27]

to highlighttheinefficiency

due

tocongestionexternalities.

One

unit oftrafficwilltravel fromorigin

A

to destination B,using either route 1or route2.

The

latency functions aregiven

by

x^ 2

^1{x)

=

—,

_h

[x)

=

-X.

Itisstraightforward to seethat theefficientallocation [i.e.,onethatminimizesthe total delay cost '^^li{xi)xi] is

xf

=

2/3

and xf

=

1/3, while the (Wardrop) equilibrium allocationthatequatesdelay

on

the

two

pathsis

x^^

w

.73

>

xf and

x'^^

«

.27

<

xf

The

source of theinefficiencyisthateachunit oftrafficdoes notinternalizethe greater increaseindelayfromtravel

on

route1,so thereis too

much

useofthisrouterelative

totheefficientallocation.

Now

considera monopolist controllingboth routes

and

setting prices for travel to

maximize

itsprofits.

We

will

show

ingreater detailbelowthat inthis case, the

monop-olist will set a price including a

markup,

x^l'^

(when

li is differentiable), whichexactly internalizesthecongestionexternality. Inotherwords,this

markup

isequivalent to the Pigovian taxthat a socialplanner

would

setin order toinducedecentralizedtrafficto

choosetheefficient allocation. Consequently, inthissimple example,

monopoly

prices

win

be

_pf

^

=

_{2/^f

+

k

and

_pf

^

=

(2/3^)

+

^, for

some

constantk.

The

resultingtraffic inthe

Wardrop

equilibriumwill beidenticaltotheefficient allocation, i.e.,

x^^

—

2/3

and

x^^^

=

1/3.

Next

considera duopolysituation,where eachrouteiscontrolledbyadifferent

profit-maximizingprovider. Inthiscase,itcanbe

shown

that equilibrium priceswilltake the

ofcongestionexternalities,allmarketstructureswouldachieveefRciency,andachange frommonopoly

Ii(x)=x73

l2(x)=(2/3)x

Figure1:

A

two

link

network

withcongestion-dependantlatency functions.

form_pf-^

=

Xi{l[

+

I'o) [seeEq. (20) in Section4],or

more

specifically,

_pf^

^

0.61

and

p^^

?» 0.44.

The

resulting equihbriumtrafficis

_xf^

w

.58

<

xf and

x^^

«

.42

>

xf

,

which

alsodiffers

from

theefficientallocation.

We

will

show

thatthisisgenerallythecase intheohgopoly equilibrium. Interestingly, while in the

Wardrop

equihbrium without

prices, there

was

too

much

traffic

on

route 1,

now

there is too little traffic because ofits greater

markup.

It is also noteworthy that although the duopoly equilibrium

is inefficient relative to the

monopoly

equihbrium, in the

monopoly

equilibrium k is

chosensuchthatallofthe

consumer

surplusiscaptured

by

themonopolist, whileinthe oligopolyequilibriumusers

may

havepositive

consumer

surplus.^

The

intuitionfortheinefficiency ofduopolyrelativeto

monopoly

can be obtainedas

follows.

There

is

now

a

new

sourceof(differential)

monopoly power

foreachduopolist,

which

theyexploit

by

distortingthe patternoftraffic:

when

provider1,controllingroute 1,chargesahigherprice,itrealizesthatthis will

push

some

trafficfromroute1toroute

2,raisingcongestion

on

route 2.

But

this

makes

thetrafficusing route1

become more

"locked-in," becausetheiroutside option, travel

on

the route2, has

become

worse."*

As

aresult,theoptimalpricethateachduopolistchargeswillincludeanadditional

markup

overthePigovian

markup. These

areXiZj forroute 1

and

X2l'i forroute2. Sincethese

two

markups

are generally different, they will distortthe patternoftraffic

away from

theefficient allocation. Naturally, however, prices are typically lowerwith duopoly, so even

though

socialsurplus declines, userswill

be

betteroffthanin

monopoly

(i.e.,they

will

command

a positive

consumer

surplus).

Although

there isalargehterature on modelsofcongestionboth in transportation

and communication

networks, very few studies have investigated the implications of havingthe"propertyrights" over routesassigned toprofit-maximizingproviders. In [3],

Basar

and

Srikant analyze

monopoly

pricingunderspecificassumptions onthe utility

and

latency functions.

He

and

Walrand

[15] study competition

and

cooperation

among

internet serviceprovidersunderspecific

demand

models. Issues ofefficientallocation of ^Consumersurplusisdefined asthedifferencebetweenusers'willingness topayandeffective costs,

Pi

+

li{xi),andisthusdifferentfromsocialsurplus(whichisthe differencebetweenusers'willingness

topay andlatencycost, li{xi),thusalsotakes intoaccount producersurplus/profits). See Mas-Colell,

Winston, and Green[21].

*Using economicsterminology,wecouldalsosay that thedemandforroute1becomesmore

"inelas-tic". Sincethisterm has adifferentmeaninginthecommunication networksliterature (see [34]),we

flowsor traffic acrossroutes

do

not arise inthesepapers.

Our

previous

work

[1] studies

the

monopoly problem and

contains theefficiencyofthe

monopoly

result(withinelastic traffic

and more

restrictiveassumptions onlatencies),but

none

of the otherresultshere.

To

minimizeoverlap,

we

discussthe

monopoly problem

onlybrieflyin thispaper. In the rest ofthe paper,

we

use the terminology of a (communication) network, though all of the analysis applies to resource allocation in transportation networks,

electricity markets,

and

othereconomic applications. Section 2 describes the basic en-vironment. Section 3 briefly characterizes the

monopoly

equihbrium

and

establishes

itsefficiency. Section 4 defines

and

characterizes the oligopoly equilibriawith

compet-ingprofit-maximizing providers. Section 5 contains the

main

results

and

characterizes theefficiency properties ofthe ohgopoly equilibrium

and

provide

bounds

onefficiency.

Section 6containsconcluding

comments.

Regardingnotation,allvectors areviewedas

column

vectors,

and

inequalitiesare to beinterpreted componentwise.

We

denote by

M^

thesetofnonnegative /-dimensional vectors. Let

d

be aclosedsubset of[0,oo)

and

let

/

:

Q

i—>

M

beaconvexfunction.

We

usedf{x) todenotethesetofsubgradientsof

/

atx,

and

f~{x)

and

f'^{x) todenote theleft

and

rightderivatives of

/

at x.

2

Model

We

consider a network with/parallel links. Let

I =

{!,...,/} denote thesetoflinks.

LetX,denote thetotalflow

on

linki,

and x

=

[xi,...,xi]denotethe vectoroflinkflows.

Each

link in thenetwork has aflow-dependentlatency functionk{xi),which measures the traveltime (ordelay) asa function of the total flowonlinki.

We

denotethe price per unit flow(bandwidth)oflinkihyPi. Let

p

=

{pi,- ,Pi]denotethe vector ofprices.

We

are interestedinthe

problem

ofroutingdunits of flow acrossthe/links.

We

as-sume

thatthisistheaggregateflow of

many

"small" users

and

thusadoptthe

Wardrop's

principle(see [41]) incharacterizing the flow distributionin thenetwork;i.e.,the flows arerouted along paths with

minimum

effective cost, defined asthe

sum

ofthelatency

at the given flow

and

the price of thatpath (seethe definitionbelow). Wardrop's

prin-ciple is used extensively in modellingtraffic behavior in transportationnetworks ([4], [9], [25])

and

communication

networks ([31], [8]).

We

also

assume

that the usershave

a reservation utility

R

and

decide not to send their fiow ifthe effectivecost exceeds the reservation utility. This implies that user preferences can be represented by the piecewisehnear aggregateutilityfunction u{-) depictedinFigure 2.^

Definition 1 For a given price vector

p

>

0,^ a vector

x^^

G

Mi

is a

Wardrop

Thissimplifyingassumptionimpliesthatallusers arc "homogeneous"inthe sense thatthey have

thesamereservationutility,R.

We

discusspotentialissues inextendingthisworkto userswithelastic

and heterogeneousrequirementsintheconcludingsection.

''Sincethe reservationutilityof usersisequal toR,wecanalso restrictattention topi<

R

forall

u(x)

Figure2: Aggregateutilityfunction.

equilibrium

(WE)

if

(1)

We

denotetheset of

WE

ata given

p by W{p).

We

adoptthe followingassumption

on

thelatency functions throughoutthe paper.

Assumption

1 For eachi6X, the latency functionli : [0,cx))i—> [0,oo]isproperclosed convex,nondecreasing,

and

satisfies /^(O)

=

0.

Hence,

we

allow the latency functions to be extended real-valued (see [7]). Let

Ci

=

{x

E

[0,oo) I li{x)

<

oo} denote theeffective

domain

ofli.

By

Assumption

1,

Ciis aclosed interval of theform [0, b] or[0,oo). Let

bd

=

sup^g,-;^x.

Without

lossof generality,

we

can

add

the constraintXj

£

C,inEq. (1). Usingtheoptimalityconditions

for

problem

(1),

we

seethat avector

x^^

€

M^

isa

WE

if

and

onlyif_X^j^jxf^^

<

d

and

there exists

some

A

>

such_{that A( X^^^^}

x^^

—

d)

=

and

foralli,

R-k{xf^)-p,

<X

ifxf'^-O,

=

A if

<

_xf"^

<

be

>A

ifxf^

=

6c..

(2)

When

thelatency functions are real-valued[i.e.,Ci

=

[0,oo)],

we

obtainthe following characterization of a

WE,

whichisoftenusedasthe definition of a

WE

intheliterature.

This

lemma

states that in the

WE,

the effective costs, defined as li{xf^^)

+

pi, are

equalized

on

alllinkswithpositiveflows(proof omitted).

Lemma

1 Let

Assumption

1 hold,

and assume

further that Ci

—

[0,oo) foralli

G

I.

Then

anonnegativevector x* E

W{p)

if

and

onlyif

li{x*)+pi

<

R,

V

iwithx*

>

0, I

2=1

with_J2i=i^*

=

c?ifniirij{/j(xj)

_+Pj}

<

R.

Example

2 below

shows

that condition (3) in this

lemma

may

not hold

when

the latency functions are notreal-valued.

Next

we

establishthe existence ofa

WE.

Proposition

1

(Existence

and

Continuity)

Let

Assumption

1 hold. Foranyprice vector

p

>

0, theset of

WE,

W{p),

is nonempty. Moreover, thecorrespondence

W

:

R^

^

M^

isupper semicontinuous.

Proof.

Given any

p

>

0,considerthe followingoptimization

problem

maximize^>o

_^^

({R

-

Pi)Xi

-

_j k[z)dz\ (4)

subject to _{2_,}^i

^

'^

i=l

Xi

€

Ci,

V

i.

Inview of

Assumption

(1) (i.e., liis nondecreasingfor all i), itcanbe

shown

that the objectivefunction of

problem

(4) isconvexoverthe constraintset, whichis

nonempty

(since G Cj)

and

convex. Moreover, the firstorderoptimahtyconditions of

problem

(4),whichare also sufRcient conditionsforoptimality,are identical to the

WE

optimality conditions [cf. Eq. (2)] .

Hence

aflowvectorx^'^'^

G

^{v)

if^^ndonlyif itisan optimal

solution of

problem

(4). Sincethe objectivefunction ofproblem(4) iscontinuous

and

the constraintsetiscompact, this

problem

hasan optimalsolution,showingthat

W{p)

isnonempty.

The

fact that

W

is

an

upper semicontinuous correspondence at every

p

follows

by

using the

Theorem

ofthe

Maximum

(seeBerge [5], chapter6) for

problem

(4).

Q.E.D.

WE

flowssatisfyintuitivemonotonicityproperties giveninthe following proposition.

The

prooffollowsfromthe optimality conditions[cf. Ecj. (2)]

and

isomitted (see[1]).

Proposition

2

(Monotonicity)

Let

Assumption

1 hold. For a given

p

>

0, let

P-j

=

\PiWj-(a) For

some p

<

p,letx GVV'(p)

and

x G W{p). Then, _X^j^jx^

>

_J^j^jXi.

(b) For

some

pj

<

Pj,letx G W{pj,p-j)

and

x GVF(pj,p_j).

Then

Xj

>

Xj

and

Xj

<

Xi,

foralliy^ j.

(c) For

some

I C

I, supposethatPj

<

pj forallj

G

X

and

Pj

=

Pj forallj ^J,

and

Ejgi^j(p)-For a given price vectorp, the

WE

need not be unique in general.

The

following

example

illustrates

some

properties ofthe

WE.

Example

2 Consider a

two

hnk

network. Letthe total flow

be

d

=

1

and

the reserva-tionutilityhe

R =

1.

Assume

thatthe latency functions aregiven

by

^

I oo otherwise.

At

the price vector (pi,P2)

=

(1,1), the set of

WE,

W{p),

is given by the set ofall

vectors (xi,X2) with

<

Xj

<

2/3

and

X^jx,

<

1.

At any

price vector {pi,P2) with

Pi>

P2

=

I,

W{p)

isgiven

by

all(0,X2) with

<

X2

<

2/3.

This

example

also illustrates that

Lemma

1 need not hold

when

latency functions arenot real-valued. Consider, forinstance, the pricevector (^1,^2)

=

(1

~

e,1

~

le) for

some

scalara

>

1. Inthiscase, theunique

WE

is(xi,X2)

=

(1/3,2/3),

and

clearly

effectivecosts

on

the

two

routes arenotequalized despite thefactthattheyboth have positive flows. This arises because the path with the lower effective cost is capacity constrained, so

no

more

trafficcan use thatpath.

Under

further restrictions

on

the k,

we

obtain the followingstandard result in the literaturewhich assumesstrictlyincreasinglatency functions.

Proposition

3

(Uniqueness)

Let

Assumption

1 hold.

Assume

further that li is

strictly increasing over Cj. For

any

price vector

p

>

0, the set of

WE,

W{p),

is a singleton. Moreover,the function

W

:W_^, >-^

M^

iscontinuous.

Proof.

Under

thegivenassumptions, for any

p

>

0, the objective function of

problem

(4) is strictly convex,

and

therefore has a unique optimal solution. This shows the uniquenessofthe

WE

atagivenp. Since thecorrespondence

W

is

upper

semicontinuous from Proposition 1

and

single-valued,itis continuous.

Q.E.D.

In general,underthe

Assumption

that/j(0)

=

for alli

E

X,

we

havethe following

result,

which

isessentialinouranalysiswithnonunique

WE

flows.

Lemma

2 Let

Assumption

1hold. For agiven

p

>

0,definetheset

I =

{i

Gl

\ 3X, X

E W{p)

withx^7^x^}. (5)

Then

h{xi)

-

0,

Vie2,

Vxe

W{p),

Proof. Consider

some

i

&T. and

x

G W{p).

Sincei

E

T, thereexists

some

x

E

W{p)

such thatXi

^

Xi-

Assume

withoutlossofgenerahtythatXt

>

Xi.

There

are

two

cases

to consider:

(a) IfXk

>

Xk for all A;

^

i, _{then Xljgi^i}

>

'^IjeJ^i-' '^^ich implies that the

WE

optimality conditions [cf. Eq. (2)] forx hold with A

=

0.

By

Eq. (2)

and

Xi

>

x,,

we

have

li{xi)

+Pi<

R,

li{xi)

+Pi>

R,

which togetherimplythat /i(xi)

=

li{xi).

By

Assumption

1 (i.e., liisconvex

and

/,(0)

=

0),itfollows that/,(x,)

=

0.

(b) IfXk

<

Xkfor

some

k,bythe

WE

optimality conditions,

we

obtain

li{Xi)

+Pi<

lk{Xk)

+Pk,

k{x,)

+f)i>

IkiXk)

+Pk-Combining

the above with Xj

>

x,

and

Xfc

<

x^,

we

seethat /j(xj)

=

li{xi),

and

hi^k)

=

4(^fc)-

By

Assumption

1, this shows that li{xi)

=

(and also that

Pz

=Pk)-Next

consider

some

i, j

€

I.

We

will

show

that pi

=

pj. Since i

E

T, thereexist X, X G

W{p)

suchthatXi

>

Xi.

There

arethree cases to consider:

• Xj

<

Xj.

Then

a similar

argument

topart (b)above

shows

thatpi

=

pj

.

• Xj

>

Xj. If Xfc

>

Xk for allk

^

i,j, then

_{X^m^m ^}

^'

™ply™g

that the

WE

optimahty conditionsholdwith A

=

0. Therefore,

we

have

k{x,)

+Pt<

R,

lj{xj)

+

Pj

>

R,

whichtogetherwithli{xi)

=

lj{Xj)

=

implythatpi

—

pj.

• Xj

=

Xj. Since j ET, there exists

some

x

E

W{p)

suchthat Xj

_^

Xj. Repeating theabove

two

stepswithXj instead ofXj yieldsthe desiredresult.

Q.E.D.

We

nextdefinethe socialproblem

and

thesocial

optimum,

whichisthe routing (flow allocation) that

would

be chosenbyaplannerthathasfullinformation

and

fullcontrol overthenetwork.

Definition 2

A

flowvector

x^

isa social

optimum

ifit is an optimalsolution of the socialproblem

niaximize2;>o

_2.

(-^

~

ki^i) jXi (6) 2=1

/

subject to

>,

^i

^

^•

Inviewof

Assumption

1,the social

problem

has a continuousobjective function

and

a

compact

constraintset,guaranteeingthe existence ofasocial

optimum,

x^. Moreover, usingtheoptimality conditionsforaconvex

program

(see [7], Section4.7),

we

seethat a vector

x^ G

M^

isa social

optimum

if

and

only if _X^j^^

xf

<

d

and

there exists a subgradient gi.

E

dli{xf) foreachi,

and

a A'^

>

such that A'^(_{X],_i}

_xf

—

d)

=

and

foreachi,

R-k[xf)-xfgi^

<X'

ifxf^O,

(7)

=

A^

iiO<xf<bc„

>

A^ if

if

=

6c..

For future reference, fora givenvectorx

6 R^,

we

define the value of the objective functioninthe socialproblem,

/

S{x)

=

Y.{R-h{x,))x,,

(8)

1=1

asthe socialsurplus,i.e., thedifferencebetweenusers'willingness to pay

and

thetotal latency.

3

Monopoly

Equilibrium

and

Efficiency

In this section,

we

assume

that a monopolist service provider

owns

the / links

and

charges a price ofp^ perunit

bandwidth on

linki.

We

consideredarelated

problem

in

Acemoglu and

Ozdaglar[1]foratomicuserswithinelastic traffic(i.e.,theutilityfunction

ofeach ofafinite set ofusersisa step function),

and

withincreasing, real-valued

and

differentiable latency functions. Here

we

show

that similar results hold for the

more

general latency functions

and

the

demand

model

consideredinSection2.

The

monopolistsetsthe prices to

maximize

hisprofitgiven

by

/

?,=i

where

x G W{p).

Thisdefinesatwo-stage

dynamic

pricing-congestiongame,

where

the monopolistsets prices anticipating the

demand

of users,

and

given the prices (i.e., in

Definition3

A

vector {p'^'^ ,x^'^^)

>

is a

Monopoly

Equilibrium

(ME)

if x^

W{p^'^)

and

n(p*^^,x^"^)

>

n(p,x),

V

p,

V

X

G

ly{p) .

Our

definition ofthe

ME

is strongerthanthestandard

subgame

perfect

Nash

equi-libriumconceptfor

dynamic

games.

With

a sHghtabuseofterminology,letus associate a

subgame

perfect

Nash

equilibrium with the on-the-equilibrium-path actions of the two-stagegame.

Definition 4

A

vector {p*,x*)

>

is a

subgame

perfect equihbrium

(SPE)

of the pricing-congestion

game

ifx*

G W{p*) and

forall

p

>

0, thereexists x

6

W

{p) such

that

U{p*,x*)

>

U{p,x).

The

followingpropositionshowsthatunder

Assumption

1,the

two

solutionconcepts

coincide. Sincetheproofis notrelevantfortherestof the argument,

we

provideit in

Appendix

A.

Proposition

4

Let

Assumption

1 hold.

A

vector [p^^^ ,x^'^) isan

ME

if

and

onlyif it isan

SPE

ofthe pricing-congestiongame.

Sincean

ME

{p*,x*)isan optimalsolution oftheoptimization

problem

I

maximizep>o,x>o

^^PiXi

(9)

subject to X

£ W{p),

it is slightly easier to

work

withthan an

SPE.

Therefore,

we

use

ME

asthe solution conceptinthispaper.

The

preceding

problem

hasan optimalsolution,whichestablishesthe existence ofan

ME.

Inthenextproposition,

we

show

that the flow allocation atan

ME

and

the social

optimum

are thesame.

Proposition

5 Let

Assumption

1hold.

A

vectorxisthe flow vector atan

ME

if

and

onlyifitisasocial

optimum.

To

provethis result,

we

firstestablishthe following

lemma.

Lemma

3 Let

Assumption

1hold. Let (p,x)bean

ME. Then

foralliwithx^

>

0,

we

have

p,

=

R-

li{x^).

Proof.

Define

I =

{ie

I

\Xi

>

0}.

By

the definition ofa

WE,

for alli

E

I,

we

have

j9j

+

li{xt)

<

R.

We

first

show

that

li{xi)

_+Pi

=

lj{xj)

+

Pj,

V

i,_J

G

X.

Suppose

li{xi)

+Pi

<

lj{xj)

+

pjfor

some

i,j

G

T. Usingtheoptimality conditionsfor

a

WE

[cf Eq. (2)],

we

seethat

x

€ W{pi

+

e,p^i) forsufficientlysmalle, contradicting

the factthat(p,x)isan

ME. Assume

next thatpi

+

li{xi)

<

R

for

some

i

E

T. Consider the price vector

p

=

p

+

e_X^j^j^i'

where

e^is thei*''unitvector

and

eissuchthat

Pi

+

e

+

li(xi)

<

R,

y

i

eJ.

Hence, x isa

WE

atpricep,

and

thereforethe vector (p,x) isfeasible for

problem

(9)

and

has astrictlyhigherobjectivefunction value, contradicting the fact that(p,x)is

an

ME. Q.E.D.

Proof

of

Proposition

5: Inviewof

Lemma

3,

we

canrewrite

problem

(9) as

maximize3;>o

2_.

{R~

hi^i)

i=l I

subject to _{2_]}^i

^ ^

This

problem

isidentical tothe social

problem

[cf

problem

(6)], completingtheproof

Q.E.D.

In addition to thesocial surplus defined above, it is also usefulto define the

con-sumer

surplus, as thedifi^erence

between

users'willingness to

pay

and

effective cost,i.e.,

Y2i=i(-^

~

hi^i)

—

Pi)xi (See Mas-Colell,

Winston and

Green, [21]).

By

Proposition5

and

Lemma

3, it isclear thateven

though

the

ME

achievesthe social

optimum,

allof thesurplusiscaptured bythemonopolist,

and

users are just indifferent

between

sending

theirinformationornot(i.e., receive

no consumer

surplus).

Our

major

motivation for the study of ohgopolistic settings is that they provide a better approximation toreality,

where

thereis typically competition

among

service providers.

A

secondary motivationisto see

whether

an

ohgopoly equilibriumwillachieve anefhcient allocationlikethe

ME,

while also transferring

some

or allofthe surplus to theconsumers.

4

Oligopoly

Equilibrium

We

suppose that there are

S

service providers, denote the set ofservice providers by

iS,

and assume

thateachserviceprovider s

£

S

owns

adifferentsubsetTs ofthelinks.

Serviceproviderscharges a pricep^per unit

bandwidth

onlinki

E

Is- Giventhe vector

ofprices oflinks

owned by

other service providers,

_{Ps =}

\pi]i^i,, theprofit of service providersis

UsiPs,P-s,x)==

_^PiXi,

for X e W{ps,p-s),

where

Ps

=

\pi\ieXs-The

objective ofeachservice provider, hke the

monopoHst

inthe previous section,

is to

maximize

profits. Becausetheir profits

depend on

the pricesset byother service providers, each service provider formsconjectures about the actions of other service providers, as

weh

asthe behaviorofusers, which,

we

assume, they do accordingtothe notion of

(subgame

perfect)

Nash

equihbrium.

We

refer to the

game

among

service providers asthe price competitiongame.

Definition5

A

vector{p'-'^,x'~'^)

>

is a(pure strategy) OligopolyEquilibrium

(OE)

if

x^^

S

W

(p?^,p°f)

and

forallsG5,

n,(pf^,pef,xO^)

>

n,fe,pef,x),

Vp,

>

0,

V

X

6 H/fe,pef).

(10)

We

refertop'-'^_asthe

_OE

_price.

As

forthe

monopoly

case,thereis acloserelation

between

apurestrategy

OE

and

a pure strategy

subgame

perfect equilibrium.

Again

associatingthe

subgame

perfect equilibrium withthe on-the-equilibrium-path actions,

we

have the following standard

definition.

Definition 6

A

vector {p*,x*)

>

isa

subgame

perfectequihbrium

(SPE)

ofthe price competition

game

ifx* E:

W

{p*)

and

thereexistsafunction x{p)

G

W

[p)suchthatfor

all

seS,

^s{P*s,P*-s^X*)>Ils{Ps.pls^x{ps,P*..,))

Vp, >0.

(11)

The

followingproposition generalizesProposition4

and

enables us to

work

withthe

OE

definition,whichis

more

convenientforthesubsequentanalysis.

The

proofparallels

that ofProposition4

and

isomitted.

Proposition

6 Let

Assumption

1hold.

A

vector {p'-'^,

x^^)

is an

OE

if

and

onlyifit isan

SPE

ofthe pricecompetitiongame.

The

pricecompetition

game

isneitherconcave norsupermodular. Therefore,classical

argumentsthat areusedto

show

the existence of apurestrategy equilibriumdo not hold

(see [13], [37]). Inthenextproposition,

we show

thatforlinearlatency functions, there

exists apurestrategy

OE.

Proposition

7 Let

Assumption

1 hold,

and assume

furtherthatthe latency functions arelinear.

Then

the pricecompetition

game

has apurestrategy

OE.

Proof.

Foralli

^

I,let/j(x)

=

aiX. Definetheset

Io^{ieI

\a,

=

0}.

Let Iq denotethe cardinality of setIq-

There

are

two

cases to consider:

• Iq

>

2:

Assume

that there exist i, j € Xq such thati

E

Ts

and

j

€

Is' for

some

s j^ s'

E

S.

Then

itcan

be

seen that a vector (p^-^,x"^-^) with

_pf

^

=

for all

i

E

Iq

and

x'-'^

E

W{p'-'^)is

an

OE.

Assume

next thatforalli

G

2o,

we

havei

E

Is

for

some

s

E

S. Then,

we

can

assume

withoutloss ofgenerality that providers

owns

asingle linki'with a^/

=

and

considerthecase/q

=

1.

• _/o

<

1: Let

Bsiplf)

be theset of

_pf^

suchthat

(pf^,x°^)Garg

max

^p^Xi.

(12)

Let B{p'-'^)

=

[Bs[p2.f)]s^s- Inview of the hnearity ofthe latency functions, it

followsthatB{p'^^)isan

upper

semicontinuous

and

convex- valued correspondence. Hence,

we

canuseKakutani'sfixedpoint

theorem

to assertthe existence ofap'-'^

suchthatB{p'-'^)

=

p'^^ (seeBerge [5]).

To

complete the proof,itremainsto be

shown

that there exists

x°^

E

W{p'^^) suchthat Eq. (10) holds.

IfIq

—

0,

we

have by Proposition3thatW{p'^^) isa singleton,

and

thereforeEq. (10) holds

and

(p°-^,

IV

(p°^)) isan

OE.

Assume

finallythat exactlyoneofthea^'s (withoutloss ofgeneralityai) isequal

to 0.

We

show

thatfor all x,

x E

W{p'-'^),

we

have Xi

=

Xi, for alli

^

1. Let

EC{x,p°^)

=

mmj{lj{xj)

_+pf^}.

Ifat leastoneof

EC{x,p°^)

<

R, or

EC{x,p°^)

<

R

holds, thenone can

show

that _X]t=i^i

—

X^i=i ^i

~

'^- Substituting Xi

=

d

—

Y2iei i^i^i ^^

problem

(4),

we

see that the objective function of

problem

(4) is

strictlyconvexinx_i

=

[x^jj^i,thusshowingthat

x

=

x. Ifboth

EC{x,p^^)

=

R

and

EC{x,p'^^)

=

R, thenXi

=

Xi

=

1^^{R

—

_pf^)

foralli

^

1,establishingour claim.

For

some

x

E

W{p'^^),consider the vector x^-^

=

(d

—

_{X^j^j Xj,x_i)} . Sincex_iis

uniquelydefined

and

Xi is chosen suchthat theprovider that

owns

link1 hasno incentive to deviate, itfollows that {p^^,x'~'^)isan

OE.

Q.E.D.

The

existence resultcannot

be

generalized to piecewise linear latency functions or to latency functions

which

are linearover their effective domain, as illustrated in the followingexample.

Example

3 Considera

two

linknetwork. Letthe total flowhe d

=

1.

Assume

that the latency functions are givenby

, ( \ n 1 f \

/O

iiO<x<S

h{x)

=

0, l2{x)

=

< ^-5

^>^^

for

some

e

>

and

6

>

1/2,withtheconventionthat

when

e

=

0,hix)

=

oofor x

>

6.

We

first

show

that thereexists

no

purestrategy oligopolyequihbrium forsmalle (i.e.,

thereexists

no

purestrategy

subgame

perfectequihbrium).

The

followinglistconsiders

allcandidateoligopoly priceequihbria{pi,P2)

and

profitable unilateraldeviationsfore

sufficientlysmall,thus estabhshingthenonexistenceofan

OE:

1. _Pi

=

P2

=

0:

A

small increase in the price of provider 1 will generatepositive

profits,thus provider1hasanincentive to deviate.

2. _pi

=

_P2

>

0: Let

x

bethe flow aflocationat the

OE.

IfXj

=

1, thenprovider 2 has

an

incentive todecreaseitsprice. IfXi

<

1, thenprovider 1has anincentive to decreaseitsprice.

3.

<

_pi

<

p2- Player 1hasanincentive toincreaseitspricesinceitsflow allocation remains the same.

4.

<

p2

<

Pi: Fore sufficientlysmall, theprofitfunction of player 2,givenpi, is

strictlyincreasing asafunction ofp2!showingthatprovider 2hasanincentive to increaseitsprice.

We

next

show

that a

mixed

strategy

OE

alwaysexists.

We

definea

mixed

strategy

OE

asa

mixed

strategy

subgame

perfectequilibrium of the pricecompetition

game

(see

Dasgupta

and

Maskin, [10]). Let

B^

be thespace ofall (Borel) probability measures

on [0,i?]". Let /<, denote the cardinality ofJj, i.e., the

number

ofhnks controlled by

serviceprovider5. Letfig

G

B^'be a probabihty measure,

and

denotethe vector of these

probabilitymeasures

by

/j,

and

the vector of these probabilitymeasures excluding s

by

Definition 7 (/i*,x*(p))isa

mixed

strategy OligopolyEquilibrium

(OE)

ifthefunction x*{p)

G

W

(p) forevery

p

G [0,R]^

and

/ ^s{Ps,P-s,x*{ps,p-s)) d{i2l{p,) Xi_i*_^{p_s)) J[o,rY

/ Us{ps,P^s,X*{ps,P-s))d{iJ-siPs) XIJ--,{P-s))

J\O.R]'

>

'[0,R]'

foralls

and

/ig

G B'\

Therefore, a

mixed

strategy

OE

simplyrequires thatthereisnoprofitabledeviation toadifferentprobability

measure

for eacholigopolist.

Example

3

(continued)

We

now

show

thatthe following strategyprofileisa

mixed

strategy

OE

for the above

game when

e -^ (a

mixed

strategy

OE

also exists

when

e

>

0,butitsstructureis

more comphcated and

lessinformative);

(

0<p<R{l-5),

Mp)-<

1-^

R{l-6)<p<R,

\ 1 otherwise, (

0<p<R{l-6),

^2{P)

=

1

1-^

R{l-5)<p<R,

[ 1 otherwise.

Notice that pi has

an

atom

equalto 1

—

5 at i?.

To

verifythat this profile isa

mixed

strategy

OE,

letji'

be

thedensity of/i,with the conventionthatjj'

=

oo

when

thereis

an

atom

atthat point. LetAfj

=

{p |

/i^_{p)

>

0}.

To

establish that _{pi,fJ-2)isa

mixed

strategyequihbrium,it sufficesto

show

thattheexpectedpayoff to playeri isconstant

for all_Pi

G

Mi when

the other player choosesp-i according to /i_i (see [24]).

These

expectedpayoffsare

n

{p, Ip-i)

=

/ U,{p^,P-i,X{p^,P-i))dfl-^(p^i),

Jo

whichareconstant forall_pi

G

Mi

for i

=

1.2. Thisestablishesthat (/^i./io) isa

mixed

strategy

OE.

The

next proposition

shows

that a

mixed

strategyequilibriumalwaysexists.

Proposition

8 Let

Assumption

1hold.

Then

the pricecompetition

game

hasa

mixed

strategy

OE,

(^^^,x°^(p)).

The

proofofthispropositionislong

and

notdirectlyrelatedtherestofthe argument, soitisgivenin

Appendix

B.

We

next provide

an

exphcit characterization ofpure strategy

OE.

Though

ofalso independent interest,these results are

most

useful forus toquantify'theefficiency loss

ofoHgopolyinthenextsection.

The

following

lemma

shows

that an equivalent to

Lemma

1 (which required

real-valued latency functions) also holds with

more

general latency functions at the pure strategy

OE.

Lemma

4 Let

Assumption

1 hold. If{p'^^,x^^) isapurestrategy

OE,

then

h{x?'^)+P?'^

=

min{/,(xH+p°^},

V?withxP>0,

(13)

j

k{x*)+pf^

<

R,

Vzwithzp^>0,

(14)

^xP

<

d, (15)

with

_J2Li

^?^

=

d^i

mmj{lj{xf^)

+

Pj}

<

R.

Proof. Let

{f^,x°^)

be an

OE.

Since

x°^

£

VF(p^^), conditions(14)

and

(15)follow bythe definition ofa

WE.

Considercondition(13). As,sumethat thereexist

some

z,j

G

J

with

xf ^

>

0,

x°^

>

suchthat

;,(xf^)+pp^</,(x^°^)+pf^.

Usingthe optimality conditionsfora

WE

[cf.Eq.(2)], thisimpliesthat xf-^

=

hc^-

Con-sider changing

_pf^

_{to pf-^}

+

e for

some

e

>

0.

By

checkingthe optimality conditions,

we

see that

we

can chooseesufficientlysmallsuch that x'^^

G H^(pf^

+

e,p?f).

Hence

serviceprovider that

owns

linkicandeviate to

_pf

-^

₊

_e

_and

_increaseitsprofits, contra-dictingthefactthat(p*^"^, x'~'^) isan

OE.

Finally,

assume

to arrive atthe contradiction thatminj{/j(x^^)-\-Pj]

<

R

and

'Ylii=\

^?^

<

'^- Usingthe optimality conditionsfora

WE

[Eq. (2)with A

=

since_J2i^i

^?^

<

"^l'^^^^

imphes

that

we must

have

xf^

=

bc^ for

some

i.

With

a similar

argument

toabove, adeviation to

_pf^

+

ekeeps x'-'^ as a

WE,

and

is

more

profitable,completingthe proof.

Q.E.D.

We

needthe following additionalassumptionforour pricecharacterization.

Assumption

2 Givenapurestrategy

OE

{p^^,x'-'^),iffor

some

i

E

2

with

_xf^

>

0,

we

have/^(xf^)

=

0,then1^

=

{i}.

Note

thatthisassumptionisautomaticallysatisfiedifalllatency functions arestrictly

increasing orifallserviceproviders

own

onlyonelink.

Lemma

5 Let {p'-^^,x'-^^) be apurestrategy

OE.

Let

Assumptions

1

and

2 hold. Let

Us

denotetheprofitofserviceproviders at {p'^^,x^^).

(a) Ifn^'

>

for

some

s'

G

5,then

n^

>

foralls

e

S.

(b) If

n,

>

for

some

s

G

5,then

_pf^xf^

>

forallj GT,.

Proof.

(a) For

some

j

G

Xy,define

K

=

_pf^

+

lj{xf^),whichispositivesince Ug'

>

0.

Assume

lis

=

for

some

s. Fork

E

Is,considerthe pricepfc

=

A'

—

e

>

for

some

small e

>

0. Itcan beseen that atthe price vector {pk,P-k)^ ^^^corresponding

WE

link

flow

would

satisfyx^

>

0. Hence,serviceprovidershasanincentive todeviate to pk atwhich hewill

make

positiveprofit, contradicting thefactthat

{p^^,x^^)

is

apurestrategy

OE.

(b) Since Ug

>

0,

we

havep'^fx'^f

>

for

some

m

G Is-

By

Assumption

2,

we

can

assume

withoutlossofgeneralitythatImi^m^)

>

(otherwise,

we

are done). Let j

G

Is

and assume

to arrive at a contradiction that

_p^^xf^

=

0.

The

profitof serviceprovidersatthepurestrategy

OE

canbewritten as

Yis

=

ns+p^fx^f,

where

lisdenotestheprofitsfrom hnksotherthan

m

and

j. Let

p^^

=

K

—lm{x'^)

for

some

K

. Consider changingtlieprices p'^^

and

p^^

suchthat the

new

profitis

n,

=

n^

+

{K -

UxZ^

-

e)){xZ^

-,)

+

e{K

-

/,(e)).

Note

that eunits of flow are

moved

fromlink

m

to link j suchthat the flows of other linksremainthe

same

at the

new

WE.

Hence, thechangeintheprofit is

n,

-

n,

=

(/„(x^,^)

-

UxZ^

-

e))x^^

+

e(/„(x^^

-

e)

-

/,(e))).

Sincelm{x^^)

>

0, ecan be chosensufficientlysmallsuchthattheaboveisstrictly

positive, contradictingthefactthat

{p^^,x^^)

isan

OE.

Q.E.D.

The

following

example shows

that

Assumption

2cannot be dispensed with for part

(b) ofthis

lemma.

Example

4

Consider athree link

network

with

two

providers,

where

provider 1

owns

links1

and

3

and

provider 2

owns

link2. Letthe total flowbe d

=

1

and

the reservation

.

utiUtybei?

=

L

Assume

thatthe latency functions are given

by

h{xi)

=

0,

l2ix2)=X2,

h{x3)=ax3,

for

some

a

>

0.

Any

price_{vector {pi,P2,P3)}

=

(2/3, 1/3,b)withb

>

2/3

and

(xi, X2,X3)

=

(2/3,1/3,0) isa purestrategy

OE,

so^3X3

=

contrarytopart (b) ofthe

lemma.

To

see

why

thisis

an

equihbrium, notethat provider 2is clearlyplaying abest response. Moreover,inthis allocation

Hi

=

4/9.

We

canrepresent anydeviation ofprovider 1

by

{pi,Ps)

=

{2/3-5,2/3-ae-5),

for

two

scalarse

and

6,whichwillinduce a

WE

of(xi, X2,X3)

=

(2/3

+

5

—

e,1/3

—

5,e) .

The

correspondingprofitofprovider 1 atthisdeviationis

Hi

=

4/9

—

5"^

_<

4/9, estab-lishingthat provider1 isalsoplayinga bestresponse

and

we

have apurestrategy

OE.

We

next establish that under an additional assumption, a pure strategy

OE

will

neverbeatapoint of non-difTerentiability of the latency functions.

Assumption

3

There

exists

some

s

E

S

suchthat k isreal-valued

and

continuously differentiableforalli

Els

Lemma

6 Let (p°^,x°^) be an

OE

with min^

{p^^

+

lj{xf^)}

<

R

and

pf^xf

^

>

for

some

i. Let

Assumptions

1,2

and

3 hold.

Then

where

l^{xf^)

and

l~{xf^) arethe right

and

leftderivatives ofthe function li at

xf^

respectively.

Since theproofofthis

lemma

islong,itisgivenin

Appendix

C.

Note

that

Assumption

3cannotbe dispensed withinthis

lemma.

Thisisillustrated inthenextexample. 17

Example

5 Consider a

two

linknetwork. Letthe total flowbe d

=

1

and

the reserva-tionutihtybei? =: 2.

Assume

thatthelatency functions aregivenby

if

<

X

<

i

h{x)-l2ix)-

,

2(x-i)

_otherwise.

Itcanbeverifiedthat the vector {Pi^,P2^)

=

(1, 1),with

(xf^,x^^)

=

(1/2, 1/2) isa purestrategy

OE, and

is atapoint of non-differentiabiUtyforbothlatency functions.

We

next provide

an

exphcit characterization of the

OE

prices, whichis essential in

ourefficiencyanalysisin Section5.

The

proofisgivenin

Appendix

D.

Proposition

9 Let {p^^,

x^^)

be an

OE

such that

pf^xf^

>

for

some

i GI. Let

Assumptions

1, 2,

and

3 hold.

a)

Assume

thatmin^

{pf^

+

lj{x'^^)}

<

R. Then,for allse

5

and

i GX,,

we

have

(

xf%{x°^),

ifl'j{x°^)

=

for

some

j

^

Is,

??""={

xf^^K^P^)+

v-^^'^'^^r , otherwise.

(^6)

b)

Assume

thatmin^{p'j^

+

Ijix'j^)}

—

R- Then, forallsG<S

and

i £Xg,

we

have

pf^>^,n-(:^?^)-

(17)

Moreover,ifthereexists

some

iG

T

suchthat

X

=

{i} for

some

s

G

5,then

p?^

<

^rc(^r)

+

_^

'

, (18)

If_{the latency functions U are} all real-valued

and

continuouslydifferentiable, then analysis of

Karush-Kuhn-Tucker

conditionsforohgopolj'

problem

[Eq. (78)in

Appendix

D] immediatelyyieldsthe followingresult:

Corollary

1 Let (p°^,

x°^)

bean

OE

suchthat

p°^x°^

>

for

some

i

G

I. Let As-sumptions1

and

2hold.

Assume

alsothatZjisreal-valued

and

continuouslydifferentiable

foralli. Then, foralls

G

5

and

i

EiTs,^^

have

r

xf

^/^(xp^), if/;(xf^)

=

for

some

j

^

J,

Pi

^

\ r..ir.

mm

J

P

_

l.f^OE\

^OEv(^OE\

, £,

<^i?

-

/,(xf ^) , x?^l'AxY^)

+

^'^^

'

\ }, otherwise.

(19)

Thiscorollaryalsoimpliesthatinthetwolinkcasewithreal-valued

and

continuously differentiable latency functions

and

with

minimum

effective cost lessthan R, the

OE

pricesare

pP

=

xf^(/;(x?^)

+

/^(xn)

(20)

asclaimedinthe Introduction.

5

Efficiency

of

Oligopoly

Equilibria

In thissection,

we

study theefficiencyproperties ofohgopolypricing.

We

take asour

measure

ofefficiencythe ratio ofthesocialsurplus oftheequilibriumflow allocation to the social surplus ofthe social

optimum,

§(x*)/§(x'^),

where

x* referstothe

monopoly

orthe oligopolyequilibrium [cf. Eq. (8)]. In Section3,

we

showed

that the flow alloca-tion at a

monopoly

equilibrium isasocial

optimum.

Hence,in congestion

games

with

monopoly

pricing, there isnoefficiencyloss.

The

following

example

showsthatthis is

notnecessarilythe caseinohgopolypricing.

Example

6 Considera

two

linknetwork. Letthe total flow

he

d

=

1

and

the reservation utihtybei?

=

1.

The

latency functions are given

by

3

li{x)^0,

kix)

=

~x.

The

uniquesocial

optimum

forthis

example

isx'^

=

(1,0).

The

unique

ME

{p^^,

x^^)

is

x^^

=

(1,0)

and

p^'^

=

(1,1).

As

expected, the flow allocations at the social

optimum

and

the

ME

are thesame.

Next

consider a

duopoly

where

each ofthese links

is

owned

by adifferentprovider. UsingCorollary1

and

Lemma

4, it followsthat the flow allocation atthe

OE,

x'^^, satisfies

/i(xf^)

+

x?^[/;(x?^)

+

Ux'i^)]

=

kix^

+

x^^[l[{x?^)

+

l',ix^%

Solvingthistogetherwith

xf^

+

x^^

=

1showsthattheflow allocation attheunique oligopoly equilibrium is x'-'^

=

(2/3,1/3).

The

socialsurplus at the social

optimum,

the

monopoly

equilibrium,

and

the oligopoly equilibrium are given

by

1, 1,

and

5/6, respectively.

Beforeproviding a

more

thoroughanalysis oftheefficiencyproperties ofthe

OE,

the nextpropositionprovesthat,asclaimedintheIntroduction

and

suggestedby

Example

6, a change in the

market

structurefrom

monopoly

to

duopoly

in a two

hnk

network typicallyreducesefficiency.

Proposition 10

Consider a

two

link network

where

each link is

owned

by a

differ-ent provider. Let

Assumption

1 hold. Let {p'^^,x^^)

be

a pure strategy

OE

such

that

_pP^xf^

>

for

some

i

G

J

and

mm,

{pf^

-f-lj{xf^)}

<

R. If/'i(xf^)/xf^ j^

l'2{x^^)/x^^, then S(xO-^)/S(x^)

<

1 .

Proof.

Combining

the

OE

priceswiththe

WE

conditions,

we

have /i(xf^)

+

xf^{i[{xf^)

+

ux^))

=

Hxr)

+

xriux'",")

+

ux^^)),

where

we

use thefactthatmin,

_{pf^

+

lj{x?^)}

<

R- Moreover,

we

canuse optimality conditions (7) toprove that a vector (xf ,xf)

>

is asocial

optimum

if

and

onlyif

/l(xf)

+

xf/;(xf)=/2(xf)+xf/^(xf),

(seealsotheproofof

Lemma

4). Since l[{x'^^)/x'^^

^

{^{x^^)j

x^^

, theresult follows.

Q.E.D.

We

next quantify theefficiency ofohgopoly equilibriaby providing a tight

bound

onthe efficiency loss incongestion

games

witholigopoly pricing.

As we

have

shown

in

Section4,such

games do

notalwayshave apurestrategy

OE.

In the following,

we

first

provide

bounds

on congestion

games

thathave purestrategyequilibria.

We

nextstudy

efficiencyproperties of

mixed

strategyequilibria.

5.1

Pure

Strategy

Equilibria

We

consider pricecompetition

games

thathave purestrategy equilibria(thissetincludes, but is larger than,

games

with linear latency functions, see Section 4).

We

consider latency functionsthatsatisfy

Assumptions

1, 2,

and

3. Let £/ denote thesetoflatency functions {Zi}ieisuchthat the associated congestion

game

hasapurestrategy

OE

and

theindividualZ^'ssatisfy

Assumptions

1,2,

and

3.^ Givena parallel linknetwork with/

links

and

latency functions {/,}iei

£

C.I-, let0£/({/i})denotethesetofflow allocations at

an

OE.

We

definetheefficiencymetricat

some

x'^^

G

OE{{li\) as

where

x^

isasocial

optimum

given the latency functions{li}iei

and

R

isthereservation

utihty. In other words, our efficiency metric is the ratio ofthe social surplus in an equilibrium relative tothe surplus in the social

optimum.

Following theliteratureon the "price ofanarchy",inparticular [19],

we

are interestedintheworstperformancein

anoligopoly equilibrium, so

we

lookfora lower

bound on

inf inf ri{{l,},x'^^).

We

first prove

two

lemmas, which reducethe set oflatency functions that needto

be consideredin

bounding

theefficiencymetric.

The

next

lemma

allows us to use the oligopoly pricecharacterization giveninProposition9

Lemma

7 Let [p'^^,x'^^) beapurestrategy

OE

such that

pf^x^^

=

for alli

G

I.

Then

x'^^isasocial

optimum.

Proof.

We

first

show

that k{xf^)

=

for alli G I.

Assume

that lj{xf^)

>

for

some

j

G

I. Thisimplies that

_{xf^ >}

and

therefore

_pf^

=

0. _{Since lj{xf^)}

>

0, itfollows

by

Lemma

2that for allx G

W{p),

we

haveXj

=

x^^. Consider increasing p*?^ to

some

smalle

>

0.

By

theupper semicontinuityof

W{p),

it followsthatfor all

X G W{e,p9.f),

we

have\xj

—Xj^\

<

5for

some

5

>

0. Moreover,by

Lemma

2,

we

have,

forall

X

GW{e,p'ilf),Xi

>

.xf foralli7^ j. Hence,theprofitofthe provider that

owns

^Moreexplicitly,Assumption2implies thatifany

OE

{p°^ ,x^^)associatedwith{i,}j(=ihas

xf^ >

andli{xf^)

=

0,thenJ,

=

{i}.

link jisstrictlyhigherat price vector _[e^p'^f) thanatp*^^, contradictingthe factthat (p^^,

^0£)

is^^

OE.

Clearly

x^^

>

for

some

j

and

hence

mmi^j{pf^

+

li{xf^)}

=

_pf^

+

lj{xf^)

=

0,

which

imphes by

Lemma

4that

_J2i&x^?^

=

d- Using/^(xP'^)

=

0,

and

G

dli{xf^) for

alli,

we

have

R

-

li{xf^)

-

xf^gi^

=R,

y

lel,

for

some

gi^

G

dli{xf^). Hence,

x^^

satisfiesthe sufficientoptimality conditions fora

social

optimum

[cf.Eq. (7) with

X^

=

R],

and

the resultfollows.

Q.E.D.

The

next

lemma

allows us to

assume

without loss ofgenerality that

R

_Yli=i

_^f

~

X]i=ih{xf)xf

>

and

_X]j^i

xf^

=

dinthesubsequentanatysis.

Lemma

8 Let {k}iex

G

Cj.

Assume

that

either (i)_{Yli=i ^ii^i)^i}

—

_•^Xli=i^2^^^^

some

social

optimum

x^,

or (ii)

_ELi

^?'^

<

dfor

some

x^^

G 0^{{h}).

Then

everyx*^^

G

OE{{li}) isasocial

optimum,

implyingthat rj{{li},x'^^)

—

1.

Proof.

Assume

_{that Ei=i^i(^f)^f}

—

_^Yli=i^i-

Since x^ is a social

optimum

and

everyx'-^^

G

OE{{li})isafeasiblesolution tothesocial

problem

[problem(6)],

we

have

=

^(/? -

k{xf))xf

> J^iR -

k{xf''))xf^.

V x^^

G ag({/0).

2=1 1=1

By

the definition ofa

WE,

we

havexf-^

>

and

/?

-

_kixf'^)

>

_{pf ^}

>

(where_pf-^is

the price of

hnk

iatthe

OE)

foralli. This

combined

withtheprecedingrelation

shows

that

x^^

isasocial

optimum.

Assume

nextthat

_Ei=i

^?^ <

dfor

some

x'-^^

G

0S({/,}). Let p'^^be theassociated

OE

price.

Assume

that

p^^x^^

>

for

some

j

&

I

(otherwise

we

are

done by

Lemma

7). Since

_Ei=i

^?^

<

d,

we

have

by

Lemma

4that m.mj^j{pj-\-lj{xf^}

=

R. Moreover,

by

Lemma

5, itfollowsthat

p^xf^

>

foralli

G

X. Hence,foralls

e

S, {{pf^)ieis,x'-'^) isan optimalsolution ofthe

problem

maximize((p,)_gj^_j.)

_Y.V^X^

subject to P^

+

l^{x,)

=

R,

Vtels,

pf^

+

/,(x,)

=

i?.

-ii^ls

Tx?''<d.

Substitutingfor {pi)iei, in the above,

we

obtain

maximize2:>o

_2.

[R

—

U{Xz))xi

subject to XiETi,

^

i _^Is,

where

T^

=

{xj

|

pf^

+

k{xi)

=

R}

iseither asingleton or a closed interval. Sincethis

isaconvex problem,using the optimality conditions,

we

obtain

R-k{xf^)-xf^gi^

=

0,

VzgX„Vsg5,

where

gi^ £ dk{xf^).

By

Eq. (7), itfollowsthat

x^^

isasocial

optimum.

Q.E.D.

Hence, infinding a lower

bound on

theefficiency metric,

we

canrestrict ourselves, without loss of generahty, to latency functions {/j} G Cj such that X]i=i'i(^f)^f

<

RY!i=\ ^f

for

some

social

optimum

x^

,

and

_Yli=i

^?^

^

d.for all

x^^ G

US{{li}).

By

the following

lemma, we

canalso

assume

_{that X]i=i}

_^f

=

d-Lemma

9 For asetoflatency functions{/j},gj,let

Assumption

1hold. Let (p*^^,

x^^)

bean

OE

and x^

bea social

optimum.

Then

iex iei

Proof.

Assume

to arrive at acontradiction that

Yli^jxf^

>

X^jgjxf. This implies that x'^^

>

Xj for

some

j.

We

alsohave lj{x^^)

>

lj{x^). (Otherwise,

we

would

have lj{xj)

=

l'j{Xj)

=

0,whichyields a contradiction

by

the optimality conditions (7)

and

thefactthat

^^^jxf

<

d). Usingtheoptimality conditions (2)

and

(7),

we

obtain

R

-

_{/,(xf ^)}

- p°^

>R-

_/,(xf)

-

_xf

_5;^

,

for

some

gi^ G dlj{x^).

Combining

the preceding with lj{x^^)

>

lj{xj)

and

pj'^

>

x^^Z~(x^^)

(cf.Proposition9),

we

see that

xf%ixf^)<x^g„

contradicting

x^^

>

xJ

and

completingthe proof.

Q.E.D.

5.1.1

Two

Links

We

firstconsider aparallel linknetwork with twolinks

owned

by twoserviceproviders.

The

next theorem providesatight lower

bound on

r2{{h},x'^^) [cf. Eq. (21)]. In the following,

we assume

without loss ofgenerality that d

=

I. Also recall that latency functions in£2satisfy

Assumptions

1, 2,

and

3.

Theorem

1 Consider a two link network where each link is

owned

by a difi'erent

provider.

Then

r2{{k},x^^)

>

^,

V

{/,},.i,2

e

£2,

x^^

e ag({/J),

(22)

and

the

bound

istight,i.e.,there exists {li}i=i^2 S

^2 and

x*^^

G

OE{{li})that attains the lower

bound

inEq. (22).

Proof.

The

prooffollowsa

number

ofsteps:

Step1:

We

are interestedinfinding a lower

bound

for the

problem

inf inf r2({/J,x«^). (23)

Given

{/j}

G

C2, letx'-'^

6

OE{{li})

and

let

x^

be a social

optimum.

By

Lemmas

8

and

9,

we

can

assume

that

5Zi=i^?^

~

Xli=i

^f

—

1- Thisimplies that there exists

some

isuchthat

xf^

<

xf. Since the

problem

is symmetric,

we

canrestrict ourselves to{li}

G £2

suchthat

xf^

<

xf.

We

claim

inf inf

_{r2{{k},x^^)>r^f,}

(24)

where

^2,1

-

mmimize,s,

(,5),>o

—

—

T5-5

—

j^-^

{E) yf,yOE^o subject to if

<

yfilfy, i

=

_{1, 2,} (25) il

+

yUil)'

=

if

+

ynify,

(26) ^f

+

yf(^f)'<^,

(27)

E^f

=

l' (28) z=l Zi

<

_;f , (29) /;

=

0, if /f

=

0. (30) li<y?^l[. ^

=

l,2, (31)

E?/?^ =

l' (32) i=l

+

{Oligopoly EquilibriumConstraints};, f

=

1,2.

Problem

(E)can be viewedasafinitedimensional

problem

thatcapturesthe

equihb-rium

and

the social

optimum

characteristics oftheinfinitedimensional

problem

givenin

Eq. (23). Thisimpliesthat instead ofoptimizingoverthe entire functionl^,

we

optimize

over the possible values ofli{-)

and

dk{-) atthe equilibrium

and

the social

optimum,

which

we

denote byk,l'^,if,(if)'[i.e., (if)' isavariable that representsallpossiblevalues ofgi^

G

dli(yf)].

The

constraints ofthe

problem

guaranteethat these valuessatisfy the necessaryoptimahtyconditions for asocial

optimum

and

an

OE.

In particular, condi-tions (25)

and

(31) capturetheconvexityassumption onli{-) byrelatingthe valuesli,l[

and

/f,(/f)' [notethat theassumption1^(0)

=

isessentialhere]. Condition(26) isthe 23