Competition and efficiency in congested markets

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COMPETITION

AND

EFFICIENCY

IN

CONGESTED MARKETS

Daron

Acemoglu

and

Asuman

Ozdaglar

Working

Paper

06-1

1

January

20,

2006

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MASSACHUSETTS

INSTITUTE

OF TECHNOLOGV

JUN

2 2006

J

Competition

and

Efficiency

in

Congested

Markets^

Daron

Acemoglu

Department

of

Economics,

Massachusetts

Institute of

Technology

Asuman

E.

Ozdaglar

Department

of Electrical

Engineering

and

Computer

Science

Massachusetts

Institute

of

Technology

January

20,

2006

Abstract

We

studythe efficiencj'ofoligopoly equilibriaincongested markets.

The

moti-vatingexamplesarethe allocation ofnetworkflowsin acommunicationnetworkor oftrafficina transportation network.

We

showthat increasingcompetition

among

oligopolists can reduce efficiency, measured as the difference between users'

will-ingness to pay anddelay costs.

We

characterize atight

bound

of5/6onefficiency

in pure strategy equilibria

when

there is zero latency at zero flow and a tight

bound

ofIs/Ji

—

2 withpositive latency atzero flow. These bounds are tight even

when

the numbers of routes andoligopolists are arbitrarily large.

*We

thank XinHuang, Ramesh Johari, EricMaskin, Eilon Solan,Nicolas StierMoses, Jean Tirole,

John Tsitsiklis, Ivan Werning,

Muhamet

Yildiz, two anonymous referees and participants at various

1

Introduction

We

analyze pricecompetitionin thepresenceofcongestion costs. Considerthe following

environment: one unit of traffic can use one of / alternative routes.

More

traffic

on

a

particular route causes delays, exerting a negative (congestion) externality

on

existing

traffic.^ Congestioncosts arecaptured

by

aroute-specific non-decreasing convexlatency

function, li(•). Profit-maximizing oligopolists set prices (tolls) for travel

on

each route

denoted

by

p,.

We

analj'ze

subgame

perfect

Nash

equilibria of this environment,

where

for each price vector, p, all traffic chooses the

path

that has

minimum

(delay plus toll)

cost, li

+

Pi,

and

oligopolists choose prices to

maximize

profits.

The

environment

we

analyze is of practical importance for a

number

of settings.

These

include transportation

and communication

networks,

where

additional use of a route (path) generates greater congestion for all users,

and

markets in

which

there are

"snob" effects, so that

goods

consumed

by

fewer other

consumers

are

more

valuable (see

for example, [53]).

The

key feature of these environments is the negative congestion

externality that users exert

on

others. This externality has

been

well-recognized since

the

work by

Pigou [40] in economics,

by

[46], [57], [5] intransportationnetworks,

and by

[36], [24], [23], [30] in

communication

networks.

More

recently, therehas

been

a growing

literature that focuses

on

quantification of efficiency loss (referred to as the price of

anarchy) that results

from

externahties

and

strategic behavior in different classes of

problems: selfish routing (e.g., [25], [45], [10], [11], [39]

and

[15]); resource allocation

by market mechanisms

(e.g., [22], [47], [31], [59]); network design (e.g., [3]);

and

two-stage competitive facihtylocation without congestion costs

and

externalities (e.g., [54]). Nevertheless, the game-theoretic interactions

between

(multiple) service providers

and

users, or the effects of competition

among

the providers

on

the efficiency loss has not

been

considered in networks with congestion (externalities). This is an important area

for analysis since in

most

networks congestion is a first-order issue

and

(competing)

profit-maximizing entities chargeprices for use. Moreover,

we

will

show

that the natm-e

ofthe analysis changes significantly in the presence ofprice competition.

We

provide a general

framework

for the analysis of price competition

among

ser-viceproviders^ in a congested (and potentially capacitated) network, studyexistence of

purestrategy

and

mixed

strategy equilibria,

and

characterize

and

quantify the efficiency

properties of equilibria.

There

are four sets of

major

results

from

our analysis.

First,

though

the equilibrium of traffic assignment without prices can be highly

inefficient (e.g., [40], [45], [10]), price-setting

by

a monopolist internahzes the negative

externality

and

achieves efficiency.

Second, increasingcompetition can increaseinefficiency. Infact, changingthe

market

structure

from

monopoly

to

duopoly

almost alwaj^s increases inefficiency. This result

contrastswith

most

existingresults intheeconomicshterature

where

greatercompetition

tends to improve the allocation ofresources (e.g. see Tirole [51]).

The

intuition for this

result,.

which

is related to congestion, is illustrated

by

the

example

we

discuss below.

^

^

An

_{externality ariseswhenthe actions ofthe playerin agame affects the payoff ofother players.}

^We

use oligopolist andserviceprovider interchangeablythroughout thepaper.

^Because,inourmodel,users arehomogeneous andhavea constantreservationutility, intheabsence

Third

and

most

important,

we

providetight

bounds on

the extentof inefficiency inthe

presence of ohgopolistic competition.

We

show

that

when

latency at zero flow (traffic)

is equal to zero, social surplus (defined as the difference

between

users' willingness to

pay and

the delay cost) in

any

pure strategy oligopoly equilibrium is always greater

than

5/6 of the

maximum

social surplus.

When

latency at zero flow can be positive,

there is a slightly lower

bound

of 2\/2

—

2

w

0.828.

These

bounds

are independent of

both

the

number

of routes, /,

which

could be arbitrarily large,

and

how

these routes are distributed across different oligopolists (i.e., of

market

structure). Simple

examples

reach thesebounds.

Finally,

we

also

show

that pure strategy equilibria

may

fail to exist. This is not

surprisingin

view

ofthefactthat

what

we

havehereisaversionofa

Bertrand-Edgeworth

game

where

pure strategy equilibria

do

not exist in the presence of convex costs of

production or capacity constraints (e.g., [14], [49], [7], [56]). Ho'vever, in our

ohgopoly

environment

when

latency functions arelinear, a purestrategyequilibrium alwaysexists,

essentiallybecause congestion externalities

remove

the payoff discontinuities inherent in

the

Bertrand-Edgeworth game.

Non-existence

becomes an

issue

when

latency functions are highly convex. In this case,

we

prove that

mixed

strategy equilibria always exist.

We

also

show

that

mixed

strategy equilibriacan lead to arbitrarily inefficientworst-case

realizations; in particular, social surplus can

become

arbitrarily small relative to the

maximum

social surplus,

though

the average

performance

of

mixed

strategy equilibria

is

much

better.

The

following

example

illustrates

some

ofour results.

Example

1 Figure 1

shows

a situation similar to the one first analyzed

by Pigou

[40]

to highlight theinefficiency

due

to congestion externahties.

One

unit oftraffic willtravel

from

origin

A

to destination B, using either route 1 or route 2.

The

latency functions

are given

by

k{x)

=

Y'

^2(2;)

=

-X.

It is straightforward to see that theefficient allocation [i.e., onethat minimizes the total

delay cost '^ili{xi)xi] is

_xf

—

2/3

and xf

=

1/3, while the

(Wardrop) equihbrium

allocation that equates delay

on

the

two

paths is _xf'^ R^ .73

>

_{xf and}

x^^

~

-27

<

_xf

.

The

source ofthe inefficiency is that each unit oftraffic does not internalize the greater increase in delay

from

travel

on

route 1, so there is too

much

use of this route relative

to the efficient allocation.

Now

consider a monopolist controlling

both

routes

and

setting prices for travel to

maximize

its profits.

We

show

below

that in this case, the

monopoHst

will set aprice including a

markup,

Xj/^

(when

_k is differentiable),

which

exactly internalizes the

con-gestionexternality. In other words, this

markup

is equivalent to the Pigovian taxthat a

socialplanner

would

set in order to induce decentralizedtrafficto choosetheefficient

al-location. Consequently, in thissimpleexample,

monopoly

prices

wiU

be_pf^-^

=

(2/3)

+k

and

p^^

—

(2/3^) -f-k, for

some

constant k.

The

resulting traffic in the

Wardrop

equi-librium will

be

identical to the efficient allocation, i.e., Xj^-^

=

2/3

and

x,^^

=

1/3.

l,(x)=x /3

I unitof

traffic

^x)=i2/3)x

Figure 1:

A

two

link network with congestion-dependant latency functions.

Finally, consider a

duopoly

situation,

where

each route is controlled

by

a different

profit-maximizingprovider. In thiscase,itcan

be

shown

that equilibriumprices willtake

the form

_pf

^

—

Xi{l[

+

I'o) [seeEq. (20) inSection4],or

more

specifically,

_pf

^

«

0.61

and

p^^

K, 0.44.

The

resulting equilibrium traffic is

_xf^

w

.58

<

_{xf and}

x^^

«

.42

>

_xf

,

which

alsodiffers

from

theefficientallocation.

We

will

show

thatthisisgenerallythe case

in the oligopoly equilibrium. Interestingly, while in the

Wardrop

equilibrium without

prices, there

was

too

much

traffic

on

route 1,

now

there is too little traffic because

of its greater

markup.

It is also noteworthy that although the

duopoly equihbrium

is inefficient relative to the

monopoly

equilibrium, in the

monopoly

equilibrium k is

chosen such that all ofthe

consumer

surplus is captured

by

the monopolist, while inthe

oligopoly equilibrium users

may

havepositive

consumer

surplus.^

The

intuition for the inefficiency of

duopoly

relative to

monopoly

is related to a

new

source of (differential)

monopoly

power

for each duopolist,

which

they exploit

by

distorting the pattern of traffic:

when

provider 1, controlling route 1, charges a higher

price, itrealizesthatthis

wiU

push

some

traffic

from

route1 toroute2, raisingcongestion

on

route 2.

But

this

makes

the traffic using route 1

become

more

"locked-in," because

their outside option, travel

on

the route 2, has

become

worse.^

As

a result, the optimal

price that each duopolist charges will include an additional

markup

over the Pigovian

markup.

These

are Xi/j for route 1

and

Xg/'j for route 2. Since these

two

markups

are generally different, they will distort the pattern of traffic

away

from

the efficient

allocation. Naturally, however, prices are typically lower with duopoly, so even

though

socialsurplus declines, users willbebetter offthanin

monopoly

(i.e.,they will

command

a positive

consumer

surplus).

There

isalarge literature

on models

ofcongestion

both

intransportation

and

commu-nication networks (e.g. [5], [38], [44], [33], [34], [45]).^ However, very few studies have

''Consumer surplusisthedifferencebetweenusers'willingness topay (reservationprice)andeffective

costs, _Pi

+

li{xi), and is thus different from social surplus (which is the difference between users'

willingness to pay and latency cost, li{xi), thus also takes intoaccount producer surplus/profits). See

[32].

^Usingeconomics terminology,we could alsosaythatthedemand forroute 1becomesmore

"inelas-tic". Since this term has a different meaning in the communication networks literature (see [48]), we

donotuseit here.

®Someofthesepapersalso use prices(ortolls) to induceflowpatterns thatoptimize overallsystem

investigated the implications of having the "property rights" over routes assigned to

profit-maximizing providers. In [4], Basar

and

Srikant analyze

monopoly

pricing under

specific assumptions

on

the utihty

and

latency functions.

He

and

Walrand

[19] study

competition

and

cooperation

among

internet service providers

under

specific

demand

models. Issues ofefficient allocation of flows or traffic across routes

do

not arise inthese

papers.

Our

previous

work

[1] studies the

monopoly

problem

and

contains the efficiency

of the

monopoly

result, but

none

of the other results here.

More

recent independent

work by

[3] builds

on

[1]

and

also studies competition

among

service providers. Using

a

different

mathematical

approach, they provide non-tight

bounds on

the efficiency loss

for the case of elastic traffic. Finally, incurrent work, [2],

we

extend

some

ofthe results

ofthis

paper

to a

network

with paraUel-serial structure.

In the rest of the paper,

we

use the terminology of a (communication) network,

though

all of the analysis applies to resource allocation in transportation networks,

electricity markets,

and

other

economic

applications. Section 2 describes the basic

en-vironment. Section 3 briefly characterizes the

monopoly

equilibrium

and

estabhshes its

efficiency. Section 4 defines

and

characterizes the oligopoly equilibria with

competing

profit-maximizing providers. Section 5 contains the

main

results

and

characterizes the

efficiency properties ofthe oligopoly equilibrium

and

provide

bounds on

efficiency.

Sec-tion 6 provides a tight efficiency

bound

when

there

may

be positive latencyat zero flow.

Section 7 contains concluding

comments.

Regarding

notation, all vectors are viewed as

column

vectors,

and

inequalities are to

be

interpreted componentwise.

We

denote

by

M^

the set of nonnegative /-dimensional

vectors. Let Ci

be

a closed subset of[0,oo)

and

let

/

: Cj i—>

M

be aconvex function.

We

use df{x) to denote the set of subgradients of _/ at x,

and

f~{x)

and

f~^{x) to denote

the left

and

right derivatives of

/

at x.

2

Model

We

consider a

network

with / parallel links. Let

X =

{1,. . .

,/} denote the set of links.

Let Xi denotethetotal flow

on

linki,

and x

=

[xi,. .

.

,xj] denotethe vector oflink flows.

Each

link in the

network

has a flow-dependent latency function li{xi),

which measures

the travel time (or delay) as a function of the total flow

on

link i.

We

denote the price

per unitflow (bandwidth) of

hnk

i

by

p,. Let

p

=

[p\, ,pi] denote the vector of prices.

We

are interested in the

problem

ofrouting d units of flow across the / finks.

We

as-sume

thatthis istheaggregate flow of

many

"small" users

and

thus

adopt

the

Wardrop's

principle (see [57]) incharacterizing the flow distribution in the network; i.e., the flows

are routed along paths with

minimum

effective cost, defined as the

sum

of the latency

at the given flow

and

the price of that

path

(see the definition below).^

We

also

as-sume

that the users have a reservation utility

R

and

decide not to send their flow ifthe

effective cost exceeds the reservation utility. This implies that user preferences can be

induceoptimal flows, with thegoal ofchoosingtolls from this set accordingtosecondary criteria, e.g.,

minimizing thetotal amount oftolls orthe numberoftolled routes; see [8], [21], [28], [27], and [20].

'''Wardrop's principle is used extensively in modelling traffic behavior in transportation networks,

u(x)

Figure 2: Aggregate utility function.

represented

by

the piecewise linear aggregate utilityfunction u (•) depicted in Figure 2.®

To

accountforadditionalsideconstraints inthetrafficequilibriumproblem, including

capacity constraints

on

thelinks,

we

usethe followingdefinition ofa

WE

(see [29], [26]).

Lemma

1

shows

that this definition is equivalent to the

more

standard definition of a

WE

used in theliterature

under

some

assumptions.

Definition

1

equilibrium

(WE)

if

For a given price vector

p

>

0,^ a vector

x^^

is a

Wardrop

WE

G

arg

max

< "V^ (i?

-

li{xf'^)

-

Pi)xj (1)

We

denote the set of

WE

at agiven

p by W{p).

Assumption

1 For each i

G

J, the latency function li convex, nondecreasing,

and

satisfies /^(O)

=

0.

[0,oo) 1-^ [0,do] is closed,^"

The

assumption

ofzero latency at zero flow, i.e., Zj(0)

=

0, implies that alllatency is

due

toflow oftraffic,

and

there are

no

fixed latency costs.^' _{It is adopted to simplify the}

discussion, especially the characterization of equilibrium prices in Proposition 9 below.

A

trivial relaxation of this assumption to li{0)

—

L

for

alH

G

J

for

some

L

>

wiU

have

no

effect

on

any of the results in the paper. Allowing for differential levels of

*This simplifying assumption impliesthat all users are "homogeneous" in the sense tha,t they have

the samereservation utility, R. The analysis belowwill show that the value ofthisreservation utility

R

has noeffect on any of the results as long as it is strictly positive.

We

discuss potential issues in

extending thiswork tousers withelastic andheterogeneous requirements intheconcluding section.

^Since the reservationutility ofusers is equal to /?, we can also restrict attention topi

<

R

for all

i. Throughoutthe paper, weusep

>

andp e [0,i?]^ interchangeably.

'"For-a function/ :

R"

i-^ (

—

00,00], wesay that / is closed ifthe level set {x

\ f{x

<

c)} is closed

for everyscalar c. Note that a functionis closed ifandonlyif it is lowersemicontinuous over

R"

(see

[9], Proposition 1.2.2),

^'Thisassumptionwouldbe agoodapproximationtocommunicationnetworkswherequeueingdelays

^j(O) complicates the analysis, but has Httle effect

on

the

major

results. This case is discussed in Section 6,

where

we

provide a slightly lower tight

bound

for the inefficiency

ofoligopoly equilibria without this assumption.

Another

feature of

Assumption

1 is that it allows latency functions to be extended

real-valued, thus allowing for capacity constraints. Let Ct

=

{x

E

[0,oo) | li{x)

<

oo}

denote the effective

domain

of _k.

By

Assumption

1,

Q

is a closed interval ofthe

form

[0, b] or [0,oo). Letbe,

=

sup^-gj;;^x.

Without

lossof generality,

we

can

add

the constraint

Xi

€

Ci in Eq. (1). Usingthe optimality conditionsfor

problem

(1),

we

seethat a vector

^WE

_g

_{]^/^ jg g^} -^/g j£

^^^

Q^Yy if _J^iei"^Y^

—

^ ^'^^ there exists

some

A

>

such that A( Y.^^z^"i^

-

d)

=

and

for all i,

i?-/,(xr^)-p,

<A

ifx|^^

=

0, (2)

=

A

ffO<xf'^<6c,,

>A

ffxf^^^c..

When

the latency functions are real-valued [i.e.,

Q

=

[0,oo)],

we

obtainthe following

characterization of a

WE,

which

is oftenusedas the definition ofa

WE

inthe hterature.

This

lemma

states that in the

WE,

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

+

pi, are

equalized

on

all links with positive flows.

Lemma

1 Let

Assumption

1 hold,

and assume

further that C,

—

[0,oo) for all i

£

I.

Then

a nonnegative vector x*

G

W{p)

if

and

only if

h{x*)

+

Pi

=

min{L(a;*) -1-p,},

V

i with x*

>

0, (3)

j

li{x*)

+

Pi

<

R,

V

i with X*

>

0,

iex

with X^,gjX*

=

difmin^ {lj{xj)

+

_Pj}

<

R.

Example

2

below shows

that condition ₍₃₎ in this

lemma

may

not hold

when

the latency functions are not real-valued.

The

existence, uniqueness,

and

continuity

prop-erties of a

WE

are well-studied (see (5], [12], [50]).

We

provide here the standard proof

for existence, based

on

establishing the equivalence of

WE

and

the optimal solutions of

a convex network optimization problem,

which

we

will refer to later in our analysis.

Proposition

1

(Existence

and

Continuity)

Let

Assumption

1 hold. For

any

price

vector

p

>

0, the set of

WE,

W{p),

is

nonempty.

Moreover, the correspondence

W

:

M^

:=i

M^

is

upper

semicontinuous.

Proof. Given any p

>

0, consider the following optimization

problem

maximizex>o

'^({R-pi)xi-

h{z)dz] (4)

subject to _{2_,}^i

^

d.

Inviewof

Assumption

(1) (i.e., _kisnondecreasingfor all i), it can be

shown

thatthe objective function of

problem

(4) is convex over the constraint set,

which

is

nonempty

(since

e

C,;)

and

convex. Moreover, the first order optimality conditions of

problem

(4),

which

arealso sufficient conditions foroptimality, areidenticaltothe

WE

optimahty

conditions [cf. Eq. (2)].

Hence

a flow vector

x^^

E

W{p)

if

and

only if it is an optimal solution of

problem

(4). Since the objective function of

problem

(4) is continuous

and

the constraint set iscompact, this

problem

has

an

optimalsolution,

showing

that

W{p)

is nonempty.

The

fact that

W

is

an upper

semicontinuous correspondence at every

_p

follows

by

using the

Theorem

of the

Maximum

(see Berge [6], chapter 6) for

problem

(4).

Q.E.D.

WE

flows also satisfy intuitivemonotonicity properties giveninthe following

propo-sition.

The

prooffollows from the optimality conditions [cf. Eq. (2)]

and

is omitted (see

[I])-Proposition

2

(Monotonicity)

Let

Assumption

1 hold. For a given

p

>

0, let

P-j

=

b»]»#j-(a) For

some p

<

p, let

x G

W{p)

and

x G W{p). Then,

_^^^x^i

_—

X^zei^*-(b) For

some

_Pj

<

Pj, let

x £ W{pj,p^j) and

x

€

W{pj,p-j).

Then

Xj

>

Xj

and

Xj

<

Xj,

for all i y^ j.

(c) For

some

J

C

T, suppose that pj

<

pj for all _j

E

I

and

pj

=

pj for all _j

_^

I,

and

let

X

G W{p)

and

x G W{p).

Then

_Yljei^i

—

^

_ei^r

For a given price vector p, the

WE

need not be unique in general.

The

following

example

illustrates

some

properties ofthe

WE.

Example

2 Consider a

two

hnk

network. Let the totalflow

he

d

=

1

and

the

reserva-tion utility he

R

=

I.

Assume

that the latencj' functions are given

by

l^^^lj^^fO

_~

ifO<.T<|

[ oo otherwise.

At

the price vector _(pi,P2)

—

(1,1), the set of

WE,

W{p),

is given

by

the set of all

vectors (xi,X2) with

<

x,

<

2/3

and

_{J^, Xj}

<

1.

At any

price vector _(pi,P2) with

Pi

>

P2

=

1,

W{p)

is given

by

afl (0,X2) with

<

X2

<

2/3.

This

example

also illustrates that

Lemma

1 need not hold

when

latency functions

are not real-valued. Consider, for instance, the price vector _(pi,P2)

=

(1

—

e,1

—

ae)

for

some

scalar a

>

1. In this case, the unique

WE

is (xi,X2)

=

(1/3,2/3),

and

clearly

effective costs

on

the

two

routes are not equalized despite the fact that they

both

have

positive flows. This arises because the

path

with the lower effective cost is capacity constrained, so

no

more

trafHc can use that path.

Under

furtherrestrictions

on

the k,the followingstandard resultfollows (proof omit-ted).

Proposition

3

(Uniqueness)

Let

Assumption

1 hold.

Assume

further that

_k

is

strictly increasing over

_d.

For

any

price vector

p

>

0, the set of

WE,

W{p),

is a

singleton. Moreover, the function T-^ :

R^

h->

M^

is continuous.

Since

we

do

not

assume

that the latency functions are strictly increasing,

we

need the following

lemma

in our analysis to deal with

nonunique

WE

flows.

Lemma

2 Let

Assumption

1 hold. For a given

p

>

0, define the set

T —

{i

e

1

\ 3 X,

X e

W{p)

with Xi -^ x^}. (5)

Then

k{x,)

=

0,

V

z

G

J,

V

X

e

_Wi-p),

Pi

=

Pj,

V

I, _J

e

i.

Proof.

Consider

some

i

E

2

and x

€

W{p).

Since i

6

X, there exists

some

x G

li'l^(p)

such that Xi

^

Xi.

Assume

without loss ofgenerality that Xj

>

Xj.

There

are

two

cases to consider:

(a) If Xfc

>

Xk for all k

_^

i, then Yljei^i -^

Yljei^J'

which imphes

that the

WE

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

x

hold with A

=

0.

By

Eq. (2)

and

Xi

>

x^,

we

have

li{xi)

_+Pi

<

R,

li(xi)

+pi

>

R,

which

together implythat li{xi)

=

li[xi).

By

Assumption

1 (i.e., /j is convex

and

^^(O) == _0), it follows that li{xi)

=

0.

(b) If Xfc

<

Xfc for

some

k,

by

the

WE

optimality conditions,

we

obtain

li[Xi)

+p,

<

lk{Xk)

+Pk,

k{ii)

+Pi>

lk{ik)

+Pk-Combining

the

above

with Xj

>

Xj

and

Xk

<

Xk,

we

see that li(xi)

=

k{xi),

and

hi^k)

=

hi^k)-

By

Assumption

1, this

shows

that li{xi)

=

(and also that

Pz=Pk)-Next

consider

some

i, j

G

I.

We

will

show

that _Pi

=

_Pj. Since i

G

Z, there exist

X,

X G

W{p)

such that Xj

>

x^.

There

are three cases to consider:

• Xj

>

Xj. If Xk

>

Xk for all k

^

i,j, then

^^Xm

<

d, implying that the

WE

optimality conditions hold with A

=

0. Therefore,

we

have

li{x^)

+Pi

<

R,

lj{xj)

+

_Pj

>

R,

which

together with li{xi)

=

lj{xj)

=

imply that _Pi

=

_pj.

• Xj

=

Xj. Sincej

€

I,

by

definition there

must

exist

some

other

x G

W{p)

suchthat

Xj 7^ Xj. Repeating the above

two

steps with Xj instead of Xj yields the desired

result.

Q.E.D.

Intuitively, this

lemma

states that if there exist multiple

WEs,

.t,x

G

W{p)

such

thatXi 7^ ij, thenthe latency function k

must

be locally flat

around

Xi (and Xi).

Given

the assumption that /i(0)

=

and

the convexity of latency functions, this immediately

implies li{xi)

=

0.

We

next define the social

problem

and

the social

optimum, which

is the routing (flow

allocation) that

would

be chosen

by

a planner that has full information

and

fullcontrol

over the network.

Definition

2

A

flow vector

x^

is a social

optimum

if it is

an

optimal solution ofthe

socialproblem maximize3.>o 2_]

[^

^

^ii^i))^i (6) iei subject to _{2_]}^i

^

d. iei

Inviewof

Assumption

1,the social

problem

has a continuousobjective function

and

a

compact

constraintset, guaranteeing the existenceof

a

social

optimum, x^

. Moreover,

using the

optimahty

conditions for a convex

program

(see [9], Section 4.7),

we

see that

a vector

x^ G

R^

is a social

optimum

if

and

only if

Yliei^i

—

^ ^^^'^ there exists a

subgradient gi.

G

dli{xf) for each i,

and

a A^^

>

such that

_^^{Yliei^f

—

d)

=

and

for each i,

R-k{xf)-xfg,,

<A^

ifa:f

=

0, (7)

=

A^

[{0<xf<bc„

>A^

iixf^bc,.

For future reference, for a given vector x

G M^,

we

define the value ofthe objective function in the social problem,

S(x)-^(i?-/,(x,))xi,

(8)

iei

as the social surplus, i.e., the difference

between

users' wilhngness to

pay

and

the total

3

Monopoly

Equilibrium

and

Efficiency

In this section,

we

assume

that a

monopoHst

service provider

owns

the /

hnks

and

charges a price of_Pi per unit

bandwidth on

hnk

i.

We

considered a related

problem

in

[1] for atomic users with inelastic traffic (i.e., the utility function of each of a finite set

of users is a step function),

and

with increasing, real-valued

and

differentiable latency functions.

Here

we show

that similar results hold for the

more

general latency functions

and

the

demand

model

considered in Section 2.

The

monopolist sets the prices to

maximize

his profit given

by

n(p,2:)

=

^PiXi,

where x

E

W{p).

This defines a two-stage

dynamic

pricing-congestion game,

where

the

monopolist sets prices anticipating the

demand

of users,

and

given the prices (i.e., in

each

subgame),

users choose their flow vectors according to the

WE.

Definition

3

A

vector

{p^^,x^^)

>

is a

Monopoly Eqmlibn.um

(ME)

if x^^^

€

W{p^^)

and

n(p^^-^,x^^^) >n(p,a;),

Vp>0,

VxG

M/(p).

Our

definition ofthe

ME

is stronger

than

the standard

subgame

perfect

Nash

equi-librium concept for

dynamic

games.

With

a slight abuseofterminology, let us associate

a

subgame

perfect

Nash

equilibrium with the on-the-equilibrium-path actions of the

two-stage

game.

Definition

4

A

vector {p*,x*)

>

is a

subgame

perfect equilibrium

(SPE)

of the pricing-congestion

game

if x*

E

W{p*)

and

for all

_p

>

0, there exists

x

E

W

(p) such that

n(p*,x*)>n(p,x).

The

fohowingproposition

shows

that

under

Assumption

1, the

two

solution concepts

coincide. Since the proof is not relevant for the rest of the argument,

we

provide it in

Appendix

A.

Proposition

4

Let

Assumption

1 hold.

A

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

an

ME

if

and

only if it is

an

SPE

ofthe pricing-congestion

game.

Since

an

ME

(p*,x*) is

an

optimal solution ofthe optimization

problem

maximizep>o, x>o ^Pi^;,: (9)

subject to

X

eW{p),

it is easier to

work

with

than

an

SPE.

Therefore,

we

use

ME

as the solution concept in

this paper.

The

preceding

problem

has an optimal solution,

which

establishes the existence of

an

ME.

Moreover,

we

have:

Proposition

5 Let

Assumption

1 hold.

A

vector

x

isthe flow vector at

an

ME

if

and

onlj' if it is a social

optimum.

Moreover, if (p,x) is

an

ME,

then for all i with Xj

>

0,

we

havepi

—

R

—

li{xi).

This proposition therefore establishes that the flow allocation at an

ME

and

the

social

optimum

are the same. Its proof is similar to

an

analogous result in [1]

and

is

omitted.

In addition to the social surplus defined above, it is also useful to define the

con-sumer

surplus, asthe diflFerence

between

users' wiUingnessto

pay and

effective cost, i.e.,

^^^j

{R

—

li{xi) —pi)xi (see [32]).

By

Proposition 5, it is clear that even

though

the

ME

achieves thesocial

optimum,

all ofthe surplus is captured

by

themonopolist,

and

users

are just indifferent

between

sending their information or not (i.e., receive

no

consumer

surplus).

Our

major

motivation for the study of oligopolistic settings is that they provide a better approximation to reality,

where

there is typically competition

among

service

providers.

A

secondary motivationistosee

whether

anoligopolyequilibriumwillachieve

an

efficient allocation like the

ME,

while also transferring

some

or all ofthe surplus to

the consumers.

4

Oligopoly

Equilibrium

We

suppose that there are

5

service providers, denote the set of service providers

by

S,

and assume

that each service provider s

G

5

owns

a different subset T^ of the links.

Service provider s charges aprice_piper unit

bandwidth on

linki

E

Xs.

Given

the vector

of prices of links

owned

by

other service providers,

p^

=

[Pili^i^, the profit of service

provider s is

ns(p,,p_^,x)

=

^PiXi,

for

X e

W{ps,p-s),

where

p,

=

\pi]zei,-The

objective of each service provider, like the monopolist in the previous section,

is to

maximize

profits.

Because

their profits

depend on

the prices set

by

other service

providers, each service provider forms conjectures about the actions of other service

providers, as well as the behavior of users, which,

we

assume, they

do

according to the

notion of

(subgame

perfect)

Nash

equilibrium.

We

refer to the

game

among

service

providers as the price competition game.

Definition

5

A

vector {p'^^,x^^)

>

is a (pure strategy) Oligopoly Equilibrium

(OE)

if

x°^

e

W

{p°^,

_p°f

)

and

for all s

e

5,

n,(p°^,pef

,x°^)

>

n,(p„pef

,x),

V

p,

>

0,

V

.X

€

_wip,,p'^f). (lo)

We

refer top*^-^ _{as the}

_OE

_price.

As

for the

monopoly

case, there is a close relation

between

a pure strategj^

OE

and

a pure strategy

subgame

perfect equilibrium.

Again

associating the

subgame

perfect

equilibrium with the on-the-equilibrium-path actions,

we

have:

Definition

6

A

vector {p*,x*)

>

isa

subgame

perfect equilibrium

(SPE)

oftlie price

competition

game

if x*

€

W

(p*)

and

there exists a function

x

:

R^

i—*

R^

such that

x{p)

G

W

(p) for all

_p

>

and

for all s

£

S,

Usip:y_„xn>Us{ps,p_,,x{ps,p*_,))

vp,

>o.

(11)

The

following proposition generalizes Proposition 4

and

enables us to

work

with the

OE

definition,

which

is

more

convenientfor thesubsequent analysis.

The

proofparallels

that of Proposition 4

and

is omitted.

Proposition

6

Let

Assumption

1 hold.

A

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

OE

if

and

only if it is

an

SPE

of the price competition

game.

The

pricecompetition

game

isneitherconcave norsupermodular. Therefore,classical

arguments

that are usedto

show

the existenceofapurestrategyequilibrium

do

not hold

(see [16], [52]). In the next proposition,

we show

that for linear latency functions, there

exists a pure strategy

OE.

The

proof is provided in the appendix.

Proposition

7 Let

Assumption

1 hold,

and assume

further that the latency functions

are linear.

Then

the price competition

game

has apure strategy

OE.

The

existence result cannot be generalized to piecewise linear latency functions or to latency functions

which

are linear over their effective

domain,

as illustrated in the following example.

Example

3 Consider a

two

link network. Let thetotal flowbe d

=

1.

Assume

that the

latency functions are given

by

1 I \ n 1 I \

Jo

ifO<x<5

for

some

e

>

and

5

>

1/2, with the convention that

when

e

=

0, hix)

=

oo for

x

>

5.

We

first

show

that there exists

no

pure strategy oligopoly equilibrium for small e (i.e.,

there exists

no

pure strategy

subgame

perfectequilibrium).

The

following list considers

all candidate

ohgopoly

price equilibria (pi,P2) arid profitable unilateral deviations for e

sufficiently small, thus establishingthe nonexistence of

an

OE:

1. _Pi

=

_P2

=

0:

A

small increase in the price of provider 1 will generate positive

profits, thus provider 1 has

an

incentive to deviate.

2. _Pi

—

_P2

>

0: Let

x be

the flow allocation at the

OE.

If Xi

=

1, then provider 2

has an incentive to decrease its price. If xi

<

1, then provider 1 has

an

incentive to decrease its price.

3.

<

pi

<

P2: Player 1 has

an

incentive to increase its price since itsflow allocation

remains the same.

4.

<

_P2

<

_Pi'- For e sufficiently small, the profit function of player 2, given pi, is

strictlyincreasing as afunction ofp2,

showing

that provider 2 has

an

incentive to

increase its price.

We

next

show

that a

mixed

strategy

OE

always exists.

We

define a

mixed

strategy

OE

as a

mixed

strategy

subgame

perfect equilibrium oftheprice competition

game

(see

Dasgupta and

Maskin, [13]). Let

S"

be the space of all (Borel) probability measures on [0,i?]". Let Ig denote the cardinality of Xg, i.e., the

number

of links controlled

by

serviceprovider s. Letfig

€

B^'

be

a probability measure,

and

denotethe vector ofthese

probability measures

by

fi

and

the vector of these probability measures excluding s

by

Definition 7

{fi*,x*{p))is a

mixed

strategy OligopolyEquilibrium

(OE)

ifthe function

x*{p)

e

W

(p) for every

p

€

[0,

RY

and

/ lis_(ps,P-5,a;* _(pa,P-s)) d(yu*(pj

X

/xljp_,))

JlQ,R]'

>

/ Ug{ps,p-s,x* (p,,p_s))d{ng (p,)

X

n*_^(p_,))

for all s

and

fig

G

'S^^

Therefore, a

mixed

strategy

OE

simplyrequires that therebe no profitabledeviation

toa different probability

measure

for each oligopohst.

Example

3

(continued)

We

now

show

that the following strategy profile is the

unique

mixed

strategy

OE

for the above

game when

e

^

(a

mixed

strategy

OE

also

exists

when

e

>

0, but its structure is

more

complicated

and

less informative):

r

0<P<R{l-6),

Mi(p)=|

1-^

R{l-5)<p<R,

( 1 otherwise, (

0<p<R{l-5),

f^^iP)={

i-^

R{l-S)<p<R,

( 1 otherwise.

Notice that fii has an

atom

equal to 1

—

5 at i?.

To

verifythat this profile is a

mixed

strategy

OE,

let /x' be the density offi, with the convention that ji'

—

oo

when

there is

an

atom

at that point. Let

_Mi

=

_{p

\ fj,'^(p)

>

0} .

To

establish that (/ii,/i2) is a

mixed

strategy equilibrium, it suffices to

show

that the expected payoff to player i is constant

for all _Pi

G

Mi when

the other player chooses p_j according to ^_j (see [37]).

These

expected payoffs are

t[{pi\li^i)= j Il,{pi,p_i,x{pt,p-i))dfj,-i{p-i). (12)

Jo

The

WE

demand

x{pi,p2) takes the simple

form

of Xi{pi,p2) == 1 if _pi

<

_P2

and

xi (pi,P2)

=

1

—

(^ifPi

>

P2

The

exactvalue of ii (pi,P2)

=

1

when

pi

=

p2isimmaterial

since this event

happens

with zero probability. It is evident that the expression in (12)

is constant for all _pi

G

Mi

for f

=

_{1,2 given pi}

and

p2 above. This establishes that

(PiiPa) is a

mixed

strategy

OE.

It can also

be

verified that there are no other

mixed

strategy equilibria.

The

next proposition,

which

is proved in

Appendix

B, establishes that a

mixed

strategy equilibrium always exists.

Proposition

8 Let

Assumption

1 hold.

Then

the price competition

game

has a

mixed

strategy

OE,

(p°^

,x"^^

(p))

.

We

next provide

an

explicit characterization of piure strategy

OE.

Though

of also

independent interest, these results are

most

useful for us to quantify the efficiency loss

of oligopoly in the next section.

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

3

Let

Assumption

1 hold. If {p'^^,x'^^) is a pure strategy

OE,

then

k{xf^)+pf^

=

imn{lj{xf^)

+

_pf^},

Viwithxp^>0,

(13)

k{x?^)+P?'^

<

R,

Vfwithxf^>0,

(14)

[x?"

<

d, (15)

E^

with

_Ezei^?^

=

^ if min,{/,(i°^)

+

_Pj}

<

R.

ProoL

Let (p^^.x"^^) be

an

OE.

Since

x°^

€

VK(p°'^), conditions (14)

and

(15) follow

by

thedefinition ofa

WE.

Considercondition (13).

Assume

that thereexist

some

i,j

G

X

with

_xf

^

>

0,

x°^

>

such that

k{xr)+pr<iA^r)+pT-Using

the optimality conditions for a

WE

[cf. Eq. _(2)], this impliesthat

_xf^

=

_bd-

Con-sider changing

_pf

^

to

_pf^

-|-e for

some

e

>

0.

By

checkingthe

optimahty

conditions,

we

see that

we

can choose e sufficientlysmall such that x*^^

G

^^{p^^

+

£,P-f)-

Hence

the

service provider that

owns

hnk

i can deviate to

_pf^

-I-e

and

increaseits profits,

contra-dicting the fact that {p^^,x'^^) is

an

OE.

Finally,

assume

to arrive at a contradiction

that minj{/j(x^^) _-1-Pj}

<

R

and

_Yliejx'^^

<

d. Using the optimality conditions for a

WE

[Eq. ₍₂₎ with

A

=

since

_Yliei^?^

"^ _{^]' ^^^^^} implies that

we

must

have

_xf^

=

_bd

for

some

i.

With

a similar

argument

to above, a deviation to

_pf^

-I- e keeps

x^^

as a

WE,

and

is

more

profitable, completing the proof.

Q.E.D.

We

need the following additional

assumption

for our price characterization.

Assumption

2

Given

apure strategy

OE

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

some

z

€

T

with

xf^

>

0,

we

have li{xf^)

^

0, then I^

=

{i}.

Note

that thisassumptionis automaticallysatisfiedifalllatency functions arestrictly

increasing or if all service providers

own

only one link.

Lemma

4

Let {p'^^,x'^^)

be

a pure strategy

OE.

Let

Assumptions

1

and

2 hold. Let

lis denote the profit of service provider s at (p°^,x°^).

(a) Iflis'

>

for

some

s'

G

S, then

H^

>

for all s

e

5.

(b) If Hs

>

for

some

s

eS,

then

_pf^xf^

>

for allj

els-Proof.

(a) For

some

j

G

J,', define

K

=

p^^

+

lj{x^^),

which

ispositive since lis'

>

0.

Assume

lis

=

for

some

s. For k

E

is, consider the pricepk

=

K

—

e

>

for

some

smaU

e

>

0. It can

be

seen that at _{the price vector {Pk,P-k)^ ^^^}corresponding

WE

link

flow

would

satisfyx^

>

0. Hence, service provider s has

an

incentive to deviate to Pk at

which

he will

make

positive profit, contradicting the fact that (p*^^,x'^^) is

a pure strategy

OE.

(b) Since lis

>

Oi

we

have

p^^x^^ >

for

some

m

E

Is-

By

Assumption

2,

we

can

assume

without loss ofgenerality that lm{x^^)

>

(otherwise,

we

are done). Let

j

G

Xs

and

assume

to arrive at a contradiction that p'^^x^^

=

0.

The

profit of

service provider s at the pure strategy

OE

can be written as

OE^OE

^s^^s^v'it^

m

'

where Hj

denotestheprofitsfromlinksother than

m

and

j. Let

p^

=

K

—Imix^)

for

some

K.

Consider changingtheprices

p^

and

p^^

such that the

new

profit is

I[s^Tls

+

{K

-

UxT

-

e))(xr

-

e)

+

e{K

-

l,{e)).

Note

that e units of flow are

moved

from link

m

to link j such that the flows of other links

remain

the

same

at the

new

WE.

Hence, the change inthe profit is

.

Hs

-

Hs

=

{IM°J) -

UxZ"

-

e))xZ^

+

eiUxZ""

-

e)

-

/,(e))).

Since lm.{x'^)

>

0, e canbe chosensufficientlysmallsuchthat the aboveis strictly

positive, contradicting the fact that (p'^®,x'^^) is

an

OE.

Q.E.D.

The

following

example shows

that

Assumption

2 cannot be dispensed with for part

(b) of this

lemma.

Example

4

Consider a three

hnk

network with

two

providers,

where

provider 1

owns

links 1

and

3

and

provider 2

owns

link 2. Let thetotal flow be d

=

1

and

the reservation

utility

be

i?

=

1.

Assume

that the latency functions are given

by

/l (Xi)

=

0,

12{X2)=X2,

h^xz)

^

axz, 15

for

some

a

>

0.

Any

price_{vector (pi,P2,P3)}

=

(2/3, 1/3,6) with6

>

2/3

and

(xi,X2,X3)

=

(2/3,1/3,0) is a pure strategy

OE,

so ^3X3

=

contrary to part (b) ofthe

lemma.

To

see

why

this is

an

equihbrium, note that provider 2 is clearly playing a best response.

Moreover, in this allocation IIi

=

4/9.

We

can represent

any

deviation of provider 1

by

(pi,P3)

=

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

for

two

scalars e

and

5 ,

which

will induce a

WE

of(xi,X2,X3)

=

(2/3

+

5

—

e,1/3

—

5,e).

The

corresponding profit ofprovider 1 at this deviation is 111

—

4/9

—

6^

_<

_4/9,

estab-lishing that provider 1 is also playing a best response

and

we

have a pure strategy

OE.

We

next establish that, under

an

additional mild assumption, a pure strategy

OE

will never

be

at a point ofnon-differentiability of the latency functions.

Assumption

3

There

exists

some

s

E

S

such that k is real-valued

and

continuously

differentiable for all i

E

Ig

Lemma

5 Let (p°^,a;°^) be

an

OE

with min^

_{pf^

+

lj{x°^)}

<

R

and

_{pf^xf^ >}

for

some

i. Let

Assumptions

1, 2

and

3 hold.

Then

C(0

=

_C(0,

V^e2:,

where

l^ixf^)

and

l^ixf^) are the right

and

left derivatives of the function li at

xf^

respectively.

Sincetheproofof this

lemma

islong, itisgivenin

Appendix

C.

Note

that

Assumption

3 cannot

be

dispensed with in this

lemma.

This is illustrated in the next example.

Example

5 Consider a

two

link network. Let the total flow he

d

—

1

and

the

reserva-tion utility he

R

—

2.

Assume

that the latency functions are given

by

^^^^)

=

'^^^)

=

_|2(x-i)

otherwise.

It can be verified that the vector

_(pf^,^^^)

=

(1, 1), with

(xf^,x^^)

=

(1/2, 1/2) is a

pure strategy

OE,

and

is at apoint of non-differentiability for

both

latency functions.

We

next provide an explicit characterization of the

OE

prices,

which

is essential in

our efficiency analysis in Section 5.

The

proof is given in

Appendix

D.

Proposition

9

Let (p'^^,x<^^)

be an

OE

such that

_{pf^xf^ >}

for

some

i

€

I. Let

Assumptions

1,2,

and

3 hold.

a)

Assume

that min^

{pf^

+

lj{xf^)}

<

R-

Then,

for all s

G

5

and

i

E

Is,

we

have '

xf^l'iix?^), if^^(2:°^)

-

for

some

_j

_i

Z„

P"""

=

{ x?^l',{x?^)+ J^-^'

^^T

, otherwise.

(1^)

b)

Assume

that mirij

{p°^

+

lj{xf^)}

=

R. Then, for all s

G

5

and

i

els,

we

have

i^'yxf^i^?").

(17)

Moreover, ifthere exists

some

i

G

I

such that Ij

~

{i} for

some

s

G

5, then

pr<xf^c(^r)+

_^

'

1 (18)

If the latency functions /j are all real-valued

and

continuously differentiable, then analysis of

Karush-Kuhn-

Tucker conditions for oligopoly

problem

[problem (82) in

Ap-pendix

D] immediately yields the following result:

Corollary

1 Let

{p°^,x°^)

be an

OE

such that

_pf^xf^

>

for

some

i

G

I. Let

As-sumptions 1

and

2hold.

Assume

alsothat_kisreal-valued

and

continuouslydifferentiable

for all i.

Then,

for all s

G

<S

and

i

E

Is,

we

have

X

l',{xY^), if _lj{xf^)

=

for

some

_j

^

J„

P^

^

mm{R-k{x^^)

, x?^i:ix?^)

+

J^'^

^'\

_{\, otherwise.}

.'.(iV-K)

(19)

This corollaryalsoimpliesthat inthe

two

linkcasewithreal-valued

and

continuously

differentiable latency functions

and

with

minimum

effective cost less

than

R, the

OE

prices are

p?^-xf''{l[ix?^)

+

l',{xr)) (20)

as claimed in the Introduction.

5

Efficiency of

Oligopoly

Equilibria

This section contains our

main

results, providing tight

bounds

on the inefficiency of

oligopolyequilibria.

We

take asour

measure

of efficiencytheratioofthesocialsurplusof

the equilibrium flow allocation to thesocialsurplus ofthe social

optimum,

§{x'')/S{x^),

where

x* refers to the

monopoly

or the oligopoly

equihbrium

[cf. Eq. (8)]. Section 3 established that the flow allocation at a

monopoly

equilibrium is a social

optimum.

Hence, in congestion

games

with

monopoly

pricing, there is

no

efficiency loss.

The

following

example shows

that this is not necessarily the case with oligopoly pricing.

Example

6 Consider a

two

linknetwork. Letthetotalflow

he

d

=

1

and

the reservation

utility he

R

—

1.

The

latency functions are given

by

3

/i(x)

=

0, kix) =^ _-X.

The

unique social

optimum

for this

example

is x'^

=

(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

AiE

are the same.

Next

consider a

duopoly

where

each ofthese

hnks

is

owned

by

a different provider. Using Corollary 1

and

Lemma

3, it follows that the flow allocation at the

OE,

x'-'^, satisfies

h{x?^)

+

x?^[/;(xf^)

+

l',{x^^)]

=

_h{xr)

+

x^Ux'^,'')

+

/;(x?^)].

Solving this together with

xf

^

+

X2^

—

1

shows

that the flow allocation at the unique

oligopoly equilibrium is x*^^

—

(2/3,1/3).

The

social surplus at the social

optimum,

the

monopoly

equilibrium,

and

the oligopoly equilibrium are given

by

1, 1,

and

5/6,

respectively.

Before providing a

more

thorough

analysis ofthe efficiency propertiesofthe

OE,

the

nextproposition proves that, as claimed inthe Introduction

and

suggested

by

Example

6, a

change

in the

market

structure from

monopoly

to

duopoly

in a

two

link

network

typically reduces efficiency.

Proposition

10

Consider a

two

link network

where

each link is

owned

by

a

differ-ent provider. Let

Assumption

1 hold. Let

{jP^ ^x^^^

be a pure strategy

OE

such

that

_pf^xf^

>

for

some

i

E

I

and

min^

{pf^

+

lj{xf^)}

<

R. If l[{x'^'^)/xf'^

^

/^(x°^)/x^^, then

S(xO^)/S(x^)

_<

1 .

Proof.

Combining

the

OE

prices with the

WE

conditions,

we

have

ii(x?^)

+

_xf

^(/;(xf^)

+

/^(x^^))

=

/2(xf ^)

+

x^^(/;(x?^)

+

/;(x°^)),

where

we

use the fact that miUj

{pf^

+

lj{xj^^)}

<

R. Moreover,

we

can use optimality

conditions ₍₇₎ to prove that avector (xfjXj)

>

is asocial

optimum

if

and

only if

/i(xf)

+

xf^;(xf)

=

hixl)

+

x%{xl).

Since

l[{x°^)/x°^

_^

/^(x^^)/x^^, the result foUows.

Q.E.D.

We

next quantify the efficiency of oligopoly equilibria

by

providing a tight

bound

on

the efficiency loss in congestion

games

with oligopoly pricing.

As we

have

shown

in

Section 4, such

games

do

not always have apure strategy

OE.

In the following,

we

first

provide

bounds on

congestion

games

that have purestrategy equilibria.

We

next study

efficiency properties of

mixed

strateg}^ equilibria.

5.1

Pure

Strategy

Equilibria

We

considerpricecompetition

games

thathave purestrategyequilibria (this set includes,

but is substantially larger than,

games

with linear latency functions, see Section 4).

We

considerlatency functions that satisfy

Assumptions

1, 2,

and

3. Let

£/

denote the set of

latency functions for

which

the associated price competition

game

has a pure strategy

OE

and

the individual k's satisfy

Assumptions

1, 2,

and

3.^^

We

refer to an element of

the set Ci

by

{li}iei-

Given

a parallel link network with /

hnks and

latency functions

{h}ieJ

£

C/i let OE{{li}) denote the set of flow allocations at an

OE.

We

define the

efficiency metric at

some

x'^^

G

OE{{li}) as

rii{k}.x"^)

=

"^'^-"^

,

^'^

'

/'s

> (21)

where x^

isa social

optimum

given_{the latency functions {kjiei}

and

R

isthe reservation

utility. In other words, our efficiency metric is the ratio of the social surplus in

an

equilibrium relative to the surplus in the social

optimum.

Following the literature

on

the "price of anarchy", in particular [25],

we

are interested in the worst performance in

an

ohgopoly equilibrium, so

we

look for a lower

bound

on

inf inf r!{{k},x°^).

We

first prove

two lemmas, which

reduce the set of latency functions that need to

be considered in

bounding

the efficiency' metric.

The

next

lemma

allows us to use the oligopoly price characterization given in Proposition 9.

Lemma

6 Let {p'-^^,x'^^) be a pure strategy

OE

such that

_p^^xf^

=

for all i

e

I.

Then

x'^^ is a social

optimum.

Proof.

We

first

show

that li{xf^)

=

for all i

6

T.

Assume

that lj{x'^^)

>

for

some

j

G

I. This implies that

xf^

>

and

therefore

pf^

=

0. Since Ijixf^)

>

0, it follows

by

Lemma

2 that for all

x E W{p),

we

haveXj

=

x^^

. Consider increasing

p^^

to

some

small e

>

0. Bj^ the

upper

semicontinuity of

W{p),

it follows that there exists

some

e

>

sufficiently small such that for all x

G

W{e,p'2f),

we

have \xj

—

x^^\

<

5 for

some

6

>

0. Moreover,

by

Proposition 2,

we

have, for all

x G

W{e,p'^f), Xj

>

_xf^

for all

i 7^ j. Hence, theprofit ofthe provider that

owns

linkj is strictlyhigher atprice vector {^,P-f)

than

atp°^, contradicting the fact that

{p°^,x°^)

is

an

OE.

Clearly

x°^

>

for

some

j

and

hence minigijpf

^

+

kix^'^)]

=

p°^

+

lj{x°^)

=

0,

which imphes by

Lemma

3 that

_J2,ex^?^

^

^- Using li{xf^)

=

0,

and

G

dli{xf^) for

all i,

we

have

R

-

_k{xf^)

-

_xf

^p,,

=

R,

V

i

G

T,

for

some

gi.

G

dl^{xf^). Hence, x'^^ satisfies the sufficient optimality conditions for a

social

optimum

[cf. Eq. (7) with A'^

=

R],

and

the result follows.

Q.E.D.

The

next

lemma

allows us to

assume

without loss of generality that

R

Yliiez

^f

~

^jg2:^j(xf)xf

>

and YlieJ^?^

=

din

the subsequent analysis.

Lemma

7 Let {U]rei

£

C/-

Assume

that

^"Moreexplicitly, Assumption 2impliesthat ifany

OE

[p'-'^,x'-'^)associatedwith{/j}igihas

xf^ >

and hixf^)

=

0, thenI^

=

{i}.

either (i)

_'^i^2kixf)xf

=

_RJ2iei^i

^'-"^

some

social

optimum

Xg,

or (ii)

_Y^iei^?^

<

dfor

some

x°^

_{€ 0^{{k}).}

Then

every x'^^

E

OE{{li}) is asocial

optimum,

implying that r/({/i},x'^^)

=

1.

Proof.

Assume

that _{Yliex^i(^i)^i}

~

_^IZiei^f-

Since

x^

is a social

optimum

and

everyx'^^

€

OE{{li}) is afeasible solution to the social

problem

[problem (6)],

we

have

By

the definition of a

WE,

we

have

_xf

^

>

and

i?

-

_{/i(xf ^)}

>

_pf

^

>

(where

_pf

^

is

the priceof

hnk

i at the

OE)

for all i. This

combined

withthe precedingrelation

shows

that x°-^ is asocial

optimum.

Assume

nextthat_YlieJ

-^P"^

<

^

^^^

some

x°^

€

OE{{li}). Letp*^^

be

the associated

OE

price.

Assume

that p'^^x^^

>

for

some

j

€

X

(otherwise

we

are

done by

Lemma

6). Since _X^iej^?'^

<

"^j

we

have

by

Lemma

3 that minjgijpj +/j(x^-^}

—

R. Moreover,

by

Lemma

4, it followsthat

ptxf^

>

for alli

G

I. Hence, foralls

e

S, i{pf^)ieis,x^^)

is

an

optimal solution ofthe

problem

maximize((p,),^j^,:,)

_^PiXj

ieXs

subject to _Pi

+

li{xi)

=

R,

y

i

elg,

iex

Substituting for _(pi)i€is inthe above,

we

obtain

maximize3;>o

_{y ^{R}

—

li{xi))xi

ieis

subject to Xi

e

T^, \/ i

^

I^,

where

Tj

=

{xi \

pf^

+

li{xi)

—

R}

is either a singleton or a closed interval. Since this

is aconvex problem, using the optimality conditions,

we

obtain

R-k{xf^)-x°''gi^=0,

VieZ„Vse5,

where

g^

G dk{xf^).

By

Eq. (7), it follows that

x°^

is a social

optimum.

Q.E.D.

This

lemma

implies that in finding a lower

bound

on

the efficiency metric,

we

can

restrict ourselves, without loss of generality, to latency functions {U}

G

Cj

such that

'^ieT^i{^i)^i

<

R-Yliei^i ^°r

some

social

optimum

x^

,

and ^^i^j^?^

~

'^ ^^^ ^^^

x'^^

G

OE{{lj}).

By

the following

lemma,

we

can also

assume

that

_X^jgjxf

=

d.