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DEWEY
HB31
.M415
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
PRICE COMPETITION
IN
COMMUNICATIONS NETWORKS
Daron Acemoglu
and
Asuman Ozdaglar
Working Paper 06-10
December 2005
Room
E52-251
50 Memorial Drive
Cambridge,
MA 021 42
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://ssrn.com/abstract=901 803
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
JUN
2
2006
LIBRARIES
Price Competition in
Communication Networks
Asuman Ozdaglar
Daron Acemoglu
Department of
Department of Economics
and Computer Science
Massachusetts Institute of Technology
Cambridge,
MA
Electrical Engineering
Massachusetts Institute of Technology
02142
Cambridge,
Email: daron@mit.edu
MA
02142
Email: asuman@mit.edu
Abstract
— We
study
the
properties
efficiency
oligopoly equilibria in congested networks.
of
Our measure
is the difference between users' willingness to
pay and delay costs. Previous work has demonstrated that
in networks consisting of parallel links, efficiency losses
from competition are bounded. In contrast, in this paper
of efficiency
we show
that in the presence of serial links, the efficiency
loss relative to the social
optimum can be
arbitrarily large
because of the double marginalization problem, whereby
each serial provider charges high prices not taking into
account the effect of
this strategy
on the
to a general
when
are no delay costs without transmission
(i.e.,
latencies at
zero are equal to zero), irrespective of the
number
there
of serial
parallel providers, the efficiency of strong oligopoly
bounded by 1/2, where strong oligopoly
which each provider plays a
response and all of the traffic is transmitted.
equilibria can be
However, even with strong oligopoly equilibria, inefficiency
can be arbitrarily large when the assumption of no delay
costs without transmission is relaxed.
that in a parallel-link
network with
always greater than or equal to
inelastic
More
5/6.
recently,
Hayrapetyan, Tardos, and Wexler [13] studied pricing
in a parallel-link
users,
network with
elastic
and homogeneous
and provided bounds on the efficiency metric.
This paper shows that the efficiency loss with competing service providers
is
considerably higher
when we
consider more general network topologies, suggesting
that unregulated
consumption
have significant costs
To
equilibria are equilibria in
strict best
show
[1]
[15] extend this result
network topology. Acemoglu and Ozdaglar
and homogeneous users, the efficiency metric with an
arbitrary number of competing network providers is
profits of other
providers along the same path. Nevertheless,
and
Huang, Ozdaglar and Acemoglu
the
illustrate
in
in general
may
networks
terms of resource allocation.
potential
inefficiencies
of price
competition in congested communication networks,
consider a parallel-serial
destination pair
we
topology where an origin-
linked by multiple parallel paths, each
is
potentially consisting of an arbitrary
number of
serial
Imks. Congestion costs are captured by link-specific non-
decreasing convex latency fionctions, denoted by
INTRODUCTION
I.
There has been growing
link
interest in pricing as a
of allocating scarce network resources (see,
[19], [25]).
Although prices may be
network objectives,
trolled
by
in practice
[17],
set to satisfy
some
many
prices are con-
for-profit service providers that
at least in part, to increase their
method
e.g.,
charge prices,
revenues and
profits.
Research to date suggests that profit-maximizing pricing
may improve
the allocation of resources in
com-
munication networks. Let the metric of eificiency be
the difference between users' willingness to pay and
delay costs in the equilibrium relative to that in the
social
optimum (which would be chosen by
planner with
full
information and
full
a a network
control over users).
Acemoglu and Ozdaglar [2] show that with inelastic and
homogeneous users, pricing by a monopolist controlling
all
links
efficiency
in
a parallel-link network always
(i.e.,
the efficiency metric
is
achieves
equal to one).
i.
Each
link
is
owned by
/»
() for
a different service provider.
All users are inelastic and homogeneous.
This environment induces the following two-stage
game: each service provider simultaneously
for transmission
Pi.
Observing
all
of bandwidth on
its
sets the price
link,
denoted by
the prices, in the second stage users
route their information through the path with the lowest
effective cost,
where
effective cost consists of the
of prices and latencies of the links along a path
sum
[i.e.,
sum of Pi + li{-ys over the links comprising a path].
Our objective is to study the efficieny properties of the
subgame perfect equilibria of this game.
The main novel aspect of this model compared to the
parallel-link case
(serial) service
is
the pricing decisions of different
providers along a single path.
particular provider charges a higher price,
it
When
a
creates a
negative externality on other providers along the
same
path, because this higher price reduces the transmission
they receive. This
li(x,),R
the equivalent of the double mar-
is
economic models with multiple
ginalization problem in
monopolies, and leads to significant degradation of the
equilibrium performance of the network.
d
most extreme form, the double marginalization problem leads to a type of "coordination failure",
In
its
whereby
units
Rescrvalion
utilily
:
R
providers, expecting others to charge high
all
prices, also charge prohibitively high prices, effectively
killing all data transmission
low
to arbitrarily
Fig.
1
A
.
network with
serial
and parallel
links.
on a given path, and leading
efficiency. This type of pathological
behavior can happen in subgame perfect equilibria (what
we
OE), but
refer to as oligopoly equilibria,
we show
cannot happen in strict subgame perfect equilibria,
OE, which follows the notion of strict equilibrium
introduced in Harsanyi [12]. In strict OE, each service
that
provider must play a
strict
best response to the pricing
of other service providers, which
strategies
is
sufficient
to rule out the pathological coordination failures
men-
we show
Nevertheless,
ginalization
OE
that strict
arbitrarily large efficiency losses
can also have
because the double mar-
problem can again prevent any transmission
on a particular path, even when such transmission
is
Instead,
we
define an even stronger notion of equi-
librium, strong
transmitted.'
OE,
We
as a strict
show
that
OE
when
in
which
all traffic
latency without any
equal to zero [i.e., k (0) = 0], there is a tight
bound of 1/2 on the efficiency of strong OE irrespective
of the number of paths and service providers in the
network. This bound is reached by simple examples. In
strong OE, the double marginalization problem is still
present, and this is the reason why the bound of 1/2 is
lower than the 5/6 bound in our previous work, [1].
Furthermore, we show that even with strong OE, when
traffic is
the assumption that
li
optimum can be
These
results
(0)
is
it
different
among
somewhat
mind that
among
much
better
is
[3],
related
who
analyze
network context under specific
utility
who
and latency functions; He
study competition cooperation
demand
study resource allocation in a wireless network
previous work,
Acemoglu and Ozdaglar
and the
[1]
Hayrapetyan, Tardos, and Wexler [13],
recent work,
consider the performance of a network with competing
No
providers.
other paper, to the best of our knowledge,
has investigated price competition in the presence of
serial providers or
more general
topologies.
Model
performance results
which
assume
is
consider a network with / parallel paths that
connect two nodes. Each path
I=
{1,
.
.
.
,
/} denote the
the set of links on path
and x
Each
always transmitted. Our model incorporates a
=
[xi,.
link in the
.
.
i.
when
this reservation
i
set
consists of
rii
of paths and
links.
Let
denote
TVi
Let xt denote the flow on path
,xi] denote the vector of path flows.
network has a flow-dependent latency
function li{xi), which measures the delay as a function
of the
total
flow on link
i
(see Figure
1).
the price per unit flow (bandwidth) of link j
— \Pj]jeK,iei denote
We are interested in
denote
by
pj. Let
the problem of routing
of flow across the / paths.
aggregate flow of
We
the vector of prices.
not necessarily the case.
All traffic will be transmitted in equilibrium
utility is sufficiently large.
[14],
closely
[5],
Internet service providers under specific
who
to the social
prices, e.g., [22] or [7],
reservation utility for users, so that this
More
[4]).
under fixed pricing. None of these papers, except our
P
traffic
Johari and Tsitsiklis
and of network design
models; as well as Acemoglu, Ozdaglar, and Srikant
i,
realistic topologies,
'Models of selfish routing without
all
al.
in a
assumptions on the
the examples
pathological, and this begs the
could be obtained in more
et.
(e.g.,
[23]);
works of Basar and Srikant
monopoly pricing
We
service providers might
has to be borne in
question of whether
that
market mechanisms
Anshelevich
are the
Perakis
[7],
of resource allocation by
[10]);
Sanghavi and Hajek
(e.g.,
arbitrarily large.
have very poor performance relative
are
and Friedman
[21],
relaxed, the efficiency
achieve satisfactory network performance in general.
optimum
Tardos [22], Correa, Schulz, and Stier-Moses
II.
=
shed doubt on the conjecture that un-
regulated competition
Nevertheless,
Koutsoupias and Papadimitriou [18], Roughgarden and
and Walrand,
socially optimal.
that
future work.
left for
related to our paper includes studies quantifying
efficiency losses of selfish routing without prices (e.g.,
[16],
tioned above.
loss
an area
Work
it
strict
is
is
many
We
assume
d
that this
units
is
the
"small" users and thus adopt
the Wardrop's principle (see [24]) in characterizing the
flow distribution
minimum
along paths with
sum of
network;
in the
i.e.,
u(x)
the flows are routed
effective cost, defined as the
the latencies and prices of the links along that
path (see the definition below). Wardrop's principle
used extensively
modelling
in
behavior
trafiic
is
in trans-
portation networks ([6], [8], [20]) and communication
networks
([22], [7]).
reservation
utility
R
if the effective cost
We
assume
also
and decide not
that users
have a
send their flow
to
exceeds the reservation
utility.
This
implies that user preferences can be represented by the
piecewise linear aggregate
fimction
utility
u()
Aggregate
Fig. 2.
depicted
utility function.
Figure 2?
in
Definition 1: For a given price vector
^WE
]^/
g
jg
^ Wardrop equilibrium
p
>
(WE)
0, a
vector
if
max
>
((i?
— y^ Pi)xi—
I
/,(z)dz)(2)
>
jeM
i^'
'^^^
^(/,(xr^)+p,)
E
R,
<
d,
<
We
R.
Q,E.D.
For a given price vector p, the
if
denote the
rainkeziZjeArJji^k") +
set of WE at a given p by
W{p).
We
adopt the following assumption on the latency
Assumption
[0,
:
tiable,
oo) I—>
1:
[0,
For each
oo)
is
i
G I,
the latency function
convex, continuously differen-
=
nondecreasing, and satisfies Zj(0)
0.
H^
:
1
W{j)),
]R^
zij
hold. For any price vector
is
R^
p >
0, the set
of
nonempty. Moreover, the correspondence
is
Ij is
strictly
W
the function
:
El
i-^
in the sense that they
all
users are
have the same reservation
W
is
discuss potential issues in extending this
and heterogeneous requirements
in the
work
to users
R.
We
with elastic
concluding section.
1
hold.
a singleton. Moreover,
is strictly
at a
p >
0, the
convex, and
shows the
given p. Since the correspon-
it is
1
continuous. Q.E.D.
next define the social problem and the social
full control
is
the routing (flow
central
allocation) that
network planner that has
and information about the network.
Definition 2:
it is
obtain:
upper semicontinuous from Proposition
and single-valued,
We
WE
A flow
vector
x^
is
a social optimum
if
an optimal solution of the social problem
'^[R-
"homoge-
utility,
we
continuous.
therefore has a unique optimal solution. This
maximizea;>o
This simplifying assumption implies that
is
objective function of problem (2)
following optimization problem
neous"
Mi
is
Proof: Under the given assumptions, for any
would be chosen by a
Proof sketch: Given any p > 0, the proof is based
on using Assumption 1 (in particular the nondecreasing
assumption on the latency functions) to show that the
set of WE is given by the set of optimal solutions of the
li,
i
WE, W{p),
0, the set of
optimum, which
upper semicontinuous.
on the
G J, there exists some j G
increasing. For any price vector
further that for all
Mi, such that
dence
sumption
WE need not be unique
further restrictions
uniqueness of the
Proposition 1: (Existence and Continuity) Let As-
WE,
Under
Proposition 2: (Uniqueness) Let Assumption
Assume
p >
functions throughout the paper except in Section IV-F.
li
^
(1)
in general.
Z^eI^i'' = d
with
Pj}
xY"^
<
with
i
yx^<d.
s.t.
xf^ > 0,
VJwithxr^>0,
V
subject to
2_. Xi
lei
<
'Y^lj{xi)\xi
d.
(3)
.
By Assumption
1,
Proposition 3: Let Assumption
the social problem has a continuous
compact constraint
objective function and a
teeing the existence of a social optimum,
set,
guaran-
x^ Moreover,
.
game has
competition
using the optimality conditions for a convex program,
we
see that a vector x'^
only
A'^(
if
€
M+
optimum if and
such that
a A^ >
a social
is
< d and there exists
= and for each i e
-d)
xf
X^jgi xf
X^f^i
R-Y^
Ijixf)
- xf J2
^'ji^f)
if
^f
=
Io
A^
if
xf >
aj
>
0.
= {ieI\
^
a,
=
0},
Then
2:
p°^ =
with
define the value of the
>
Jo
(4)
we
some
the set of
is
There are two cases
0.
I.
For a given vector x € M+,
for
ajX
Iq
the price
Gl
i
Define the set
=
such that aj
e Mi). Let Iq denote the cardinality of set Jq.
for all j
=
=
Proof: Let lj{x)
(or equivalently,
0'
Then
a pure strategy OE.
I,
< ^^
hold and assume
1
latency functions are linear.
that the
W{p'^^)
it
to consider:
can be seen that a vector (p*^^,
for all
€
i
Iq,
x'-'^)
e Mi and x°^ e
j
an OE.
is
objective fimction in the social problem,
II.
S(a;)
= ^(i?iei
as the social surplus,
users' willingness to
III.
Y,ljix,)yi,
(5)
i.e.,
pay and the
other service providers, p_j
the profit of service provider j with j G
nj(pj,p-j,x)
in the
Mi
=
\pk\k^j,
per semicontinuous and convex-valued correspondence.
game among
exists
x^^ 6
If
refer to
we
that W{p'-'^) is
an OE.
Assume
we have
=
finally that 7o
generality 1
Xi
e Iq-
= x,,
We
show
for all
i
EC{x,p°'')
1,
and
that without loss
/
= mini
1.
of
x € W{p^^),
that for all x,
Let
^
+pO^}.
/,(xfc)
Nash
equilibrium.
We
If at least
one of
service providers as the price
vector
as the
have by Proposition 2
which they do according
perfect
(p*^^, x*^^)
>
all
i
is
a
e I, j e Mi,
Vpj>o, yx€W{pj,p^f).
We
0,
B{jP^) = p^^
remains to show that there
it
service provider forms
n>r'Pef,x°^)>n,(p,,p?f,x),
p^^
=
Jo
such that
W{p'^'^) such that (6) holds.
EC{x,p°'^)
and for
p'~'^
a singleton, and therefore (6) holds and {p^^, ^{p'^^))
(pure strategy) Oligopoly Equilibrium (OE) if x*^^ e
W {pf^,p'2f)
of a
To complete the proof,
competition game.
A
can use Kakutani's fixed point theorem to
assert the existence
conjectures about the actions of other service providers,
subgame
we
Hence,
is
as well as the behavior of users,
(7)
[Bj{p^f)\. In view of the linearity of
the latency fiinctions, it follows that B{jP^) is an up-
is
= PiXi,
by other service providers, each
pjXi.
=
B(p°^)
Let
where x e W{j[)j,p-j).
The objective of each service provider is to maximize
profits. Because their profits depend on the prices set
Definition 3:
the set
total latency.
network. Service provider j charges a price pj per unit
bandwidth on link j £ Mi. Given the vector of prices of
refer to the
Bj{p°f) be
iSWCpj.pO^)
of which owns one of the links on the paths
to the notion of
let
max
{p^^,x°^)eaig
the difference between the
Oligopoly Pricing and Equilibrium
owned by
For some j € Mi,
1:
such that
jeM.
We assume that there are multiple service providers, each
links
<
/o
p°^
of
<R,
holds, then one can
EC{x,p°^)
or
show
that Y^^^i Xi
Substituting x\
=
d
convex
=
[xi]i^i, thus
=
<R
=
Y^{^i Xj
d.
—
"^i^x, i^i ^i '" problem (2), we
see that the objective function of problem (2) is strictly
both
in
x_i
EC{x,p°^) = R
(6)
Yl
and
hi.^')
showing
that
EC{x,p^^) =
= Yl
x
=
x. If
R, then
':»(*')'
OE price.
which, by the assumption that
The next proposition shows
that
functions, there exists a pure strategy
for linear latency
for
OE.
establishing our claim.
some
j
G Mi, implies
Ij
that Xi
is
=
strictly increasing
Xj for all
i
^
1,
For some x £ W{p'^^), consider the vector
—
{d
xi
Since x-i
^i^^-i)
chosen such that the providers on link
X]i=^i
is
incentive to deviate,
it
x'-'^
=
uniquely defined and
is
follows that (p^'^, x'-^^)
A.
of OE
Inefficiency
have no
1
is
an OE.
we
In this section,
and
Q.E.D.
Efficiency Analysis
IV.
strict
study the efficiency properties of OE,
OE
and strong
The existence
result cannot
be generalized
convex latency functions as shown
to general
following
the
in
optimum, E>{x'^^)/E>{x^) [cf
social
(this set includes,
Example
flow be
Consider a two link network. Let the
1:
=
rf
1.
Assume
total
that the latency functions are
given by
games
but
is
=
< a; <
x>5.
if
(°
hix)
0,
-
and 6
=
^2(2:)
>
1^^
or
games with
Given a
III).
consider
OE
strict
linear
parallel-link
and latency
OE{{lj}) denote the set of
i,
OE
x^^ = [xf^ji^j at an OE (or strict
depending on the context). We define the efficiency
some x^^ e OE{{lj})
at
as
1/2, with the convention that
00 for
verified that there exists
at the
flow allocations
(5
metric
some e >
when e = 0,
for
OE
larger than,
latency functions, see Section
We
(5)].
have
that
network with / paths, Ui links on path
functions {li}(jeN, iei)'
h{x)
take as our
of the social surplus
ratio
equilibrium flow allocation to the social surplus at the
price competition
example.
We
(defined below).
measure of efficiency the
>
a;
no pure
5.
can be easily
It
OE
strategy
for small
e (see [1] for details).
(8)
Nevertheless, a mixed strategy
define a
mixed
strategy
OE
as a
OE
always
mixed
perfect equilibrium of the price competition
Dasgupta and Maskin,
(Borel) probability measures on
total
Let
number of
€ B he
fij
B
Let
[9]).
[0,
We
exists.
strategy
subgame
game
(see
be the space of
all
T denote the
T = ^j^i nt.
R\ Let
.
links in the network,
i.e.,
where x^
/x
we
equilibrium, so
Equilibrium
(p) for every
(OE)
p e
[0,
mixed
a
is
the
if
strategy
(pj)
X
rK{Z,},xO^).
inf
show
first
that the
performance of an
2:
n*_j {p_j))
and pj e B, where Ilj{p,x*{p)) is the profit of
service provider j at price vector p and x*{p) e W{p).
strategy
OE
thus requires that there
profitable deviation to a different probability
is
no
measure for
each oligopolist. The existence of a mixed strategy
can be established along the lines of the analysis in
which leads to the following result (proof omitted):
1
with identically
OE
[1],
Proposition 4: Let
competition
(/iO^,xO^(p)).
Assumption
game has
a
1
hold.
mixed
Then
strategy
latency functions
and one link on path 2 with latency function l{x2)
A;
>
0.
Let the total flow be d
(1,0).
Now
this
=
example
Each of the three service providers on path
by
i
=
1, 2,
and
3,
charge price p\
=
provider on path 2 charges p2
that there
1
is
is
no deviation
that
=
1,
1/2.
is
1
,
the
x^
=
denoted
while the service
It
can be verified
profitable for any of
the service providers. First, consider the serial providers
on path
1
;
given the prices of two of the serial service
providers, there will always be zero traffic on path
OE,
=
and the
consider the following strategy combination.
the remaining service provider
price
can be
Consider a two path network, which has
reservation utility be i? = 1.
Uj{pj,p_j, X* {pj,p-j))d {pj (pj) X p*_- {p_j))
The unique social optimum for
mixed
OE
arbitrarily bad.
Example
for all j
A
are
function
R]^ and
nj{pj,p^j,x* {pj,p-j))d {p*
/
we
[18]),
of an oligopoly
fi-j.
kx2, where
>
Following the
look for a lower bound on
inf
3 links on path
/
the latency fiinctions
utility.
interested in the worst performance
We
Definition 4: {iJ,*,x*{p))
W
optimum given
the reservation
is
on the "price of anarchy," (see
literature
and the vector
of these probability measures excluding j by
Oligopoly
R
a probability measure, and denote the
vector of these probability measures by
x*{p) e
a social
is
and
{Ij}
is
would lead
(since any price for this provider
profits).
are not
Moreover,
it
1,
so
playing a best response
to zero
can be verified that these strategies
weakly dominated, since
if
i
=
1,2 were to play
=
and the provider on path 2 were to set a high
= 3 would choose to play pf = 1. (This also
establishes that the OE will be trembling hand perfect,
p\
see [11], pp. 351-356 for a definition). Finally,
Given the
consider the provider on path
2.
the serial providers on path
and a fixed
1
let
us
strategies
of
fc
>
0,
be verified that the optimal strategy of this provider
set
P2
=
1/2.
The
can
it
is
to
resulting equilibrium flow allocation
is
OE and Price
B. Strict
enough p2,i
Characterization
In this paper, rather than allowing coalitions to form,
we
study a stronger concept of equilibrium, Harsanyi's
equilibrium (see [12], or [11], pp. 11-12), which
strict
requires each player's best response to be unique. Re-
OE
Nash equilibrium and our
standard
that the
call
concept only require each player,
weak
service provider, to play a
particular each
in
We now
best response.
strengthen this condition.
„0£
-[»^U
A
Definition 5:
vector (p°-^, x"^^)
(Oligopoly Equilibrium)
which involves routing
2 (though not
all
of the
all
the admitted traffic
on path
for all
i
€l,
£
j
if
>
Afi,
admitted).
traffic is necessarily
Therefore, the efficiency metric for this example
n,(pf^,pef,xO^)>n,(p„p2f,x),
is
Vp,
r2({/,},x^^)
which goes
= Eli^?^-l2{x§^)^§^ _
as
to
Example 2
This result
is at
why
—
some
is
strategy
OE
with the
inefficient.
much
1
inefficiency
is
The
because
charge unreasonably high
a best response (even weakly undominated)
It is
them
for
to
do
because other providers
so,
also charge unreasonably high prices, so there
is
no
transmission on this path and they suffer no adverse
We may
expect this pathological situation not to arise
number of reasons.
firms
First,
may
not coordmate
on such an equilibrium (especially when other
exist). In this case, for
realize that if they all
all
make
we may
example,
higher profits and would
on a path
to
still
be playing equi-
we may even
expect providers
It
to the results
particular to the tight
Nash equilibrium, where
is
of our previous work,
bound on
because, once serial providers along a path form a
would be to behave
path, thus removing the
double marginalization problem. In
in our previous
of 5/6.
work
[1] applies
both equilibria,
in
we
{^j}(jeA/'i, igi).
have not changed the behavior of
the users given by the
also that
we
since as
is
WE
(as in Definition 1).
have removed the qualifier "pure
well known,
strict equilibria
Notice
strategy,"
always have to
by
equilibria,
definition, involve players being indifferent
among
the
which they are mixing). Therefore, there
which a mixed strategy OE exists, but
strategies over
strict
OE does not.
Moreover,
are also situations in
but a
We
strict
OE
1
can be verified that there
this as a serious
exists,
shortcoming, since,
above showed, even pure strategy
for linear latency functions.
proof of Proposition
3,
we have
The proof
and therefore
Proposition 5: Let Assumption
1
OE
do
the following result
is
is
similar to the
omitted.
hold.
ther that the latency functions are linear
i
OE
does not.
do not view
Example
it
which a pure strategy
not always exist. Moreover,
[1], in
efficiency of 5/6. This
subcoalition, their optimal strategy
as a single provider along that
of
for a
best response, while the former does not. Notice that
their
only providers along a given path can form coalitions,
would lead
OE
at a strict
The difference between Definitions 3 and 5 is obvious.
The latter requires service providers to play a strict
as
can be verified that a special form of
coalition-proof subgame perfect
OE
strict
to denote the set
\x^^\i^i
network with latency functions
equilibria
expect them to
form a "coalition" and coordinate
pricing decisions.
=
flow allocations x'^^
focus on
(9)
)•
are situations in
reduced their prices, they would
librium actions. Second,
we
be pure strategy (because mixed strategy
consequences from charging unreasonable prices.
for a
refer to
and we use the notation 0£({Zj})
levelpathological, however.
so
service providers on path
prices.
We
oo.
topology can be arbitrarily
there
>0,Pj^ vT^ ^ ^ ^ w^fe-,p-f
p^^ as the strict OE price.
l/Ak
In the remainder of this paper,
+
establishes that pure strategy
parallel-serial link
reason
fc
OE
a strict
is
a;°^ G w(pf'^,p'^f'] and
Assume
and
fur-
for all
e I, there exists some j G A/i such that Ij is strictly
Then the price competition game has a strict
increasing.
OE.
this case, the analysis
and leads
to a
bound
We
strict
next provide an explicit characterization of the
OE
prices,
which
will be essential for the subse-
,
quent efficiency analysis. The following
lemma
Lemma
sumption
Let {p^'^,x^^) be a
1:
1
hold Then
pf^xf^ >
strict
for
estab-
OE
X and
E
i
The
Assume
Proof:
pOE^OE ^
to
Q fpj
arrive
gQjjjg
any price pj with pj
Z
> pf^, we
nj{pj,pOf,x)
contradiction
a
at
G
j
and j €
have
that
example shows
OE
Defini-
(cf.
shown
OE
Inefficiency
C.
Example
while others are making positive
profits.
We
n
make
Example 2 shows
R=
non-strict
OE
profit,
no longer holds
for the parallel-serial topology.
on the other hand, ensures
that
it
holds for
of
all
OE
Consider a one path network, which has
=
d
= 1/2, with a
= 1/4. Using the
x^
x/n. Let
utility
be
this
example
corresponding social
surplus
price characterization given in
WE,
Proposition 6 and the definition of a
charge the price p'^^
1
and
=
and the reservation
1
there exists a unique strict
Lemma
strict
efficiency losses
OE
of Strict
4).
for
OE,
strict
For any n, the unique social optimum for
any
Lemma
3:
even with
l.
S(x'^)
positive profits (see [1],
that this result
providing bounds on
large.
the total flow be
have
in [1] that for parallel-link topology, if at
one of the providers makes positive
in
links with identical latency fianctions l{x)
is
the providers
that
can be arbitrarily
As shown in Example 2, the result of the preceding
lemma does not extend to non-strict OE prices, i.e., there
may be OE in which some of the providers make zero
profit
be useful
will
it
a gen-
and as
the inefficiency of price competition. However, the next
Q.E.D.
tion 9).
is
at
= n,ipf^p1f,xO^),
contradicting the definition of the strict
price characterization in Proposition 6
eralization of the price characterization in [1],
in that paper,
Then,
Afi.
.OE\
r.OE
Pf
OE. Let Asall
by
prices are given
OE.
lishes that all path flows are positive at a strict
flow
is
x'^^
example
is
=
l/(n
=
1).
-f-
OE,
in
+
1),
l/(n
The
it
which
follows that
all
providers
and the equilibrium
efficiency metric for this
therefore
allows us to write the optimization problems for each
provider
We
in
terms of equality and inequality constraints.
first
order optimality conditions to
obtain an explicit characterization of the
i
n+ljn+l
y'-
can then use the
Proposition 6: Let Assumption
G X, j € Mi, we have
1
strict
ri{{lj},x^n
(l-i)i
OE prices.
hold. Then, for all
which goes
as
to
An
(-+1)-
n
—
>
cxd.
This example establishes that even with
(a)
strict
OE, which
rules out the pathological coordination failures discussed
above, efficiency losses can be arbitrarily large. The rea-
fceM
son for
this is again the
double marginalization problem,
which increases the cost of transmission so much that
there is no transmission in equilibrium along certain
(b)
Pf'
paths
n —
some
>
for
ky^j, s
/
(e.g.,
along the single path in the example as
oo). This type
of behavior
level, especially
when we
also pathological at
is
think of networks where
i,
the reservation utility, R, of users
high enough. This
is
leads us to define an even stronger notion of equilibrium,
R-EkexJkix?"^]
strong OE.
^f^[E.eM/;K°^) +
—
'
Definition 6:
E,^.
OE
otherwise.
^jgjxp^ =
(10)
In particular, for
two
links,
tive cost is less than R, for
z
when
=
the
1,2, j
minimum
e
M,
effec-
the strict
A
vector {p^^,x'^^)
(Oligopoly Equilibrium)
strong
d.
OE price
In this case,
if
it
we
and denote the
is
>
a strong
is
a strict
OE, and
refer to p'-'^ as the
set
of strong
allocations in a network with latency functions
by m\{lj}).
OE
flow
by {Ij}
difference between Definition 5 and Defini-
The only
tion 6
is
we
that in the latter
require
utility,
R, of users
is
of the potential
be the case when the
flow, d, to be transmitted. This will
reservation
all
some
for
that
OE
with
Two Paths
=
i
where each
1,2,
link
is
owned by
Assume
even with strong
illustrates that
was shown
n
links
4: Consider a
on path
1
latency functions and
—
x^jl.
utility
be
The unique
OE
social
optimum
By
the definition of a
for this
example
x^
is
Using Proposition 6 and the definition of a
price of link j
G
OE)
A/i at the
and
(where
for all
R—
the
is
pf^
This combined
i.
with the preceding relation shows that a;*^^
=
WE,
WE, we have xf^ >
EjeMM^?"") > E.-eM^r ^
is
a social
Q.E.D.
The following lemma provides
total flow admitted
at
OE
an
and
a relation
between the
optimum.
at a social
fiows x'^^ must satisfy
Lemma
For
3:
a
be an
E h^-T) + ^?^ E
^;(-?'^)
[
+
'K-?'')]
+ x^^ =
strong
OE
1
shows
E-:
•
latency
1
hold. Let
^E^
ieJ
unique
that
involves
Proof:
Assume
to
arrive
Eiei^f^ > Eiei^ffor
some
at
the social
strong
We
OE
to (0, 1) as
optimum
is
1,
»
We
surplus at
while the social surplus
goes to 1/2 as
n
—
>
at the
oo.
G
y je^f^.
>
we would have
also have lj{xf^)
lj{xf) for
lj{xf)
definition of the
we
=
some j e
I'Axf)
=
Afi.
for all
which yields a contradiction by the optimality
Afi,
conditions (4) and the fact that
next present two lemmas, which will be useful in
that
xf^ > xf
Hence,
i.
[Otherwise,
j
contradiction
a
This implies that
lj{x?'=)>lj{xf
cf),
+ 2'n + 2/'
n — oo. The social
Vn
which goes
fiinctions
{p^^,x^^)
optimum. Then
and x^ be a social
OE
E
Substituting for the latency functions and solving the
above together with a;f^
OE
of
set
Assumption
{lj}{jeM'i, iei)' '^t
=
we have
U'R-J2lj{xf))xf
optimum.
R=\.
(1,0).
(3)],
in [1]).
and the reservation
1
problem [problem
optimum and every
a social
is
a feasible solution to the social
networks (which
one link on path 2 with latency function l{x2)
=
=
T,zel {T,jeMjji=^i)) ^i
is
which
two path network, which has
with identically
Let the total flow be d
every
implying
the efiiciency loss
in parallel link
be bounded below by 5/6
to
Example
OE
that
x^^ € OE{{lj})
a different
provider. First, consider the following example,
can be worse than that
optimum,
I.
RY^i^jxf. Since x^
We now characterize the efiiciency properties of strong
OE. We start with a two path network, with rii links on
path
=
ri{{lj},xOE)
Then
Xg-
social
a
is
large enough.
Proof:
D. Efficiency of Strong
optimum
social
OE{{lj})
G
x*^^
WE
Eiei ^f <
^-l
Using the
and the optimality conditions
(4),
obtain
providing a bound on the efficiency metric for strong
OE. Note that these lemmas
The first lemma allows us
generality that
are valid for
to
all
OE
assume without
RYlLi ^f - ELi h{xf)xf >
as well.
loss
fl-E
in the
Combining
subsequent analysis.
j
Lemma
2:
Given
a
set
of
latency
0.(^?^)-P?^)
>fi-E
{iAxf)-xfi^{xf)).
of
£
Afi,
the preceding with lj{xf^)
with
functions
strict
inequality for
some
>
j,
lj{xf) for
all
and
j^'^>xY'^l',{xY'')>xfl^{xf),
{^j}jeM, ieT, assume that
[using Proposition 6(a) and the fact that xl'{x)
E(E '.•(-f)K = ^E
iei
iei
creasing,
Q.E.D.
cf Assumption
1],
we
is
nonde-
obtain a contradiction.
•5 ^y+yfl
The next theorem provides a tight lower bound on
r2{{lj}, x°^) [cf. (8)] for a strong OE. In the following,
we assume
without loss of generality that
Theorem
on path
links
=
i
where each
1,2,
link
E2/f =
function
Suppose
Ij.
that
i.
e Ni has a latency
Assumption 1 holds and the
h,j
+ h{yf-y?^)<ilv
=
y<y?%,
i
game has
r2({/,},x°^)>i,
Moreover, the bound
x'^^ e
O^
a strong
V
OE. Then
a;°^ G ag'{0,}).
{{I]]) that attains the lower
bound
and
+
problem
mum
are interested in finding a lower
bound
for
Strict
OE
Constraints.
r2{{lj},x~- ')
^'''xOBeOE
(12)
Ij
({/,})
let
The
conditions for a social
are the
same
to {Ij} for
which
symmetric,
xf^ <
j
€
We
xf.
i
we
such that
can
xf
restrict
<
constraints of the
optimum and a
by
lj{-)
here]. Condition (15)
xf.
social
ourselves
we
at the
denote by
problem guar-
OE
strict
is
claim that
since
and (19) capture the convexity
=
is
and
essential
the optimality condition for the
optimum. Condition (18) uses the nondecreasing
we
we must
(13)
(which
OE). In
relating the values kjj'ij
and the convexity assumption on the latency
inf
G X, we
i
A/i,
and Z'()
lj{-)
ifj, {ifjY [note that the assumption lj{0)
i=l
is
some
as the conditions for a strong
assumption on
txo-=f:xf=i.
This implies that there exists some
opti-
problem
antee that these values satisfy the necessary optimality
particular, conditions (14)
Since the problem
for
equilibrium and the social optimum, which
lij,l'-j,lf,,(lfjy.
{Ij},
and social
characteristics of the infinite dimensional
optimize over the possible values of
x°^ e 0^{{lj}) and let x^ be a
social optimum. By Lemma 3 and the fact that x'^^ €
OE {{Ij}) (i.e., it is a strong OE), we have
Given
dimensional
finite
that captures the equilibrium
over the entire fionction
jnf^
(19)
given in (12). This implies that instead of optimizing
the problem
inf
jsM,
l,2,
Problem (E) can be viewed as a
We
(18)
in (11).
Proof: The proof follows a number of steps:
Step 1:
^ieM,
(11)
there exists {Ij}
is tight, i.e.,
(17)
owned by
is
a different provider, and link j
price competition
(16)
d=l.
Consider a two path network, with n,
1:
Y.{llj)')<R,
fiinctions;
x^^ <
are focusing on {lj{-)} such that
x'1'
have
h,j+l[,j{yf-y?'')<llj,
where
for all j
^2
-
mmimize
,s
G A/i
.
Finally, the last set
of constraints are the
necessary conditions for a pure strategy OE. In particular,
for a
„OEs
OE
two path network, using Proposition
6,
the Strict
Constraints are given by
R-y?''{'i:,eM.h,)-y^'={T.jeMM)
(E)
R-yf{T.j^^fA,i)-y|{T.j
>ieM.
jeK
h,j)
n2y§^
subject to
[
tj<yiiii,j)',
i
=
E
jeM
1,2,
jG^f^,
jeU2
^i.
+
jeJ^i
E^+E
j6A^2
'^-
jeAf^
(14)
[and therefore ni and n2 are also decision variables in
-I-
)
^f(E(C)')
problem
(E)].
Note
problem
(12),
we
(EC)+^f(E(0')'
any feasible solution of
same objective function value. Therefore,
optimum value of problem (E) is indeed a lower
bound on the optimum value of problem (12).
(E) with the
(15)
that given
have a feasible solution for problem
the
Step 2: Consider the following change of variables for
problem (E)
mi = myf^ and
Next, using the transformation
7712
= n2y2^
to write:
OE =
mm
To
^2^2^^
1
h =
2_^ ^iJ'
m2>0
mi,
= 2^
^2
'2J1
h <
subject to
I2
jeM
we
though
OK = mmimize
if ,(if )'>o
T2
iE')
Now
'2%
(21),
By
Finally,
=
Example 4 shows
3,
2=1
Note
that this
1
problem has a very similar structure
of
[1]
to
for parallel-link networks.
Let
^{ify Ji,l'i,yi ^yf^) denote the optimal solution of
problem (E'). We have shown in [1] that if = Q for
{Ji
1
=
1,2.
Step 3: Using Tf
we
for
=
i
1, 2,
and k
=0,l[=
0,
strong
OE
1-
1
>
-
-.
2
is tight, i.e.,
^
i
2'
^2y£f
(20)
R
focus on strong OE, there exists
1/2. In contrast to the
ensures that
which
considered in
is
all
of the
case in Example
traffic is
transmitted
the key to the existence of a
This
[1].
is
again because of the double
marginalization problem: each provider along path
a greater incentive to increase
all
provider), because
along path
1
,
its
1
has
price (relative to the
these links are
it
owned by
the
same
does not internalize the reduction
of the other link owners along the same
path. Consequently, in
min
is
bound on the inefficiency of equilibrium.
The bound with strong OE is nonetheless worse
than the efficiency bound in the parallel-link topology
in the profits
see that
r^^ =
r.OE\
r2{{lj},x^
when we
bound of
benchmark where
=
OE-
bound
that this
mm
in equilibrium,
Constraints.
the finite-dimensional problem considered in the proof
of Theorem
also
Q.E.D.
Therefore,
OE
it is
l,2,
a tight
Strict
thus
1,
ihi x0^eOE{{ij})
h + l[{yf-y?'^)<lf,
+
r2{{lj},x
inf
min
i
>
ni,n2
R,
2
=
(13), this implies that
+ yf{ify,
+ yfiify <
li<y?^l'^,
satisfies
it
The corresponding optimum value
inf
if
integers.
an optimal solution to the program
is
1, 2, 1)
and moreover,
1/2.
if
1,
and n2 are
that n\
a solution to (20).
+ yUi2y =
.OE
can verified that (l2,l'2^yi'^ ,y2^ ,'fhi,m2)
it
(^, YiO,
vf.yr'^>°
subject to
il
R.
also have to ensure that the solution to this
program ensures
I'lVi
<
7711^2'
i=\
and rewrite problem (E) as
ti
+ m2l'2 =
E=
jeATj
,";>o
y^'^1'2,
mil'2
J6JV2
j€A/i
(21)
R
l2.'2
and
this
Example
4, there are
higher prices
induces greater fraction of users to
2, increasing inefficiency. To see the role of
more clearly, consider a modified version of
Example 4, where all n links along path 1 are owned by
the same service provider. This would make the example
choose path
serial links
subject to
I2
<
2/2*^^2'
nij/f^/^
1=1
<
R.
equivalent to a parallel-link topology. In this case the
OE
xf ^ = 2/3 and
unique
strict
X2^ =
1/3, and this example reaches the 5/6
[1] rather
flows are given by
than 1/2 bound of
Example
4.
bound of
E.
We
OE
of Strong
Efficiency
with Multiple Paths
next consider an I path network, with Ui links
provider.
OE
properties of a strong
in
Example
5: Consider an
on path
1
I
path network, which has
one link on each of the paths
=
latency function l{x)
=
x{I
and the reservation
1
Clearly, the unique social
of a
2,
.
.
R=
be
[1, 0,
WE,
unique
.
.
,
.
optimum
it
OE
arbitrarily
l
high even
that involves parallel
example
for this
is
at the
=
1
the
In
7:
2/n)^
OE
owned by
+ 2/n)
(i.e., all
same
the
„OE
2
\l
which goes
n
to 1/2 as
—
>
3e
if e
)
'
(1,0),
^
0,
1.
inefficient equilibrium is a result
oo.
>
is
similar to that of
Theorem
generalizes
Theorem
and
1
1.
The proof
6,
we
have
that x'^^ —*
Therefore, the highly
of the parallel-serial
there
Theorem
m
links
Ij.
which
Suppose
that
game
Assumption
1
holds and the
has a strong OE. Then
we
In this paper,
presented an analysis of price com-
communication networks with congestion.
petition in
V x^^ G Og'CO,}).
s tight, i.e.,
competition in networks with the serial-parallel topology.
there exists {Ij] and
parallel-link topology studied in [1], the parallel-serial
bound
in (22).
OE
Congestion
bound on the
does not generalize once
that li{Q)
relax the assumption
= 0.
OE
the social optimal. In particular,
due
can
now be
to an
extreme
(pathological) form of double marginalization,
whereby
arbitrarily inefficient.
efficiency loss of strong
we
of pure
result is that contrary to the case
topology leads to significant efficiency losses relative
to
Unfortunately, the
the efficiency implications of price
(22)
{{Ij}) that attains the lower
Positive Latency at
Conclusions
V.
Our major
Moreover, the bound
E
congestion,
2 on efficiency,
quite close to, but slightly lower than 5/6.
is
The focus has been
rj{{h},x°^)>\,
oS
positive
i
price competition
x°^ €
—
again a tight bound of 2\/2
£ J, where each link is owned by
provider, and link j, j G Mi, has a latency
on path
a different
function
2: Consider a general / path network, with
is
is
congestion. In fact, [1] shows that with
latency at
parallel topology, but positive latency at
omitted.
is
1
have
otherwise.
—> 1 and e
and r2({/j},x°^) -^
Consequently, 6
+ 2/n)
links along path
topology, not of the assumption that there
The next theorem
network
assumption
same example with
n
we would
provider),
is
1
2
for a
with the parallel-serial
useful to note that in the
is
It
example
1-i
OE
inefficient.
I (1,0),
r/({/,},^°^)
we
presence of positive latency
zero congestion, strong
1
efliciency metric for this
oo,
>
serial links if the
the parallel-link topology
Hence the
—
and n
1
and the efficiency metric
at a strong
and
topology can be arbitrarily
(/-l)(l
_
relaxed. This establishes:
is
Proposition
at
+ 2/n'(/-l)(l +
2)
definition
is
2/n
„0£
^
+
1.
can be seen that the flow allocation
strict (strong)
en
e(n
'
This example shows that the efficiency loss could
be
Using Proposition 6 and the
0].
(0, 1),
>
unique
at the
by
+b
+ 2)
2e
xP^ —
The flows
(1,0).
hj^/n. Then, as 6
l)/2. Let the total flow
utility
=
r2(0,},xO^)^0.
/ with the same
,
.
=
e
have that
Zj(0)
=
a;'^
-
n
latency functions and
x^
are given
e(n
an / path network.
with identically
is
OE
x^^
Let
be d
(strong)
strict
link is owned by a different
i, where each
The following example illustrates the efficiency
on path
links
optimum
social
serial
all
This
is
partly
providers on a particular path charge pro-
on
that path to
the concept of strict
OE, which
hibitively high prices expecting others
do so as well.
Example
n
links
6:
on path
Consider a two path network, which has
1
with identically
latency functions and
one link on path 2 with latency function l{x2)
for
some
scalars e
>
and 6
>
0.
=
ex2
+b
Again the unique
We
showed
requires
removes
loss
all
that
service providers to play strict best responses,
this pathological behavior, but the efficiency
of strict
OE
is
also
unbounded because of the
related
double marginalization problem. In particular, the
cost of transmission on a path consisting of
many
total
providers can be sufficiently high that most of the users
do not transmit
when
Yet,
may
[9;
we
this,
we
that
a strict
is
long as there
as
zero congestion, there
OE
with
is
J.,
Game
Theory. The
MIT
Press,
no such
tight
[14
examples of extreme
boimd even
inefficiency, there
a flavor of pathological results, however. Therefore,
ture research
and methods similar
is
Acknowledgments:
Muhamet
We
thank
[20
[21
[22;
efficiency in
[23
D., and Ozdaglar, A.,"Flow control, routing, and
performance from service provider viewpoint," LIDS report,
[24;
WP1696, May 2004.
user principle for resource allocation in wireless networks,"
IEEE Conference on Decision and
J.,
stability
E., Dasgupta A., Kleinberg
and Roughgarden T., "The price of
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Wexler
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Basar, T. and Srikant R., "Revenue-maximizing pricing and
capacity expansion in a many-users regime," Proceedings of
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[6]
Beckmann, M., Mcguire, C. B., and Winsten, C. B., Studies in
the Economics of Transportation. Yale University Press, 1956.
[7]
Correa,
J.
R.,
Schulz, A.
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Dafermos,
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91-118, 1969.
Low,
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and Lapsley, D.
[25
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Netherlands, 1994.
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Sanghavi
S.
and Hajek
B.,
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Wardrop, J. G., "Some theoretical aspects of road traffic
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good
Acemoglu,
T.,
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Methods,
congested markets," submitted for publication, January 2005.
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an area for fu-
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suspect that these worst-case results are not informative
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Braess'
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we
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arbitrarily inefficient.
In all the
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Rationale for
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game
with strong OE, and the equilibrium can once again be
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[15
[5]
E.,
Fudenberg, D. and Tirole
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zero latency at
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[12;
a tight boimd of 1/2 on the
removed, however, there
[4]
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[11
all
from price competition.
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Once
defined a stronger notion of
OE, which
transmitted in equilibrium. For strong
traffic
we showed
[3]
E.,
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of the
[2]
Friedman,
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R and Maskin
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53