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DEWEY
HB31
.M415
u
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
COMPETITION AND EFFICIENCY
IN
CONGESTED MARKETS
Daron Acemoglu
and
Asuman Ozdaglar
Working Paper 06-1
January 20, 2006
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,corn/abstract=898938
MASSACHUSETTS INSTITUTE
OF TECHNOLOGV
JUN
2
2006
J
LIBRARIES
Competition and Efficiency
in
Congested Markets^
Daron Acemoglu
Department of Economics,
Massachusetts Institute of Technology
Asuman
Department
E. Ozdaglar
of Electrical Engineering
and Computer Science
Massachusetts Institute of Technology
January
20,
2006
Abstract
We study the efficiencj' of oligopoly equilibria in congested
markets.
The moti-
vating examples are the allocation of network flows in a communication network or
We show that increasing competition among
measured as the difference between users' willingness to pay and delay costs. We characterize a tight bound of 5/6 on efficiency
in pure strategy equilibria when there is zero latency at zero flow and a tight
bound of Is/Ji — 2 with positive latency at zero flow. These bounds are tight even
when the numbers of routes and oligopolists are arbitrarily large.
of traffic in a transportation network.
oligopolists can reduce efficiency,
*We thank Xin Huang, Ramesh Johari, Eric Maskin, Eilon Solan, Nicolas Stier Moses, Jean Tirole,
Tsitsiklis, Ivan Werning, Muhamet Yildiz, two anonymous referees and participants at various
John
seminars and conferences
for useful
comments.
Introduction
1
We
analyze price competition in the presence of congestion costs. Consider the 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
Congestion costs are captured by a route-specific non-decreasing convex latency
function, li (•). Profit-maximizing oligopolists set prices (tolls) for travel on each route
traffic.^
denoted by
p,.
We
analj'ze
subgame
for each price vector, p, all traffic
+ Pi,
perfect
Nash
equilibria of this environment,
chooses the path that has
minimum
where
(delay plus
toll)
maximize profits.
li
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
cost,
example,
for
and
oligopolists choose prices to
[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
economics, by
[40] in
[46], [57], [5] in
transportation networks, and by
communication networks. More recently, there has been a growing
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 twostage competitive facihty location 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 charge prices for use. Moreover, we will show that the natm-e
[36], [24], [23], [30] in
literature that focuses
of the analysis changes significantly in the presence of price competition.
We
provide a general framework for the analysis of price competition
among
ser-
vice providers^ in a congested (and potentially capacitated) network, study existence of
pure strategy 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
inefficient (e.g., [40], [45], [10]), price-setting
externality
and achieves
assignment without prices can be highly
by a monopolist internahzes the negative
efficiency.
Second, increasing competition can increase inefficiency. In
fact,
changing the market
monopoly to duopoly almost alwaj^s increases inefficiency. This result
contrasts with most existing results in the economics hterature where greater competition
tends to improve the allocation of resources (e.g. see Tirole [51]). The intuition for this
result,. which is related to congestion, is illustrated by the example we discuss below.
structure from
An externality arises when the actions of the player in a game affects the payoff
^We use oligopolist and service provider interchangeably throughout the paper.
^
^Because, in our model, users are homogeneous and have a constant reservation
of congestion externalities,
all
market structures would achieve
efficiency,
of other players.
utility, in
the absence
and a change from monopoly
Third and most important, we provide tight bounds on the extent of inefficiency
We
presence of ohgopolistic competition.
is
show that when latency
in the
at zero flow (traffic)
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 these bounds.
we
Finally,
show that pure strategy equilibria may fail to exist. This is not
what we have here is a version of a Bertrand-Edgeworth
also
surprising in view of the fact that
game where pure
strategy equilibria do not exist in the presence of convex costs of
production or capacity constraints
(e.g., [14], [49], [7],
environment when latency functions are
essentially because congestion externalities
[56]).
Ho'vever, in our ohgopoly
a pure strategy equilibrium always
linear,
remove the payoff discontinuities inherent
when
the Bertrand-Edgeworth game. Non-existence becomes an issue
are highly convex.
We
also
In this case,
we prove that mixed strategy
show that mixed strategy
exists,
in
latency functions
equilibria always exist.
equilibria can lead to arbitrarily inefficient worst-case
can become arbitrarily small relative to the
though the average performance of mixed strategy equilibria
realizations; in particular, social surplus
maximum
is
social surplus,
much better.
The following example
Example
illustrates
some
of our results.
shows a situation similar to the one first analyzed by Pigou [40]
to highlight the inefficiency due to congestion externahties. One unit of traffic will travel
from origin A to destination B, using either route 1 or route 2. The latency functions
are given by
Figure
1
1
k{x)
It is
=
=
^2(2;)
Y'
straightforward to see that the efficient allocation
-X.
[i.e.,
one that 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 of the inefficiency
is
that each unit of traffic 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.
including a markup,
We
Xj/^
show below that in this case, the monopoHst will set a price
(when k is differentiable) which exactly internalizes the con,
gestion externality. In other words, this
social planner
would
markup
location. Consequently, in this simple example,
and p^^
—
(2/3^)
-f-
equivalent to the Pigovian tax that a
is
set in order to induce decentralized traffic to choose the efficient al-
k, for
some constant
k.
monopoly
The
prices
wiU be
pf^-^
=
(2/3)
Wardrop
=
=
librium will be identical to the efficient allocation,
resulting traffic in the
i.e.,
to duopoly, for example, would have no efficiency consequence.
Xj^-^
2/3 and x,^^
+k
equi-
1/3.
l,(x)=x /3
I
unit of
traffic
^x)=i2/3)x
Figure
1:
A
two
link
network with congestion-dependant latency functions.
duopoly situation, where each route is controlled by a different
it can be shown that equilibrium prices will take
^
—
the form pf
Xi {l[ + I'o) [see Eq. (20) in Section 4], 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 also differs from the efficient allocation. We will show that this is generally the case
Finally, consider a
profit-maximizing provider. In this case,
in the oligopoly equilibrium.
Interestingly, while in the
Wardrop equilibrium without
was too much traffic on route 1, now there is too little traffic because
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 of the consumer surplus is captured by the monopolist, while in the
oligopoly equilibrium users may have positive consumer surplus.^
prices, there
of its greater
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, it realizes that this wiU push some traffic from route 1 to route 2, raising congestion
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
These are
markup.
Xi/j for route 1
and
Xg/'j
route
for
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
social surplus declines, users will
be better
off
than
in
monopoly
(i.e.,
they
will
though
command
a positive consumer surplus).
There
is
a large literature on models of congestion both in transportation and commu-
nication networks
(e.g.
[5],
[38],
[44],
[33],
[34], [45]).^
However, very few studies have
is the difference between users' willingness to pay (reservation price) and effective
and is thus different from social surplus (which is the difference between users'
willingness to pay and latency cost, li{xi), thus also takes into account producer surplus/profits). See
''Consumer surplus
costs, Pi
+
li{xi),
[32].
^Using economics terminology, we could also say that the demand for route 1 becomes more "inelasSince this term has a different meaning in the communication networks literature (see [48]), we
do not use it here.
tic".
®Some
objective,
of these papers also use prices (or tolls) to induce flow patterns that optimize overall system
and a number of studies have characterized the
"toll set",
i.e.,
the set of
all
tolls
that
investigated the implications of having the "property rights" over routes assigned to
profit-maximizing providers. In
specific
[4],
Basar and Srikant analyze monopoly pricing under
assumptions on the utihty and latency functions. He and Walrand [19] study
among internet service providers under specific demand
competition and cooperation
models. Issues of
efficient allocation of flows or traffic across routes
do not
arise in these
Our previous work [1] studies the monopoly problem and contains the efficiency
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, in current work, [2], we extend some of the results
papers.
of the
of this paper to a network with paraUel-serial structure.
In the rest of the paper,
though
all
we use the terminology
of a (communication) network,
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
efficiency.
its
Section 4 defines and characterizes the oligopoly equilibria with competing
profit-maximizing providers. Section 5 contains the main results and characterizes the
and provide bounds on
efficiency properties of the oligopoly equilibrium
tion 6 provides a tight efficiency
bound when
may be
there
efficiency. Sec-
positive latency at 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 a convex 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.
:
Model
2
We
X=
consider a network with / parallel links. Let
Let Xi denote the total flow on link
Each
link in the
i,
and x
=
[xi
,
.
.
,
.
{1,
xj]
.
,
.
.
/} denote the set of links.
denote the vector of link flows.
network has a flow-dependent latency function
the travel time (or delay) as a function of the total flow on link
li{xi),
i.
We
which measures
denote the price
by p,. Let p = [p\,
,pi] denote the vector of prices.
We are interested in the problem of routing d units of flow across the / finks. We assume that this is the aggregate flow of many "small" users and thus adopt the Wardrop's
per unit flow (bandwidth) of hnk
i
principle (see [57]) in characterizing the flow distribution in the network;
are routed along paths with
at the given flow
sume that the
minimum
and the price
effective cost, defined as the
of that
path
users have a reservation utility
effective cost exceeds the reservation utility.
induce optimal flows, with the goal of choosing
minimizing the total amount of
the
tolls
number
is
i.e.,
R
and decide not to send
the flows
of the latency
(see the definition below). ^
We
also as-
their flow
if
the
This implies that user preferences can be
from this set according to secondary
of tolled routes; see
[8],
used extensively in modelling traffic behavior
and in communication networks, e.g., [45], [10].
'''Wardrop's principle
e.g., [5], [12], [38], [50],
tolls or
sum
criteria, e.g.,
[21], [28], [27],
in
and
[20].
transportation networks,
u(x)
Figure
2:
Aggregate
utility function.
represented by the piecewise linear aggregate utility function u
To account
1
WE used
shows that
in
this definition
following definition of a
problem, including
WE (see
more standard
equivalent to the
is
For a given price vector p
(WE)
[29], [26]).
definition of a
denote the set of
Assumption
>
0,^
(i?
-
a vector
x^^
is
a Wardrop
if
WE
We
we use the
2.®
the literature under some assumptions.
Definition 1
equilibrium
depicted in Figure
for additional side constraints in the traffic equilibrium
capacity constraints on the links,
Lemma
(•)
G arg
max
<
"V^
WE at a given p by
1 For each
i
li{xf'^)
/^(O)
Pi)xj
(1)
W{p).
G J, the latency function
convex, nondecreasing, and satisfies
-
=
li
1-^
[0,
oo)
0,
implies that
[0,
do] is closed,^"
0.
The assumption of zero latency at zero flow, i.e., Zj(0) =
due to flow of traffic, and there are no fixed latency costs. ^'
It is
all
latency
is
adopted to simplify the
discussion, especially the characterization of equilibrium prices in Proposition 9 below.
A
trivial relaxation of this
have no
effect
on any
assumption to
same
— L
of the results in the paper.
*This simplifying assumption implies that
the
li{0)
reservation utility, R.
The
all
for
alH G J
Allowing
some L >
for
wiU
for differential levels of
users are "homogeneous" in the sense tha,t they have
analysis below will
show that the value
of this reservation utility
R
has no effect on any of the results as long as it is strictly positive. We discuss potential issues in
extending this work to users with elastic and heterogeneous requirements in the concluding section.
^Since the reservation utility of users
is
equal to
/?,
we can
also restrict attention to pi
< R
for all
Throughout the paper, we use p > and p e [0,i?]^ interchangeably.
'"For- a function /
R" i-^ — 00,00], we say that / is closed if the level set {x f{x < c)} is closed
for every scalar c. Note that a function is closed if and only if it is lower semicontinuous over R" (see
i.
:
(
\
[9],
Proposition
1.2.2),
^'This assumption would be a good approximation to communication networks where queueing delays
are
more substantial than propagation
delays.
^j(O)
complicates the analysis, but has Httle effect on the major results.
This case
is
where we provide a slightly lower tight bound for the inefficiency
of oligopoly 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,
is a closed interval of the form
[0, b] or [0, oo). Let be, = sup^-gj;;^ x. Without loss of generality, we can add the constraint
Xi € Ci in Eq. (1). Using the optimality conditions for problem (1), we see that a vector
^WE g ]^/^ jg g^ -^/g j£ ^^^ Q^Yy if
such that
J^iei "^Y^ — ^ ^'^^ there exists some A >
=
and for all i,
A( Y.^^z ^"i^
d)
discussed in Section
6,
|
Q
i?-/,(xr^)-p,
<A
ifx|^^
=
ffO<xf'^<6c,,
A
>A
When the latency
This lemma states that
equalized on
Lemma
Then
all
which
(2)
[i.e.,
Q=
[0,
oo)],
we obtain the
often used as the definition of a
WE,
in the
following
WE in the hterature.
the effective costs, defined as li{x^^)
+
are
pi,
with positive flows.
links
Assumption
1 Let
is
0,
ffxf^^^c..
functions are real-valued
WE,
characterization of a
=
1
hold,
h{x*)
+ Pi =
and assume further that C,
if and only if
—
[0,
oo) for
all
i
£ I.
W{p)
a nonnegative vector x* G
V
min{L(a;*) -1-p,},
i
with x*
>
0,
(3)
j
li{x*)
+ Pi <
V
R,
>
with X*
i
0,
iex
with X^,gjX*
Example
=
d
min^ {lj{xj)
if
+ Pj} <
below shows that condition
2
latency functions are not real-valued.
erties of
a
WE
for existence,
R.
are well-studied (see
The
based on establishing the equivalence of
a convex network optimization problem, which
Proposition
:=i
1
we
not hold
when
the
and continuity prop-
provide here the standard proof
WE and the optimal solutions of
will refer to later in
our analysis.
(Existence and Continuity) Let Assumption 1 hold. For any price
Moreover, the correspondence
is nonempty.
W
> 0, the set of WE, W{p),
M^ is upper semicontinuous.
vector p
M^
We
[12], [50]).
(5],
lemma may
(3) in this
existence, uniqueness,
Proof. Given any p
>
0,
:
consider the following optimization problem
maximizex>o
subject to
'^({R-pi)xi2_, ^i
Xi
e
^
Ci,
d.
V
i.
h{z)dz]
(4)
In view of
Assumption
e
(since
C,;)
conditions
[cf.
is
nondecreasing for
is
all i), it
convex over the constraint
and convex. Moreover, the
which are also
(4),
k
(1) (i.e.,
objective function of problem (4)
can be shown that the
set,
which
sufficient conditions for optimality, are identical to the
Eq.
(2)].
Hence a flow vector
nonempty
is
order optimality conditions of problem
first
x^^
E W{p)
and only
if
WE optimahty
an optimal
if it is
solution of problem (4). Since the objective function of problem (4)
continuous and
is
problem has an optimal solution, showing that W{p)
is an upper semicontinuous correspondence at every p
is nonempty. The fact that
follows by using the Theorem of the Maximum (see Berge [6], chapter 6) for problem
the constraint set
compact,
is
this
W
Q.E.D.
(4).
WE flows also satisfy intuitive monotonicity properties given in the following propoThe proof
sition.
follows from the optimality conditions
[cf.
(Monotonicity) Let Assumption
hold.
Eq.
(2)]
and
is
omitted
(see
[I])-
Proposition 2
P-j
=
some
(b) For
for all
i
For some
(c)
let
Pj
<
p, let
<
Pj, let
x £ W{pj,p^j) and x € W{pj,p-j). Then
JC
T, suppose that pj
p, the
some properties
illustrates
<
pj for
all
WE
R=
I.
Assume that the
price vector (pi,P2)
vectors (xi,X2) with
1,
W{p)
<
—
<
x,
given by
is
are not real- valued.
scalar a
pj
Under
ted).
and
Xj
<
Xj,
=
>
pj for all j
^ I, and
The
following
^ ei^r
in general.
oo
2/3 and J^, Xj
(0,X2) with
afl
Lemma
=
1
and the
reserva-
by
otherwise.
WE,
<
<
1
1.
X2
W{p), is given by the set of all
At any price vector (pi,P2) with
<
2/3.
need not hold when latency functions
Consider, for instance, the price vector (pi,P2) = (1 — e, 1 — ae)
1. In this case, the unique
is (xi,X2) = (1/3,2/3), and clearly
WE
on the two routes are not equalized despite the
positive flows. This arises because the path with the lower
constrained, so no more trafHc can use that path.
effective costs
Xj
ifO<.T<|
(1,1), the set of
This example also illustrates that
some
and
latencj' functions are given
[
=
>
Xj
WE.
~
P2
—
I
E
need not be unique
of the
l^^^lj^^fO
At the
j
Consider a two hnk network. Let the total flow he d
2
tion utility he
for
let
y^ j.
X G W{p) and x G W{p). Then Yljei^i
Example
>
0,
x G W{p) and x G W{p). Then, ^^^x^i — X^zei^*-
For a given price vector
example
Pi
>
For a given p
b»]»#j-
For some p
(a)
1
both have
fact that they
effective cost
is
capacity
further restrictions on the k, the following standard result follows (proof omit-
(Uniqueness) Let Assumption
Proposition 3
singleton. Moreover, the function T-^
:
R^
M^
h->
we do not assume that the latency
Since
lemma
the following
Lemma
in
is
T—
{i
1
e
3 X, X e
\
WE,
the set of
0,
W{p),
functions are strictly increasing,
p >
hold. For a given
1
further that k
is
is
a
continuous.
our analysis to deal with nonunique
Let Assumption
2
>
For any price vector p
d.
strictly increasing over
Assume
hold.
1
0,
W{p) with
we need
WE flows.
define the set
x^}.
-^
Xi
(5)
Then
=
=
k{x,)
Pi
Proof. Consider some
such that Xi
^
Xi.
E
i
2
V
0,
V
Pj,
G J, V X e Wi-p),
z
I,
J
e i.
and x € W{p). Since
Assume without
6 X, there
i
>
loss of generality that Xj
some x G li'l^(p)
There are two cases
exists
Xj.
to consider:
(a)
>
If Xfc
Xk for
all
^
k
optimality conditions
i,
[cf.
then Yljei^i
Eq.
(2)] for
-^
Yljei^J' which imphes that the
x hold with A
=
0.
By
Eq. (2) and Xi
WE
>
x^,
we have
which together imply that
^^(O) == 0), it follows
(b)
If Xfc
<
Xfc
for
some
li{xi)
that li{xi)
k,
by the
li{xi)
+Pi < R,
li(xi)
+pi > R,
=
=
li[xi).
By Assumption
1
(i.e., /j is
convex and
0.
WE optimality conditions,
li[Xi)
+p, <
lk{Xk)
+Pk,
k{ii)
+Pi>
lk{ik)
+Pk-
we obtain
Combining the above with Xj > Xj and Xk < Xk, we see that li(xi) = k{xi), and
(and also that
hi^k) = hi^k)- By Assumption 1, this shows that li{xi) =
Pz=Pk)-
X,
Next consider some i, j G I. We will show that Pi = Pj. Since
X G W{p) such that Xj > x^. There are three cases to consider:
• Xj
<
Xj.
Then a
similar
argument to part
(b)
above shows that
i
G Z, there
pi
~ pj.
exist
• Xj
>
Xj.
If
Xk
>
Xk for
k
all
^
optimality conditions hold with A
which together with
• Xj
=
Xj
7^
li{xi)
=
^^Xm
i,j,
then
=
Therefore,
0.
li{x^)
+Pi < R,
lj{xj)
+ Pj >
lj{xj)
=
<
implying that the
d,
WE
we have
R,
imply that
Pi
=
pj.
Since j € I, by definition there must exist some other x G W{p) such that
Xj. Repeating the above two steps with Xj instead of Xj yields the desired
Xj.
result.
Q.E.D.
lemma states that if there exist multiple WEs, .t,x G W{p) such
then the latency function k must be locally flat around Xi (and Xi). Given
and the convexity of latency functions, this immediately
the assumption that /i(0) =
Intuitively, this
that Xi
7^ ij,
implies li{xi)
We
=
0.
next define the social problem and the social optimum, which
allocation) that
would be chosen by a planner that has
full
is
the routing (flow
information and
full
control
over the network.
A
Definition 2
social
flow vector x^
is
a social optimum
if it is
an optimal solution of the
problem
maximize3.>o
2_]
[^ ^
^ii^i))^i
(6)
iei
subject to
2_] ^i
^
d.
iei
In view of
Assumption
a compact constraint
set,
1,
the social problem has a continuous objective function and
guaranteeing the existence of a social optimum, x^
using the optimahty conditions for a convex program (see
a vector x^ G
R^
subgradient
G dli{xf)
for
each
gi.
is
a social
for
optimum
each
i,
and only
if
and a
A^^
>
if
[9],
Section 4.7),
—
^
^^^'^
Yliei^i
such that ^^{Yliei^f
.
Moreover,
we
see that
there exists a
—
d)
=
and
i,
R-k{xf)-xfg,,
<A^ ifa:f = 0,
= A^ [{0<xf<bc„
>A^ iixf^bc,.
(7)
For future reference, for a given vector x G M^, we define the value of the objective
function in the social problem,
S(x)-^(i?-/,(x,))xi,
(8)
iei
as the social surplus,
latency.
i.e.,
the difference between users' wilhngness to pay and the total
Monopoly Equilibrium and
3
Efficiency
we assume that a monopoHst service provider owns the / hnks and
bandwidth on hnk i. We considered a related problem in
In this section,
charges a price of Pi per unit
[1]
for
atomic users with inelastic
of users
traffic (i.e.,
the utility function of each of a finite set
a step function), and with increasing, real- valued and differentiable latency
is
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
functions. Here
= ^PiXi,
n(p,2:)
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
A
Definition 3
W{p^^) and
{p^^,x^^) >
vector
is
Our
definition of the
ME
is
stronger than the standard
librium concept for dynamic games.
With a
(ME)
a Monopoly Eqmlibn.um
Vp>0, VxG
n(p^^-^,x^^^) >n(p,a;),
WE.
slight
if
x^^^ €
M/(p).
subgame
perfect
abuse of terminology,
let
Nash
equi-
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*)
pricing-congestion
game
if
x*
>
is
a subgame perfect equilibrium (SPE) of the
E W{p*) and
for all
p >
0,
there exists x
E
W
(p)
such
that
n(p*,x*)>n(p,x).
The fohowing
coincide.
proposition shows that under Assumption
Since the proof
is
1
,
the two solution concepts
not relevant for the rest of the argument,
we provide
in
it
Appendix A.
Proposition 4 Let Assumption 1 hold. A vector
it is an SPE of the pricing-congestion game.
Since an
ME
(p*,x*)
is
subject to
easier to
is
an
ME
if
and only
if
an optimal solution of the optimization problem
maximizep>o, x>o
it is
{p^^^ ,x^'^^)
^Pi^;,:
(9)
X eW{p),
work with than an SPE. Therefore, we use
ME
as the solution concept in
this paper.
The preceding problem has an optimal
an ME. Moreover, we have:
solution,
10
which establishes the existence of
Let Assumption
Proposition 5
onlj'
is
if it
we have
pi
hold.
1
a social optimum. Moreover,
—R—
A
vector x
if (p,
x)
is
the flow vector at an
is
an ME, then
for all
i
optimum
if
and
>
0,
li{xi).
This proposition therefore establishes that the flow allocation at an
social
ME
with Xj
are the same.
Its
proof
is
ME
similar to an analogous result in
and the
[1] and is
omitted.
In addition to the social surplus defined above,
it
is
also useful to define the con-
sumer surplus, as the diflFerence between users' wiUingness to 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 the social optimum, all of the surplus is captured by the monopolist, and users
are just indifferent between sending their information or not (i.e., receive no consumer
surplus).
Our major motivation
for the
a better approximation to
reality,
providers.
an
A
is that they provide
where there is typically competition among service
secondary motivation is to see whether an oligopoly equilibrium will achieve
efficient allocation like
the
study of oligopolistic settings
ME,
while also transferring some or
all of
the surplus to
the consumers.
Oligopoly Equilibrium
4
We
5
suppose that there are
service providers, denote the set of service providers
5 owns
S, and assume that each service provider s G
by
a different subset T^ of the links.
charges a price pi per unit bandwidth on link i E Xs. Given the vector
of prices of links owned by other service providers, p^ = [Pili^i^, the profit of service
Service provider
provider s
s
is
ns(p,,p_^,x)
for
X e W{ps,p-s), where
The
is
to
p,
=
= ^PiXi,
\pi]zei,-
objective of each service provider, like the monopolist in the previous section,
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,
notion of (subgame perfect) Nash equilibrium.
we assume, they do according to the
We refer to the game among service
providers as the price competition game.
Definition 5
if
x°^ e
A
vector {p'^^,x^^)
W {p°^, p°f
)
and
for all s
>
is
a (pure strategy) Oligopoly Equilibrium (OE)
e 5,
n,(p°^,pef ,x°^) > n,(p„pef ,x),
We
refer to p*^-^ as the
As
for
OE
V
p,
>
0,
V
.X
€ wip,,p'^f).
(lo)
price.
the monopoly case, there
is
OE and
Again associating the subgame perfect
a close relation between a pure strategj^
a pure strategy subgame perfect equilibrium.
equilibrium with the on-the-equilibrium-path actions,
11
we
have:
Definition 6 A vector {p* ,x*) > is a subgame perfect equilibrium (SPE) of tlie price
competition game if x* €
R^ i—* R^ such that
(p*) and there exists a function x
>
all
and
for
all
for
s
£
S,
x{p) G
p
(p)
W
W
:
vp, >o.
Usip:y_„xn>Us{ps,p_,,x{ps,p*_,))
The
OE
following proposition generalizes Proposition 4 and enables us to work with the
more convenient
Proposition 4 and is omitted.
definition,
that of
(11)
which
for the
is
Proposition 6 Let Assumption 1 hold.
an SPE of the price competition game.
A
subsequent analysis. The proof parallels
vector {p^^ ,x'^^)
is
an
OE
if
and only
if it
is
The price competition game
is
neither concave nor supermodular. Therefore, classical
arguments that are used to show the existence of a pure strategy equilibrium do not hold
(see [16], [52]). In the next proposition,
OE. The proof
exists a pure strategy
Proposition 7 Let Assumption
we show
that for linear latency functions, there
provided in the appendix.
and assume further that the latency functions
Then the price competition game has a pure strategy OE.
are linear.
The
is
1
hold,
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 the total flow be d
=
Assume that
1.
the
latency functions are given by
1
I
n
\
1
I
ifO<x<5
Jo
\
and 5 > 1/2, with the convention that when e = 0, hix) = oo for x > 5.
e >
show that there exists no pure strategy oligopoly equilibrium for small e (i.e.,
there exists no pure strategy subgame perfect equilibrium). The following list considers
for
some
We
first
all
candidate ohgopoly price equilibria (pi,P2) arid profitable unilateral deviations
sufficiently small, thus establishing the nonexistence of
1.
Pi
=
2.
Pi
—
=
P2
0:
A
small increase in the price of provider
thus provider
profits,
P2
>
0:
1
3.
<
pi
<
1
will
Xi
=
generate positive
has an incentive to deviate.
Let x be the flow allocation at the OE.
has an incentive to decrease
to decrease
for e
an OE:
its price.
If
xi
<
1,
If
then provider
1,
then provider 2
1
has an incentive
its price.
P2: Player 1
has an incentive to increase
remains the same.
12
its
price since its flow allocation
<
4.
<
P2
For
Pi'-
e
sufficiently small, the profit function of player 2, given pi,
a function of
strictly increasing as
increase
We
OE
its price.
next show that a mixed strategy
OE
always
exists.
We
define a
mixed strategy
as a mixed strategy subgame perfect equilibrium of the price competition
Dasgupta and Maskin,
on
p2,
Let
[0, i?]".
service provider
[13]).
Definition 7
e W
/
lis
Let S" be the space of
denote the cardinality of Xg,
s.
Let
€
fig
{fi*,x*{p))
(p) for every
(ps,P-5,
a;*
(see
i.e.,
the
(Borel) probability measures
number
B^' be a probability measure,
and the vector
fi
all
game
of links controlled by
and denote the vector of these
these probability measures excluding s by
Ig
probability measures by
x*{p)
is
showing that provider 2 has an incentive to
is
p €
of
a mixed strategy Oligopoly Equilibrium (OE)
[0,
RY
if
the function
and
d(yu*(pj X /xljp_,))
(pa,P-s))
JlQ,R]'
>
for all s
and
fig
G
Ug{ps,p-s,x* (p,,p_s))d {ng
/
(p,)
X
n*_^ (p_,))
'S^^
OE simply
Therefore, a mixed strategy
requires that there be no profitable deviation
to a 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
Mi(p)=|
(
1-^
0<P<R{l-6),
R{l-5)<p<R,
1
otherwise,
(
i-^
f^^iP)={
Notice that
fii
has an atom equal to
strategy
OE,
an atom
at that point. Let
let
/x'
be the density of
strategy equilibrium,
for all Pi
otherwise.
1
(
it
Mi =
1
fi,
—
5 at
To
i?.
verify that this profile
with the convention that
ji'
—
is
a mixed
oo when there
is
To establish that (/ii,/i2) is a mixed
(p) > 0}
show that the expected payoff to player i is constant
{p
suffices to
0<p<R{l-5),
R{l-S)<p<R,
.
\
fj,'^
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).
Jo
13
(12)
WE
demand x{pi,p2) takes the simple form of Xi{pi,p2) == 1 if pi < P2 and
The
xi (pi P2) = 1 — (^ if Pi > P2 The exact value of ii (pi P2) = 1 when pi = p2 is immaterial
since this event happens with zero probability. It is evident that the expression in (12)
,
,
is
f
=
1,2 given pi and p2 above.
a mixed strategy OE.
It
can also be verified that there are no other mixed
is
proved in Appendix B, establishes that a mixed
constant for
is
G Mi
all pi
for
(PiiPa)
strategy equilibria.
The next
proposition, which
This establishes that
strategy equilibrium always exists.
Proposition 8 Let Assumption
strategy OE, (p°^ x"^^ (p) )
Then
hold.
1
the price competition
game has a mixed
,
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
k{xf^)+pf^ = imn{lj{xf^) + pf^},
with
k{x?^)+P?'^
<
R,
E^[x?"
<
d,
Ezei^?^ =
ProoL
by the
^
if
definition of a
>
0,
min,{/,(i°^)
WE.
x°^ >
Viwithxp^>0,
(13)
Vfwithxf^>0,
(14)
(15)
+ Pj} <
Let (p^^.x"^^) be an OE. Since
with xf ^
OE, then
a pure strategy
R.
x°^ €
Consider condition
and (15) follow
Assume that there exist some i,j G X
VK(p°'^), conditions (14)
(13).
such that
k{xr)+pr<iA^r)+pTUsing the optimality conditions
for a
WE
[cf.
Eq.
(2)], this
xf^ = bd- Conoptimahty conditions, we
implies that
pf ^ to pf^ e for some e > 0. By checking the
we can choose e sufficiently small such that x*^^ G ^^{p^^ +
provider that owns hnk i can deviate to pf^ -I- e and increase
the fact that {p^^,x'^^) is an OE. Finally, assume to arrive
sider changing
-|-
see that
service
dicting
that minj{/j(x^^) -1-Pj}
WE
for
[Eq. (2)
some
and
WE,
We
i.
is
with A
=
< R and
Yliejx'^^
<
d.
since
14
at a contradiction
Using the optimality conditions for a
implies that we must have xf^ = bd
Yliei^?^ "^ ^]' ^^^^^
With a similar argument to above, a deviation to pf^
more profitable, completing the proof. Q.E.D.
need the following additional assumption
£,P-f)- Hence the
its profits, contra-
for
-I-
e
keeps
x^^
our price characterization.
as a
Assumption
we have
^
li{xf^)
Note that
increasing or
Lemma
lis
Given a pure strategy
2
0,
assumption
this
If lis'
(b)
If
Hs
for
some
is
automatically satisfied
own only one
>
for
some
s
if all
some
€
z
eS,
T
with xf^
>
0,
latency functions are strictly
link.
1
and
2 hold.
Let
(p°^,x°^).
s at
G S, then H^ >
s'
x'~^^), if for
be a pure strategy OE. Let Assumptions
{p'^^,x'^^)
>
,
{i}.
denote the profit of service provider
(a)
{p^^
=
service providers
if all
4 Let
then I^
OE
e 5.
for all s
then pf^xf^
>
els-
for all j
Proof.
some
(a) For
=
lis
>
e
G
j
It
K = p^^ + lj{x^^), which
is
positive since
K
lis'
>
Assume
smaU
0.
—
For k E is, consider the price pk =
e >
for some
can be seen that at the price vector {Pk,P-k)^ ^^^ corresponding
for
0.
J,', define
some
s.
WE link
>
flow would satisfy x^
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'^^)
a pure strategy OE.
(b) Since
lis
>
we have p^^x^^ >
Oi
assume without
for
loss of generality that
m
some
lm{x^^)
E
Is-
>
By Assumption
(otherwise,
j G Xs and assume to arrive at a contradiction that p'^^x^^
service provider s at the pure strategy
OE
we
=
2,
is
we can
are done). Let
The
0.
profit of
can be written as
OE^OE
^s^^s^v'it^ m
where Hj denotes the
for
profits
Note that
e
Hs
+ {K -
prices
-
Hs
Since lm.{x'^)
p^
UxT - e))(xr -
The
e)
+ e{K -
>
= {IM°J) - UxZ" 0, e
e))xZ^
+ eiUxZ"" -
1
utility
e)
-
following example shows that
Assumption
is
of
/,(e))).
can be chosen sufficiently small such that the above
is
strictly
an OE. Q.E.D.
2 cannot be dispensed with for part
lemma.
Example 4
links
l,{e)).
moved from link m to link j such that the flows
same at the new WE. Hence, the change in the profit is
positive, contradicting the fact that (p'^®,x'^^)
(b) of this
p^
units of flow are
other links remain the
.
= K —Imix^)
m and j. Let
and p^^ such that the new profit is
from links other than
some K. Consider changing the
I[s^Tls
'
Consider a three hnk network with two providers, where provider 1 owns
and 3 and provider 2 owns link 2. Let the total flow be d = 1 and the reservation
be i? = 1. Assume that the latency functions are given by
/l
(Xi)
=
0,
12{X2)=X2,
15
h^xz)
^
axz,
for
some a >
(2/3, 1/3,0)
why
see
Any price vector
0.
(pi,P2,P3)
a pure strategy OE, so ^3X3
is
two scalars
e
and
The corresponding
5
2/3 and (xi,X2,X3)
=
To
,
which
will
induce a
WE of (xi, X2, X3)
1 is
also playing a best response
=
is 111
+ 5 — e, 1/3 — 5,e)
4/9 — 6^ < 4/9, estab-
(2/3
—
.
and we have a pure strategy OE.
OE
next establish that, under an additional mild assumption, a pure strategy
never be at a point of non-differentiability of the latency functions.
Assumption
3 There exists some
differentiable for all
Lemma
5
some
i.
for
>
= (2/3-5,2/3-ae-5),
profit of provider 1 at this deviation
lishing that provider
We
6
to part (b) of the lemma.
is
(pi,P3)
will
with
an equihbrium, note that provider 2 is clearly playing a best response.
this allocation IIi = 4/9. We can represent any deviation of provider 1 by
this
Moreover, in
for
= (2/3, 1/3,6)
= contrary
i
E
s
E
S
such that k
is
real-valued and continuously
Ig
Let (p°^, a;°^) be an
Let Assumptions
1,
2
OE with
{pf^
and 3 hold. Then
min^
C(0 = C(0,
where l^ixf^) and l^ixf^) are the
right
and
+
< R and pf^xf^ >
lj{x°^)}
V^e2:,
left
derivatives of the function
li
at
xf^
respectively.
Since the proof of this
lemma is
long,
3 cannot be dispensed with in this
Example
5
tion utility he
it is
given in Appendix C. Note that Assumption
lemma. This
is
illustrated in the next example.
Consider a two link network. Let the total flow he d
R—
2.
Assume that the latency
^^^^)
=
'^^^)
—
and the reserva-
1
functions are given by
= |2(x-i)
otherwise.
can be verified that the vector (pf^,^^^) = (1, 1), with (xf^,x^^) = (1/2, 1/2) is a
pure strategy OE, and is at a point of non-differentiability for both latency functions.
It
We
next provide an explicit characterization of the
our efficiency analysis in Section
5.
The proof
Proposition 9 Let (p'^^,x<^^) be an
Assumptions 1,2, and 3 hold.
a)
OE
is
such that
Assume that min^ {pf^ + lj{xf^)} < R- Then,
'
xf^l'iix?^),
P"""
=
{
if
x?^l',{x?^)+ J^-^' ^^T
16
,
OE
prices,
which
essential in
is
given in Appendix D.
pf^xf^ >
for all s
^^(2:°^)
otherwise.
-
G
5 and
for
some
for
i
E
some
Is,
j
i
€ I. Let
we have
i Z„
(1^)
b) Assume that mirij {p°^
+
lj{xf^)}
=
R. Then, for
all s
G
5 and
i
els, we have
i^'yxf^i^?").
Moreover,
if
there exists
some
i
G
I
such that Ij
pr<xf^c(^r)+
the latency functions
If
/j
are
all
pendix D] immediately yields the following
Corollary
Let {p°^,x°^) be an
1
{i} for
^
some
s
G 5, then
(18)
'
1
real-valued and continuously differentiable, then
Karush-Kuhn- Tucker conditions
analysis of
~
(17)
OE
for oligopoly
problem [problem
Ap-
(82) in
result:
such that
pf^xf^ >
for
some
i
G I. Let As-
sumptions 1 and 2 hold. Assume also that k is real-valued and continuously differentiable
i. Then, for all s G <S and i E Is, we have
for all
X
P^
^
if
l',{xY^),
mm{R-k{x^^)
x?^i:ix?^)
,
+
J^'^ ^'\
\,
=
lj{xf^)
for
some
j
otherwise.
.'.(iV-K)
(19)
in the two link case with real- valued and continuously
and with minimum effective cost less than R, the OE
This corollary also implies that
differentiable latency functions
prices are
p?^-xf''{l[ix?^)
+
l',{xr))
(20)
as claimed in the Introduction.
5
Efficiency of Oligopoly Equilibria
This section contains our main
oligopoly equilibria.
We take as
results,
providing tight bounds on the inefficiency of
our measure of efficiency the ratio of the social surplus of
the equilibrium flow allocation to the social surplus of the social optimum, §{x'')/S{x^),
where x*
refers to the
monopoly
or the oligopoly
equihbrium
Eq.
[cf.
(8)].
Section 3
monopoly equilibrium is a social optimum.
congestion games with monopoly pricing, there is no efficiency loss. The
established that the flow allocation at a
Hence,
in
following example shows that this
Example
utility
he
is
not necessarily the case with oligopoly pricing.
6 Consider a two link network. Let the total flow he d
R—
1.
The
latency functions are given by
/i(x)
=
3
kix)
0,
17
=^
-X.
=
1
and the reservation
^ J„
example is x'^ = (1, 0). The unique ME (p^^, x^^)
As expected, the flow allocations at the social
is x'^^
(1,0) and
(1)1)optimum and the AiE are the same. Next consider a duopoly where each of these hnks
is owned by a different provider.
Using Corollary 1 and Lemma 3, it follows that the
The unique
optimum
social
=
flow allocation at the
h{x?^)
OE,
for this
=
p^'^^
x'-'^, satisfies
+ x?^[/;(xf^) +
Solving this together with
oligopoly equilibrium
is
+ X2^ —
xf ^
x*^^
l',{x^^)]
—
= h{xr) + x^Ux'^,'') +
1
(2/3,1/3).
/;(x?^)].
shows that the flow allocation at the unique
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 of the efficiency properties of the OE, the
next proposition proves that, as claimed in the 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 different 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^)
Proof. Combining the
ii(x?^)
OE
fact that
1
.
prices with the
+ xf ^(/;(xf^) +
where we use the
<
miUj
/^(x^^))
{pf^
WE conditions,
=
+ lj{xj^^)} <
conditions (7) to prove that a vector (xfjXj)
/i(xf )
Since l[{x°^)/x°^
We
on the
^
+
xf^;(xf)
=
>
R. Moreover,
is
hixl)
+ x^^(/;(x?^) +
/;(x°^)),
we can use optimality
a social optimum
if
and only
if
+ x%{xl).
/^(x^^)/x^^, the result foUows. Q.E.D.
next quantify the efficiency of oligopoly equilibria by providing a tight bound
efficiency loss in congestion
Section
/2(xf ^)
we have
4,
games with oligopoly
pricing.
As we have shown
such games do not always have a pure strategy OE. In the following, we
provide bounds on congestion games that have pure strategy equilibria.
efficiency properties of
mixed
We
in
first
next study
strateg}^ equilibria.
Pure Strategy Equilibria
5.1
We consider price competition games that have pure strategy equilibria
(this set includes,
games with linear latency functions, see Section 4). We
consider latency functions that satisfy Assumptions 1, 2, and 3. Let £/ denote the set of
but
is
substantially larger than,
latency functions for which the associated price competition
18
game has a pure
strategy
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
OE
efficiency metric at
some
x'^^
G OE{{li})
rii{k}.x"^)
as
= "^'^-"^
^'^
,
/'s
'
R
optimum given the latency
is the reservation
functions {kjiei and
In other words, our efficiency metric is the ratio of the social surplus in an
where x^
utility.
is
a social
equilibrium relative to the surplus in the social optimum.
we
[25],
an ohgopoly equilibrium, so we look
for a lower
inf
bound on
r!{{k},x°^).
inf
prove two lemmas, which reduce the set of latency functions that need to
first
be considered in bounding the
The next lemma
efficiency' metric.
oligopoly price characterization given in Proposition
Lemma
Then
6 Let
x'^^
is
We
Following the literature on
are interested in the worst performance in
the "price of anarchy", in particular
We
(21)
>
{p'-^^,x'^^)
be a pure strategy
OE
allows us to use the
9.
such that
p^^xf^ =
for all
i
e I.
a social optimum.
=
for all i 6 T. Assume that lj{x'^^) >
for some
and therefore pf^ = 0. Since Ijixf^) > 0, it follows
G I. This implies that xf^ >
by Lemma 2 that for all x E W{p), we have Xj = 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, the profit of the provider that owns link j is strictly higher at price vector
{^,P-f) than at p°^, 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
Proof.
first
show that li{xf^)
j
.
R-
k{xf^)
some gi. G dl^{xf^). Hence,
social optimum [cf. Eq. (7) with
for
The next lemma
^jg2:^j(xf)xf
Lemma
>
- xf ^p,, =
R,
V
i
G T,
x'^^ satisfies the sufficient optimality conditions for a
A'^
=
and the
R],
result follows.
Q.E.D.
allows us to assume without loss of generality that
and YlieJ^?^
= din
R Yliiez ^f ~
the subsequent analysis.
7 Let {U]rei £ C/- Assume that
^"More explicitly, Assumption 2 implies that
and hixf^) = 0, then I^ = {i}.
if
any
19
OE
[p'-'^ ,x'-'^)
associated with {/j}igi has
xf^ >
either
or
(i)
(ii)
'^i^2kixf)xf
Y^iei^?^ < d
Then every
=
RJ2iei^i
for
optimum
social
Xg,
some x°^ € 0^{{k}).
E OE{{li})
x'^^
some
^'-"^
is
=
a social optimum, implying that r/({/i},x'^^)
1.
Proof. Assume that Yliex^i(^i)^i ~ ^IZiei^f- Since x^ is a social optimum and
every x'^^ € OE{{li}) is a feasible solution to the social problem [problem (6)], we have
By
have xf ^ >
and i? - /i(xf ^) > pf ^ >
(where pf ^ is
at the OE) for all i. This combined with the preceding relation shows
the definition of a
the price of hnk
that x°-^
is
i
WE, we
a social optimum.
Assume next that YlieJ -^P"^ < ^ ^^^ some x°^ € OE{{li}). Let p*^^ 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 follows that ptxf^ > for all i G I. Hence, for all s e S, i{pf^)ieis,x^^)
is
an optimal solution of the problem
maximize((p,),^j^,:,)
^PiXj
ieXs
subject to
Pi
+
=
li{xi)
y
R,
i
elg,
iex
Substituting for (pi)i€is in the above,
maximize3;>o
we obtain
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
is a convex problem, using the optimality conditions, we obtain
Since this
interval.
\
VieZ„Vse5,
R-k{xf^)-x°''gi^=0,
where g^ G dk{xf^). By Eq.
This
lemma
that
implies that in finding a lower
restrict ourselves,
'^ieT^i{^i)^i
(7), it follows
<
x'^^ G OE{{lj}).
without
is
a social optimum.
bound on the
^°r
some
social
the following lemma,
,
also
we can
{U} G Cj such that
optimum x^ and ^^i^j^?^ ~
we can
20
Q.E.D.
efficiency metric,
loss of generality, to latency functions
R-Yliei^i
By
x°^
assume that X^jgjxf
=
'^
d.
^^^ ^^^
Lemma
be an
8 For a set of latency functions {k}iei,
and x^ be a social optimum. Then
Assumption
let
1
hold. Let {p^^,x'^^)
OE
E^>?"<E-?Proof. Assume to arrive at a contradiction that J2iei^i
-^ J2iex^i- '^^^^^ implies
we would have
for
some
We
also
have
lj{x'j'^)
>
lj{xj).
(Otherwise,
>
Xj
j.
xf^
=
—
lj{Xj)
I'Axj)
0, which yields a contradiction by the optimality conditions (7) and
the fact that X^^gi^f < d). Using the optimality conditions (2) and (7), we obtain
that
R - l,{xf)
for
some
G
gi^
xf^l~{Xj'^)
Combining the preceding with
9), we see that
dlj{x^).
(cf.
-vT>R- hi^') - ^j9i„
>
lj{x'^^)
lj{Xj)
and pf^
>
Proposition
xf-l-ixf^)<x^g,,
contradicting
xf^ >
Two
Links
5.1.1
We
first
Xj
and completing the proof. Q.E.D.
consider a parallel link network with two links
The next theorem provides a
tight lower
Starting with the two-link network
is
owned by two
service providers.
bound
of 5/6 on r2{{li},x^^)
useful
two reasons:
first,
[cf.
Eq.
(21)].
the two-link network
avoids the additional layer of optimization over the allocation of links to service providers
in characterizing the
bound on
general case by reducing
it
inefficiency;
and second, we
prove the result for the
will
to the proof of the two-link case.
Although the details of the proof of the theorem are involved, the structure is straightThe problem of finding a lower bound on r2{{li}, x^^) is an infinite-dimensional
forward.
problem, since the minimization
is
over latency functions.
The proof
first
lower-bounds
the infinite-dimensional problem by the optimal value of a finite-dimensional optimization problem using the relations between the flows at social
and convexity
equilibrium,
then shows that the solution
will involve one
and the price characterization
reduces the problem of characterizing the bound on inefficiency
of the latency functions. It
of the finks having zero latency.
Finally, using this fact
from Proposition 9, it
to a simple minimization problem, with optimal value 5/6.
is
optimum and
An
intuition for this value
provided below.
In the following,
we assume without
loss of generality that
latency functions in C2 satisfy Assumptions
Theorem
provider.
1 Consider a
two
link
1, 2,
and
d
=
1.
Also recall that
3.
network where each
link
is
owned by a
different
Then
r2({/z},x°^)
>
I
V
{/a,=i,2
21
G £2, x°^ G ag({U),
(22)
and the bound
tight,
is
i.e.,
there exists {li}i=io G C2 and
x'-'^
G OE{{li}) that attains
the lower bound in Eq. (22).
Proof. The proof follows a number of
Step
We
1:
steps:
are interested in finding a lower
inf
bound
for the
problem
r2i{k},xO^).
inf
(23)
Given {/J e £2, let x*^^ E OE{{li}) and let x^ be a social optimum. By Lemmas 7
and 8, we can assume that X]i=i^f^ = Y^'i=i^i — !• This implies that there exists
some i such that xf^ < xf. Since the problem is symmetric, we can restrict ourselves
to {li} € £2 such that xf^ < xf, i.e., we restrict ourselves to {k} G Co such that
xf ^ < xf — e for some e > 0. We claim
inf
r2({/a,x°^)>
inf
inf r,^f(6),
(24)
where we define problem (E') as
...
OF^
r2,j(e)=
.
mmimize,s,
„S),>o
y?
,
'
—
'1 i/l
L, ;'>o
^l
R - hyf^ - hy?^
T5-5c
^_ g
(E
)
''2i/2
yQ^>Q
—
^2
subject to
l'f<yi{l'z)\
J
k<y?''l'i,
il
Zf
1
= l,2,
= 1,2,
(25)
(26)
+ y^{ily = i'i+yf{ify,
+ 2/f(/f)'<i^,
(27)
(28)
2
+
^i
(30)
^i(y2''''-yf)<^f,
'
J/2°^>yf
+
e,
(31)
2
E yrOS =
+
i,
(32)
{Ohgopoly Equilibrium Constraintsjt,
i
=
1,2.
Problem (E^) can be viewed as a finite dimensional problem that captures the equilibrium and the social optimum characteristics of the infinite dimensional problem given in
Eq. (23). This implies that instead of optimizing over the entire function k, we optimize
over the possible values of /j(-) and dli{-) at the equilibrium and the social optimum,
which we denote by
ues of
gi^
G
dli{yf)].
li,l'i,lf
The
,
{if)' [i.e., (if)' is
a variable that represents
the necessary optimality conditions for a social
ditions (25)
and
all
possible val-
constraints of the problem guarantee that these values satisfy
(26) capture the convexity^
22
optimum and an OE.
assumption on
li{-)
by
In particular, conrelating the values
lij'i
and
if,
=
{ify [note that the assumption /^(O)
is
essential here]. Conditions (27)
from the optimality conditions for the social optimum. Condition
follows by the convexity of the function /i(-), which implies the relation
and
(28) follow
(30)
/i(xf)>;i(xf^)+g,,(a:f-xf^),
where
G dli{xf^).
gi^
Using the relation Yl'i=i^?^
—
X]?=i ^f
—
1'
'^^ write
the
preceding constraint as
Zi(xf)>/i(xf^)+5jx°^-2:f),
which turns out to be more convenient in the analysis of the optimality conditions (see
Step 3). Similarly, condition (31) follows by the facts that we are considering {li} such
and Yli=i ^?^ — X^Li ^f ~ ^- Note that we use
that xf ^ < xf — € for some e >
the relaxed constraint ^j^j xf < 1 in the optimization problem (which provides a lower
bound to the original problem) since this makes the analysis of the optimality conditions
easier.
Finally, the last set of constraints are the necessary conditions for a pure strategy
These are written separately for i = 1,2, for the two cases characterized
9, giving us two bounds, which we will show to be equal.
More exphcitly, the Oligopoly Equihbrium constraints are given by:
For
t
—
1:
[corresponding to a lower bound for pure strategy OE,
in
OE.
Proposition
{p'''^ ,y^^),
with
mmj{pf^ + l,{yf^)}<R],
h+y?''[l[
/i
where
/;
For
=
t
=
2;
min,{pf^
/^yf^),
^^
-
+
yP''[/'i
+
+
l'2{y?'') [cf.
l2]
=
l2
/2]
<
R,
Eq.
[corresponding to a lower
r?f{e)
l[
^
=
lt{y?^) and
we have a
Therefore, the
optimum
(16)].
bound
for
pure strategy OE,
{p'^^ ,y'^^),
with
l'^
=
^9~(y°^)
>
y?^l'2,
<
y?^[i[
[cf.
(34)
+
i2\,
Eqs. (17), (18)].
We
will
show
in
Step 4 that
riiie).
Note that given any
that
(33)
+ l,{yf^)}^R],
R-h
R-h
where
+ y?^[l[+l2i
feasible solution of
feasible solution for
optimum value
problem
problem
(E'^)
(23), there exists
some
e
>
such
with the same objective function value.
of problem inf^^orofie)
is
indeed a lower bound on the
value of problem (23).
Step 2: Let (;f,yf),=i,2 satisfy Eqs. (25)-(29).
^f 2/f
+
iM
23
<
We
R-
show that
(35)
Using Eqs.
(27), (28),
and
(29),
ifyf
If
+
{yf?ilfy
{yl?illy
we have using Eq.
Next,
straints
>
+
ily^
lf=0
let {li,yf^)i=i^2 satisfy
[i.e.,
+
{y!r{if)'
+ {ymi!y<R-
follows. If {yffilf)' + {yDHlfY
again showing the result.
then the result
0,
(25) that
we obtain
for all
i,
=
0,
then
Eq. (32) and one of the Oligopoly Equilibrium con-
Eqs. (33) or (34)]. Using a similar argument,
+
ky^'^
ky^''
we can show that
<R.
(36)
Step 3: Let {if ,(Jf)' ,TiJ[,y? ,yf^) denote an optimal solution of problem (E).
show that
[f
=
for
i
=
We
1, 2.
assign the Lagrange multipliers //f > 0,A-^,7'^ >
to Eqs. (25), (27), (28),
respectively, and 0"^ >
to Eq. (29). Using the first order optimality conditions, we
We
obtain
y-f
'
^rf U "rft? +
[R-lfyS-llySf
^.^
/^f
+
=0 if/I>0
A^
>
-A^M +
A^yf
=0
if(rf)'>0
>
if
-/ifyf-A^yf + 7^yf
^f
^^f4lV^M?
{R-lfyf-Ily
2^2
A^f (^T)'
-
if
=0
if
>
if
A^(/T)'
=
{ify
/f
0,
(38)
0,
(/"f)'
>
{ify
-
+ 7¥f +
)'
=
(37)
(39)
0,
^^
=0 ifyf>0
(40)
.
>
We
that yf
first
>
show that [f = 0. If yf =
and (/|)' > 0. By Eq. (38),
in this case Eq. (37)
cannot be equal to
We
that yf
next show that if
>
and
this in Eq. (40)
(if)'
>
0.
and using
—
0.
By
^-^
If
yf
—
=
0.
—
=
We
Assume to arrive at a contradiction that it
we have /if + A'^ < 0, which is a contradiction
0.
or (if)'
=
0,
we
/f
=
0.
are done by Eq. (25).
Assume
Eq. (39), this imphes that -fif-X^+jf = 0. Substituting
> together with Eq. (36), we obtain if = 0.
Step 4: Since [f = 0, in view of Eq. (30), we have [i
= 0. Using in addition if
e, by Eq. (30), we have
yf
+
S
yf
or ([f )'
0, we are done by Eq. (25). Assume
claim that
this implies that A"'
/if > 0.
Using Step 2 and the fact that y| > 0,
and shows that Eq. (37) is strictly positive. This establishes that
is.
if
l'^
24
= 0. Moreover, since y2^ >
— 0, we see that for i = 1, 2
and
all e
>
0,
r?f{e)
>
minimize
subject to
^^
-
1
,,4>o
I2
<
E
(41)
2/2*^^21
=
yf ^
1,
that satisfies Eqs.
and l[ =
which follows because any vector iy^^,k,Q with /i =
(33) or (34) is a feasible solution to the above problem. It is straightforward to show that
the optimal solution of this problem is (^2, [2, y?^, y2^) — (f ^) fi 3)) ^"d therefore it
>
follows that r^f{e)
=
5/6
for all
=
f
1,2 and
inf
all e
>
0.
By
Eq. (24), this implies that
r2({/J,x«^)>|.
inf
We next show that this bound is tight. Consider the latency functions li{x) = 0, and
As shown in Example 6, the corresponding OE flow vector is x'~'^ — (31 !)>
h{x) —
and the social optimum is x^ = (1,0). Hence, the efficiency metric for these latency
functions is r2{{li},x'-'^) = 5/6, thus showing that
f-'C-
min
min
r2({/,},x'^^)
=
5
Q.E.D.
It is
instructive to briefly consider the intuition underlying the 5/6 bound.
The
maximized when as much of the traffic as possible goes tlirough route 2
and when the latency on route 2 is as high as possible, i.e., when X2l2{x2) is maximized.
But these two requirements are in conflict in the sense that when the latency on route 2 is
high, there will be less traffic on that route, because in a WE we must have p\ + /i(2;i) =
P2 + '2(^2). Moreover, with zero latency on route 1, equilibrium prices will satisfy
Pi = 2:1/2(2^2) and P2 = X2l'2{x2)- So the problem is to maximize X2l2{x2) while satisfying
Xil'2{x2) — X2^2(^2) + ^2(2^2)- This Constraint immediately implies that Xj > X2, and
since ^2(2:2) < R, the efficiency loss can never exceed 1/2. But the bound is in fact
efficiency loss
much
is
tighter than this.
Since Xi
harmful
for the objective, since
Xi/2(x2)
=
+
X2^2(^'2)
happen when
I2
is
We
it
X2, convexity of
I2,
i.e.,
a greater
/j
given
I2,
is
tends to increase xi (as the inspection of the condition
This reasoning suggests that the worst case will
shows).
hnear, which
efficiency loss of 1/6
5.1.2
^2(3^2)
>
is
and the bound
exactly the case in our
Example
6,
leading to the
of 5/6.
Multiple Links
next consider the general case where we have a parallel link network with / links
and S service providers, and provider
s
owns a
25
set of links Is
C
I.
It
can be seen by
augmenting a two
link
network with links that have latency functions
X
if
^
=
0,
otherwise,
cx)
that the lower bound in the general network case can be no higher than 5/6. However,
this
is
OE, the
a degenerate example in the sense that at the
latency functions given above are equal to
network which has positive flows on
Example
We
0.
at the
all links
flows of the links with
next give an example of an / link
OE
and an
efficiency metric of 5/6.
7 Consider an / link network where each link is owned by a different provider.
= 1 and the reservation utility be i? = 1. The latency functions
Let the total flow be d
are given
by
h{x)
The unique
social
=
optimum
k{x)^^{I-l)x,
0,
for this
flow allocation at the unique
OE
is x"^
=
[1, 0,
.
.
.
,
0].
It
can be seen that the
is
x°^ =
Hence, the efficiency metric
example
1^2,...,!.
1
1
_3'3(/-l)'''''3(/-l)
for this
example
is
The next theorem generalizes Theorem 1 to a parallel link network with
The new feature here is not only the existence of more than two links, but
/
>
2 links.
also the fact
that to find the worst-case bound, we have to optimize over the allocation of links across
service providers.
program
The
strategy of the proof
is
again to reduce the infinite-dimensional
to a finite-dimensional optimization problem,
and then show that
the case in
Theorem
Theorem
2 Consider a general parallel fink network with / links and
where provider
s
owns a
and the bound is tight,
bound in Eq.
set of links I^
>
I
C
T.
there exists {li\iex S
i.e.,
S service providers,
Then
V {k},^x e
£;,
x°^ e ag({/J),
l^i ^iid
x^^ G 0S({/,})
(42)
that attains
(57).
Proof. The proof again follows a number
Ste'p 1:
reduces to
1.
r/({/a,x°^)
the lower
it
of steps:
Consider the problem
inf
inf
r;({/J,a;°^).
^'•>^^'xOEeaB({;.})
26
(43)
Given {k} 6 Cj, let x^^ G OE{{k}) and let x^ be a social optimum. By Lemmas 7 and
— 1- Hence there
8, we can assume without loss of generality that Yliei^?^ ~ IZi^-r^f
JiEl
^
< xf Without loss of any generality, we restrict ourselves
exists some i such that a;f
.
£
to the set of latency functions {h}iei
Proposition
1, it
the
is
<
xf. Similar to the proof of
r,({/J,x°^)>infr,°f(e),
inf
inf
where rff{e)
we denote by
such that xf^
-C/
can be seen that Problem (43) can be lower bounded by
optimum value
of the following finite dimensional problem,
which
(E^):
OB
r?f{e)
= mmmuze
(E^)
(f,(lf)'>0
l,.l'.>0
yf,vf^>o
Is CI
subject to
If
if
<
yf(ify,
+
yf{ify
(44)
1,
=
if
+
..SnS\i
2,
yfii^)'
yf<i
(45)
(46)
1^1
..OE
1,
Xs
+
The new
=
{1}
for
some
s
ii
if
—
(47)
0,
i-
{Oligopoly Equilibrium Constraints}^,
feature relative to the two link case
is
I/s
the presence of
1,2.
as choice variables
to allow a choice over possible distribution of links across service providers (with the
constraint IJ^Is
— ^
left implicit).
again written separately for
In addition,
by
Lemma
t
=
The
oligopoly equihbrium constraints, which are
1,2 for the two cases in Proposition
we have added constraint (47) to impose Assumption 2
{If, {If
y,k,lyf,y?^) be an optimal
.
.
.
.
also T^
(recall
X^'s.
that xf ^ >
solution of the preceding problem.
Note that the constraints that involve {if, {lfy,yf) ioi i — 2,
have the same structure as in problem (E'^). Therefore, by the
to show /f =
in Step 3 of the proof of Proposition 1, one can
each i = 2, ... ,1 Similarly, one can extend the same argument
Proposition 1 to show that if = 0.
Step 3:
depend on
4).
Step 2: Let
and
9,
,
I are decoupled and
same argument used
for
show that If —
given in the proof of
Since [f = 0, it follows that h =
and I[ =
[d. Eqs. (45) and (46)],
{1}. Therefore, using the price characterization from Proposition 9, the
—
27
structure of the problem simplifies to
minimize
rff>
,„,;>o, .=2
^>0,
i
/
=l
1
- ^^=y^i
subject to
k
<
k
+ yf%<R,
^^g^
R
/
..OEii
l'^.
Vi
1
I
=
=
2,..., I,
(49)
2,..., I,
(50)
„.OE
->R,
^'
(51)
E
where we have
The
first set
constraints (given
Let
4-'
0, for
f
l[
=
0,
see the
{{li, l[)i=2,...j
^
OE
2,
that
.
/ij
.
Assume
I.
,
.
=
.
,
.
.
/.
price characterization in Proposition 9).
{y?^)i=i,...,i)
denote an optimal solution of the preceding
fii
>
the contrary,
i.e.,
<
l^
>
0,Xi
9 consecutively to the constraints of the problem.
=
2,
due to the convexity assumptions on the li. Similar
the second set of constraints are due to the ohgopoly equilibrium
problem. Assign the Lagrange multiphers
i
=
of constraints are
to the two link case,
Step
=
also used the fact that if
We
yf^l'^ for
0, i
=
2,
.
.
,
.
I,
and 7
>
and
will show that li = y^^l'i for all
some i = 2, ... ,1 This imphes
.
Using the optimality conditions, we have
0.
^
+
=
if
y°^ >
>
if
yf
=0
if
>
if
^
.
(52)
=
OB
R
-| +
+
A,/:.
\i
/"
r,
>
-
(53)
0,
+ e -0 ifyf^>0
>0
ifyf^
=
(54)
0.
By feasibility [cf. Eq. (51)], we have yf^ > 0. Moreover, by our assumption [Fj < yf^l'^,
we have yf ^ > and /' > 0. Eq. (52) imphes that ^ > 0. We also have from Eq. (53)
that Aj > y^^ jR, which when substituted in Eq. (54) yields a contradiction in view of
>
0.
Hence, for alH
constraint (50)
is
=
2,
...,/,
we have
li
=
y^^l'i-
It is
also straightforward to see that
binding at the optimal solution (otherwise
decrease the objective function value), which implies that
alH
=
2,
.
.
,
.
/.
By
using the transforination of variables
28
li
=
it
would be possible to
and yf^/^ = i?/2 for
i?/2
can be seen that the optimal value of problem (48)
of the following problem:
it
the
is
same
as the optimal value
lyOE
minimize
1
i,i'>o
subject to
I
I
;r—
<
y'^^l',
+
y°^/'
y?^l'
which
all €
is
>
identical to
problem
>
(41) in the two-link case,
<
R,
R.
showing that
for
alH
=
1,
2
and
0,
rffi^)
>
inf
ri{{li},x^^)>^.
I-
Hence, we have
inf
Finally,
Example
7
shows that the preceding bound
min
is
rj({li},x^^)
rnin
tight,
=
i.e.,
5
-
Q.E.D.
A
notable feature of Example 7 and this theorem
inefficiency
feature as
5.2
is
independent of the number of links
much
inefficiency as
Mixed Strategy
As we illustrated
(cf. Example 3).
/.
is
that the (tight) lower
Thus
bound on
arbitrarily large networks can
smaU networks. ^^
Equilibria
in Section 4,
pure strategy oligopoly equilibrium
Nevertheless, as
shown
may
fail
to exist
games always have a
we discuss the efficiency properties of mixed
in Proposition 8, such
mixed strategy equilibrium. In this section,
OE.
Although there has been much less interest in the efficiency properties of mixed
strategy equilibria, two different types of efficiency metrics present themselves as natural
strategy
candidates.
The
first
considers the worst realization of the strategies, while the second
focuses on average inefficiency across different realizations of
mixed
strategies.
We
refer
'^^This result superficially contrasts with theorems in the economics literature that large oligopolistic
markets approach competitive behavior (e.g., [41], [17], [35], [55], [56]). These theorems do not consider
arbitrary large markets, but replicas of a given market structure. In our model as well, if we take a
given network and replicate it n times (i.e., increase d to nd and the number of service providers by n),
then as n — oo, the efficiency metric tends to 1. In fact, in Example 6, replicating the network once,
i.e., n = 2, achieves full efficiency, because of Bertrand competition between two oligopolists with zero
»
latencies.
29
to the
first
second as
metric as worst-realization metric, and denote
average metric, and denote
tlie
Given a
set of latency functions {li}iei
some
equilibria. For
in
Example
3
[in
€ OM({/j}),
fi
We
=
{x
by rf'dli}), and to the
it
rf{{li}).
OM{{li}) denote the
set of
denote the support of
Mi —
X £ W{p),
\
let
,
let Mi{fi)
particular, recall that
OEm{{li},t^)
by
it
{p
\
fi'^ip)
some p
for
>
s.t.
mixed strategy
as defined before
/Xj
0}]. Further, let
pi
G Mi^p,)
for all i}.
define the worst-realization efficiency metric as
fr({Za)=.
where
r/ is
inf
Jnf
inf
r,({?a,x^^),
given by Eq. (21).
Similarly, the average efficiency metric
r/({/J)=
inf
we show
defined as
/..•
inf
{i,}ec, ti.eOM({i,}]
In the next example,
is
[ rj{{h},x''^
{p))dfi,
df^s-
J
J
that the worst-realization efficiency metric for games
with no pure strategy equilibrium can be arbitrarily low.
3 (continued) Consider the prices pi = R and p2 = R{1 - 5) that satisfy
for the unique mixed strategy equilibrium given in Example 3 as e —* 0. The
Example
Pi
€ Mi
WE at these prices
is
given by
x''^
and the worst-realization
= il-
efficiency metric
is
rr'({/a)
which as 5 —>
On
1
goes to
6,5),
=
1
-
<5^
0.
the other hand, as
e
-^
0,
the average efficiency metric, r/ ({/,})
given by
rR
pR
^~f{{h])=
is
r {pi,p2) dpi
/
/
X dp2,
J(\-5)rJ{\-6)R
the inefficiency at the price vector {pi,P2) at the unique mixed strategy
characterized above. Therefore;
where r {pi,P2)
OE
is
1
r{p.,P2)
=
^
Pi < P2
ifp,>p2
if
1-fcM
'
and thus,
rtm) -
1
- /'
/''
(l-6)R J (\-6)R
30
^-^^^dpi
-n-
X dp2
Thus to
we have
we need
calculate rf{{l^}),
A =
(Pi
R
compute the
to
-P2)dfii X
last integral.
Denoting
by A,
this
dfl2
{l-5)R Jp2
R
Pi
6_
P2dH\ X
Pldfl2 X di^i
R
d/i2
{l-6)RJp^
IJ(1-6)R J{1-S)R
S_
R
Now
recall that
l{l-S)R
7(l-<5)i?
yUi
\^
\"
has an atom equal to
A = ^ R{l-5)+
^
L
Jn-5)R
J(\-5)R
J
/
^P^
"P^
1
—
J at R, so
1-5)
I
J(i-6)R
/(l-5)i?
i?\
m
pi
\(>
= (1 - 5)2 - (1 - ^) + (1 - 5)[\nR= -(1-5) 5 -(1-5) In (1-
dpi
- R{1 -
5)
P\
ln((l
-
5)/?)]
(5)
can be calculated that
It
A
reaches a
maximum
Therefore, in this example, ff{{li}) reaches 0.84
We
5/6).
of approximately 0.16 for 5
?»
5/6
conjecture, but are unable to prove, that 5/6
is
also a lower
average efficiency metric, ff{{li}), in mixed strategy OE. This
question.
In this section,
flow.
To
is left
as
bound
0.8.
than
the
for
an open research
^^
Bound
6
w
(in fact, slightly greater
for Positive
we
Latency at Zero Flow
—
relax the assumption li{0)
simplify the exposition in this section,
latency functions, but as our previous analysis
0,
and allow positive latency
at zero
we focus on continuously differentiable
indicates, the main result. Theorem 3,
holds for general convex latency functions.
Assumption 4
We
first
i E I, the latency function
and nondecreasing.
For each
differentiable, convex,
li
:
[0,
oo)
h-> [0,
oo)
is
continuously
provide an equilibrium price characterization, which generalizes Corollary
Proposition 11
Let {p"^,x"'^) be an
OE
such that
p^x^^ >
for
some
i
e
1.
J.
Define the index set
M={jEl\pf^ + k{xn
'''As
<
pf"^
+
/,(0)}.
pointed out by one of our anonymous referees, the intuition provided following
(55)
Theorem
1
suggests that even in the case of a mixed strategy equilibrium, the average efficiency metric should
not
fall
below 1/2. Nevertheless, proving
conditions for a mixed strategy
OE
this conjecture has not
are considerably
31
been possible because the equilibrium
for a pure strategy only.
more involved than those
Let Assumptions 2 and 4 hold. Then, for
r
all s
5 and
G
i
E Is and
xf ^/^(xp^),
if
min{i?-/i(xf^)
,
xf^/:.(xf^)
+
L
„^^'^^-"^
i
^ Af, we have
I'^ixf^)
=
for
some
otherwise.
(56)
The proof
theorem follows immediately from the proof of Corollary 1. In
all latencies where xf ^ — 0, so that any i E
can be
discarded when considering the individual optimization problem of each service provider.
In what follows, let L*i denote the set of latency functions for which the associated price
competition game has a pure strategy OE and the individual /j's satisfy Assumptions 2
and 4.
particular,
of this
N
is
M
the set of
Theorem 3 Consider a general parallel link network with /
where provider s owns a set of links X^ C X. Then
r/({;a, x°^)
and the bound
is
tight,
> 2^2 -
i.e.,
V {/Jiei e
2,
there exists {/j}jgi G
C,\
links
and S service providers,
x°^ € Ot{{k)),
-C*,
(57)
and x'^^ G C)E{{1^') that attains
the lower bound in Eq. (57).
Proof. The proof follows those of Theorems 1 and 2 closely. Once again, the problem
(23) is lower-bounded by a modified version of the finite dimensional problem (E}) (see
which we introduce additional variables l\ > 0, which
Zj(-) at 0. Using the convexity of the latency
functions, we replace constraint (44) by
the proof of
Theorem
2),
in
represent the value of the latency function,
if<yfiify +
Following the same line of argument,
bounded below by a problem identical
it
i^.
can be seen that problem (23) can further be
is replaced by
to (48) except that constraint (49)
Using a similar transformation, this problem can be seen to be equivalent to
mmimize
i,i',(0>o
;r—
1
R
yOB'yOE^O
subject to
/
<
+
1°,
I
+ y°^l' <
R,
y^^l'
y?^i'
—
R.
solution ofthis problem is (r,r,/"°,yp^,y°^) = (2-^2, V2, 3-2\/2, %/2/2,
and
the corresponding optimal value is 2\/2 — 2.
\/2/2)
The optimal
1
>
32
We
next show that this bound
each link
utility
is
he
owned
R—
/i(x)
=
1.
a different provider. Let the total flow he d
bj'
The latency
0,
The corresponding
and
OE
x°^ =
functions
is
=
flow vector
'%/2
optimum
social
r/({/i},x'^^)
- l)V2x +
1
and the
reser\'ation
1
[3
-
\/2\
2,...,/.
2
V
/
1
1
'"'/-l
x^
min
min
Vf =
2 V2),
is
/
1
'/-
{I
= (1,0). Hence, the efficiency
= 2\/2 — 2. thus showing that
is
—
functions are given by
li{x)
2
and the
Consider an / hnk parallel network where
tight.
is
metric for these latency
^ 2^2 -
ri{{k},x°^)
2.
Q.E.D.
It is
interesting to note that 2\/2
sumption
=
li{0)
OE. In terms
—
w
2
.828
<
Therefore, relaxing the as-
5/6.
has a small effect on the worst-case performance of a pure strategy
we provided
of the intuition
positive allows us to increase
I2
for
Theorem
slightly for a given
/g)
1,
the fact that /2(0) can be
leading to a small deterioration
in performance.
Conclusions
7
we presented an analysis of competition in congested networks. We esnumber of results. First, despite the potential inefficiencies of flow-routing
In this paper,
tablished a
by a monopolist always achieves the social optimum. Second, and in contrast to the monopoly result, oligopoly equilibria where multiple service
providers compete are typically inefficient. Third and most importantly, when latency at
zero flow is zero, there is a tight bound of 5/6 on inefficiency in pure strategy oligopoly
equilibria. When latency at zero flow can be positive, the bound is shghtly lower at
2\/2 — 2 « .828. These bounds apply even for arbitrarily large parallel link networks.
without
prices, price-setting
A number
•
of concluding
comments
are useful:
Our motivating example has been the
flow of information in a communication net-
work, but our results apply equally to
traffic
assignment problems and oligopoly
in product markets with negative externalities, congestion or snob effects (as originally suggested
•
Our
by Veblen
[53]).
analysis has been quite general, in particular, allowing for constant latencies
and capacity
specialize the
constraints.
Some
of the analysis simplifies considerably
when we
network to increasing and real-valued (non-capacity constrained)
latencies.
33
On
the other hand, our analysis has been simphfied by our focus on parallel hnk
networks.
We
have started extending this analysis in ongoing work
gies consisting of parallel-serial structure.
rules out
many
[2]
for topolo-
This parallel-serial topology, however,
interesting cases, including those that could potentially lead to
Braess' paradox, and the analysis for
more general topologies
is
an open area
for
future research.
One
simplifying feature of our analysis
is
the assumption that users are "homo-
geneous" in the sense that the same reservation
is
utility,
R, applies to
all users.
It
possible to conduct a similar analysis with elastic and heterogeneous users (or
but this raises a number of new and exciting challenges. For example,
monopoly or oligopoly providers might want to use non-hnear pricing (designed
as a mechanism subject to incentive compatibility constraints of different types of
traffic),
users, e.g., [58]). This is an important research area for understanding equilibria in
communication networks, where users often have heterogeneous quality of service
requirements.
•
While we have established that worst-reahzation efficiency metric in mixed strategy
ohgopoly equilibria can be arbitrarily low, a bound for average efficiency metric is
an open research question.
34
Appendix A: Proof
8
If
{p'"'^,x^'^)
Assume
an ME, then
is
of Proposition 4
SPE by
an
it is
Let {p'^"^,x'''^) be an SPE.
definition.
some p >
to arrive at a contradiction that there exists
and x G W{p) such
that
n(p^^,a:^^)<n(p,.x).
If
W{p)
is
(58)
a singleton, we immediately obtain a contradiction. Assume that W{p)
not
is
a singleton and X^j^i^;, = Yli^j^i f°^ ^^^ x, x 6 W{j)). By Lemma 2, it follows that
n(p, i) = n(p, x) for all X G l-^Cp), which contradicts the fact that (p^^,x^^^) is an
SPE.
Assume
finally that
W[p)
<
^Xj
/
we have
p^
[cf.
Eq.
[cf.
i
ei.
(5)].
Eq.
To
(2)]
= R
for all
i
00
for
implies that Xj
=
be, for all
We
some
show that given
(^
(59)
G I, where
VK(p) with x,
see this, note that since Yliei^i
hold with A
=
If bci
x € W{p).
x,
iex
X={z6X|3x, xG
X
some
for
2^2;,,
iex
For this case,
not a singleton and
is
=
'^ ^^ ^^^^
7^ x,},
^^ optimality conditions for
Assume that p < R. By Lemma 2, li{xi) = for
El, we get a contradiction by Eq. (2). Otherwise, Eq.
i
>
i
0.
G
0,
/.
Since Xj
there exists
=
x^ for all
some
e
U{p',x')>U{p,x)-6,
>
^ 2", this contradicts Eq.
such that
i
yx'eWip'),
all
(2)
(59).
(60)
where
"={«-. iei
The preceding
relation together with Eq.
an SPE, thus establishing our claim.
We
first
(«"
(58) contradicts the fact that {p'^'^^x^'^) is
show that
Y.^l>Y.ir.
iex
Assume
(62)
iex
to arrive at a contradiction that
E^^<E^^iei
(63)
iex
This implies that there exists some j & 2 such that x^- < Xj (which also implies that
xj < bcj). We use the
optimahty conditions [Eq. (2)] for x and x' to obtain the
WE
following:
•
There
exists
some A >
such that for some
i
G
2",
R-k{xi)-pi = 0>~X,
35
where we used the
i
i
E
[cf.
Eq.
=
facts that li{xi)
(58)].
=
Since A
0,
we
pi
,
have, for
<0
>0
R-li{xi)-p,
There
•
exists
some
A*^
>
=R
[cf.
all
Lemma
2]
and
>
Xi
for
^ I,
i
•
ifx^<6c,,
if.T,
=
(64)
6c,.
such that
e-lj{x'j)<y,
(since xj
<
Xj
and
pj
—R—
R-
If A^
=
0,
e),
i
^
T.
is
and
li{xl)
for all
a contradiction.
If A*^
>
0,
>
e
i
lj{xj)
>
=
(65)
^ I,
<X'
> A^
-Pi
then by Eq. (65) and the fact that
Ijix^j)
which
some
if
if
—
xl
-
0,
xl
>
0.
(66)
(Lemma
2),
we obtain
lj{xj),
then '}2iei^i
~
ifi
i^i
^-
Assume
first
that xl
<
Xi for all
Then
iei
which yields a contradiction by Eq.
Eqs. (64) and (66), we have
(63).
«£X
Assume next that
i?-/fc(x^)-p,
x],
>
Xk for some k
^T. By
>A^
R-Ik[xk)-Pk<^,
which together imphes that
lk{xk)
>
hi^l), yielding a contradiction and proving Eq.
(62).
Since W{p) is an upper semicontinuous correspondence and the i*'' component of
W{p) is uniquely defined for all i ^ X, it follows that Xi(-) is continuous at p for all
i ^ T. Together with Eq. (5), this implies that
> Y^piXi + '^{R-e)xi + ^p^{xl-Xi)
lei
iei
i(^j
lei
where the
last inequality holds for sufficiently small
the proof.
Q.E.D.
36
e,
establishing (60), and completing
Appendix B: Proof
9
For
all
e I,
i
let li{x)
—
a^x. Define the set
=
lo
Let
of Proposition 7
{i
&2
ai
\
—
0}.
denote the cardinality of set Iq. There are two cases to consider:
Iq
>
Assume that there exist i, j € Tq such that i Els and j G Is' for some
for aU i G 1q
s ^ s' e S. Then it can be seen that a vector {p°^,x'^^) vnth pf-^ =
and x*^-^ G I'r(p'^^) is an OE. Assume next that for all i G Tq, we have i G 2^ for some
s G 5. Then, we can assume without loss of generality that provider s owns a single
Case
link
1: /q
with
i'
Case
2:
=
aj/
h<l:
5;
and consider the case
/q
Let Bs{p?.f) be the set of
=
1.
pf^ such that
max
(p°^,x°^) Garg
Ps>0
a:€W(ps,p?f
=
Let S(p°^)
B{p^^)
is
By
\Bs{v°f)]seS-
Theorem
the
Vp^x^.
':—
(67)
iG^s
)
Maximum ([6]), it follows that
We next show that it is convex-
of the
an upper semicontinuous correspondence.
valued.
Lemma
9 For
all s
G 5 and p5f >
0,
the set Bs(p°f )
is
a convex
set.
Proof. For some s G 5 and p°^ > 0, let ps G Bs{p°f) and p, G Bs{p?.f) such that (ps, x)
and {ps,x) are optimal solutions of problem (67). Denote x^ = [xjjigi^ and x^ = [xj],gj^.
If pjxs = pjxs = 0, then the vector jp + (1 — 7)p G Bs{p?.f) for all 7 G [0, 1], and we
are done.
Assume
= pjxs > 0. We will show that Ps = Ps- Using a similar argument
Lemma 4(b), it can be seen that Xj > for all i E Is and x^ >
We claim that Pi = p for all G X, and Pi — p for all E Is- This can
that pjxs
as in the proof of
for all
i
G
Tj.
i
be seen by checking the
order optimahty conditions of problem (67) and follows by
first
the hnearity of the latency functions.
have
Pi
+
=
a^Xi
pi
+
aiXi
=
Assume
pjxs
=
R, or
=
holds, then the fact that Ps
strictly
pi
Using this
+
to arrive at a contradiction that
that Xi
<
i
G
UiXi
+
Xi for all
Is-
p >
+
also see that
a^x^
<
i?.
.
If
the
which implies, by
first
we
either
first
case
(67) in this case reduces to a
Assume next that the second
p,
The
pi
problem
ps follows since
set.
we can
fact,
< R and
a,Xj
concave program over a polyhedral
p'^Xs-,
i
Lemma
case holds.
and the fact
order conditions of problem (67) in
2
this case yields
Pi
=
(see the proofs of Corollary 1
• Pi
+
fljXj
>
pi
AT]
.
for
I
and Proposition
i
G
Vzg2„
,
11).
(68)
There are two cases to consider:
I5:
N M
M
it can be seen that
[see Eq. (55) for the definitions of sets
d
Using this fact (together with the monotonicity relation on x, and Xi) in
In this case,
and
+ a^Xi
y"
X^'^^'
the price characterization (68), we obtain pi
37
<
pi,
yielding a contradiction.
+
• Pi
ttiXi
<pi +
In this case,
it
UiXi for
i
els'.
can be seen from the equivalent characterization of the
> A, implying that J2iei^^ ~
that the corresponding multiphers satisfy A
can also immediately see that Xj
^^gj
>
"^
(2)
'^'^^
'^-
^ Tg. Together with X^j^j Xi <
X^igi^ii which contradicts the fact that
Xj for
^i> this implies that X^iei^*
WE
all
j
Q.E.D.
Proof of Proposition
correspondence, we can
7: Since
B{p^^)
is
an upper semicontinuous and convex- valued
use Kakutani's fixed point theorem to assert the existence of a
— p'~'^
(see [6]). To complete the proof, it remains to show that
x°^ G W[p°^) such that Eq. (10) holds.
If Jo = 0, we have by Proposition 3 that W{p°^) is a singleton, and therefore Eq.
(10) holds and (p°^, W{jP^)) is an OE.
Assume finally that exactly one of the a^'s (without loss of generality ai) is equal
to 0. We show that for all x, x 6 W(jP^), we have Xi — Xi, for all i 7^ 1. Let
EC{x,p°^) = mm.j{lj{xj) +p°^]. If at least one of
p"-'^
such that B{p^^)
there exists
EC{x,p°'^)
problem
(4),
R,
EC{x,p°^) <
or
R
Yliei^i ~ ^- Substituting xi — d — '^^^j^ ^^j x,
we see that the objective function of problem (4) is strictly convex in
holds, then one can
in
<
show that X^ig^ii
=
x_i = [xiji^^i, thus showing that x = x. If both EC{x,p'^^) — R and EC{x,p'^^) = R,
then Xi = Xi = 1^^{R — pf^) for all i ^ I, establishing our claim.
Since x_i
For some x € ]V{p^^), consider the vector x^^ = {d — X],^i Xj, x_i)
is uniquely defined and Xi is chosen such that the provider that owns link 1 has no
.
incentive to deviate,
follows that {p'^^,x'^^)
Appendix C: Proof
10
We
it
will
an OE. Q.E.D.
is
of Proposition 8
prove Proposition 8 using Theorem 5* of Dasgupta and Maskin
by stating a
slightly simphfied version of this theorem. Consider
the strategy space of player
s,
an
denoted by P,, be a closed interval of
S
We
[13].
start
player game. Let
M"'' for
some n^ G N,
s
and
its
P_s
=
payoff function by 7rs(ps,P-s)-
We
also denote
p
=
{ps,p-s),
P =
Yl ^s,
and
s
Yl
Definition
-Ps-
Al
To
Let
state
7r(p)
Theorem
=
5* in
we need the
[13],
Xlses ^«(P^'?'-^)-
"""
(P) ^^
following three definitions.
upper semicontinuous
in
p
if for
all p,
limsup7r(p) <n{p).
Definition
A2 The
profit function i^s{Ps,P-s)
38
is
weakly lower semicontinuous in ps
if
E
for all Ps
A G
Ps, there exists
AHm mf
such that
[0, 1]
+
TrsiPa,P-s)
-
(1
k
jt s
with
A3
1
<
mf
A) lim
e P-3,
>
tTs{Ps,P-s)
t^s{Ps,P-s)-
Ps ]Ps
Ps IPs
Definition
Ps
for all
< D < Dg and each
s, let Ds G N. For each D with
one-to-one,
continuous
function.
Let
P{s) be a subset
be
a
f^
For each player
/c
<
5, let
of P, such that
Pis)
=
{(pi,...,P5)
In other words, P{s)
measure
zero).
eP\3k^s,3D,0<D<Ds
is
P
a lower dimensional subset of
Theorem 5*
p^
s.t.
(which
=
is
f,1 (p.)}
also of
.
Lebesgue
in [13] states:
Theorem Al (Dasgupta-Maskin) Assume
that iTsiPsTP-s)
continuous in p except
is
on a subset P**of P{s), weakly lower semicontinuous in p^ for all s and bounded, and
that 7r(p) is upper semicontinuous in p. Then the game [(Ps,7rs) s = 1,2,
',5] has a
mixed strategy equilibrium.
•
;
We
show that our game
7r,(p,,p_s)
=
Wardrop
all
x G W{p),
that
lim inf
Ps^ps
Given p
>
0,
will select
a
and X^jXi(p)
'
[Q,R]
G 5,
s
<
d
for all p,
bounded.
an upper semicontinuous correspondence, we select x* {) such
-Ks{Ps-,P-s) is clearly
W {ps,p-s)
Since
V
Il,{p,,p^,,x* (p,,p_,)),
—
We
W (ps,p_s), such that
equilibria,
that will satisfy these hypotheses. First, since P^
and
Theorem Al.
the hypotheses of
satisfies
function x* (ps,p_s) from the set of
is
V
=
x*(p„p_,)
^
.
=
since pj
V
,
jeij
J
pk for
all j,
V
x*(p„p_,),
p,
>
0,
V p_, >
0.
(69)
^
els
/c
G X, where
T
is
defined in Eq. (5) in
V
p^
Lemma
2, it
follows that
=
lim inf n^ips^Ps)
^s{P3,P-s),
>
V
0,
p-^
>
0,
PsTps
hence ensuring that 7ra(ps,p_s)
We claim that we have
=
ns(p5,p_5, x* {Ps,P~s))
5^x;(p) > ^.x,(p),
j
Assume the
Vp >
0,
is
weakly lower semicontinuous.
VX G
iy(p).
(70)
j
contrary. This implies that there exist
some p >
0, s
G 5, and x E W{p)
such that
5^x,(p)>^x;(p).
jeis
jeis
39
(71)
By
we have that X]jgi^ a;*(p",p_5) -^ ^^^^^ x*(ps,P-s)
Combined with Eq. (71), this imphes that
Eq. (69),
{p"}
T Ps-
jeis
some
for
Ps
Next,
<
jeXs
monotonicity of
Ps, contradicting the
we show that
some sequence
for
is
7Ts{Ps,P-s)
WE by Proposition
2.
continuous in p except on a set P**.
We
define the
set
P**
By the upper semicontinuity
Lemma
Moreover, by
P—
which
is
{p
\
of
{p
Pj
—
\
W{p)
not a singleton}.
is
W{p), we see that
2, it follows
pk, for
a lower dimensional
Finally,
=
set.
C
that P**
some j
k}
y^
TTs{ps,p-s)
is
continuous at
all
p ^ P**.
P, where
U
{p
\
Pj
—
R, for some j},
This establishes the desired condition
for
Theorem Al.
we show that
seS
leJ
Given some p > 0, define X as in Eq. (5) of Lemma 2. If X = 0,
then we automatically have that n is continuous at p. Assume that I ^ $. Since xf ^(•)
is continuous at p for all i ^ T and Pj = p^ for all j, k E T, it is sufhcient to show that
is
continuous at
Yliei-'^i^P) ^^
all p.
continuous at
i.e.,
p,
for a
sequence {p"} with p" G [0,M]^ and p"
—
>
p,
we
show that
= Y.xnp)\imY^x:{pn
^—
^—^
n— CO
iei
iei
Define
d{pn
Since Xi{-)
two
is
continuous at p for
all i
=
J2x;{p-).
^ J, we have d(p") -^ d{p)
=
J2iai^iip)- Consider
cases:
• YlieT^c^
>
^
~
n
Since x*{p)
d{p).
Eq. (70)] and li{x*)
=
for all
sufficiently large Yliei^ziP^)
• Y^iex^Ci
that
^
d
—
^jgj ^*i (p)
d[p).
~
By
i
the
is
G J,
~
^'
maximum
/i-norm element of
this implies that Yli^jX*{p)
=
[cf.
establishing the claim.
the same reasoning as in the previous part, this implies
Yliex ^c,
Moreover,
for all e
>
0,
there exists some n sufficiently
large such that
establishing the claim.
The preceding enable
W{p)
d and for aU
us to apply the theorem, completing the proof.
40
Q.E.D.
Appendix D: Proof of
11
We
first pro\'e
Lemma
5
the following lemma:
Lemma
10 Let (jP^,x°^) be an OE such that min^ {p°^ + lj{xf^)} < R. Let Assumptions 1 and 2 hold. U p^^xf^ > for some j e J, then W{p^^) is a singleton.
for some j e J, it follows, by Lemma 4, that pf^xf^ >
Proof. Since pf^x'^^ >
i e I.
We first show that for all x G W{p'^^), we have Xj < xf^ for all i. If
hixf'^) > 0, then by Lemma 2, Xi = xf^ for all x E iy(p°^). If k{xf^) = 0, then
and Assumption 2, which implies that
by the fact that xf^ >
Xs = {i} for some
Xi < xf^ by the definition of an OE (cf. Definition 10).
Since miuj {p°-^ + lj{x^^)} < R, we have J2ieJ^?^ ~ ^- Moreover, the fact that
Xi < xp-^ for all X 6 W{p
) implies that miuj {p'j^ + lj{xj)} < R as well, and therefore
J2i^jXi = d, showing that Xi = xf^ for all x G W{p°^), for all i G I. Q.E.D.
for all
.s
Proof of
Lemma
5.
We
first
prove this result for a network with two links. Assume
to arrive at a contradiction that
>
/+(x°^)
Let
{e''"}
be a scalar sequence with
e''
/2-(x°^).
(72)
Consider the sequence
I 0.
WE given price vector
+
{xi(e''')}
By
where
Xi{e'')
1 and
(pf^
correspondence W{p) is upper-semicontinuous and W{p'^^) is a
singleton. Therefore, it follows that Xi(e''') -^ xf^. Define
is
the load of fink
Lemma
10, the
1
at a
e''
,P2^).
Proposition
WE
fc-co
dpi
e'=
WE
given price vector (pf^ —
Xi(— e''") be the load of link 1 at a
Since W{p'^^) is a singleton, we also have xi(— e'^) —> xf^. Define
Similarly, let
a-Xi(pf^,pO^)
Since min_, {p^^
+
lj{x^^)}
<
R,
xf^-Xi(-6^-)
^
fci™
dpi
it
dp,
^
e^-
can be seen using
a+xi(pf^,pO^)
e''",p^^).
Lemma
'
3 that
-1
^
-ii{x?'^)+in^?''y
and
5-Xi(pf^,p^^)
dp,
Since /j'"(xf^)
=
^r(^?'^)
-1
_^
-
by Assumption
iti^?'')+i2i^?''y
3, this
combined with Eq.
a+xi(pf^,p^^)
d-x,{p?^,pO^)
dpi
dpi
41
(72) yields
.__.
Consider the profit of service provider
1,
ni(pi
jjj
)
=
Vi
x^
Define
.
fe
fc-tcx)
(9pX
Since
pf^
is
a
maximum
of ni(-,p^^),
a+ni(pf^,p^^)
^3ifM!)
dpi
when combined,
_ OE ^
nsd^^MM^ <-"'n
^^'
-'"'
dp,
which,
we have
(7f^^
^^^>
d^,
. .o. ^ ^o.^I^M!^ >
'
(77)
0,
dpi
yields
a+xi(pf^,p^^) ^ a-xi(pf^p^^)
~
'
5pi
which
5pi
a contradiction by Eq. (75), thus showing that we have l2{x2^) = /^(x^^).
next consider a network with multiple links. As in Eqs. (73) and (74), we define
is
We
for all
i
e J,
d^x,{pO^)
dpi
-rOE
= hm
^k
fc^oo
)
[-^')
d-Xiip'^^)
dpi
Using the same
fine of
argument
as ab(Dve,
we obtain
-1
1
dpi
1
'+K"^'>
-1
d-xiipo^) ^
dpi
'9)
1
Itixo^
i
<r(xps)
Let
1
G Xj and without loss of any generality, assume that
Assumption 3). For all i e Is, i ^ 1, we obtain
,
(recall
9+x,(p°^)
1
^
d^xdp^^
1
42
all
li 's
for
i
€ Xs are smooth
To
arrive at a contradiction,
assume that l'^{x^^) >
l^
(x^^)
for
some
^
j
Is-
Then
the preceding two sets of equations imply that
dpi
opi
for all
e
i
Is-
Next, Eqs. (76) and (77) for multiple link case are given by
a-n.(p°^)
a-x.(p°^)
^
which are inconsistent with Eq.
for the multiple link case.
Appendix
12
We
V
^
dp,
dpi
_^^_^j
p?-?:^>o.
ap,
(81)
a contradiction. This proves the claim
(80), leading to
Q.E.D.
E: Proof of Proposition 9
assume that miuj {pf^ + ljix^^)} < R- Consider service provider s and assume
loss of generality that 1 E Is- Since p^^x^^ >
for some j G Is' and s' e S,
it follows by Lemma 4 that pf^xf^ >
for all i G I- Together with Lemma 3, this
implies that {{pf^)i^2^,x'-'^) is an optimal solution of the problem
first
without
am '2e((p,),£i..^)>o
(82)
ieis
subject to
ll{Xl)
+Pl=
li{x^)+Pi,
I
els-
{1}
i^Is,
h{xi)+Pl=li{Xi)+pf^,
hixi)+Pi <R,
(83)
J2^^^dwe have that li is continuously differentiable in a neighborhood of xf ^
for all i (since the gradient mapping of a convex function is continuous over the set the
function is differentiable, see Rockafellar [42]). Therefore, by examining the KarushKuhn- Tucker conditions of this problem, we obtain
By Lemma
5,
pf^
= xf^/;(xr)-0,
VzeJ,,
(84)
where
if
),
_ ^Eje x,^r
showing the result in Eq.
^
l'^{xf^)
otherwise,
(16).
43
=
for
some
j
^
Is,
(S^)
We next assume that min^ {pf^+lj{x^^)} — R.
I and Lemma 3, this imphes that
for some j
Using the assumption that p'^^Xj'^
>
E.
p?^^R~k{xr),
and thus
for all s
€ 5, x'^^
is
vi,
an optimal solution of
maximize
subject to
/.(-^ ~ U[xi))xi
Xi
E
Ti,
^
i
^X^
^Xi<d,
(86)
where Ti = {x,; pf-^ + li(x^ — K\ is either a singleton or a closed
is a convex problem, using the optimality conditions, we obtain
interval.
Since this
|
i?-i,(xp^)-xf^5;,
where
^^
>
is
=e„
ViGZ,,
the Lagrange multiplier associated with constraint (86), and
dli{xf^). Since /"(xf^)
P?'^
<
gi^,
gi^
€
the preceding implies
= R-h{x?^)>x^\{xf%
VzeJ,
proving (17).
To prove (18), consider some i € X with T^ = {i} for some s and the sequence of price
vectors {p''"} with p^ = [pf^ — e'',p^f). Let {x'^} be a sequence such that x*^ € W{p^)
for all k. By the upper semicontinuity of W{p), it follows that x'"' —> x with x 6 W{p'^^)
and X
<
for all
/c,
(77)
and
Lemma 10). Moreover, by Lemma 2, we have xf > xf^
Xi > xf^, showing that xf -^ xf^. We can now use Eqs.
substituting i instead of 1 and using X^ = {i}) to conclude that
x*^^ (see the proof of
which implies that
(81) (by
pT <
xf'^iti^?')
Q.E.D.
44
+
'^'
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