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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
COMPETITION AND EFFICIENCY
CONGESTED MARKETS .
IN
Daron Acemoglu
Asuman
E. Ozdaglar
Working Paper 05-06
February 7, 2005
1
Room
E52-251
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Cambridge, MA 021 42
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'>3SACHUSETTS INSTITUTE
OF TECHNOLOGY
MAR
1
2005
LIBRARIES
1
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
February
17,
2005
Abstract
We study the
efficiency 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. This bound is tight even when the number of routes
and oligopolists is arbitrarily large. We also study the efficiency properties of
mixed strategy equilibria.
of
traJfi-C
in a transportation
oligopolists
can reduce
network.
efficiency,
'We thank Xin Huang, Ramesh Johari, Eric Maskin, Nicolas Stier Moses, Jean Tirole, John Tsitsiklis,
Muhamet Yildiz and participants at the INFORMS conference, Denver 2004 for useful
Ivan Werning,
comments.
Introduction
1
We
analj'ze 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) externahty on existing
Congestion costs are captured by a route-specific non-decreasing convex latency
traffic.^
function,
li
(•).
denoted by
pi.
Profit-maximizing oligopohsts set prices
We
each price vector,
for
(tolls) for travel
on each route
analyze subgame perfect Nash equilibria of this environment, where
p, all traffic
chooses the path that has
minimum
(toll
plus delay)
maximize profits.
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
h+Pi, and oligopohsts choose
The environment we analyze is
cost,
example,
for
The key
[38]).
prices to
feature of these environments
externality that users exert on others.
is
the negative congestion
This externality has been well-recognized since
[33], Wardrop [41], Beckmann et al.
and by Orda et. al. [23], Korilis et. al. [18], Kelly [17],
Low [20] in communication networks. More recently, there has been a growing literature
that focuses on quantification of efficiency loss (referred to as the price of anarchy) that
results from externaUties and strategic behavior in different classes of problems; selfish
routing (Koutsoupias and Papadimitriou [19], Roughgarden and Tardos [31], Correa,
Schulz, and Stier-Moses [8], Perakis [26], and Friedman [12]), resource allocation by
market mechanisms (Johari and Tsitsiklis [16], Sanghavi and Hajek [32]), and network
the work by Pigou
[27] in
economics, by Samuelson
in transportation networks,
[4]
design (Anshelevich
et.al.
[2]).
Nevertheless, the game-theoretic interactions between
and users, or the effects of competition among the providers
has not been considered. This is an important area for analysis
(multiple) service providers
on the
efficiency loss
since in
most
Moreover, we
applications, (competing) profit-maximizing entities charge prices for use.
will
show that the nature of the analysis changes
significantly in the
presence of price competition.
We provide a general framework for the analysis of price competition among providers
in
a congested (and potentiaUy capacitated) network, study existence of pure strategy
and mixed strategy equihbria, and characterize and quantify the efficiency properties of
There are four sets of major results from our analysis.
First, though the equilibrium of traffic assignment without prices can be highly
inefficient (e.g., [27], [31], [8]), price-setting by a monopolist internalizes the negative
externality and achieves efficiency.
Second, increasing competition can increase inefficiency. In fact, changing the market
structure from monopoly to duopoly almost always increases inefficiency. This result
contrasts with most existing results in the economics literature where greater competition
equilibria.
tends to improve the allocation of resources
result
is
(e.g.
driven by the presence of congestion and
see Tirole [36]).
is
illustrated
The
intuition for this
by the example we discuss
below.^
^
An
externality arises
when
the actions of the player in a
game
^Because users are homogeneous and have a constant reservation
affects the payoff of other players.
utility in
our model, in the absence
we provide a tight bound on the extent of inefficiency in the presence of
which apphes irrespective of the number of routes, /. We show that 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
Simple examples reach this 5/6 bound. Interestingly, this bound is
social surplus.
Third,
pricing,
obtained even
when
the
number
of routes, /,
Fourth, pure strategy equilibria
the fact that
what we have here
is
may
fail
is
arbitrarily large.
This
to exist.
is
not surprising in view of
a version of a Bertrand-Edgeworth game where pure
strategy equilibria do not exist in the presence of convex costs of production or capacity constraints (e.g.,
Edgeworth
[11],
Shubik
Benassy
[35],
[6],
Vives
[40]).
However,
environment we show that when latency functions are linear, a pure
strategy equihbrium always exists, essentially because some amount of congestion externahties remove the payoff discontinuities inherent in the Bertrand-Edgeworth game.
in our oligopoly
Non-existence becomes an issue in our environment
convex. In this case,
we prove that mixed
when
latency functions are highly
strategy equilibria always exist.
We
also
show
that mixed strategy equilibria can lead to arbitrarily inefficient worst-case realizations;
in particular, social surplus
maximum
can become arbitrarily small relative to the
social
surplus.
The
following example illustrates
Example
1
Figure
1
some
of these results.
shows a situation similar to the one first analyzed by Pigou [27]
due to congestion externalities. One unit of traffic will travel
to highlight the inefficiency
from origin A to destination B, using either route
are given by
x^
^1
It is
= —,
{x)
h
or route
1
[x)
=
straightforward to see that the efficient allocation
2.
The
latency functions
2
-X.
[i.e.,
one that minimizes the total
2/3 and xf = 1/3, while the (Wardrop) equilibrium
allocation that equates delay on the two paths is x^^ w .73 > xf and x'^^ « .27 < xf
delay cost '^^li{xi)xi]
The
xf
is
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.
olist will set
We
will
show
below that in this case, the monop(when li is differentiable), which exactly
in greater detail
a price including a markup,
x^l'^
internalizes the congestion externality. In other words, this
markup
is
equivalent to the
Pigovian tax that a social planner would set in order to induce decentralized
traffic to
choose the efficient allocation. Consequently, in this simple example, monopoly prices
win be
in the
pf ^ = {2/^f + k and pf ^ =
Wardrop equilibrium will be
(2/3^)
+ ^,
for
some constant
k.
The
identical to the efficient allocation,
resulting traffic
i.e.,
x^^ —
2/3
and x^^^ = 1/3.
Next consider a duopoly situation, where each route is controlled by a different profitmaximizing provider. In this case, it can be shown that equilibrium prices will take the
all market structures would achieve
would have no efficiency consequence.
of congestion externalities,
to duopoly, for example,
efRciency,
and a change from monopoly
,
Ii(x)=x73
l2(x)=(2/3)x
Figure
form pf-^
p^^
=
?» 0.44.
A
1:
Xi
{l[
The
two link network with congestion-dependant latency functions.
+
I'o)
Eq. (20) in Section
[see
resulting equihbrium traffic
4],
or
xf^
more
w
specifically,
.58
pf^
^
< xf and x^^ «
and
0.61
.42
>
xf
which also differs from the efficient allocation. We will show that this is generally the case
in the ohgopoly equilibrium. Interestingly, while in the Wardrop equihbrium without
prices, there was too much traffic on route 1, now there is too little traffic because
of its greater markup. It is also noteworthy that although the duopoly equilibrium
is
monopoly equihbrium, in the monopoly equilibrium k is
consumer surplus is captured by the monopolist, while in the
oligopoly equilibrium users may have positive consumer surplus.^
is
inefficient relative to the
chosen such that
The
follows.
all
of the
intuition for the inefficiency of duopoly relative to
There
is
now a new
which they exploit by distorting the pattern of
1,
charges a higher price,
2,
raising congestion
it
monopoly can be obtained
2.
But
this
"locked-in," because their outside option,
when provider
traffic:
realizes that this will
on route
push some
traffic
1,
controlling route
from route
1
to route
makes the traffic using route 1 become more
travel on the route 2, has become worse."* As
a result, the optimal price that each duopolist charges will include an additional
over the Pigovian markup. These are XiZj for route
two markups are generally
as
source of (differential) monopoly power for each duopolist,
1
and
X2l'i
markup
for route 2. Since these
different, they will distort the pattern of traffic
away from
the efficient allocation. Naturally, however, prices are typically lower with duopoly, so
even though social surplus declines, users
will
command
will
be better
off
than in monopoly
(i.e.,
they
a positive consumer surplus).
Although there is a large hterature on models of congestion both in transportation
and communication networks, very few studies have investigated the implications of
having the "property rights" over routes assigned to profit-maximizing providers. In
Basar and Srikant analyze monopoly pricing under specific assumptions on the
[3],
utility
and latency functions. He and Walrand [15] study competition and cooperation among
internet service providers under specific demand models. Issues of efficient allocation of
^Consumer surplus is defined as the difference between users' willingness to pay and effective costs,
li{xi), and is thus different from social surplus (which is the difference between users' willingness
to pay and latency cost, li{xi), thus also takes into account producer surplus/profits). See Mas-Colell,
Winston, and Green [21].
*Using economics terminology, we could also say that the demand for route 1 becomes more "inelastic". Since this term has a different meaning in the communication networks literature (see [34]), we
Pi
+
do not use
it
here.
do not arise in these papers. Our previous work [1] studies
monopoly problem and contains the efficiency of the monopoly result (with inelastic
traffic and more restrictive assumptions on latencies), but none of the other results here.
To minimize overlap, we discuss the monopoly problem only briefly in this paper.
In the rest of the paper, we use the terminology of a (communication) network,
flows or traffic across routes
the
though
of the analysis applies to resource allocation in transportation networks,
all
markets, and other economic applications. Section 2 describes the basic en-
electricity
Section 3 briefly characterizes the monopoly equihbrium and establishes
vironment.
Section 4 defines and characterizes the oligopoly equilibria with compet-
its efficiency.
ing profit-maximizing providers. Section 5 contains the
main
results
and characterizes
the efficiency properties of the ohgopoly equilibrium and provide bounds on efficiency.
Section 6 contains concluding comments.
Regarding notation,
all
vectors are viewed as
be interpreted componentwise.
vectors. Let
We
d be a closed subset of
oo)
[0,
column
M^
denote by
and
vectors,
and inequalities are to
the set of nonnegative /-dimensional
let
/
:
Q
M
i—>
We
be a convex function.
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
consider a network with / parallel links. Let
Let X, denote the total flow on link
Each
i,
=
and x
I=
[xi,
.
.
.
{!,...,/} denote the set of links.
denote the vector of link flows.
,xi]
network has a flow-dependent latency function k{xi), which measures
i. We denote the price
link in the
the travel time (or delay) as a function of the total flow on link
per unit flow (bandwidth) of link
We
i
hy
Let p
Pi.
=
denote the vector of prices.
,Pi]
{pi,-
are interested in the problem of routing d units of flow across the / links.
sume that
this
the aggregate flow of
is
many
"small" users
principle (see [41]) in characterizing the flow distribution in the network;
are routed along paths with
at the given flow
ciple
is
minimum
and the price
used extensively
in
effective cost, defined as the
traffic
and communication networks
a reservation utility R and decide not
[9],
[25])
the reservation
utility.
as-
sum
i.e.,
the flows
of the latency
path (see the definition below). Wardrop's prin-
of that
modelling
We
and thus adopt the Wardrop's
([31],
to
behavior in transportation networks
[8]).
We
send their fiow
([4],
assume that the users have
also
if
the effective cost exceeds
This implies that user preferences can be represented by the
piecewise hnear aggregate utility function u{-) depicted in Figure 2.^
Definition 1
For a given price vector
p >
0,^
a vector
x^^ G Mi
is
a Wardrop
This simplifying assumption implies that all users arc "homogeneous" in the sense that they have
utility, R. We discuss potential issues in extending this work to users with elastic
and heterogeneous requirements in the concluding section.
the
same reservation
''Since the reservation utility of users
i.
Throughout the paper, wc use p >
is
equal to R,
and p €
[0,
RY
we can
also restrict attention to pi
interchangeably.
<
R
for all
u(x)
Figure
equilibrium
(WE)
2:
Aggregate
utility function.
if
(1)
We
denote the set of
We
WE
at a given
p by W{p).
adopt the following assumption on the latency functions throughout the paper.
Assumption
1 For each
i
6 X, the latency function
convex, nondecreasing, and satisfies
/^(O)
=
li
:
[0, cx))
Hence, we allow the latency functions to be extended
Ci
=
for
problem
i—>
[0,
oo] is
proper closed
0.
real- valued
(see
[7]).
Let
{x E [0, oo)
li{x) < oo} denote the effective domain of li. By Assumption 1,
Ci is a closed interval of the form [0, b] or [0, oo). Let bd = sup^g,-;^ x. Without loss of
generality, we can add the constraint Xj £ C, in Eq. (1). Using the optimality conditions
I
and there
(1),
exists
we see that a vector x^^ € M^ is a WE if and only if
some A > such that A( X^^^^ x^^ — d) = and for all
R-k{xf^)-p, <X ifxf'^-O,
= A if < xf"^ <
>A ifxf^ = 6c..
When
the latency functions are real-valued
WE,
characterization of a
This
lemma
equalized on
Lemma
which
states that in the
all links
1 Let
[i.e.,
Ci
=
[0,
d
(2)
be
oo)],
we obtain the following
often used as the definition of a
WE,
<
WE in the literature.
the effective costs, defined as li{xf^^)
+
pi,
all
i
are
with positive flows (proof omitted).
Assumption
Then a nonnegative
is
X^j^j xf^^
i,
1
hold,
vector x* E
li{x*)+pi
=
and assume further that Ci
if and only if
—
[0,
oo) for
G I.
W{p)
mm{lj{x*)
+ pj},
V
i
with X*
>
0,
(3)
li{x*)+pi
<
V
R,
with x*
i
>
0,
I
2
with J2i=i ^*
Example
=
2
c?
=1
if niirij
{/j(xj)
+Pj} <
R.
below shows that condition
latency functions are not real- valued. Next
lemma may
(3) in this
we
not hold
establish the existence of a
when
the
WE.
(Existence and Continuity) Let Assumption 1 hold. For any price
1
> 0, the set of WE, W{p), is nonempty. Moreover, the correspondence
M^ is upper semicontinuous.
Proposition
W
vector p
R^
^
Proof. Given any p
>
0,
:
consider the following optimization problem
maximize^>o
^^
subject to
({R -
2_, ^i
^
-
Pi)Xi
j
k[z)dz\
(4)
'^
i=l
Xi
In view of
Assumption
(1) (i.e.,
li
is
€
Ci,
V
i.
nondecreasing for
can be shown that the
all i), it
which is nonempty
G Cj) and convex. Moreover, the first order optimahty conditions of problem
optimality
(4), which are also sufRcient conditions for optimality, are identical to the
conditions [cf. Eq. (2)]
Hence a flow vector x^'^'^ G ^{v) if ^^nd only if it is an optimal
solution of problem (4). Since the objective function of problem (4) is continuous and
the constraint set is compact, this problem has an optimal solution, showing that W{p)
is nonempty. The fact that
is an upper semicontinuous correspondence at every p
follows by using the Theorem of the Maximum (see Berge [5], chapter 6) for problem
objective function of problem (4)
is
convex over the constraint
set,
(since
WE
.
W
(4).
Q.E.D.
WE flows satisfy intuitive monotonicity properties given in the following proposition.
The proof
follows from the optimality conditions
(a)
=
Ecj. (2)]
(Monotonicity) Let Assumption
Proposition 2
P-j
[cf.
1
and
hold.
is
omitted (see
For a given p
>
[1]).
0,
let
\PiWj-
For some p
(b) For some pj
for all
(c) For
let
i y^
<
<
p, let
x G
Pj, let
VV'(p)
and x G W{p). Then, X^j^j
>
x^
J^j^j
x G W{pj,p-j) and x G VF(pj,p_j). Then Xj
>
Xi.
Xj
and
Xj
<
Xi,
j.
some I C I, suppose that Pj <
X G W{p) and x G W{p). Then
X
and Pj
=
pj for all j
G
T,jei^'jiP)
^ Ejgi^j(p)-
Pj for all j
^ J, and
For a given price vector
example
Example
p,
the
WE
some properties
illustrates
need not be unique in general. The following
WE.
of the
Consider a two hnk network. Let the total flow be d
2
tion utility he
R=
Assume
1.
^
oo
I
At the
<
Pi>
=
P2
W{p)
I,
is
Xj
given by
all (0,
are not real- valued.
some
scalar a
>
<
X2) with
This example also illustrates that
for
Lemma
1
<
X2
is
given by the set of
all
2/3.
Consider, for instance, the price vector (^1,^2) = (1 ~ e, 1 ~ le)
In this case, the unique
is (xi,X2)
(1/3,2/3), and clearly
WE
1.
on the
further restrictions
which assumes
Proposition 3
reserva-
need not hold when latency functions
=
on the two routes are not equalized despite the
positive flows. This arises because the path with the lower
constrained, so no more traffic can use that path.
Under
and the
price vector {pi,P2) with
fact that they
effective costs
literature
1
otherwise.
= (1,1), the set of WE, W{p),
< 2/3 and X^j x, < 1. At any
price vector (pi,P2)
vectors (xi,X2) with
=
that the latency functions are given by
k,
we
effective cost
both have
is
capacity
obtain the following standard result in the
strictly increasing latency functions.
(Uniqueness) Let Assumption 1
For any price vector p
strictly increasing over Cj.
singleton. Moreover, the function
W
:
M^
W_^, >-^
is
Assume
hold.
>
the set of
0,
further that
WE,
W{p),
is
li
is
a
continuous.
Proof. Under the given assumptions, for any p > 0, the objective function of problem
This shows the
(4) is strictly convex, and therefore has a unique optimal solution.
uniqueness of the
at a given p. Since the correspondence
is upper semicontinuous
from Proposition 1 and single- valued, it is continuous. Q.E.D.
W
WE
In general, under the Assumption that
result,
which
Lemma
2
is
/j(0)
essential in our analysis with
Let Assumption
I=
{i
1
=
hold. For a given
Gl
\
for all
nonunique
p
>
0,
i
E X, we have the following
WE flows.
define the set
3 X, X E W{p) with x^
7^ x^}.
Then
h{xi)
-
Pi
==
0,
Pj,
Vie2, Vxe
V
z,
j e T.
W{p),
(5)
.
Proof. Consider some
such that
^
Xi
Xi-
&T. and x G W{p). Since
i
Assume without
E T, there
i
>
generahty that Xt
loss of
some x E W{p)
There are two cases
exists
Xi.
to consider:
(a)
If
>
Xk
Xk for
all
A;
^
optimality conditions
>
then Xljgi^i
i,
Eq.
[cf.
(2)] for
'^IjeJ^i-' '^^ich implies
x hold with A
=
0.
By
Eq.
1
(i.e.,
that the
and
(2)
Xi
WE
>
x,,
we have
which together imply that
=
/,(0)
(b)
If
Xk
<
0), it follows
Xk for
some
k,
that
by the
+Pi<
R,
li{xi)
+Pi>
R,
=
/i(xi)
/,(x,)
li{xi)
=
li{xi).
By Assumption
li
is
convex and
0.
WE optimality
conditions,
li{Xi)
+Pi<
lk{Xk)
+Pk,
k{x,)
+f)i>
IkiXk)
+Pk-
we obtain
Combining the above with Xj > x, and Xfc < x^, we see that /j(xj) = li{xi), and
(and also that
hi^k) = 4(^fc)- By Assumption 1, this shows that li{xi) =
=Pk)-
Pz
X,
Next consider some i, j € I. We will show that pi = pj. Since
X G W{p) such that Xi > Xi. There are three cases to consider:
• Xj
<
• Xj
>
Xj
.
Then
Xj.
If Xfc
i
E T, there
a similar argument to part (b) above shows that pi
>
Xk for
all
k
^
optimahty conditions hold with A
which together with
li{xi)
=
X^m^m ^
i,j,
then
=
Therefore,
0.
k{x,)
+Pt<
R,
lj{xj)
+ Pj >
R,
lj{Xj)
=
imply that
^'
exist
= pj
™ply™g
that the
WE
we have
pi
— pj.
= Xj. Since j E T, there exists some x E W{p) such that Xj ^ Xj. Repeating
the above two steps with Xj instead of Xj yields the desired result.
• Xj
Q.E.D.
We
next define the social problem and the social optimum, which
allocation) that
would be chosen by a planner that has
over the network.
full
is
the routing (flow
information and
full
control
A
Definition 2
social problem
flow vector x^
is
a social
niaximize2;>o
optimum
2.
(-^
~
an optimal solution of the
if it is
ki^i) jXi
(6)
2=1
/
>, ^i ^
subject to
In view of Assumption
1,
^•
the social problem has a continuous objective function and
a compact constraint set, guaranteeing the existence of a social optimum, x^. Moreover,
using the optimality conditions for a convex program (see
a vector x^ G
subgradient
for each
M^
gi.
E
is
a social optimum
dli{xf) for each
and only
if
and a
i,
>
A'^
if
[7],
Section 4.7),
X^j^^ xf
such that
<
A'^(
we
see that
d and there exists a
X],_i
xf
—
d)
=
and
i,
R-k[xf)-xfgi^
<X' ifxf^O,
= A^ iiO<xf<bc„
> A^ if if = 6c..
For future reference, for a given vector x 6 R^,
we
(7)
define the value of the objective
function in the social problem,
/
S{x)
= Y.{R-h{x,))x,,
(8)
1=1
as the social surplus,
i.e.,
the difference between users' willingness to pay and the total
latency.
3
Monopoly Equilibrium and
In this section,
we assume
that a monopolist service provider owns the / links and
charges a price of p^ per unit bandwidth on link
Acemoglu and Ozdaglar
[1]
for
of each of a finite set of users
i.
We
considered a related problem in
atomic users with inelastic
is
a step function)
differentiable latency functions.
general latency functions and the
The monopolist
Efficiency
,
traffic (i.e.,
the utility function
and with increasing,
demand model considered
sets the prices to
maximize
and
more
real- valued
Here we show that similar results hold
for the
in Section 2.
his profit given
by
/
?,=i
where x G W{p). This defines a two-stage dynamic pricing-congestion game, where the
monopolist sets prices anticipating the demand of users, and given the prices (i.e., in
each subgame), users choose their flow vectors according to the
WE.
.
Definition 3
A
vector {p'^'^ ,x^'^^)
>
a Monopoly Equilibrium
is
(ME)
if
x^
W{p^'^) and
n(p*^^, x^"^)
Our
definition of the
ME
>
V
n(p, x),
p,
V X G ly
stronger than the standard
is
librium concept for dynamic games.
With a sHght abuse
{p)
subgame
perfect
Nash
equi-
of terminology, let us associate
a subgame perfect Nash equilibrium with the on-the-equilibrium-path actions of the
two-stage game.
Definition 4
A
>
vector {p*,x*)
pricing-congestion
game
if
x*
is
a subgame perfect equihbrium (SPE) of the
G W{p*) and
for all
p
>
0,
there exists x 6
W
such
{p)
that
U{p*,x*)
The
>
U{p,x).
following proposition shows that under
coincide.
Since the proof
Assumption
1
,
the two solution concepts
not relevant for the rest of the argument, we provide
is
in
it
Appendix A.
Proposition 4
it is
an
SPE
ME
Since an
Let Assumption
1
hold.
of the pricing-congestion
{p*,x*)
is
A
vector [p^^^ ,x^'^)
is
ME
an
if
and only
if
game.
an optimal solution of the optimization problem
I
maximizep>o, x>o
subject to
it
is
slightly easier to
^^PiXi
(9)
X £ W{p),
work with than an SPE. Therefore, we use
ME
as the solution
concept in this paper.
The preceding problem has an optimal solution, which establishes
ME. In the next proposition, we show that the flow allocation at an
optimum are the same.
Proposition 5
only
if it is
3
1
hold.
A
vector x
is
and the
the flow vector at an
ME
social
if
and
a social optimum.
To prove
Lemma
Let Assumption
the existence of an
ME
this result,
we
first
Let Assumption
1
establish the following
hold. Let (p,x) be
an ME. Then
have
p,
=
R- li{x^).
10
lemma.
for all
i
with x^
>
0,
we
Proof. Define
+
j9j
li{xt)
<
R.
I=
We
{i
e
first
I
\
Xi
li{xi)
Suppose
a
WE
+Pi <
li{xi)
[cf
Eq.
the fact that
(2)],
(p,
the price vector
x)
p
lj{xj)
we
is
and has a
is
a
WE
By
0}.
+Pi =
+ pj
see that
the definition of a
for
lj{xj)
some
x € W{pi
+ Pj,
V
WE,
for all
i
E I, we have
at price p,
'
+ e + li(xi) <
i,
J
G
X.
G T. Using the optimality conditions
i,j
+ e,p^i)
an ME. Assume next that pi
where e^ is the
= p + e X^ j^j ^i
Pi
Hence, x
>
show that
for sufficiently small
+ li{xi) < R
y
R,
i
and therefore the vector
for
contradicting
some i E T. Consider
and e is such that
for
unit vector
i*''
e,
eJ.
(p,
x)
is
feasible for
problem
strictly higher objective function value, contradicting the fact that (p, x) is
(9)
an
ME. Q.E.D.
Proof of Proposition
5:
In view of
Lemma
maximize3;>o
2_.
we can
3,
rewrite problem (9) as
{R~ hi^i)
i=l
I
subject to
This problem
is
identical to the social
2_] ^i
problem
^^
problem
[cf
(6)],
completing the proof
Q.E.D.
In addition to the social surplus defined above,
sumer
it
is
also useful to define the con-
between users' willingness to pay and effective cost, i.e.,
~ hi^i) — Pi)xi (See Mas-Colell, Winston and Green, [21]). By Proposition 5
Y2i=i (-^
and Lemma 3, 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
surplus, as the difi^erence
their information or not
(i.e.,
Our major motivation
receive
for the
a better approximation to reality,
no consumer
surplus).
study of ohgopolistic settings is that they provide
where there is typically competition among service
A secondary motivation is to see whether an ohgopoly equilibrium will achieve
an efhcient allocation like the ME, while also transferring some or all of the surplus to
providers.
the consumers.
4
Oligopoly Equilibrium
We
suppose that there are S service providers, denote the set of service providers by
and assume that each service provider s £ S owns a different subset Ts of the links.
Service provider s charges a price p^ per unit bandwidth on link i E Is- Given the vector
iS,
11
owned by other
of prices of links
provider s
service providers,
Ps =
\pi]i^i,,
the profit of service
is
==
UsiPs,P-s,x)
^PiXi,
X e W{ps,p-s), where Ps = \pi\ieXsobjective of each service provider, hke the monopoHst in the previous section,
is to maximize profits. Because their profits depend on the prices set by other service
providers, each service provider forms conjectures about the actions of other service
providers, as weh as the behavior of users, which, we assume, they do according to the
notion of (subgame perfect) Nash equihbrium. We refer to the game among service
providers as the price competition game.
for
The
Definition 5
if
x^^ S
A
{p'-'^
vector
W (p?^,p°f)
and
n,(pf^,pef,xO^)
We
refer to
p'-'^
as the
,
>
x'~'^)
for all s
is
a (pure strategy) Oligopoly Equilibrium (OE)
G 5,
> n,fe,pef,x),
Vp, >
0,
V X 6 H/fe,pef).
(10)
OE price.
OE and
Again associating the subgame perfect
equilibrium with the on-the-equilibrium-path actions, we have the following standard
As
for the
monopoly
case, there
is
a close relation between a pure strategy
a pure strategy subgame perfect equilibrium.
definition.
A vector
Definition 6
competition game
if
x*
{p*, x*)
E:
W
>
{p*)
is a subgame perfect equihbrium (SPE) of the price
and there exists a function x{p) G
[p) such that for
W
seS,
all
^s{P*s,P*-s^X*)>Ils{Ps.pls^x{ps,P*..,))
The
OE
following proposition generalizes Proposition 4
definition,
which
is
more convenient
that of Proposition 4 and
Proposition 6
is
an
SPE
is
for the
Vp, >0.
(11)
and enables us to work with the
subsequent analysis. The proof parallels
omitted.
Let Assumption
1
of the price competition
hold.
A
vector
{p'-'^ ,
x^^)
is
an
OE
if
and only
if it
game.
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 [13], [37]). In the
next proposition, we show that for linear latency functions, there
exists a pure strategy
OE.
Proposition 7 Let Assumption 1 hold, and assume further that the latency functions
are linear. Then the price competition game has a pure strategy OE.
12
Proof. For
all
i
^ I,
let /j(x)
=
aiX. Define the set
Io^{ieI
=
\a,
0}.
Let Iq denote the cardinality of set Iq- There are two cases to consider:
>
Assume that there exist i, j € Xq such that i E Ts and j € Is' for some
for all
E S. Then it can be seen that a vector (p^-^, x"^-^) with pf ^ =
i E Iq and x'-'^ E W{p'-'^) is an OE. Assume next that for all i G 2o, we have i E Is
for some s E S. Then, we can assume without loss of generality that provider s
owns a single link i' with a^/ = and consider the case /q = 1.
• Iq
2:
s j^ s'
• /o
<
Let Bsiplf) be the set of
1:
pf^ such
max
(pf^,x°^)Garg
Let B{p'-'^)
=
[Bs[p2.f)]s^s-
that
^p^Xi.
(12)
In view of the hnearity of the latency functions,
it
an upper semicontinuous and convex- valued correspondence.
Hence, we can use Kakutani's fixed point theorem to assert the existence of a p'-'^
such that B{p'-'^) = p'^^ (see Berge [5]). To complete the proof, it remains to be
follows that B{p'^^)
shown that there
If
Iq
—
0,
(10) holds
Assume
is
x°^ E W{p'^^) such
exists
we have by Proposition
and
(p°-^,
IV (p°^))
finally that exactly
is
that Eq. (10) holds.
3 that W{p'^^)
is
a singleton, and therefore Eq.
an OE.
one of the
a^'s
(without loss of generality ai)
We
show that for all x, x E W{p'-'^), we have
EC{x,p°^) = mmj{lj{xj) +pf^}. If at least one of
to
0.
EC{x,p°^) < R,
holds, then one can
Y2iei i^i^i ^^
show that
(4), we
problem
or
Xi
=
Xi, for all
EC{x,p°^) <
i
is
^
equal
1.
Let
R
~
—
'^Substituting Xi = d
X^i=i ^i
X]t=i ^i
see that the objective function of problem (4)
—
is
convex in x_i = [x^jj^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 ^ 1, establishing our
strictly
claim.
For some x E W{p'^^), consider the vector x^-^
uniquely defined and Xi
incentive to deviate,
it
is
=
(d
— X^j^j Xj,x_i)
.
Since x_i
chosen such that the provider that owns link
follows that {p^^, x'~'^)
is
1
is
has no
an OE.
Q.E.D.
The
existence result cannot be generalized to piecewise linear latency functions or
to latency functions which are linear over their effective domain, as illustrated in the
following example.
13
Example
3 Consider a two link network. Let the total flow he d
=
Assume that the
1.
latency functions are given by
=
(
\
h{x)
,
n
1
f
\
l2{x)
0,
=
/O
<
^-5
iiO<x<S
^>^^
e >
and 6 > 1/2, with the convention that when e = 0, hix) = oo for x > 6.
show that there exists no pure strategy oligopoly equihbrium for small e (i.e.,
there exists no pure strategy subgame perfect equihbrium). The following list considers
all candidate oligopoly price equihbria {pi,P2) and profitable unilateral deviations for e
sufficiently small, thus estabhshing the nonexistence of an OE:
for
some
We
first
1.
=
Pi
2.
=
P2
0:
A
small increase in the price of provider
thus provider
profits,
= P2 > 0: Let x be the flow aflocation at the OE. If Xj =
has an incentive to decrease its price. If Xi < 1, then provider
pi
to decrease
<
3.
pi
<
generate positive
will
1
has an incentive to deviate.
1
1,
then provider 2
1
has an incentive
its price.
has an incentive to increase
p2- Player 1
its
price since its flow allocation
remains the same.
<
4.
p2
<
Pi:
For
sufficiently small, the profit function of player 2, given pi, is
e
strictly increasing as
increase
a function of
p2!
showing that provider 2 has an incentive to
its price.
OE always exists. We define a mixed strategy
mixed strategy subgame perfect equilibrium of the price competition game (see
Dasgupta and Maskin, [10]). Let B^ be the space of all (Borel) probability measures
on [0, i?]". Let /<, denote the cardinality of Jj, i.e., the number of hnks controlled by
service provider 5. Let fig G B^' be a probabihty measure, and denote the vector of these
probability measures by
and the vector of these probability measures excluding s by
We
OE
next show that a mixed strategy
as a
/j,
Definition 7
x*{p)
G
/
J[o,rY
W
(/i*,
(p) for
x*(p))
is
every p G
a mixed strategy Oligopoly Equilibrium (OE)
[0,
^s{Ps,P-s,x* {ps,p-s))
the function
if
R]^ and
d{i2l{p,) X i_i*_^{p_s))
>
Us{ps,P^s,X* {ps,P-s))d
/
{iJ-s
iPs)
X
IJ--,
{P-s))
J\O.R]'
'[0,R]'
for all s
and
/ig
G B'\
Therefore, a mixed strategy
to a different probability
OE
measure
simply requires that there
for
each oligopolist.
14
is
no profitable deviation
Example
OE
strategy
e
>
0,
but
3 (continued) We now show that the following strategy profile is a mixed
above game when e -^
(a mixed strategy OE also exists when
structure is more comphcated and less informative);
its
for the
(
Mp)-<
\
(
=
^2{P)
1-^
0<p<R{l-5),
R{l-6)<p<R,
1
otherwise,
0<p<R{l-6),
R{l-5)<p<R,
1-^
1
otherwise.
1
[
Notice that pi has an atom equal to 1 — 5 at i?. To verify that this profile is a mixed
strategy OE, let ji' be the density of /i, with the convention that jj' = oo when there is
an atom at that point. Let
strategy equihbrium,
for all Pi
it
Afj
=
{p
G Mi when the other
/i^
|
{p)
>
0}
To
.
establish that {pi,
show that the expected payoff
suffices to
player chooses p-i according to
fJ-2)
to player
/i_i
is
i is
a mixed
constant
(see [24]).
These
expected payoffs are
n {p,
I
p-i)
=
U,{p^,P-i,X {p^,P-i))dfl-^ (p^i)
/
,
Jo
which are constant
strategy
for all pi
G Mi
The next proposition shows
i
=
1.2.
This establishes that
(/^i./io) is
a mixed
that a mixed strategy equilibrium always exists.
Proposition 8 Let Assumption
strategy OE, (^^^,x°^(p)).
so
for
OE.
The proof of this proposition
given in Appendix B.
hold.
1
is
Then
game has
the price competition
a mixed
long and not directly related the rest of the argument,
it is
We
next provide an exphcit characterization of pure strategy OE.
independent
interest, these results are
most useful
Though
of also
for us to quantify' the efficiency loss
of oHgopoly 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
Lemma
OE.
4 Let Assumption
h{x?'^)+P?'^
1
hold.
If
{p'^^,x^^)
is
= min{/,(xH+p°^},
a pure strategy
OE, then
V?withxP>0,
(13)
Vzwithzp^>0,
(14)
j
k{x*)+pf^
<
R,
^xP
<
d,
(15)
15
^?^ = d^i mmj{lj{xf^) + Pj} < R.
with J2Li
Proof. Let {f^,x°^) be an OE. Since x°^ £ VF(p^^), conditions (14) and (15) follow
by the definition of a WE. Consider condition (13). As,sume that there exist some z, j G J
such that
with xf ^ > 0, x°^ >
;,(xf^)+pp^</,(x^°^)+pf^.
Using the optimality conditions
sider changing
we
see that
pf^
to pf-^
we can choose
service provider that
owns
+
e
e
(p*^"^, x'~'^)
that minj{/j(x^^)
Pj]
WE
some
WE, and
We
Eq.
[cf.
e
>
(2)], this
By
0.
i
i.
is
< R and
'Ylii=\
^?^ <
Given a pure strategy
0, then 1^ = {i}.
2
proof.
Note that
Lemma
this
assumption
if all
5 Let
Us denote the
OE
,
pf^ +
e
keeps
for a
=
x'-'^
bc^
as a
Q.E.D.
for
{p^^
we must have xf^
our price characterization.
x'-'^), if for
some
i
E
2
with xf^
>
0,
=
/^(xf^)
increasing or
Con-
hc^-
Using the optimality conditions
'^-
=
With
more profitable, completing the
Assumption
=
G H^(pf^ + e,p?f ). Hence
can deviate to pf -^ + e and increase its profits, contrais an OE. Finally, assume to arrive at the contradiction
need the following additional assumption
we have
implies that xf-^
checking the optimality conditions,
since J2i^i ^?^ < "^l' ^^^^ imphes that
a similar argument to above, a deviation to
[Eq. (2) with A
for
WE
some
sufficiently small such that x'^^
link
dicting the fact that
-\-
a
for
for
is
automatically satisfied
service providers
{p'-^^,x'-^^)
own only one
if all
latency functions are strictly
link.
be a pure strategy OE. Let Assumptions
1
and 2 hold. Let
profit of service provider s at {p'^^ ,x^^).
(a)
If
n^'
>
for
some
s'
(b)
If
n,
>
for
some
s
G 5, then n^ >
for all s
G 5, then pf^xf^ >
e S.
for all j
G T,.
Proof.
(a) For
some
lis
e
>
=
0.
j
K
= pf^ + lj{xf^), which is positive since
G Xy, define
some s. For k E Is, consider the price pfc = A' — e >
for
It
Ug'
>
for
0.
Assume
some small
WE
can be seen that at the price vector {pk,P-k)^ ^^^ corresponding
link
> 0. Hence, service provider s has an incentive to deviate to
flow would satisfy x^
pk at which he will make positive profit, contradicting the fact that {p^^,x^^)
a pure strategy OE.
is
> 0, we have p'^fx'^f > for some m G Is- By Assumption 2, we can
assume without loss of generality that Imi^m^) >
(otherwise, we are done). Let
j G Is and assume to arrive at a contradiction that p^^xf^ = 0. The profit of
(b) Since Ug
service provider s at the pure strategy
Yis
OE
can be written as
= ns+p^fx^f,
16
.
where
for
lis
some
Consider changing
.
=
n,
Note that
e
n^
n,
tlie
prices p'^^
+ {K - UxZ^ -
moved from link m to
same at the new WE. Hence,
- n, =
>
Since lm{x^^)
-
(/„(x^,^)
0, e
1
= L Assume
some a >
{p^^,x^^)
Assumption
is
is
strictly
an OE. Q.E.D.
2 cannot be dispensed with for part
this
=
link 2. Let the total flow be d
1
1 owns
and the reservation
that the latency functions are given by
=
Any price vector
0.
is
(2/3, 1/3, 0)
why
is
Consider a three link network with two providers, where provider
and 3 and provider 2 owns
h{xi)
see
link j such that the flows of
the change in the profit
lemma.
utiUty be i?
for
/,(e)).
can be chosen sufficiently small such that the above
following example shows that
Example 4
links
+ e{K -
UxZ^ - e))x^^ + e(/„(x^^ - e) - /,(e))).
positive, contradicting the fact that
The
e)){xZ^ -,)
units of flow are
other links remain the
(b) of this
K
m and j. Let p^^ = —lm{x'^)
and p^^ such that the new profit is
denotes the profits from hnks other than
K
0,
h{x3)=ax3,
l2ix2)=X2,
{pi,P2,P3)
a pure strategy OE, so ^3X3
= (2/3, 1/3, b)
= contrary
with b
>
2/3 and
(xi, X2, X3)
to part (b) of the
=
lemma. To
an equihbrium, note that provider 2 is clearly playing a best response.
= 4/9. We can represent any deviation of provider 1 by
is
Moreover, in this allocation Hi
= {2/3-5,2/3-ae-5),
{pi,Ps)
for
two scalars
e
and 6 which
The corresponding
,
lishing that provider 1
We
induce a
will
WE of (xi, X2, X3)
is
=
(2/3
+
5
—
e,
1/3
—
5,e)
4/9 — 5"^ < 4/9, estabalso playing a best response and we have a pure strategy OE.
profit of provider 1 at this deviation
is
Hi
=
OE
next establish that under an additional assumption, a pure strategy
will
never be at a point of non-difTerentiability of the latency functions.
Assumption
3 There exists some s E
differentiable for all
Lemma
6
some
i.
for
i
S
such that k
is
real-valued and continuously
Els
OE
Let (p°^, x°^) be an
Let Assumptions
1,
2
and
with min^ {p^^
3 hold.
where l^{xf^) and l~{xf^) are the right and
+
lj{xf^)}
< R and pf^xf ^ >
Then
left
derivatives of the function
li
at
xf^
respectively.
Since the proof of this
lemma
is
3 cannot be dispensed with in this
long,
it is
given in Appendix C. Note that Assumption
lemma. This
17
is
illustrated in the next example.
.
Example
5
be
tion utihty
Consider a two link network. Let the total flow be d
Assume
i? =: 2.
< X <
if
h{x)-l2ix)It
=
and the
1
reserva-
that the latency functions are given by
2(x-i)
,
i
otherwise.
can be verified that the vector {Pi^,P2^) = (1, 1), with (xf^,x^^) = (1/2, 1/2) is a
is at a point of non-differentiabiUty for both latency functions.
pure strategy OE, and
We
next provide an exphcit characterization of the
our efficiency analysis in Section
Proposition 9
Assumptions 1,
a)
Assume
The proof
5.
Let {p^^, x^^) be an
2,
and
that min^
(
??""={
OE
is
OE
prices,
which
essential in
is
given in Appendix D.
pf^xf^ >
such that
some
for
i
G I. Let
3 hold.
{pf^
+
lj{x'^^)}
<
R. Then, for
xf%{x°^),
if
xf^^K^P^)+
b) Assume that min^ {p'j^
+
v-^^'^'^^r
Ijix'j^)}
—
,
e
all s
l'j{x°^)
=
5
and
for
G
i
some
X,,
j
^
we have
Is,
(^6)
otherwise.
R- Then, for
all s
G
<S
and
i
£ Xg, we have
pf^>^,n-(:^?^)Moreover,
if
there exists
some
p?^ <
i
If
G
T
such that
^rc(^r) +
(17)
X = {i}
^
for
some
s
G 5, then
(18)
'
,
the latency functions U are all real-valued and continuously differentiable, then
Karush-Kuhn-Tucker conditions for ohgopolj' problem [Eq. (78) in Appendix
analysis of
D] immediately yields the following result:
Let (p°^, x°^) be an OE such that p°^x°^ >
for some i G I. Let Asand 2 hold. Assume also that Zj is real- valued and continuously differentiable
Then, for all s G 5 and i EiTs,^^ have
Corollary
sumptions
for all
i.
r
Pi
^
\
1
1
xf ^/^(xp^),
mm
r..ir.
P_
J i?
<^
l.f^OE\
/,(xf ^)
if
^OEv(^OE\
,
x?^l'AxY^)
£,
+ ^'^^
,
'
\
},
/;(xf^)
=
for
some
j
otherwise.
(19)
This corollary also implies that in the two link case with
differentiable latency functions
and with minimum
real- valued
effective cost less
and continuously
than R, the OE
prices are
pP = xf^(/;(x?^) + /^(xn)
as claimed in the Introduction.
18
(20)
^ J,
+
5
Efficiency of Oligopoly Equilibria
we study the
In this section,
measure of
efficiency properties of
ohgopoly pricing.
We
take as our
efficiency the ratio of the social surplus of the equilibrium flow allocation to
the social surplus of the social optimum, §(x*)/§(x'^), where x* refers to the
or the oligopoly equilibrium
[cf.
Eq.
In Section
(8)].
3,
we showed
monopoly
that the flow alloca-
monopoly equilibrium is a social optimum. Hence, in congestion games with
monopoly pricing, there is no efficiency loss. The following example shows that this is
tion at a
not necessarily the case in ohgopoly pricing.
Example
6 Consider a two link network. Let the total flow he d
utihty be i?
=
1
The
.
li{x)^0,
The unique
is
x^^ =
optimum
social
for this
=
and p^'^
(1,0)
=
1
and the reservation
latency functions are given by
=
kix)
3
~x.
is x'^ = (1, 0). The unique ME {p^^, x^^)
As expected, the flow allocations at the social
example
(1,1).
optimum and the ME are the same. Next consider a duopoly where each of these links
is owned by a different provider. Using Corollary 1 and Lemma 4, it follows that the
flow allocation at the OE, x'^^ satisfies
,
/i(xf^)
+
+
x?^[/;(x?^)
Solving this together with
oligopoly equilibrium
Ux'i^)]
xf^ + x^^ =
is x'-'^
=
= kix^ + x^^[l[{x?^) + l',ix^%
shows that the flow allocation at the unique
1
(2/3,1/3).
The
social surplus at the social
the monopoly equilibrium, and the oligopoly equilibrium are given by
1,
1,
optimum,
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 hnk 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 {p'^^,x^^) be a pure strategy OE such
that pP^xf^ >
for some i G J and mm,
{pf^ -f- lj{xf^)} < R. If /'i(xf^)/xf^ j^
l'2{x^^)/x^^, then S(xO-^)/S(x^)
Proof. Combining the
/i(xf^)
where we use the
OE
1
.
prices with the
xf^{i[{xf^)
fact that
<
WE conditions,
+ ux^)) = Hxr) + xriux'",") + ux^^)),
min, {pf^
+ lj{x?^)} <
conditions (7) to prove that a vector (xf ,xf)
/l(xf)
we have
+
>
is
R- Moreover, we can use optimality
a social optimum
xf/;(xf)=/2(xf)+xf/^(xf),
19
if
and only
if
(see also the
proof of
Lemma
4).
Since l[{x'^^)/x'^^
^
{^{x^^) j x^^ the result follows.
,
Q.E.D.
We
next quantify the efficiency of ohgopoly equilibria by providing a tight bound
on the efficiency
Section
loss in
congestion games with oligopoly pricing.
As we have shown
such games do not always have a pure strategy OE. In the following, we
4,
provide bounds on congestion games that have pure strategy equilibria.
efficiency properties of
mixed strategy
We
next study
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).
latency functions that satisfy Assumptions 1, 2, and 3. Let £/ denote the
but
in
first
larger than,
is
We
consider
set of latency
game has a pure strategy OE and
Assumptions 1,2, and 3.^ Given a parallel link network with /
and latency functions {/,}iei £ C.I-, let 0£/({/i}) denote the set of flow allocations
functions {Zi}iei such that the associated congestion
the individual
links
at
an OE.
where x^
utihty.
We
is
Z^'s
satisfy
define the efficiency metric at
some
x'^^
G OE{{li\)
as
a social optimum given the latency functions {li}iei and R is the reservation
is the ratio of the social surplus in an
In other words, our efficiency metric
equilibrium relative to the surplus in the social optimum.
the "price of anarchy", in particular
[19],
an oligopoly equilibrium, so we look
for
inf
we
Following the literature on
are interested in the worst performance in
a lower
bound on
ri{{l,},x'^^).
inf
We first prove two lemmas, which reduce the set of latency functions that need to
be considered in bounding the efficiency metric. The next lemma allows us to use the
oligopoly price characterization given in Proposition 9
Lemma
Then
7 Let
x'^^
is
[p'^^, x'^^)
be a pure strategy
OE
such that
pf^x^^ =
for all
i
G I.
a social optimum.
We first show that k{xf^) = for all i G I. Assume that lj{xf^) > for
and therefore pf^ = 0. Since lj{xf^) > 0,
G I. This implies that xf^ >
it follows by Lemma 2 that for all x G W{p), we have Xj = x^^.
Consider increasing
p*?^ to some small e > 0. By the upper semicontinuity of W{p), it follows that for all
X G W{e,p9.f), we have \xj —Xj^\ < 5 for some 5 > 0. Moreover, by Lemma 2, we have,
for all X G W{e,p'ilf), Xi > .xf
for all i 7^ j. Hence, the profit of the provider that owns
Proof.
some
j
^More explicitly, Assumption 2 implies that
and li{xf^) = 0, then J, = {i}.
if
any
20
OE
{p°^ ,x^^) associated with
{i,}j(=i
has
xf^ >
link j
is
strictly higher at price vector [e^p'^f)
(p^^, ^0£)
x^^ >
Clearly
some
for
Lemma 4
which imphes by
all i, we have
j and hence
that J2i&x^?^
Rfor
than at
-
li{xf^)
mmi^j{pf^ +
=
d-
xf^gi^
Using
some gi^ G dli{xf^). Hence, x^^ satisfies the
optimum [cf. Eq. (7) with X^ = R], and the
The next lemma
>
X]i=i h{xf)xf
either
or
(i)
and
ELi ^?'^ <
x*^^
X]j^i xf^ =
—
•^Xli=i
and
lj{xf^)
=
G dli{xf^)
0,
for
y lel,
sufficient optimality conditions for a
Q.E.D.
result follows.
R Yli=i ^f ~
subsequent anatysis.
d in the
some
^2^ ^^^
optimum
social
is
a social optimum, implying that rj{{li},x'^^)
Assume that Ei=i^i(^f)^f — ^Yli=i^i- Since x^
x'-^^
G OE{{li})
2
is
k{xf))xf
> J^iR 1
the definition of a
i
WE, we
at the
is
a social
a feasible solution to the social problem [problem
=1
the price of hnk
x^,
d for some x^^ G 0^{{h}).
G OE{{li})
= ^(/? By
= pf^ +
0,
allows us to assume without loss of generality that
Yli=i ^ii^i)^i
Then every
Proof.
=
8 Let {k}iex G Cj. Assume that
(ii)
every
li{xf^)}
/^(xP'^)
=R,
social
Lemma
p*^^, contradicting the fact that
^^ OE.
is
OE)
k{xf''))xf^.
—
1.
optimum and
(6)], we have
V x^^ G ag({/0).
=1
have xf-^ >
and /? - kixf'^) > pf ^ >
(where pf-^ is
i. This combined with the preceding relation shows
for all
x^^ is a social optimum.
Assume next that Ei=i ^?^ < d
OE price. Assume that p^^x^^ >
that
7).
by
is
Since Ei=i
Lemma 5,
^?^ <
d,
for
we have by Lemma 4 that m.mj^j{pj-\-lj{xf^} = R. Moreover,
p^xf^ >
an optimal solution of the problem
it
some x'-^^ G 0S({/,}). Let p'^^ be the associated
some j & I (otherwise we are done by Lemma
for
follows that
maximize((p,)_gj^_j.)
subject to
for all
i
G X. Hence,
P^
+
l^{x,)
=
R,
=
/,(x,)
i?.
Tx?''<d.
maximize2:>o
e S, {{pf^)ieis
Y.V^X^
pf^ +
Substituting for {pi)iei, in the above,
for all s
we obtain
2.
21
[R — U{Xz))xi
Vtels,
-ii^ls
,
x'-'^)
,
subject to
E
Xi
^
Ti,
i
^
Is,
where T^ = {xj pf^ + k{xi) = R} is either a singleton or a closed
is a convex problem, using the optimality conditions, we obtain
interval. Since this
|
R-k{xf^)-xf^gi^ =
where
£ dk{xf^). By Eq.
gi^
VzgX„Vsg5,
0,
(7), it follows
that
x^^
is
a social optimum.
Q.E.D.
Hence, in finding a lower bound on the efficiency metric, we can restrict ourselves,
without
loss of generahty, to latency functions {/j}
for
RY!i=\ ^f
some
social
the following lemma,
Lemma
be an
optimum x^ and
,
we can
also
and x^ be a
social
G Cj such that X]i=i 'i(^f )^f <
^ d. for all x^^ G US{{li}). By
^?^
=
assume that X]i=i ^f
9 For a set of latency functions
OE
Yli=i
{/j},gj, let
d-
Assumption
1
hold. Let (p*^^,
x^^)
optimum. Then
iex
iei
Proof. Assume to arrive at a contradiction that Yli^jxf^ > X^jgjxf. This implies
that x'^^ > Xj for some j. We also have lj{x^^) > lj{x^). (Otherwise, we would have
lj{xj) = l'j{Xj) = 0, which yields a contradiction by the optimality conditions (7) and
the fact that ^^^jxf < d). Using the optimality conditions (2) and (7), we obtain
R-
/,(xf ^)
- p°^
>R- /,(xf
)
- xf 5;^
some gi^ G dlj{x^). Combining the preceding with lj{x^^) >
x^^Z~(x^^) (cf. Proposition 9), we see that
for
lj{xj)
and pj'^
>
xf%ixf^)<x^g„
contradicting
5.1.1
Two
x^^ > xJ and completing
the proof.
Q.E.D.
Links
We first consider a parallel link network with two links owned by two service providers.
The next theorem provides a tight lower bound on r2{{h},x'^^) [cf. Eq. (21)]. In the
following, we assume without loss of generality that d = I. Also recall that latency
functions in £2 satisfy Assumptions
Theorem
provider.
1
1, 2,
and
3.
Consider a two link network where each link
is
owned by a
difi'erent
Then
r2{{k},x^^)
>
^,
V
{/,},.i,2
22
e £2, x^^ e ag({/J),
(22)
—
,
and the bound
the lower
is
bound
tight,
i.e.,
there exists {li}i=i^2 S ^2 and x*^^ G OE{{li}) that attains
in Eq. (22).
Proof. The proof follows a number of
Step
1:
We
steps:
are interested in finding a lower
inf
inf
bound
for the
problem
r2({/J,x«^).
(23)
Given {/j} G C2, let x'-'^ 6 OE{{li}) and let x^ be a social optimum. By Lemmas 8
and 9, we can assume that 5Zi=i^?^ ~ Xli=i ^f — 1- This implies that there exists
some i such that xf^ < xf. Since the problem is symmetric, we can restrict ourselves
G £2 such that xf^ < xf.
to {li}
We
inf
claim
r2{{k},x^^)>r^f,
inf
(24)
where
^2,1
-
mmimize,s,
yf,
(,5),>o
——
T5-5
{E)
j^-^
yOE^o
subject to
if
<
yfilfy,
i
=
(25)
1, 2,
+ yUil)' = if + ynify,
^f + yf(^f)'<^,
(26)
il
E^f =
z
l'
(28)
=l
Zi
<
;f
/;
=
0,
(29)
if
li<y?^l[.
E?/?^ =
i
(27)
= 0.
^ = l,2,
(30)
/f
(31)
l'
(32)
=l
+
{Oligopoly Equilibrium Constraints};,
f
=
1,2.
Problem (E) can be viewed as a finite dimensional problem that captures the equihbrium and the social optimum characteristics of the infinite dimensional problem given in
Eq. (23). This implies that instead of optimizing over the entire function l^, we optimize
over the possible values of li{-) and dk{-) at the equilibrium and the social optimum,
which we denote by k,
if, (if)' [i.e., (if)' is a variable that represents all possible values
of gi^ G dli(yf)]. The constraints of the problem guarantee that these values satisfy the
necessary optimahty conditions for a social optimum and an OE. In particular, conditions (25) and (31) capture the convexity assumption on li{-) by relating the values li,l[
and /f (/f )' [note that the assumption 1^(0) = is essential here]. Condition (26) is the
l'^,
,
23
,
optimality condition for the social optimum. Condition (29) captures the nondecreasing
assumption on the latency functions; since we are considering {li} such that x^^ < xf
we must have li < if. Condition (30) captures the relation of li{-) and l[{-) at xf^;
since
is
xf^ <
xf, the fact that li{xf^)
=
h{xf)
=
by Assumption
implies,
1,
that
the unique element of dli{xf'^). Finally, the last set of constraints are the necessary
OE. These
conditions for a pure strategy
cases characterized in Proposition
are written separately for
i
=
we
giving us two bounds, which
9,
the two
show to be
1, 2, for
will
equal.
More
For
t
=
exphcitly, the Oligopoly Equilibrium constraints are given by;
[corresponding to a lower
1:
/;
For
=
t
=
l[{y?^),
I',
=
l,{yf^)}
l2]
= k + y^'^il'i +
l'2]
<
l',{y^'') [cf.
Eq.
where /; = /+(j/P^) and
^OE _ „OE
~
ip'~'^,y'^^),
a
with
(33)
R:
(16)].
bound
for
pure strategy OE, (p^^,y°^), with
= Ri
R-h
R-h
'2,1
pure strategy OE,
+ y?''[l[ +
[corresponding to a lower
2:
mm,{pf^ +
for
h+y?^[l[ +
h
where
bound
l'^
=
l:;{y^^)
> y^%,
< y?"^[/'i +
(34)
/2],
Eqs. (17), (18)].
[cf.
We
will
show
in
Step 4 that
'2,2
Note that given any feasible solution of problem (23), we have a feasible solution for
problem (E) with the same objective function value. Therefore, the optimum value of
problem (E) is indeed a lower bound on the optimum value of problem (23).
We
Step 2: Let {lf,yf)i=i,2 satisfy Eqs. (25)-(28).
^fyf
Using Eqs.
(26), (27),
and
(28),
+
ihi
If
(yf)-(^f
)'
+
iyl^iliy
we have using Eq.
Next,
straints
>
ilvl
+
(yf )'(^f )'
lf—0
for all
+
{yimy
z,
<
R-
If (yf)2(/f)'
+ {yDHllY =
0,
then
again showing the result.
Eq. (32) and one of the Oligopoly Equilibrium conwe can show that
Eqs. (33) or (34)]. Using a similar argument,
hy?''
Step 3:
(35)
we obtain
let (/i,yf'^)i=i,2 satisfy
[i.e.,
R-
then the result follows.
0,
(25) that
+ ihl <
show that
To
solve
i^fy Q'l)' JiJ'i^yf ^y?^)
without constraint
problem
(E),
we
+ l2y?''<Rfirst
(36)
relax the last constraint [Eq. (30)].
Let
denote the optimal solution of the relaxed problem [problem (E)
(30)].
We
show that
[f
=
24
for
i
=
1, 2.
—
Assign the Lagrange multipliers
/if, A'^,7'^
multiplier 9^ to Eq. (28). Using the
y^
first
(^ _ jsys _ jsysy
-^M + A^yf
-/xfyf-A^yf
to Eqs. (25), (26), (27), respectively,
order optimaht}' conditions,
+ A^2 +
=0
>0
A
-
If /^
>
>
if /?
=
(37)
0,
if(rf)'>0
if(rf)'
+ 7^yf =0
>0
if
=
(38)
0,
)'
([f
if([f)'
>
=
(39)
0.
^f^^F4&^M?-/^faT)'-A"(^"f)' + 7"(rf)' + ^^ =0 ifyf>0
{R-lfyf-lM?
>
^'
^^R-rn'm?
'
^'^^"^^
and
we obtain
if
yf
^ ''^^"^^ ^ '' =0 :fyf>0
>0
ify|
=
=
(40)
0,
(41)
0.
show that [f = 0. If yf = or (Ff)' = 0, we are done by Eq. (25). Assume
and (/|)' > 0. By Eq. (38), this imphes that A"^ = |uf > 0. We claim that
in this case Eq. (37) cannot be equal to 0. Assume to arrive at a contradiction that it
is. Using Step 2 and the fact that
yf > 0, we have //f + A^ < 0, which is a contradiction
and shows that Eq. (37) is strictly positive. This establishes that /f = 0.
We next show that if — 0. If yf = or (Jf)' = 0, we are done by Eq. (25). Assume
that yf >
and [if)' > 0. Assume to arrive at a contradiction that if > 0. Since Tf = 0,
by Eq^ (26), we have that yf (/"f )' < yf (^f )', and therefore both yf >
and (ff )' > 0.
Since /f = 0, these imply that Eq. (25) is slack, and hence /xf — 0- By Eq. (38), this
shows A'^yf = 0, and hence A'^ = 0. Furthermore, since iJ,2 = ^^ = 0, from Eq. (41)
we have 6^ = 0, and from Eq. (39), we have /if = 7'^. Plugging this all in Eq. (40), we
obtain a contradiction, showing that ff = 0.
We
first
that yf
>
=
view of Eq. (29), we have li = 0. Hence we can impose the
problem (E) without changing the optimal function value, and
we can rewrite the constraint in Eq. (30) as l[ = 0. Using in addition /f = 0, we see
Step 4: Since [f
extra constraint
that for
t
=
/j
=
0, in
in
1,2,
OF
r^t
9,0s
> mmmiize
i^y^
subject to
25
1
I2
-
<
7
^"^^2
^—
vi^l'i-,
<j?% > R.
y?
Ey?'
1,
and l[ =
that satisfies Eqs.
which follows because any vector {yf^,h,l'i} 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
follows that
= r?f =
r?f
5/6.
By
inf
{k}eC2
We
and
social
=
|x.
optimum
r2{{k},x'-^^)
=
As shown
is x'^
=
in
'R 3R
and therefore
it
r2({;a,x^^)
inf
sOEau})
next show that this bound
l2{x)
~
2/?^' 2/2*^)
is {I2, ^2'
Eq. (24), this imphes that
Consider the latency functions li{x)
tight.
is
Example
6,
OE
the corresponding
is
x"^^
—
(|,
i),
=
0,
and the
(1,0). Hence, the efficiency metric for these latency functions
is
5/6, thus showing that
mm
mm
r2({/J,x«^)--.
rOEeOEf'-^
•4{l,})
Q.E.D.
5.1.2
We
General Case
next consider the general case where we have a parallel link network with / links
and S service providers, where provider s owns a set of links I^ C I. It can be seen by
augmenting a two link network with links that have latency functions
if
-
X
0,
l{x)
00
bound
that the lower
is
in the general
otherwise,
network case can be no higher than
|.
However,
this
a degenerate example in the sense that at the OE, the flows of the links with latency
functions given above are equal to
which has positive flows on
Example 7
utility
be
i?
0.
We
links at the
next give an example of an / link network
OE
and an
efficiency metric of 5/6.
Consider an / link network. Let the total flow be d
=
1.
The
social
=
1
and the reservation
latency functions are given by
/i(x)
The unique
all
=
optimum
/,(x)
0,
for this
flow allocation at the unique
OE
= -(/-
example
l)x,
is x'^
=
i
[1, 0,
=
.
1
_3'3(/- 1)''"'3(/-1)_
26
,
.
is
1
J-
2,
.
0].
It
can be seen that the
Hence, the efficiency metric for
tliis
example
The next theorem generahzes Theorem
1
is
to a parallel link network with /
Theorem 2 Consider a general parallel link network with /
where provider s owns a set of links Is C I. Then
r7({Zj,x°^)
and the bound
the lower
is
bound
tight,
i.e.,
>
V
I
{k},ei e Cj,
2 links.
and 5 service providers,
x°^ G 0S{{k}),
(42)
there exists {kji^i G C[ and x*^^ € OE{{li}) that attains
in Eq. (42).
Proof. The proof again follows a number of
Step
links
>
steps:
Consider the problem
1:
inf
inf
r;({^a,x^^').
(43)
{li\ € C2, let x°^ € OSd/j}) and let x^ be a social optimum. By Lemmas 8 and
we can assume without loss of generality that Yl,i=i ^?^ ~ Si=i ^i^ = 1- Hence there
exists some i such that xf^ < xf Without loss of any generality, we restrict ourselves
Given
9,
.
S £/ such that xf ^ < xf. Similar to the proof of
can be seen that Problem (43) can be lower bounded by the optimum
value of the following finite dimensional problem:
to the set of latency functions {li}iei
Proposition
1, it
mmimize
(44)
js,(,S)/>o
R-T:=^ityf
yf,yQ^>0
'
XjCX
subject to
if<yHify,
^
+ yf{i!y = if +
lf + yf{lfy<R,
if
yf{ify,
^
=h
=^
,
n,
(45)
n,
(46)
(47)
(48)
h <
(49)
If,
= 0,
if If = 0,
= 0,
J3 = {1}
for somes
if
^f
+ {Oligopoly Equilibrium Constraints}
(50)
l[
27
(51)
j,
t
=
1,2.
The new
feature relative to the two
hnk case
the presence of X^'s as choice variables
is
to allow a choice over possible distribution of links across service providers (with the
=
constraint [j^Ig
T- left
we have added
In addition,
by
Lemma
Ste]) 2:
The
implicit).
again written separately for
t
=
oligopoly equihbrium constraints, which are
1,2 for the two cases in Proposition
depend on
9,
constraint (51) to impose Assumption 2 (recall that
Jj's.
xf ^ >
5).
Let
{if, (if)' ,liJ[,yf ,y^^)
the preceding problem
[i.e.,
be an optimal solution of the relaxed version of
without constraint
(50)].
,n are decoupled
Note that the constraints that involve {if, {lf)'-,yf) for i = 2,
and have the same structure as in problem (E). Therefore, by the same argument used
in Step 3 of the proof of Proposition 1, one can show that if —
for
to show r| =
each i = 2,
n. Similarly, one can extend the same argument given in the proof of
Proposition 1 to show that if = 0.
.
.
.
.
Step 3: Since if
from Proposition
9,
=
0, it
l[
follows that
—
and Xi
=
=
fi
i=2
,,,i;.>o,
'~^„'
1
/
..OEji
< yY^'l,,
k+y?^l[<R,
subject to
k
^'
>
=
=
also used the fact that /f
2,
0, for i
of constraints are due to the convexity
case, the
constraints (given
4-'
Let
l\
=
=
2,
.
.
,
.
/,
i-2,...,/,
(52)
.
,
.
.
/.
assumptions on the
li.
Similar
second set of constraints are due to the oligopoly equilibrium
see the
OE
price characterization in Proposition 9).
denote an optimal solution of the preceding
{{liJ'i)i=2,...j, {yi'^)i=i,...,i)
problem. Assign the Lagrange multipliers
of the problem.
i
R,
first set
two link
Step
'
R
ypB>0, ,= 1,...,/
where we have
and we can assume without
[Eq. (49)],
{1}. Therefore, using the price characterization
the structure of the problem simplifies to
minimize
rff>
to the
.
,
loss of generality that
The
.
/x^,
Xi,-y
and 6 consecutively to the constraints
Using the optimality conditions, we have
7
2^j=2
^'
R
=
if
>0
ifyp^^O,
yY^ >
(53)
I'
+ ^, +
A,
>
28
if
if
/j
>
/,
=
(54)
0,
and
for
i
=
2,
.
.
,
.
/,
-ftsr+7sr7^^^^+A.sr =0
if':>o
(55)
(e;=4)
>0
By
we have yp^ >
feasibility,
for
alH
=
2,
.
0.
=
it can also be seen that y^^
which together with Eq. (55) implies that Aj
0,
.
Moreover, by Eq. (54), we have fii > 0, showing that k — yf^l'i for each
(i.e., the corresponding constraint is not slack). This implies that
k
Hence k
(52)
is
= R/2
>
0.
=
Hi
/.
,
.
=
Using symmetry,
0.
Hence, by Eq. (53), we have 7
if/l
and
not slack at
y^^l'i
+ y?% =
= R/2
V
R,
each
for
i
=
=
2
2,
.
.
2,
,
.
.
.
.
/.
=
2,
.
.
,
.
/
/.
,
We
(otherwise,
{(li, l'i)i=2,...,i, {yi^^)i=i,...,i)
i
can also see that constraint
we can improve the objective
function value), implying that
y?^
R.
Combining with the
feasibility constraint
2
z^
^ j=2
yf^
=
1
—
/'
A^
'J
j=2
X^^^o y]
we
,
see that
/'
S
from which we obtain
/
Hence we have
Rl
^OE
^
2 3
^
R
5
6'
showing that
Finally,
Example
7 shows that the preceding
min
min
bound
ri{{li\,x
is
tight,
i.e.,
'5
OE\
)
Q.E.D.
Although Example 7 and Theorem
large network
is
not the case
if
increasing d to nd). In this case,
oligopoly models
tends to
2
show that the
efficiency loss in
an arbitrarily
with / -^ 00) can be as high as in the two link network, the same
we start with a given / link network and replicate it n times (also
(i.e.,
(e.g.,
[28],
[14],
it
can be shown, as
[22],
[39],
[40]),
1.
29
in analyses of the limit
that as
n
^
behavior of
00, the efficiency metric
5.2
Mixed Strategy
As we
illustrated in Section 4, congestion
Equilibria
games with latency functions that satisfy
and 3 may not have a pure strategy oligopoly equilibrium (cf. Example
3). Nevertheless, as shown in Proposition 8, such games always have a mixed strategy
equihbrium. In this section, we discuss the efficiency properties of mixed strategy 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
Assumptions
candidates.
1, 2,
The
first
considers the worst realization of the strategies, while the second
mixed
focuses on average inefficiency across different realizations of
to the first metric as worst-realization metric,
second as the average metric, and denote
it
and denote
by ff{{li}).
We
strategies.
refer
by ff^Hk}), and to the
We discuss both of these
it
briefly.
Given a set of latency functions {kjiei let OM{{li}) denote the set of mixed strategy
some fi € OM{{li}), let Mi{fi) denote the support of fii as defined before
,
equilibria. For
in
Example
3
[in
Mi = {p
particular, recall that
OErn{{U}^^J)
=
X G W{p),
{x
for
I
We
\
iJ,[{p)
some p
>
s.t.
Further,
0}].
Pi
6
let
Mi{fj,) for all i}.
define the worst-realization efficiency metric as
fY{{h})=M
where
rj is given
by Eq.
Jnf
inf
(21).
Similarly, the average efficiency metric
mi
rf{{k})=ini
{i,}eC! fieOMilh})
In the next example,
rj{{k},x^^),
is
[
J
we show that the
defined as
[ rj{{k},xO^ {p))d^,,
d^^s.
J
worst-realization efficiency metric for
games
with no pure strategy equilibrium can be arbitrarily low.
Example 3 (continued)
Pi
G Mi
WE
for the
Consider the prices pi
= R
and p2
=
i?(l
—
6) that satisfy
unique mixed strategy equilibrium given in Example 3 as
at these prices
is
e
—
>
0.
The
given by
x^^ = (l-(5,5),
and the worst-realization
efficiency metric
is
rT({/a)
= l-5^
which as 5 ^> 1 goes to 0.
Next consider the average efficiency metric, ff{{li}). Once again, consider the limit
as e — 0. Recall that we have characterized the unique mixed strategy OE above, and
>
define r (pi,P2) as the inefficiency at the price vector (pi,P2), so that
pR
rf{{Q)=l
pR
r[pi,p2)dni-xdn2
I
J{\-5)RJ{1-5)R
30
Wardrop equilibrium
Clearly, using the
flow vector derived in
Example
3 before,
we have
that
r(p.P2)
=
|^_^(,^
ifp.>p,
'
and thus,
'
Thus to
we have
r
-K
rR
compute the
to
Denoting
last integral.
by A,
this
{P1-P2) dni X
dii2
J(1-S)rJp2
{l-5)RJp2
X
djli
—
pj-fii
/
P2dlJ,\
X
d//2
J(1-6)rJp2
(\.-5)R J (\-5)R
^^~/^^
\(>
(1-6)R
rR
rR
rPi
p-[dfl2
recall that
dii2
j-R
I
R
Now
dni X
R
(\-5)R J (\-5)R
we need
calculate rf{{li}),
A = —
-P2]
(pl
rf[{m
m
has an atom equal to
1
) dp,
J
—
/''
-
(pi)
(1
-
6)
Rdp2
iP2)
J{l-5)R
5 at R, so
A =
= (1 - 6f - (1 - 5) + (1 - 5)[\nR= -{l-5)5-{l-5)\n{l-6)
It
can be calculated that
A
reaches a
ln((l
maximum
Therefore, in this example, ff{{li}) reaches 0.84
5/6).
We
-
6)R)]
of approximately 0.16 for 5
«
^
(in fact, slightly greater
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
is left
as an
bound
0.8.
than
for the
open research
question.
Conclusions
6
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
without prices, price-setting by a monopolist always achieves the social optimum. Second, and in contrast to the
monopoly
result, oligopoly equilibria
providers compete are typically inefficient. Third, there
ciency in pure strategy oligopoly equihbria. This
large parallel
hnk networks.
Mixed strategy
inefficient realizations.
number
of concluding
bound
comments
are useful:
31
where multiple service
a tight bound of 5/6 on
is
obtained even
Finally, pure strategy equilibria
latency functions are highly convex.
A
is
may
fail
ineffi-
for arbitrarily
to exist
when some
equilibria can lead to arbitrarily
Our motivating example has been the
flow of information in a communication net-
work, but our results apply equally to
in
traffic
assignment problems and oHgopoly
product markets with negative externalities, congestion or snob
inally suggested
by Veblen
efi^ects (as orig-
[38]).
Our analysis has been quite general, in particular, allowing for constant latencies
and capacity constraints. Some of the analysis simplifies considerably when we
specialize the network to increasing and real-valued (non-capacity constrained) lais important for our
tencies. On the other hand, the assumption that /i(0) =
efficiency
bounds. This can be relaxed to derive a slightly lower bound on the
ciency in pure strategy
of the
OE, but
form of equilibrium prices
in
OE
(unless
are increasing rather than non-decreasing).
One
simplifying feature of our analysis
is
we assume that
We
all
latency functions
leave this for future work.
the assumption that users are "homo-
geneous" in the sense that the same reservation utihty, R, apphes to
is
possible to conduct a similar analysis with elastic
traffic),
effi-
this significantly complicates the characterization
all
users. It
and heterogeneous users
but this raises a number of new and exciting challenges.
(or
For example,
monopoly or oligopoly providers might want to use non-linear pricing (designed
as a mechanism subject to incentive compatibility constraints of different types of
users, e.g., [42]). 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-realization efficiency metric in mixed strategy
oligopoly equilibria can be arbitrarily low, a bound for average efficiency metric is
an open research question.
32
Appendix A: Proof of Proposition 4
7
If
{p^"^,x^'^)
Assume
an ME, then
is
it is
SPE by
an
definition.
Let (/'^^,x*^-^) be an SPE.
some p >
to arrive at a contradiction that there exists
and x G W(j)) such
that
n(p^^^,x^^^)<n(p,f).
If
W{p)
is
a singleton,
we immediately obtain
(56)
a contradiction.
Assume that W{p)
not
is
a singleton and X]i=i ^i — X^i=i ^i fo'^ ^^1 x, x € W{p). By Lemma 2, it follows that
n(p, x) = n(p, x) for all X € l'^(p), which contradicts the fact that {p^^^ ,x^'^^) is an
SPE.
Assume
finally that W{ji) is
I
not a singleton and
I
y^X; <
y_]xi,
i=l
i=l
For this case, we have pi
=R
for all
i
some
for
[cf.
Eq.
[cf.
i
E I.
(5)].
Eq.
To
(2)]
see this, note that since Yli=i ^i
—
cxD
for
some
implies that x^
=
bd
for all
We
If bci
=
hold with A
show that given
<5
i
>
^(p^
x^
there exists
x')
with
Xj 7^ Xj},
(57)
<
^^
the
WE optimality conditions for
Assume that p < R. By Lemma 2, li{xi) =
for
we get a contradiction by Eq. (2). Otherwise, Eq.
0.
E I,
i E L Since
0,
x G W{p).
E T, where
T = {iGl\3x, xE W{p)
X
x,
>
u{p,
=
x^ for all
some
x)-5,
e
>
y
i
^ I,
all
(2)
this contradicts Eq. (57).
such that
x'
E w{p'),
(58)
where
The preceding
relation together with Eq. (56) contradicts the fact that
an SPE, thus establishing our claim.
We
first
show that
Y^xt>Y^x,.
iez
Assume
{p^'^^ ,x'^'^^) is
(60)
iei
to arrive at a contradiction that
Y,xi<Y.^^-
(^1)
i€l
This implies that there exists some j E I such that xj < Xj (which also impUes that
Xj- < hcj)We use the
optimality conditions [Eq. (2)] for x and x' to obtain the
WE
following:
33
There
•
exists
some A >
such that for some
E I,
i
R-k{x^) -p^ = 0>\,
where we used the
T
G
i
[cf.
Eq.
facts that li{xi)
(56)].
=
Since A
0,
=
we
0, p,
=
i? [cf.
have, for
R-k{x^)-p, <0
There
exists
some
>
A*^
<
Xj
and
=R—
Pj
e),
and
for all
R-li{x\)-p,
If
A"^
=
0,
i
is
a contradiction.
If
A^
=
iix,
X;
>
for
some
(62)
bc,.
< A^
>
0,
>
e
>
<X'
> A^
=
(63)
^ I,
i
ifxj
=
0,
if
>
0.
=
then by Eq. (63) and the fact that lj{xj)
/j(xp
which
and
such that
e-/j(a;;.)
(since xj
2]
ifx, <6c.,
>0
•
Lemma
^ T,
all i
xJ
(64)
(Lemma
2),
we obtain
lj{xj),
then X^^^j xJ
—
d.
Assume
first
that x\
<
x, for all
^ T. Then
J2^t =
d-J2xl>d-J2x,>J2^^^
i€l
lil
which yields a contradiction by Eq.
Eqs. (62) and (64), we have
(61).
1^1
Assume next
i€l
that x^
>
Xfc
some k ^ 1. By
for
R-lk{xl)-pk>y,
R - kixk) - Pk < 0,
which together implies that lk{ik)
>
^^(^^fc))
yielding a contradiction
and proving Eq.
(60).
Since W{p) is an upper semicontinuous correspondence and the i"* component
W{p) is uniquely defined for all i ^ 2", it follows that Xi{-) is continuous at p for
i ^I. Together with Eq. (5), this implies that
^(p^x')
of
all
= ^piXi + ^pi{xl-x^) + ^{R-e)xl
i^l
i^t
>
^PiXi + ^(/2-
i&X
e)xj
+ ^Pj(x- -
Xj)
I
= ^PiX^-eY^x^ +
i=l
Y^p^{x'i-x,)
zGJ
ifX
I
>
y^^PiXj
-
6,
1=1
where the
the proof.
last inequality holds for sufficiently small
Q.E.D.
34
e,
establishing (58),
and completing
Appendix B: Proof
8
We
of Proposition 8
prove Proposition 8 using Theorem 5* of Dasgupta and Maskin
will
by stating a
the strategy space of player
s,
S
an
slightly simplified version of this theorem. Consider
denoted by P^, be a closed interval of M""
We
[10].
start
player game. Let
for
some
jig
€ N,
s
and
payoff function by ns{ps,P-s)-
its
We
also denote
p
—
{ps,p-s),
P =
Yl Ps, and
s=l
5
=
P_s
To
Ps-
Y\
Theorem 5*
state
we need the
in [10],
following three definitions.
k=l,k^s.
Al
Definition
Let
=
7r(p)
t^
Y1s^sT^s{Ps,P-s)-
[p) is
upper semicontinuous
in
p
if
for
allp,
<
limsup7r(p)
A2 The
Definition
for all Ps
E
profit function tTs{Ps,P-s)
A 6
Ps, there exists
Alim
+
is
such that for
[0, 1]
inf Trs{Ps,P-s)
7r(p).
(1
-
weakly lower semicontinuous
all
Ps
in ps
if
€ P-s,
>
A) lim inf tTs{Ps,P-s)
TTsiPs^P-s)-
Ps]Ps
PslPs
D
< Dg and each
Definition A3 For each player s, let Ds E N. For each D with
<
k ^ s with 1 < A; < 5, let /^ be a one-to-one, continuous function. Let P{s) be a subset
of P, such that
P(s)
=
In other words, P[s)
measure
zero).
EP\3k^s,3D, 0<D<Ds
{(pi, ...,ps)
is
a lower dimensional subset of
Theorem 5*
P
s.t.
pk
(which
= fg
is
fe)}
.
also of Lebesgue
in [10] states:
Theorem Al (Dasgupta-Maskin) Assume
that tTs{ps,P-s) is continuous in p except
on a subset P**of P{s), weakly lower semicontinuous in ps for all s and bounded, and
that k{p) is upper semicontinuous in p. Then the game [(P^, tt^) s = 1, 2,
5] has a
mixed strategy equilibrium.
•
;
We
show that our game
satisfies
function x* {ps,p-s) from the set of
ns{ps,P-s)
=
the hypotheses of
Wardrop
equilibria,
Theorem
AL We
•,
will select a
W (ps,p_s), such that
V
Us{ps,p-s,3:* {ps,p-s)),
s
e 5,
— [0,P] ' and ^iXi{p) < d for all p,
x G W{p), Trs{ps,P-s) is clearly bounded.
Since
{ps,p-s) is an upper semicontinuous correspondence, we select x* {) such
that will satisfy these hypotheses. First, since P,
and
all
W
that
lim inf
PsTpq
Y] x*{ps,p-s) = Y]
— xUps,P-s),
—
jeis
J
els
35
V
p^
>
0,
V p.^ >
0.
(65)
^
f
Given p
>
= pk
since pj
0,
I
k & I, where
for all j,
is
defined in Eq. (5) in
V
Ps
Lemma
2, it
follows that
lim
mf
=
TTsips^P-s)
hence ensuring that 7rs{ps,P-s)
claim that we have
=
T^siPs.P-s),
^s{Ps,P~s, x*
P-s))
iPs^
>
is
0,
V Ps >
0,
weakly lower semicontinuous.
We
Y^ x*ip) > J2 ^j(p)'
j
Assume
Vp >
V x e W{p).
0,
(66)
j
the contrary. This implies that there exist
some p >
6 5, and x € W{p)
0, s
such that
J^x,(p)>5]x*(p).
jeis
By
we have that X^,£i^ 2:*(p",p_s) — X^,gi^
Combined with Eq. (67), this implies that
Eq. (65),
{p"}
I ps-
some
Ps
<
3
els
WE by Proposition
monotonicity of
Ps, contradicting the
Next, we show that 7rs(p5,p_s)
is
some sequence
a^j(Ps,P-s) for
*
jeis
for
(67)
jeXs
2.
continuous in p except on a set P**.
We
define the
set
P**
By
Moreover, by
Lemma
P=
which
is
{p
I
2, it
Pj
=
pfc,
W{p)
is
for
set.
not a singleton}.
I
we
see that ns{ps:P-s)
some
C
is
continuous at
all
p ^ P**.
P, where
j 7^ k}
U
{p
\
pj
=
R, for some j},
This establishes the desired condition
for
Theorem Al.
we show that
7r(p)
continuous at
all p.
=
5]]7r,(p,,p_s)
Given some p
then we automatically have that
is
{p
follows that P**
a lower dimensional
Finally,
is
=
the upper semicontinuity of W{p),
vr is
>
0,
=
X^Pe2;*(p),
2"
define
as in Eq. (5) of
continuous at
p.
Assume
that
Lemma 2. If J = 0,
X 7^ 0. Since xf^{-)
continuous at p for all i ^ T and pj = pk for all j, /c G 1", it is sufficient to show that
i^ continuous at p, i.e., for a sequence {p"} with p" G [0,]R]^ and p" -^ p, we
Yliiex^lip)
show that
lim
n—too
V<(p") = Vx*(p).
-^^
—
—
•'^
iei
lEi
Define
J(p")
Since
two
Xi(-) is
continuous at p for
all i
=
5^x*(p").
^ J, we have
cases:
36
ci(p")
-^
(i(p)
=
Ylr^i
^-iip)
Consider
>
• X^jgibc,
d
—
n
sufficiently large
• X^iei^c,
^
—
d
Since x*{p)
d{p).
=
Eq. (66)] and li{x*)
i
—
By
the
d-,
maximum
the
is
El,
—
X^iex^KP")
d{p).
that Yliej^iip)
large such that
for all
^i-norm element of W{p)
this implies that
Yliei^iiP)
~
^
^^'^
[cf.
^°'^ ^^^
establishing the claim.
same reasoning
J2iei ^d- Moreover, for
as in the previous part, this implies
all e
>
0,
there exists
some n
sufficiently
\j2x:{pn-J2bc]<e,
establishing the claim.
The preceding enable
us to apply the theorem, completing the proof.
Appendix C: Proof
9
We
first
Lemma
of
Q.E.D.
6
prove the following lemma:
Lemma 10 Let (j)°^,x^^) be an OE such that min^ {pf^ + lj{xf^)} < R. Let Assumptions 1 and 2 hold. If pf^xf^ > for some j E X, then W{p'^^) is a singleton.
for some j E X, it foUows, by Lemma 5, that p^^xf^ >
show that for all x E W{p^^), we have Xi < xf^ for all i. If
then by Lemma 2, Xi = xf^ for aU x e W{p°^). If ii(xp^) = 0, then
some s by the fact that xf-^ >
and Assumption 2, which implies that
>
Proof. Since p^^x'^^
for all
i
/i(xf^)
E 1.
>
0,
Ts
=
Xi
< xf^ by
{i} for
We
the definition of an
Since min^Xi
< xf^
first
+
{pf^
x E VF(p
OE
<
lj{x^^)}
(cf.
Definition 10).
R, we have
YlLi^?^ =
^-
Moreover, the fact that
impHes that min^ {pf^ + lj{xj)} < R as well, and therefore
showing that Xi = xf^ for all x E W{p^^), for alH 6 X. Q.E.D.
for all
ELi ^i =
^.
Proof of
Lemma
)
W^e
6.
first
prove this result for a network with two hnks. Assume
to arrive at a contradiction that
/+(x°^)
Let
is
{e*^}
be a scalar sequence with
the load of link
Lemma
10, the
1
WE
| 0.
l^{x^^).
it
follows that xi{e'^)
—
»
xf^. Define
Similarly, let xi{—e^) be the load of link 1 at a
a singleton,
we
also
e'=
WE given price vector
have Xi(— e*^) -^
d-x,{p0^pO^)
dpi
xi(6^)-xf^
j.^
fc^oo
dpi
is
Xi(e'^)
WE
d^x.jpo^pr) ^
Since W{p'-'^)
(68)
Consider the sequence {xi(e'^)} where
given price vector {pf^ + e^,P2^). By Proposition 1 and
correspondence W(jp) is upper-semicontinuous and W{p'^^) is a
at a
singleton. Therefore,
e*^
>
^
^^_^
k-^oo
37
xf-^. Define
x?^-x,i-e')
e^
(pf"^
—
e^,p2^).
+
Since mirij [p^
lj{Xj
)}
<
R,
it
can be seen using
d+x,{p?^,p§^)
4 that
-1
^
-
dpi
Lemma
+ lt{x?''y
liix?'')
and
-1
d-x,{p?^,p§^)
^
- irix?'')+i2ix?''y
dp,
Since l^i^i^)
=
^i~(2^i''^)
by Assumption
3, this
Consider the profit of service provider
5pi
Since
pf^
is
a
maximum
1,
of Ili{-,P2^),
= pf^xf^.
ni(pf-^, pj*^)
^.^ ^l(pf^
k^oo
_
(68) yields
dpi
dpi
5+ni(pf^,p?^)
combined with Eq.
Define
+ 6^pp^)-^l(pf^,p^^)
ffc
we have
5+ni(pf^p^^)
^^
a-ni(pr,p^^)
o^
o^ a^xibf^p?^) ^
„
,„„,
and
o^ a-xi(pf^p?^)
dpi
which,
when combined,
dpi
yields
d^xM^P^'^) <
-
d-x,{pO^,pO^)
dp,
which
is
We
for all
a contradiction by Eq.
api
(71), thus
^
'
showing that we have l2{x2^)
=
^2^(2;^'^).
next consider a network with multiple links. As in Eqs. (69) and (70),
i
G X,
5+x^(p^_
^—
api
Using the same
line of
argument
x^(e^)-xf^
fc-^oo
dpi
-
= hm
^
as above,
^.
r^^
/o^oo
e*^
we obtain
-1
a+Xi(pO^)
dpi
e'=
-/r(x?^)
38
+ ^
'
.
'
we
define
,
Let
1
G 2s, and without loss of any generality, assume that
Assumption 3). For all i € Z^, i 7^ 1, we obtain
k^s for
all
i
E Is are smooth
(recall
a+x,(p°^)
1
^
<
To
arrive at a contradiction,
assume that l'^{x^^)
>
l~{xf^)
for
some j ^
l^.
Then
the preceding two sets of equations imply that
""^
•
dp,
dpi
for all
i
e
^
Is-
Next, Eqs. (72) and (73) for multiple link case are given by
qi(P^ , ^o. ^ ,o. a-^.(P°^)
dp,
which are inconsistent with Eq.
for the multiple
10
hnk
case.
V
^
dp,
_^^^^
(76), leading to
>
pP^ ^'-f"^'
dp,
0,
(77)
a contradiction. This proves the claim
Q.E.D.
Appendix D: Proof
of Proposition 9
We first
assume that miuj {p'?^ + lj{xj'^)} < R. Consider service provider s and assume
loss of generality that 1 € Tj. Since pf^xf^ >
for some j G Is' and s' G 5,
it follows by Lemma 5 that pf^xf^ >
for all i E I. Together with Lemma 4, this
implies that {ipf^)ieis:^'^^) is an optimal solution of the problem
without
maximize((p.),gj^_i)>o
^PiXi
(78)
ieis
subject to
h{xi)
h{xi)
+ Pi =
+ Pi ^
li{xi)
k{x^)
li{xi)+Pi<R,
y^^xj
<
d.
39
+ Pi,
+ pf^
iEls —
{l},
l^Is,
(79)
By Lemma
for all
i
6,
we have that li is continuously differentiable in a neighborhood
mapping of a convex function is continuous over the
(since the gradient
function
is
Therefore, by examining the Karush-
differentiable, see Rockafellar [29]).
Kuhn- Tucker conditions
of xf-^
set the
we obtain
of this problem,
V^GJ„
pr = xf^/:(xf^)-^,
(80)
where
(
if
0,
9^1 _ ^^261^4^,
-
lj{xf^)
for
some
j
^ I„
(^1)
otherwise,
showing the result in Eq. (16).
We next assume that min^ {pf^+lj{x'^^)} = R. Using the assumption thatp^^x^^
for some j E I and Lemma 4, this implies that
pfs
and thus
for all s
G 5, x'^^
=
>
Vi,
i?-/,(xf^),
an optimal solution of
is
maximize
subject to
/.(-^
Xi
E
~
h{xi))xi
V
Ti,
i
^ Is
Y^Xi<d,
(82)
where Tj = {xi pf^ + k{xi) = R} is either a singleton or a closed
is a convex problem, using the optimality conditions, we obtain
interval. Since this
\
R-k{x'='^)-xf^gi^
where
^s
>
is
=
9s,
"it
els,
the Lagrange multipher associated with constraint (82), and
dli{xf^). Since l~{x^^)
<
gi^,
gi^
G
the preceding implies
p^ = R- h{x?^)
> x?\{xf^),
V
^
e J,
proving (17).
To prove
p''
=
some
Z with Xg =
some s and the sequence of price
be a sequence such that x'^ € W{p'')
for all k. By the upper semicontinuity of W{p), it follows that x*^ —v x with x G W{p^^)
and X < x'-'^ (see the proof of Lemma 10). Moreover, by Lemma 2, we have x\ > xf-^
for all k, which imphes that x, > xf^, showing that xf —> xf^. We can now use Eqs.
(73) and (77) (by substituting i instead of 1 and using Is = {?'}) to conclude that
(18), consider
vectors {p''} with
{pf^
—
i
G
{i} for
e'',P-f)- Let {x*^}
Q.E.D.
40
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