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IB31
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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
PRICE AND CAPACITY COMPETITION
Daron Acemoglu
Kostas Bimplkis
Asuman Ozdaglar
Working Paper 06-37
December 12, 2006
Room
E52-251
50 Memorial Drive
Cambridge,
MA 021 42
This paper can be downloaded without charge from the
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Research Network Paper Collection
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1
at
Dewey
Cr
w
^i^»
'
i.uVf
U§RARlE S
Price and Capacity Competition
Daron Acemoglu* Kostas BimpikisJ and Asuman Ozdaglar*
December
12,
2006
Abstract
We
study the efficiency of oligopoly equilibria in a model where firms compete over capacities
and prices. The motivating example is a communication network where service providers invest in
capacities and then compete in prices. Our model economy corresponds to a two-stage game. First,
firms (service providers) independently choose their capacity levels.
levels are observed,
their
demands
they set prices. Given the capacities and
across the firms.
We
first
Second, after the capacity
(consumers) allocate
prices, users
establish the existence of pure strategy
and characterize the
subgame
perfect
These equilibria feature pure
strategies along the equilibrium path, but off-the-equilibrium path they are supported by "mixed
strategies. We then investigate the efficiency properties of these equilibria, where "efficiency" is
equilibria (oligopoly equilibria)
set of equilibria.
defined as the ratio of surplus in equilibrium relative to the
the worst oligopoly equilibria of this
game can be
first best.
arbitrarily low.
We show
However,
if
that efficiency in
the best oligopoly
(among multiple equilibria), the worst-case efficiency loss is 2(y/~N—l)/(N —
bound is tight. We also suggest a simple way of implementing the best
1)
oligopoly equilibrium. With two firms, this involves the lower-cost firm acting as a Stackelberg
leader and choosing its capacity first. We show that in this Stackelberg game form, there exists
a unique equilibrium corresponding to the best oligopoly equilibrium. We also show that an
alternative game form where capacities and prices are chosen simultaneously always fails to have
a pure strategy equilibrium. These results suggest that the timing of capacity and price choices
in oligopolistic environments is important both for the existence of equilibrium and for the extent
equilibrium
with
N
is
selected
and
firms,
this
of efficiency losses in equilibrium.
Keywords:
JEL
capacity, competition, efficiency loss, industry structure, investment oligopoly.
Classification: C72, L13.
'Department
of
Economics, Massachusetts Institute of Technology, daron@mit.edu
^Operations Research Center, Massachusetts Institute of Technology, kostasb@mit.edu
* Department
asuman@mit.edu
of
Electrical
Engineering
and Computer
Science,
Massachusetts
Institute
of
Technology,
1
Introduction
This paper studies oligopoly competition in the presence of capacity investments. Our motivation
comes from
large-scale
communication networks, particularly the Internet, which has undergone
major decentralization since the mid 1990s. These changes have spurred interest
ized network protocols
new
decentral-
and architectures that take into account the noncooperative interactions
A
between users and service providers.
is
in
key question in the analysis of these new network structures
the extent of efficiency losses in the decentralized equilibrium relative to the efficient allocation
Most
of resources.
of the
work
from
in this literature investigates the efficiency losses resulting
the allocation of users and information flows across different paths or administrative domains in
an already established network
(see, for
example,
[13],
[2],
[3],
[1],
[7]).
Arguably, the more im-
portant economic decisions in large-scale communication networks concern the investments in the
structure of the network and in bandwidth capacity. In fact, the last 20 years have witnessed significant investments in
is
to
broadband, high-speed and optical networks. Our objective in
model price and capacity competition between service providers and investigate the
properties of the resulting equilibria. Following previous research in this area,
bounds on the
efficiency losses
Our model
fixed
this
amount
by these
consists of
of flow
firms.
by providing various worst-case performance
N firms
(service providers)
paper
efficiency
we provide
explicit
results for equilibria.
and a mass of consumers wishing
to send a
from a fixed source origin to a given destination using subnetworks operated
Each user has an
inelastic
demand with
a reservation utility R.
Firms face
a linear and potentially different cost of investing to expand the capacity of their subnetwork.
For simplicity,
we assume
that once capacity
is
installed, there
is
no additional cost of allowing
consumers to use the subnetwork. In our baseline model, firms play a two-stage game. They
first
choose the level of capacity in their subnetwork, and then set prices for consumers to use their
subnetwork. This game has an obvious similarity to Kreps and Scheinkman's well-known model
of quantity
demand
is
precommitment and
price competition for
two
firms,
[9],
but
it
is
simpler because
inelastic.
For expositional purposes, we start with the special case with two firms. For this case,
characterize the set of pure strategy
equilibrium always exists.
As
in
[9],
subgame
subgame
perfect equilibria
we
fully
and prove that a pure strategy
perfect equilibria in which firms use pure strategies
along the equilibrium path are nonetheless supported by mixed strategies off the equilibrium path.
As part of our equilibrium
analysis,
we
also provide a complete characterization of the set of
mixed
strategy equilibria following any choices of capacities by firms.
We
then investigate the efficiency properties of equilibria
in
the worst-case scenarios.
We
quantify efficiency as the ratio of social surplus in equilibrium relative to the
social surplus (in the hypothetical first best). Since the
equilibria, there are
game
maximum
typically has multiple pure strategy
two possible approaches to quantifying worst-case scenarios. The
referred
first,
computer science and previous network economics
to as "the Price of Anarchy" in the
value of
literature,
looks at the worst-case scenario in terms of the possible values of the parameters and selects
the worst equilibrium
The second,
there are multiple equilibria.
if
referred to as "the Price of
and then looks
Stability" selects the best equilibrium for any given set of parameters
worst-case values of the parameters (see
Our
first result is
[8],
[2]).
that even in the simplest structure with linear costs, the Price of Anarchy
equal to zero, meaning that the equilibrium can be arbitrarily
is
that once
that
if
we
focus on the Price of Stability there
the (socially) best equilibrium
capacity competition
is
is
form
in
between two firms
also suggest a simple
way
if
is
of
games
which the firm with the lower cost
show that
also
is
Our second major
2v2 —
2
inefficiency that
~
the "incorrect" equilibria arise, there
show that our main
2
~
is
selected, capacity
equal to zero.
two
in reverse order
according to then
firms, this corresponds to a situation
of capacity investment acts as the Stackelberg leader. This
it
may
of regulation, for
example by giving a
arise as the focal point in the
game.
5/6.
results generalize to the
is
game with
N
firms.
For this case, we
again a
bound on the
The
bound
is
the Stackelberg
again
l)/(N
—
1),
and
also tight.
differences in the structure of equilibria
game and
is
Price of Stability (the combination of
N—
this
show that
and worst equilibrium)
r
worst-case parameters with the selection of best equilibrium), equal to 2(\/
we show that
We
(pure strategy) equilibrium and inefficiency in this
(the combination of worst-case parameters
Moreover, there
and
implementing the best equilibrium, by considering a game
game has a unique
bounded by 2\/2 —
Anarchy
from
result
characterize the pure strategy equilibria using a slightly different argument, and then
the Price of
meaning
5/6,
may
result
a high degree of efficiency.
sufficient to ensure
game may be implemented by some type
this Stackelberg
equilibrium
of
the "appropriate" equilibrium
first-mover advantage to lower-cost firms, or
We
if
which firms make their capacity choices sequentially,
"Stackelberg"
bound
maximum
costs of investing in capacity. In the special case with
in
inefficient.
is
no more than approximately 1/6 of the maximal social surplus. These
could be very large inefficiencies, but
We
a tight
is
selected, the
results suggest that even in the simplest capacity
price competition
for the
game
and the extent
of inefficiency
suggest that the timing of moves
is
between our baseline
an important determinant of
the extent of inefficiency in this class of games. This raises the natural question of
how
the set of
equilibria will be affected
when
We
and capacity decisions are made simultaneously.
pricing
show
that in this case there never exists a pure strategy equilibrium, which starkly contrasts with the
result that a pure strategy equilibrium always exists in the sequential
game. This nonexistence of
equilibrium results from the ability of the firms to deviate simultaneously on their capacities and
prices. In contrast, in the sequential
and then
its rivals
of events in
and then prices are
first
capacity
first,
set later
is
more reasonable
(in
constitutes a better approximation to a situation in which prices can change at
it
higher frequencies than capacities),
suggests that
its
could also respond by adjusting their prices to this deviation. Since the sequence
which capacities are chosen
the sense that
much
game, a firm could only deviate by changing
it is
important
we do not view
for industries
this result as negative. Nonetheless,
it
with major capacity investments to choose structures
of regulation that do not allow simultaneous deviations
on capacities and
prices.
In addition to the newly-burgeoning literature on competition and cooperation between users
communication networks,
paper
and firms
in
literature
on capacity competition. Classic contributions here include Levitan and Shubik,
Kreps and Scheinkman,
rationing rule
when
is
closely related to the industrial organization
and Davidson and Deneckere
[9],
total
this
demand exceeds
capacity.
[5],
A
key issue in these papers
Our simpler framework with
avoids this issue and enables us to provide a complete characterization of the
Nash
perfect
Most
closely related to our
and study the
very
which
all
when
it
decisions to the
is
the recent
model
on how investments
firms are symmetric
exists,
paper
is
always
may
as
efficient.
The
demand
subgame
paper
is
affect congestion costs,
[15],
but only focus on the case in
this case,
non-symmetric case (where
of
Anarchy
result)
and
is
an equilibrium,
to consider
inefficiencies are
and
indeed
also to introduce capacity
most communication networks.
organized as follows.
defines the price-capacity competition
and Van Roy,
Weintraub, Johari, and Van Roy put
distinguishing feature of our work
shown by our unbounded Price
rest of the
Johari,
and there are no capacity constraints. In
constraints, which are a realistic feature of
The
the
of price competition with congestion externalities in
fully characterize the equilibria in the general
important
work by Weintraub,
efficiency properties of oligopoly equilibria.
restriction
little
full set of
is
equilibria.
who add investment
[1]
inelastic
[11],
game and
Section 2 describes the model.
Section 3
the oligopoly equilibria in this game. Section 4
characterizes the continuation price equilibria and the profits in the capacity subgames. Section
5 focuses
of the
results
on the special case with two firms and characterizes pure strategy oligopoly equilibria
game
(as well as the
mixed strategy off-the-equilibrium
and provides various
efficiency
bounds
for
play). Section 6 contains our
main
the set of pure strategy oligopoly equilibria for
Section 7 generalizes the existence and efficiency results to an arbitrary
the case of two firms.
number
the best oligopoly equilibria can be implemented by a
how
Section 8 shows
of firms.
Section 9 analyzes a
multi-stage game, where the low-cost firm acts as the Stackelberg leader.
related
game with simultaneous
capacity-price decisions and shows that this
game never has a
pure strategy equilibrium. Section 10 concludes.
Model
2
We
operating
=
(x\, ...,xn) to
"flows".
c
=
(ci,
...,
where
is jiCi,
c/v).
>
7,
We
denote total flow
denote the vector of flows.
flow allocated to firm
by
i
cannot exceed
for
i
G
{1,
...,
We
capacity,
is
We
for firm
i
we
<
C{.
G {l,...,N} by
We
costly. In particular,
—
demands
refer to the
assume that firm
i.e., x-i
of as a service provider
i
>
Xj
has a capacity
and use
0,
Cj
for the
>
0,
and
denote the vector of capacities
the cost of capacity
c;
loss of generality),
denote the price charged by firm
for firm
we
(per unit flow)
i
"small" users. 2
We
by
pi
(p\, ...,pw).
we
set d
—
the lowest cost firm whenever there
is
< R
alH G
for
We
assume that
unused capacity with
the lowest available cost exceeds the reservation
that 7j
1.
{1, ...,N}.
This
is
utility.
without
this firm
Further,
N
start
is
the aggregate flow of
utility
R; they choose
and do not participate
if
we assume throughout the paper
any firm with
loss of generality, since
have no incentive to be active and can be excluded from the set
We
this
assume that the users have a reservation
also
i
ignore
are interested in the problem of allocating d units of aggregate flow between these
firms and without loss of generality,
will
this reason,
N}. 1 For simplicity (and without
and denote the vector of prices by p
many
its
Investing in capacity
additional costs of servicing flows.
We
Each firm can be thought
own communication subnetwork. For
its
firms' services as
x
N firms.
model with
start with the general
i
G
{1,
...,
7$
> R
N}.
with a definition of flow equilibrium given a vector of capacities c and a vector of
prices p.
[Flow Equilibrium] For a given capacity vector
Definition 1
vector x*
is
a flow equilibrium
'Alternatively,
we could assume
,2
>
and price vector p >
"ft
>
0,
max
1} " (R - p
{ )
:r ?
)
a
with essentially the same results, but
slightly, since there
(1)
.
in this
case Propositions 7 and 13
could be excess capacity in some equilibria.
In the presence of additional congestion costs, this small users assumption would lead to the
commonly used
0,
if
x* G arg
below need to be modified
c
communication and transport networks
Wardrop
principle,
where flows are routed along paths with
minimum effective cost (see, for example, [10], [1]). In our context, there is no need to introduce this concept and
it suffices to observe that users will choose a lower cost provider whenever this is possible.
in
(see [14]),
We
denote the set of flow equilibria at a given p and
c
by W\p,
c].
This definition captures the simple notion that users will allocate their
price firm
up
to the point
users, they will allocate their capacity to the
any more
are
where the capacity constraint of
equilibrium
for
i
This
and only
if
if
YaLi x i -
problem
for
*
is
to the lowest
reached. After
this, if
there
second lowest price firm (as long as
its
and so on.
price does not exceed their reservation utility, R),
Using the optimality conditions
this firm
demand
it
(1),
anc^ there exists A
follows that a vector x*
>
>
~
such that A (J2i=i x\
is
l)
=
a flow
an d
e {1,...,7V},
A
if
—
A
if
> A
if
x*
=
<
x*
0,
<
x*
=
(2)
C{,
Ci
a convenient representation of the flow equilibrium, which will be used in the analysis
is
below.
R - Pi <
The
following result on the structure of flow equilibria
an immediate consequence
is
of this
characterization (proof omitted):
Proposition
1
Suppose that
for
that
Let c
Pm+i — +°°
=
some
if
(ci,...,c/v)
M<
be a capacity vector and p
=
(pi,...,pw)
< pu <
-R
< PM+i (with
N, we have
M — N).
<
p\
Then, there
x\
exists
—
<
p2
...
be a price vector.
the convention
a unique flow equilibrium x £ W\p,
c]
given by
min{ci, 1},
and
m— 1
x m = min
I
cm
,
max JQ,1
- y^ xA
i=i
for all 2
<
Remark
m<M
1
equilibrium
If
is
instead of pi
22
=
when
min{c2,
We
1
P2
<
< Pm <
R, we have pi
—
pj for
some
i
^
j,
the flow
not necessarily unique, since users would be indifferent between allocating their
flow across these two firms.
states that
<
—
p\
<
p^
Note
<
also that in the special case with
R, the unique flow equilibrium
N = 2, this proposition simply
will involve x\
—
min{ci,l} and
x\}.
next define the social optimum, which
chosen by a planner that has
full
is
the capacity and flow allocation that would be
information and
full
control over the allocation of resources.
Since there
social
no cost of servicing flows beyond the capacity
is
optimum
follows immediately.
A
Definition 2
costs, the following definition for a
s s
capacity-flow vector (c ,x )
is
a social optimum
if it is
an optimal solution of
the social problem
N
N
R^Xi -£7,
maximize x >o, c >o
(3)
«=1
i-l
N
/J^i <
subject to
Xj
The
optimum
anteeing the existence of a social
—
cf
=
xf
xf
for
,
i
i£
Cj,
{l,...,iV}.
problem has a continuous objective function and a compact constraint
social
have cf
<
lj
for
,
£
{1,
i
£
{1,
...,
JV}.
We
(c
refer to c
iV}, the social capacity
-...,
s ,x s
is
from the preceding that we
also clear
It is
).
guar-
set,
s as the social capacity. In view of the fact that
given as the solution to the following maximization
problem:
•'
c
s
max
£ arg
c>0,
2i=l N Ci<l
N
f
{
I
\
}(R
~
-
%)ci
>
(4)
.
I
For future reference, for a given capacity vector
§(c)
i.e.,
>
0,
we
define the social surplus as
N
Y,(R~li)*,
(5)
the difference between the users' utility and the total capacity cost.
next consider the two-stage competition game
firms
compete
The
Game
Price and Capacity Competition
3
We
=
c
in
which capacities are chosen
first
and then
in prices as outlined in the previous section.
price-capacity competition
chooses
game
is
as follows.
First, the TV firms
At the second
i.e.,
firm
i
capacities set at the
first
stage, simultaneously choose prices,
their capacities,
c\
at cost 7j&j.
stage, firms, having observed the
i.e.,
the price vector of other firms, denoted by p_;, the profit of firm
±HlPi>P—ii X,
where x £ W\p,
c]
is
Ci,
C—i\
= ViX%
simultaneously choose
firm
i
i
charges a price
pt.
is
"JiCi,
a flow equilibrium given the price vector p and the capacity vector
objective of each firm
is
to
maximize
profits.
We
Given
refer to the
c.
The
dynamic game between the two
subgame
firms as the price- capacity competition game, and look for the
Since the capacities set in the
of this game.
vector c
=
first
perfect equilibria (SPE)
stage are observed by
all firms,
subgame, and subgame perfection requires that
{c\,...,cn) defines a proper
subgame, the continuation equilibrium strategies constitute a Nash equilibrium.
subgame, we
first
exist in
As we
some capacity subgames. For
Let
equilibria.
Hi
B
denote the space of
this reason,
all
by H-i the product measure p excluding
i
m
[Price Equilibrium] Let c
We
\pi(c)
l
p,
£
BN
(i.e., /i_j
>
=
is
For each capacity
will also refer to as
the product measure p\ x
p\ x
x ^i-\ x Mi'+i x
...
be a capacity vector.
subgame
if
x(c)
will fail to
£ W[p(c),
c
••
x Hn> an d
•••
x Wv)-
A
vector \p(c),x(c)]
c]
and
Vpj>0, Vx £
p-i(c),x(c),c]>ILi \pi,P-i(c),x,c],
vector [pf,x c (p)}
for all
i
£
c]
is
{1,
a pure
...,
W[pi,p^(c),c\.
N},
(6)
by PE(c).
a mixed strategy Price Equilibrium in the capacity subgame
and the function x c (p) £ W[p,
each
and mixed strategy price
define both pure
denote the set of pure strategy price equilibria at a given
A
in
(Borel) probability measures on the interval [0,R]. Let
strategy Price Equilibrium in the capacity
n
we
3
pure strategy equilibria
will see below,
£ B be a probability measure, and denote by
Definition 3
we
define the price equilibrium between the firms, which
the (continuation) Price Equilibrium.
every capacity
if
c
N
p £ &
every p, and
for
c
C
d\ji i{Pi) x /iii(P-i)J
ni[pi,p_j,a; (pi,p_i),c]
[0,R]'
c
/
'{o,R} N
for alii
£
{1, ...,N}
and
\ii
£ ^-
We
Ikipi,P-i,x (pi,P-i),c]d(fj,i(pi) x fJti(p-i)
v
denote the set of mixed strategy price equilibria at a given c
by MPE{c).
In the following, with a slight abuse of notation,
strategy equilibria.
4
Note that here x
from the correspondence W(p,c).
3
A
subgame
is
identified
(•) is
We
we
£
will write [/u,x(-)]
not a vector, but a function of p,
denote the profits
for firm
i
MPE{c)
i.e.,
in the
x(p)
is
for
mixed
a selection
mixed strategy
price
with the public history (of previous moves). Hence, the SPE notion requires that the
is optimal given the other player's strategies, after every history; see, for
action prescribed by each player's strategy
example,
[6], [12].
also that the pure strategy Price Equilibrium notion here may appear slightly stronger than the standard subgame perfection, since it requires that a strategy profile yields higher profits for each player for all
x €
\pi,p-i (c) ,c], rather than for some such x. Nevertheless, [1] shows, for a more general game, that this
Note
W
definition of equilibrium coincides with the standard pure strategy
convenient to work with). Given this relation, we have PE{c)
there exists
selection
\(x,
from
G
MPE(c)
such that p
]p,c]
with x(p)
=
x{-)\
W
x.
is
subgame
C MPE(c),
perfect equilibrium (but
in
is
slightly
more
the sense that for every \p,x\ £ PE(c),
the degenerate measure with
/J.{{p})
—
1,
and
x(-)
is
an arbitrary
subgame by
equilibria in the capacity
II j
[a*,
x(-), c],
[
Ik[pL,x('),c}=
Ui\p,x(p),c]diJ.(jp).
J\Q,R]
We
will also use the notation
when
U
i
finrij_uses_the_degerierate
the mixed strategy
\p.l
st rategy
(7)
N
,ii-i,x{-),c} for
mixed
i.e.,
m
some
G [0,R] to denote the expected profits
pi
with Hi(pi)
=
1,
while the. remaining firms use_
\i-i.
profit functions are discontinuous in prices,
Note that since the
it
not obvious that each
is
capacity subgame has a mixed strategy price equilibrium. In the next section, we will show that
subgame
in each capacity
We
we
a pure or mixed strategy price equilibrium always exists.
c,
next define the subgame perfect equilibrium of the entire game. For notational convenience,
focus on the actions along the equilibrium path to represent the
Definition 4
(OE)
Oligopoly Equilibrium
n
for all
a >
0,
A
[Oligopoly Equilibrium]
and
i
[p(c
for all
{p{c
if
0£;
),x( C
[f.i,x(-)}
Note that pure strategy
OE ),x(c OE
e
0£;
OE
s
),( C p
£
OE may involve
OE ,p(cOE ),x(c OE
)
,cef)]>n
i
x
[/
I
for all
is
)}
a (pure strategy)
{1, ...,N},
<=
i
perfect equilibrium.
x(.),(c i ,cef)],
refer to
'c
OE
as the
(8)
OE
capacity.
pure strategies along the equilibrium path, but mixed
some
refers to equilibria
[c
PE{c OE and
MPE(ci,c° E ). We
strategy continuation price equilibria in
per, pure strategy
)}
vector
subgame
off-the-equilibrium subgames.
Throughout the pa-
where pure strategies are used along the equilibrium
path.
4
Our
first
task
Price Equilibria
is
in
the Capacity Subgame With
to characterize the entire set of
expositional purposes,
we
start with the case
where
subgame
N—
2,
We
number
Firms
perfect equilibria in this game.
which enables us to provide an
characterization of the equilibria and the extent of the efficiency losses.
results to an arbitrary
Two
We
For
explicit
generalize our
main
of firms in Section 7 below.
consider an arbitrary capacity subgame, and then prove the existence of pure or mixed
strategy price equilibria and provide a characterization of these equilibria.
this characterization to
properties. Since in this
We
will
then use
determine the form of oligopoly equilibria and analyze their efficiency
and
in the next section
to these two firms using the indices
i
and —i.
we consider only two
firms,
we sometimes
refer
Existence of Pure and Mixed Strategy Price Equilibria With
4.1
Let c be a capacity vector such that
Proposition 2
there exists a unique Price Equilibrium in the capacity
for?:
=
equation
c\
+
i
<
1, it
all
p
c2
that for
(2)]
a price pi, firm
<
c2
subgame
q >
and
1
[p, x]
for
i
=
=R
such that Pi
<E
by the equivalent characterization
follows
[0,R]
2
,
the flow allocation (ci,c 2 ) £ W(p,
Then
1,2.
and
xi
—
C{
p-i G
so that pi
and
Proposition 3
some
Proof.
z.
This shows that
[0, i?].
=R
of a flow equilibrium
c).
[cf.
Therefore, by charging
can make a profit of
Ui{pi,p-i,x,c]
for
+
Firms
1,2.
Proof. Since
for all
c\
Two
Then
Xi
—
C{, i
=
1,2,
pi
is
—R
strictly
=p
t
Ci,
dominates
other price strategies of firm
all
the unique Price Equilibrium.
Let c be a capacity vector such that
c.\
+
c2
>
1,
i,
Q.E.D.
Q >
for
=
i
1,2 and
c,
<
1
there exists no pure strategy Price Equilibrium in the capacity subgame.
Suppose there
exists a pure strategy Price Equilibrium (p,x).
The
following
list
con-
and profitable unilateral deviations from each, thus
siders all candidates for a Price Equilibrium
establishing the nonexistence of a pure strategy Price Equilibrium:
•
Suppose
p\
<
p2-
Then the
profit of firm
1 is IIi[p,
p\ will increase firm l's profits, thus firm
•
Suppose p\
If
•
x\
=
—
P2
>
If
0.
X\
<
min{ci, 1}, then, since
Suppose pi
=
increase
price
its
P2
=
0.
x,
c]
= p\
+
c2
>
small increase in
an incentive
to decrease its price.
firm 2 has an incentive to decrease
1,
Since by assumption
and make positive
A
has an incentive to deviate.
1
inin{ci, 1}, then firm 1 has
c\
min{cj, 1}.
a <
1 for
some
i,
firm
—i
its price.
has an incentive to
profits.
Q.E.D.
Proposition 4
for
some
i.
Then
Let c be a capacity vector such that C\
there exists a
mixed strategy Price Equilibrium
Proof. The subgame following any capacity choice
on
[4],
Proposition 4.3
any such subgame.
When ci, c 2 >
1,
We
in
[1]
+ Oi >
1, Cj
the capacity
c is a special case of the
is
i
—
model
1,2 and
Cj
<
1
subgame.
in
[1].
Building
mixed strategy equilibrium
this proof here to avoid repetition.
subgame
for
in the capacity
establishes that there always exists a
do not repeat
>
in
Q.E.D.
an uncapacitated Bertrand price competition between
two firms. Thus, we immediately have the following result (proof omitted).
9
Proposition 5 Let
we have
\p,x],
p,
>
be a capacity vector such that Ci,c 2
for
—
i
both firms make zero
1, 2, i.e.,
Then,
1.
for all Price Equilibria
profits.
Characterization of Mixed Strategy Price Equilibria
4.2
We
next provide an explicit characterization of the mixe d strat egy price equilibria and_the_prqnts^
in each capacity
=
Let c
+
c%
—
c
c2
>
subgame.
be a capacity vector. Throughout
(ci,C2)
1, (H
>
for
i
=
strategy Price Equilibrium
1,2
<
--and Cj
=
ui
k
By
i.
we
focus on the case where
Proposition
the capacity subgame. Let
[fi,x(-)] in
and k denote the lower support of m,
jj,i,
some
1 for
this section,
4,
there exists a mixed
denote the upper support of
1/4
i.e.,
inf
jp
= sup|p
:
ji {
— lj,
:
Pi({p>p})
=
{{p <p})
lj.
Let (F\,F2) denote the corresponding cumulative distribution functions for the measure (pi,p 2 ),
i.e.,
=
Fi(p)
m{{jp
Recall that
<
p}), for
[p,,x(-)] is
=
i
1,2.
a mixed strategy Price Equilibrium
n
for all
p e
[0,
R],
and there
will
P ,/i-i,s(:),ft]<nf
C
/m, X
(•)
:
We
|
exists a set Pi
n,
(see, e.g., [12]).
i
now
[p,
c]
and only
= nf
if
(9)
I
such that m{Pi)
[k,ui\
,
if
for all
p e
=
1
and
A
(10)
use this property of mixed strategy equilibria to derive three
lemmas
that will allow us to explicitly characterize the unique mixed strategy Price Equilibrium in the
capacity subgame.
Lemma
i
=
Assume that
1
1,2, the
c\
+
mixed strategy
c2
>
1, C{
>
for
fn cannot have
—
i
all its
1,2 and
c,
<
1
for
mass concentrated
some
i.
Then,
at a single point,
for
any
i.e.,
p;
cannot be degenerate.
Proof.
pi
is
By
Proposition
3,
both
degenerate at some p\ £
Consider p\
for firm 2,
>
0.
which
fi^s
[0,
R]
cannot be degenerate. To obtain a contradiction, assume that
(i.e.,
Charging the price p 2
is
=
strictly decreasing in
cannot have an atom at p
=
p\.
Suppose
pi({p
p\
e,
it
—e
=
Pi})
for
=
some
1)e
We
>
then there
10
show that
p.2({p
yields a profit of (pi
showing that p 2 ({p
does;'
first
is
<
Pi})
=
0.
We
—
<
e)
Pi})
=
0.
minjco, 1}
next show that p 2
a positive probability of both firms
charging the price p\.
5
>
xzipuPi) <
mm{c 2
generates higher profits for firm
same
applies to player
the price
=
p2
Pi
+
e
strictly increasing in
,Uj's
If
1,
showing that
p, 2
< R-
p\,
<
for
e,
If
2.
thus
e
min{c 2
,
yields a profit of (pi
Finally,
.
+ e)
mass concentrated
all its
for
<5
+
1}, then, since c\
cannot have an atom at p 2
should have
/j 2
=
x 2 {p\,P\)
it
-
then charging a price of p\
1},
,
(1
-
at
p2
some small
>
c2
1,
the
firm 2 charges
if
min{ci, 1}), which
is
=
R- However both
i.
Then:
cannot be degenerate, thus we arrive at a contradiction. Q.E.D.
Lemma
Assume that
2
c\
+ c% >
>
1, <H
for
(i)
Fi and
F2
have the same lower support,
(ii)
Fi and
F2
have the same upper support,
(iii)
Fi and
F2
are strictly increasing over
—
i
Cj
— 1% =
i.e., l\
i.e.,
and
1, 2
ui
=
some
1 for
L
=
u2
<
u.
[l,u].
Proof.
(i)
Assume
that
<
l\
by Proposition
l2
follows that for
1, it
<
Let pi e Pi, p\ G Pi be two prices such that p\
.
=
x € W[(p, p 2 ),c] satisfies Xi
all
P2
p2 £
and p
Thus the
min{ci,l}.
=
p\ or
p
=
<
p\
l
2
Then,
.
any flow equilibrium
p[,
profits of firm 1 at prices pi, p[ are
given by
<
ni[Pi,p 2 ,x,c] =pixi
nifpi,p 2 ,x,c] =Pi^i,
contradicting (10).
(ii)
Assume
that u\
Proposition
p
= p'j,
1
> u2
.
and the assumption
c\
+
c2
>
1, it
any flow equilibrium x £ W[(p,p 2 ),c]
of firm 1 at prices pi
,
<
Let p\ £ Pi, Pi £ Pi be two prices such that u 2
p[ are given
follows that for
satisfies x\
=1—
pi
P2
p2 £
all
min{c 2
,
<
1}.
p[.
Then, by
and p
= pi
Thus the
or
profits
by
<
ni[pi,p 2 ,z,c] =P\Xi
Iii\p'l)
jd,
2
,x,c]
=p[x
1
,
contradicting (10).
(iii)
Assume
Pi) Pi
same
p2
,
£
to arrive at a contradiction that Fi
[l,u]
interval.
p2 £
P2
with pi
<
Suppose
such that pi
We
p'j.
F2
this implies
F2
is
[pi,p'i] for
some
constant over the
not constant over this interval, which implies the existence of
is
£ W\p,
i
\
X 2[P,P2)
show that
will first
< p2 < p 2 <
for all flow equilibria x(p)
constant over the interval
is
=
c],
p\-
By
Proposition
and the assumption
1
we have
i\
i
X2{p,p
2)
=
\
<
y
11
l-mm{ci,l}
ifp<pi,
\,
mm{c 2 ,l)
ifp>p
.
f
'
..
,
1
.
c\
+c2 >
1,
Since
F
t
is
constant over [pi,pi], this implies that the profits of firm 2 at p'2 are higher than
those at po, which, since P2,
F2 are constant over
By
[pi,Pi]-
= p\
higher profits than P2
and
P2, leads to a contradiction. Hence, the distributions F\
G
p'2
a similar argument as above,
for firm 2, leading to
follows that P2
it
= V\
yields
another contradiction and establishing this
part of the lemma.
Q.E.D.
Lemma
(i)
Assume
3
The
that
+ C2 >
c\
= 1,2,
distribution Fi, i
>
1, C{
for
=
i
1,
q <
and
2
1 for
some
i.
Then:
does not have any atoms except possibly at the upper support
u.
(ii)
(iii)
Both distributions
Fi cannot have an
The upper support u
is
atom
upper support
at the
u.
equal to R.
Proof.
(i)
Without
any p G
at
p G
[l,u]
(l,u),
we consider
loss of generality,
i.e.,
(which
Assume
{l,u).
F\ (p+)
satisfies
I
show that F\ cannot have an atom
first
to arrive at a contradiction that there exists
>
F\ (p—).
<
u
By Lemma
view of
in
small such that the prices p
1,
We
F\.
—
e
Lemma
and p
+
e
2(iii),
1).
F2
an atom
at
some
strictly increasing over the interval
is
Thus, there exists some
belong to P2, and p
+
e
<
e
>
sufficiently
P. Using Proposition
the profits of firm 2 at these two prices can be written as
U 2 [p-e,fj,i,x(-) y -
-
e) (p
-
-
min{ci, 1})
+
(1
- F
-
e)) (p
-
e)
min{c2
Fi (p.+
e) (p
+ e) (1 -
min{ci, 1})
+
(1
- Fi (p+
e)) (p
+
e)
min{c 2
it
follows that for small
Fi (p
c]
e) (1
x
(p
,
1},
.
1}.
and
n 2 [p +
e,
pi, x{-),
Since Fi (p+)
>
c]
=
1
- min{ci,
1},
f>Mi»2'(-). c ]
> n 2 [p +
e,^i,x(-),c],
Fj (p— ) and min{c2, 1}
>
enough
e,
we have
B2[pyielding a contradiction.
We
next show that F\ cannot have an atom at p
support must satisfy
I
>
0.
If
I
=
0,
since
I
12
G P»
=
I.
for
i
We first prove that the common lower
= 1,2 [cf. Lemma 2 (iii)], this implies
that the profits of either firm at any price vector are equal to
some
for
Hence,
small
i,
follows that
it
/
>
=
p
the profits of firm -i at
u cannot be equal to
U2 +
[l
there
-
[l
=
e,/ii,a;(-J,c]
e /ii,x(-),c]
)
is
= {1-F
l
an atom at
I,
+
{l
e))(l
>
Fi(l+)
i.e.,
from the preceding two relations that
Tl 2
contradicting equation
Assume that both
[F\
(I
-
e
for
some
sufficiently
for firm 1
(u, u)
-
<
[F2 (u+)
-
inin{ci, 1}.
2.
By
R.
which belongs to
e,
min{c 2
e)
,
+F
1}
then since
>
e,/i a ,z(-),c]
atom
F2 (u— >
Uo[l
at
0,
)]
Then charging
than charging a price of p\
Assume that u <
0,
+
1
1 (1
-
+
+
e)(l
min{ci,l}
+
P2
:
e)(l
- min{ci,l».
< min{c 2 ,l},
follows
it
e,
e,/ii,x('),c],
(9).
the same applies to player
say player
[l
+
I
- e)min{c 2 ,l}.
for sufficiently small
distributions have an
{u+) — Fi {u—)\
Suppose X\
(iii)
1
e,
Consider next the profits of firm 2 at the price
(ii)
I
a<
any flow equilibrium.
at
Consider the profits of firm 2 at price
0.
Tl 2
If
Since by assumption
0.
=
u.
If
p
—
u.
Then,
both firms
=
part
(ii), it
follows that there
—
follows that with probability
be charging a price of p
u—
e
u.
cj
+
c2
>
1,
lemma.
no atom at u
is
=
generates higher profits
min{ci, l}, then, since
establishing this part of the
2,
will
a price of p\
x\ (u,u)
it
for
one of the players,
Then
Uj[u,id 2 ,x(-),c}
—
u
(1
- min{c 2 ,l}) <
showing that the upper support u cannot be
Ui[R,
p, 2
,x(-),
strictly less
c]
=
R(l - min{c 2
,
1}),
than R.
Q.E.D.
The next
proposition characterizes the expected profits of the two firms in capacity subgames
with continuation mixed strategy price equilibrium.
Proposition 6 Let
Ci
<
1 for
some
The expected
i.
c
Let
=
+
(ci,c 2 ) be a capacity vector with c\
[/j,,x(-)j
>
be a mixed strategy Price Equilibrium
profits Hi[p, x(-),c]
t
c2
t
for
i
=
R{1 —
c-i)
13
f
—
7,c,',
c,;
>
for
i
in the capacity
1,2 are given by
R(l—Cj)cj
{
1,
^
otherwise.
=
1,2 and
subgame
c.
Proof.
Assume that
subgame
c
by
<
c\
c2
We
.
nf We will now
.
denote the equilibrium profits of player
use the characterization of mixed strategy equilibrium in equation
(10) to explicitly characterize the equilibrium distributions Fi
and
m
G
and F%. Note that we have k € Pi
Pf.
By Lemma
3(i),
upper support R.
Using the Flow Equilibrium characterization given in Proposition
ni[pi,/^2,2:(-),c]
Similarly, for all
atom except possibly
the distributions F\ and F2 do not contain an
write the expected profits of firm
p2 G
P2
,
1 for
PiCi(l
=
nf.
^
and P2
and
at the
we can
1,
as
-mm{c2) l})F2 (pi) ~ 1\C\,
- F2 (pi)) +pi(l
we have
R,
p 2 min{c 2 ,l}(l - Fi(p 2 ))
=
nf = nf + 71C1
^R
any p\ G Pi, p\
=
n 2 [p 2 ,/ii,:r(-),c] =
Let
in the capacity
i
+ p 2 (l -
-
Ci)Fi(p 2 )
I2C2,
nf.
nf = nf + 72 c 2
F2 (p)
Solving for F\(p) and
.
preceding relations,
in the
we obtain
^1^)=
^(pH
Let
Fi(Z)
I
denote the
= F2 (/) =
0, it
common
ci
/2
+ mm{c
Ci
nf
-"j
+ mm{c 2
,
,
n
lj -
VP
'
i
Cl
I
=
11
2
,
p^i?.
1
,
lower support of
follows that
Vp£F
n
1} — T
,
GFl!P ^i?,
1
and
ji\
flf/min{c2
nf = nf
,
1},
,u 2
,
i.e.,
lj
=
l
2
=
I
Lemma
(cf.
2).
Using
and
Cl
min{c 2 1}
,
By Lemma
atom
at
arrive at
u
2
=
and
Lemma
3(iii),
we have
=
u\
u2
=
R.
We
R, and therefore the characterization in (11)
distributions cannot have an
atom
nf
it
,
at the
use the characterization in (11) for p
=R
F1 {R) =
is
also valid for
i.e.,
Fi(R-) <
follows that
F2 (R— <
a contradiction that F\ has an atom
preceding relation between Elf and
next show that F\ does not have an
at R,
)
1.
1.
p
=
Then, using
But, by
Assume
R.
c\
<
Lemma
c2
to
and the
3(ii),
both
upper support, yielding a contradiction. Hence, we can
to write
min{c2 ,l}
1
Ci
+
14
- ti.il
min{c 2 1} ,
1
which shows that
nf =
^E =
fl(l-Ci)Ci
1
The argument
>
for c\
c 2 is
we prove
We
Equilibria.
terization,
min{c 2
,
1}
Q.E.D.
similar and completes the proof.
Two
Oligopoly Equilibria With
5
In this section,
fl(i-ci),
the existence and characterize the properties of pure strategy Oligopoly
OE
provide a characterization of pure strategy
first
Firms
Using
capacities.
this charac-
we show that the price-capacity competition game always has a pure strategy Oligopoly
Equilibrium.
We
then use
this characterization to
study the efficiency properties of pure strategy
Oligopoly Equilibria in the next section.
Proposition 7 Assume that
capacity
if
and only
c\
if
+
c-i
< R
7j
=
1
for
some
=
i
1,2.
A
capacity vector c
=
(ci,c 2 )
is
an
OE
and
R-n
<
2R - 7
<
(H
c-i,
(12)
,
for
some
%
=
1,2,
We
Proof. (Sufficiency)
since c\
+
c2
<
1,
first
show that
c\
+ c2 =
1
together with (12)
Proposition 2 implies that the profits of firm
U
i
\p(c),x,c}
= (R-j )c
i
i
£
= 1,2
profit for firm
are
i
is
c,-
/
Cj
by firm
TLi\p(ci,c-i) ,x, (c;,c_i)]
no profitable deviations with
Next consider
Cj
>
Cj.
so that the deviation
or c_j
<
1.
equilibrium
•
If Cj
i.
is
Clearly,
cx
<
if
6i,c_j
=
<
Cj,
(R —
<
OE
capacity First,
are
(13)
Proposition 2
7i)c,
an
,
where p(c) denotes the continuation equilibrium price vector, which
Consider a deviation
is
in this case
still
applies
is
(R,R).
and the resulting
n,[p(c) ,x,c], establishing that there
C{.
>
1,
Proposition 5 applies and
not profitable. So suppose that
q,
+ c_{ >
1, Cj,
IIj[p (cj, c_i)
c_^
>
and
,
x, (cj, c_j)] =
either
c"i
<
1
Proposition 4 applies and the deviation will induce a mixed strategy continuation
/i.
There are two cases to consider:
Suppose that
Cj
>
c_j,
Cj
>
c_j
and
c,
<
c_j.
which by Proposition 6 implies that the deviation
TJi\p,x{-) (ci,C-.i)]
)
=
i2(l-c_j)-7iCi,
=
(i?
<
n,;[p(c),X,c]
15
-
7i) Cj
-
7,
(^
-
Cj)
profits of firm
i
are
where the second
equilibrium profits from (13) together with
deviations with
t%
>
Suppose that
<
c_,.
c,
+ C2 —
line exploits the fact that c\
>
c,
and the third
1
line uses the definition of
c_j, establishing that there are
no profitable
C-%.
Then by Proposition
tUM),
6,
we have
= ftidHi} ~ 7i5i>
(it, c-i)}
(14)
Let c?wx denote the capacity that maximizes (14), given by
_
=
-max
Ci
<
di
Cj
<
<
<
Cj
C-i,
C2
=
1
(Necessity) Clearly, any
c\
+
ci
<
R
cj
can increase
that there exists an
Without
OE
together with (12)
<
is
Similarly,
Cj.
Therefore, for
all
capacity equilibrium with c\
we assume
that C\
+
C2
>
c%-
>
Consider the deviation to
c\
=
1
—
[
Co
<
c\
by firm
fact that 71
1
capacity, since the firm
cannot be a pure strategy
1, Ci
>
for
i
=
Then Proposition
1,2 and
cj
<
1 for
6 implies that
7 iCi.
(16)
which by Proposition
2 yields profits
= i?(l-c2 )-7ici,
ni[/x,x(-),(ci,c 2 )]
where the inequality exploits the
1,
>
OE
Suppose, to obtain a contradiction,
0.
n M ,x(.),c]=i?(l-c 2 )1
c_j.
capacity.
any c\,C2
both firms are equal to
loss of generality,
an
<
cannot be a pure strategy
1
by raising
profits
capacity, since the profits of
i.
C{.
n,[p(c) ,x,c],
OE
+
some
<
Cj
This proves that any
OE
(£i,c-j)]
establishing that there are no profitable deviations with
7j
c™ 1 <
(12) that
}
we have
fti[/ix,a;(-),
with
,
[
2#
2
we obtain from equation
c_j,
'
'
'
Since
min{c-j,l}7i
1
-
>
i?(l
=
nil>4,a;(-),c],
>
c2 )
-71C1,
and establishes that such an equilibrium cannot
exist.
Next, to obtain a contradiction, suppose that there exists an equilibrium with
(12)
is
violated.
Without
loss of generality,
assume that
« <
fe
16
c\
<
Co,
C\
+ c2 =
1,
but
so that
<">
Now
consider a deviation by firm
from Proposition
To
6,
1
to c\
= cf ax
by
as given
(15). In
view of (17), cf
ax
>
C\
and
the deviation profits are given by
>
Rc\ - 71 ci
=
IIi[/i,x(-),c].
we consider the function
see that the inequality holds,
f
,
(i?-(l-
v
Cl )
7i)
2
4ft(l-ci)(/fci-7ici)'
for ci
(for ci
7^
decreasing in
=
c\ for c\
0,
<
Note that the function f(c\)
the inequality holds trivially).
Therefore, for
ofl~1i
all cj
<
^"ll^
is
strictly
,
This implies that
2
(i?-(l-
Cl ) 7l
)
>
i?ci
-71C1.
4fl(l-ci)
The
right
hand
side in the preceding relation
that there cannot be any equilibrium
OE
is
equal to
capacity with
Ili[//, x(-),c]
c\
+
c<i
—
1
by
(13).
This establishes
that does not satisfy (12),
completing the proof. Q.E.D.
Since
we have
c\
+ C2 =
1 for all
OE
capacities, the relation in (12) can equivalently be written
as
"-»
S
2R - 7i -
*
*S
~ 2H - 72
(18)
and
c2
Note that
and
(19).
<
7j
<
i?, i
=
(19)
1,2, the capacity vector c
=
(1/2, 1/2) satisfies equations (18)
Thus, we immediately obtain the existence of a pure strategy Oligopoly equilibrium as
a corollary:
Theorem
for all
= l-Q.
»
1
The
price-capacity competition
6
In this section,
efficiency to
game has
a pure strategy Oligopoly Equilibrium.
Efficiency of Oligopoly Equilibria
we quantify the
be the ratio of the
efficiency losses of Oligopoly Equilibria.
We
take the measure of
OE
to the social surplus
social surplus of the equilibrium capacity c
17
of the social capacity c
,
equation
We
(5)].
71
and
72, either for the worst equilibrium
among
efficiency metric at
c
s
is
some c OE 6
C({ji}) denote the set of
72, let
bound on
investigate the worst-case
the set of oligopoly equilibria or for the best equilibrium
Given capacity costs 71 and
where
[cf.
problem instances characterized by
this metric over all
among
s S(c OE )/S(c s
)
OE
the set of equilibria.
We
capacities.
define the
C({7?.}) as
the social capacity given the capacity costs 7; and reservation utility
R
[cf.
(4)].
Following the literature on the efficiency losses of equilibria, we are interested in the perfor-
mance
both the worst and the best
of
In particular,
we
first
look for a lower
OE capacity
bound on the worst performance
commonly
is
C ({-n})
Anarchy
referred to as the Price of
in a capacity equilibrium,
r(H},c OE ),
inf
inf
{0< lt <R}cOE e
which
equilibria of price-capacity competition games.
in the literature (see
[8]).
We
then study
the best performance in a capacity equilibrium given an arbitrary price-competition game, and
thus provide a lower bound on
which
commonly
is
Example
<
e
<
1
r({7},c os ),
sup
inf
{0< 7! <fl}
c
OB g
referred to as the Price of Stability in the literature (see
Consider a price-capacity competition
min{l, R}, 72
— R—
2
e
.
The unique
7, it
s
=
)
oe =
oe
OE
(cf cf)
,
e.
OE )= (
2
{c 1
v
;
R
e
\R +
'
an
is
firms,
follows that the capacity vector
c
is
game with two
social capacity
S(c
Using Proposition
(20)
C({ 7i })
e
'
R+
e
capacity with social surplus
+ R)
R+e
eHl
{OE)=
K
Therefore, as
e
—
>
0,
J
the efficiency metric gives
limr({7,;},c^)
=
limi^l=0.
18
=
and
[2]).
71
— R—
t
for
some
(1,0) with social surplus
Recall that
7i
—
>
72 (as
e
when
—
>
71
=
72, §(c
2
S(c 5 ). Instead in the preceding example
the efficiency metric converges to
0),
The preceding example
Theorem
OE ) =
we have
that as
0.
implies the following efficiency result:
Consider the price-competition game with two firms. Then
r({7i}, cP
inf
inf
B =
)
0,
{0<-n<R}cOE<z c({ 7l })
i.e.,
the Price of Anarchy of the price-capacity competition
We
game
is 0.
next provide a non-zero lower bound on the Price of Stability of a price-capacity competi-
tion game.
Theorem
i
e
3
{1, ...,N},
Consider the price-competition game with two firms.
Then,
<
for all
74
<
R,
we have
sup
inf
r({7;}, c
OE ) > 2V2 -
2,
{0<7i<«} c°S€ C({ 7? })
i.e.,
the Price of Stability of the price-capacity competition
game
—
bound
is
Then, the capacity vector (cf cf )
=
is
2\/2
2
and
this
tight.
Proof.
(1,0)
is
We
assume without
loss of generality that 71
a social capacity (unique social capacity
if
71
<
72.
<
72),
,
with social surplus S(c s )
=R—
71.
Using the definition of the efficiency metric r({jl },x OE ), we consider the following optimization
problem:
oC QE
Ho_ WiOE _~2*£2_,
K~n
sup
c
Since for aU c OE G
clearly attained at
OE e C({ 7l })
C({ ji}), we have cf E
,
some c OE £
equations (18)-(19), the
+
c® E
c® E
value of
=
1,
the
supremum
maximum
C({-yi}) with the
maximum
(21)
fl
at an
OE
in the
value of c®
capacity
is
E
.
above expression
By
Proposition 7 and
given by
^ = 37^Substituting cf E
optimal value
is
=
9fi _
and c® E
=
or-
m
12
the objective function in (21),
given by
o _
-"-
.R71
2A-72
_ (fl-72)72
2R--y 2
R-11
19
is
<
we
22 >
see that the
We
we
are interested in finding a lower
bound on the preceding over
<
all
< R with
ji
<
71
i.e.,
72,
consider the following optimization problem:
(fl-72b:2
2fl- 72
Rli
R
2A-72
inf
R — 7!
0<7,<i?
71<72
This problem has a compact constraint set and a lower semicontinuous objective function [note
=
that for 71
R, the efficiency metric
For 72
solution (71,72).
j2
^
=
satisfies r({ji}, c
R, the objective function
is
strictly increasing in 71,
=
the optimal solution of the preceding problem
is
(2
is
has an optimal
<
showing that 72
1,
=
showing that 71
— V^jR
it
0.
It
For
R.
all
follows then
attains the infimum, showing that
given by (71,72)
=
- \/2)R\ with optimal
(0, (2
2V2-2.
value
Finally, to see that the
=
with 71
>
5
and 72
relative to social
We now
bound
=
optimum
—
(2
—
of 2\/2
\/2~)
2
R. As 5
limits to 2\/2
—
2.
tight, consider the best
is
—
>
0,
OE
capacity of the
game
the surplus in the best oligopoly equilibrium
Q.E.D.
Equilibria and Efficiency
7
With
N
Firms
generalize the results on the characterization and existence of pure strategy Oligopoly
Equilibria
(cf.
Section 5) and the efficiency bounds
argument
results provided so far generalize, the
is
(cf.
Section 6) to
slightly different,
characterizing the expected profits of the firms for
all
N
firms.
and does not
While
rely
the
all
on explicitly
mixed strategy Price Equilibria.
Preliminaries
7.1
The next
G {1,
N}
...,
(since
Proposition 8
Then
=
Note that
in
our analysis of mixed
on capacity subgames
in
which
set of results generalize Propositions 2-5 of Section 4.
strategy price equilibria,
Xi
Therefore
1].
)
R, the objective function value
that the unique stationary point given by 72
i
OE =
if c$
it
is
= 0,
sufficient to focus
profits are equal to
for
i
G
Let c be a capacity vector such that
5~Jj=i
R) N
ci
^
1>
p e
firm
can make a profit of
,
^
follows
1 anc^ °i
subgame
[p,
x]
>
^ov
*
^
...,
.V}.
=R
and
{1,
such that pi
by the equivalent characterization of a flow equilibrium that
the flow allocation
for all
i
^—
{l,...,iV}.
Proof. Since JZi=i
[0,
for all
for that firm).
there exists a unique Price Equilibrium in the capacity
Cj
>
Cj
(ci
,
Co,
...,
c^) G W(p,
Ri\Pi,P-i,x,c\ =PiCi,
20
c).
Therefore, by charging a price
pi,
for all
p-i e
=R
so that pi
i,
N~
R}
[0,
l
=R
This shows that p*
.
=
and Xi
€
(%, i
{1,
N},
...,
strictly
dominates
assume that there
some
exists
with ]Cili
j
<H
Q.E.D.
the unique Price Equilibrium.
is
Let c be a capacity vector such that ^2i=i
Proposition 9
other price strategies of firm
all
Q>
— Cj <l. Then
>
1, c;
for all
€
i
{1,
...,
AT} and
there exists no pure strategy Price
Equilibrium in the capacity"subgame.
exists a pure strategy Price Equilibrium (p,x).
Proof. Suppose there
<
suppose that p\
p =
1
{j
—
Pj
:
Let P\ be the set of players whose price
pj, for all j.
The
pi}.
Without
following
considers
list
all
is
loss of generality
equal to p\,
i.e.,
candidates for a Price Equilibrium and
provides profitable unilateral deviations from each, thus establishing the nonexistence of a pure
strategy Price Equilibrium:
• p\
<
miiij^ipj,
=
P\
i.e.,
Then
{1}:
small increase in p\ will increase firm
9
Pi
=
set Pi. If
—
Cp <
Let
0:
Cp =
1
T2 ie p
=
minj^ipj
c%
than the reservation
utility R.
Firm
Pi
—
<
R, and make positive
minj^i pj
<
Then
R:
firm
flow allocation and thus increase
Cp >
x
—
x\
we consider the
1,
<
has an incentive to deviate.
Then, since by assumption Yli=\
R:
firm j
e
1
A
pimin{ci,l}.
be the sum of capacities of the firms that belong to
> R and
<
thus firm
—
H\\p, x,c]
is
1
^ Pi, such that pj
with
If
l's profits,
1
then we have to consider the following two cases:
1,
1
=
p\
j
—
>
miiij^! pj
the profit of firm
1
making zero
is
j can change
its
>
ci
there exists firm
1>
j,
profits, since its price is greater
= R—
price to pj
e,
for
some
e
profits.
can increase
slightly its price
without affecting
its
its profits.
following two cases:
min{ci,l}: Firm
1
can decrease
price slightly,
its
and increase
its
flow
and
its
profits.
—
x\
Xj
Pi
=
=
<
min{ci,l}:
=
0.
profits. Let's consider
increase
~
t
1,
minjcj, 1}, which can decrease
minj^ipj
Yli=i c i
Cp >
Since
cj
its
<
1-
price
If
Cp <
1
1,
Note that
j
its
and increase
price
then firm
next the case when
1
Cp >
1
can increase
1.
profits.
21
1,
Cp <
x
such that j £ P\ and
its profits.
its
By assumption
£ Pi, since otherwise
and make positive
^
there exists firm j
price
and make positive
there exists
J2i-i c %
~
cj
<
1-
some
Firm
j
with
j can
Q.E.D.
Similar to Proposition
4,
the next proposition establishes the existence of a mixed strategy
Price Equilibrium in capacity subgames with no pure strategy price Equilibrium (proof follows
from Proposition
4.3 in
[1],
and therefore
omitted).
is
Let c be a capacity vector such that ]Ci=i
Proposition 10
and suppose that there
with YliLi
exists j
fy
-
Cj
<
Then
1.
>
ci
1,
>
c;
there exists a
for
G {1,...,N}
z
mixed strategy Price
Equilibrium in the capacity subgame.
Proposition 11 Let
>
and
Ci
i.e.,
all
for
firms
i
G
{1,
make
...,
N}. Then,
for
—
each j G {1,...,N}, J2i=i ci
for all Price Equilibria \p,x],
we have
pi
=
for
G
i
cj
>
1
{1, ...,iV},
zero profits.
The proof
Proof.
be a capacity vector such that
c
follows
from a Bertrand price competition argument among the
N
firms.
c,
where
Q.E.D.
we
In the remainder of this section,
c
such that Yli=i
is
>
<H
1, fy
>
for
consider a
G
i
subgame defined by a capacity vector
{1, ...,N},
and there
exists j
with J2i=i
°i
~
cj
< !
Proposition 9 implies that there does not exist a pure strategy Price Equilibrium in this subgame.
However, Proposition 10 implies that a mixed strategy Price Equilibrium
probability measure of prices used by firm
of Hi
by
lemmas regarding the structure
Lemma
4 Let
c
l
t
i.e.,
(i)
(ii)
strategies,
i.e.,
/
=
tG p,Ct
For
all
i
>
1.
G
Pi,
°i
min,; e
Pi=={i£{l,..,,N}
E
of the
We
:
k
~
r
1
<
cj
2
=
...
i
)
l}.
1-
./v}
U-
Let
/
Let in denote the
denote the (essential) support
Next,
JFJ.
we
will provide a
mixed strategy Price Equilibrium.
be a capacity vector such that YLi=\
that there exists j with X^i=i
mixed
in this equilibrium.
and the corresponding cumulative distributions by
[Zj,Ui]
series of
i
exists.
^ >
denote the
1,
Q >
for
minimum
i
G
{1,
...,
N}
and assume
of the lower supports of the
Let P; denote the set of firms whose lower support
is
Then:
the distribution F; does not have an
atom
at
I.
Proof.
(i)
Suppose to obtain a contradiction that Xwgp
22
ci
—
!•
Let
I'
=
min^p,
U.
Then
consider firm
j e Pi deviating to
such that the new cumulative distribution Fj
p,j
p<V
(0
(
where
is
F
is
YlieP c
i
p>V
Fjip)
Essentially
the original cumulative distribution.
shifted to
1)
all
the mass between
I
unchanged
In such a deviation profile, the flow equilibrium remains
I'.
^
given by:
is
and
I'
since,
Dut tne P r ces charged for positive flows by firm j have increased, thus
its
i
profits increase, leading to a contradiction.
(ii)
We
first
show that jePj. Assume
to arrive at a contradiction that j ^ Pi.
i^i
te.Pi
where the second inequality holds by the assumption that ^;=i
contradicts part
We
showing that
(i),
next show that
>
I
0.
j
Assume
£
c»
~
cj
< L
But
to arrive at a contradiction that
[see
I
=
0.
This implies that
the characterization of mixed
equations (9)-(10)]. However, by the assumption that Yli=i c i~ cj
cf.
this
Pi.
the profits of firm j at any price vector are equal to
strategy equilibria;
Then, we have
<
L,
there exists a price vector and a flow equilibrium at which the profits of firm j are nonzero,
thus showing that
Note that the
>
I
0.
cannot consist of only one firm, since then this firm has an incentive
set Pi
to increase its price.
Let
m/nbe
contradiction that distribution
prices
that
l—e,
Fm
I
+ e, and
Fm
two firms that belong to
has an atom at
I,
By
Assume
to arrive at a
considering the profits of firm n at
using a similar argument as in the proof of
cannot have an atom at
Pi.
Lemma
3[ii], it
can be seen
I.
Q.E.D.
Lemma
5 Let c be a capacity vector such that J2i=\ c i
that there exists j with ^Zj=i c i
the mixed strategies,
>
ck
(i)
(ii)
Ci,
for all
i
e
{1,
u
i.e.,
.
,
.
.
"
cj
<
1<
>
Let u denote the
— max^/j ^ r ..,N\
>
f° r
maximum
^ {!>•>
*
of the
u i- Let k be a firm with the
N} and assume
upper supports of
maximum
capacity,
N}. Then:
At most one distribution Fi can have an atom at the
If
1> c i
the distribution Fj has an
atom
at u,
then
23
a =
c^.
maximum upper
support
u.
i.e.,
Ill,
The maximum upper support u
equal to R.
is
Proof.
(i)
We
first
show that
at
most one distribution can have an atom
= {»
Patom
€
with probability HiePatom (Fi(u+)
p
Let
—
Catom =
.
.
•
-F
z
N}
,
>
(u-))
atom
Fi has an
:
Patom
at a contradiction that
Suppose to arrive
a price of
{1,
at u.
We
define the set
at u}.
has more than one element.
0, all
follows that
It
firms that belong to Patom will charge
u.
Ci
J2iePatom
D res =
and
max{0,
- Ei0Pa4om
1
We
c*}-
have
r
Catom
where the
price
its
some h €
tou-e
Patom-,
Pa tom
firms in
all
X] ^ > maX
strict inequality follows
there exists
which
= Yl Ci ~
i
~
5Z
<
mia{ch,
1},
e
>
(since firm h
'
f
by the assumption that X)i=i
such that Xh{p u )
Dres
Ci
ci
>
where p"
charge the price u. Then, firm h can increase
some
for
l
'
)
This implies that
!•
the price vector for
is
undercutting the rest of the firms in
is
=
This show that there exists at most one distribution, which has an atom at p
(ii)
at u.
We
will
Let Pi denote the set of firms whose lower support
is I,
i.e.,
Assume that the
We
firm
first
i's
Yli=\ c i
show that
profits
~
cj
<
G
i
Pi.
when he
Suppose that
<
c^.
Let
YLi
FL,
Using
i,
Lemma
Then
show that
since
=
P;
{i
J2heP
u are equal to
0.
G
ch
C;
—
{1,
>
.
Patom)-
u.
Cfe.
.
,
.
N}
c f-
1
[
Zj
:
Lemma
=
I}.
4(i)],
Using the assumption that
to that in the proof of
Lemma
4(ii), it
can also
Pi.
and
respectively at the
where
Pi.
=
charges the price p
Suppose to obtain a contradiction that
ci
£
i
and an argument similar
1j
be seen that k &
atom
distribution Fi has an
by reducing
its profits
Sfc
for the only firm
with atom at
u, firm
i,
we have
denote the expected profits (plus the capacity costs) of firms
mixed strategy Price Equilibrium
(i.e.,
fL
=
II,
+7^
and
Sfc
i
and k
= Hk+lk ck>
denotes the equilibrium profits of firm j in the mixed strategy Price Equilibirum).
k € Pi, and the fact that Fi, Fk do not have an
4(ii)], it
can be seen that fL
=
cj and
n =
t
flfc
frA
Cfc
24
=
c^l,
atom
at the lower support
I
[cf.
which implies
(23)
upper support
since the
Next note that
of Fi
is
u and no other firm has an atom
at u,
it is
also the case that
using equation (23), this impl ies
n =R
fc
Now
e
>
£ :
-
= R-e
consider a deviation for firm k to charging a price p
The expected
0.
Since c^
> a and
=
l
l
Therefore, for
e
>
J2j=i cj
i-E c
with probability
1
for
some
profits for firm k following this deviation satisfy
ti k
(iii)
1
(
1>
>(R-e) \1-J2
we h ave that
-£ ^
c
sufficiently
cfc
)
>
i
f
=
Cj
The proof
similar to the proof of
is
-s^+^)^(
enough, the deviation for firm k
diction and proving that
of this part
1
c;;
is
,Cfc
i
-i:^
profitable, yielding a contra-
.
Lemma.
and
3 (hi)
is
omitted.
Q.E.D.
Proposition 12 Let
be a capacity vector such that J2i=\
c
~
suppose that there exists j with X3i=i
c»
maximum
mixed
of the upper supports of the
the expected profits
Ilj[/i,
x(-),c] are given
i?
Tljlfj,,
Proof.
Let
IIj
an atom at the
x(-) ,
=
Uj
c]
+
=
^
(
1
+
c
—
I
1
+c—
Let c
=
i.e.,
1, Cj
>
for
maxj^,.,.^ q.
u
=
?'
e
{1,
...,
TV}
and
Let u denote the
maxwi^,.,, N\Ui. For firm
j,
by
C
J2i=i c i)
5Zi=i c
<
j'
jjCj as in the proof of
maximum upper
1-
strategies,
„
>
R
<
ci
Q >
support, then
)
f~ 7j cj>
^
^j
—
^
^j
7j cj>
Lemma
Lemma
25
5.
If
h as no atom
at u,
an atom
at u.
^
ias
the distribution of firm
5 implies that Fj
is
j, Fj,
has
the only distribution
having an atom at the
price
p
=R
maximum upper
=R
support u
and, moreover, c ;
=
Firm
c.
is
charging
We
consider
j
with positive probability and
II,- [i?, //_.;,
= ZJll-^Ci
z(0,c]
AT
-£
= R 1+5
i=l
Thus,
Ilf
=
/?
(
1
+
—
c
X^ili c
'
an d the expected profits of firm j are given by
)
IIj[/i 1
s(0,c]
=R
+ c-J^c,
1
I
-7jCj,
J
as claimed in the proposition.
Suppose next that F, does not have an atom
maximum
at the
upper support.
the following two cases:
•
None
atom
of the distributions have an
zation given in Proposition
are given
at u.
Then, using the Flow Equilibrium characteri-
we have that the equilibrium
1,
profits for firm j at price pj
=R
by
*&
\
Note that firm
j has to satisfy Cj
Then, as argued
belong to set Pi
Lemma
in
=
Assume otherwise and
c.
both firm
4,
—
(recall that Pi
{i
:
=
k
the Flow Equilibrium characterization
which
j,
I},
let
such that X^i=i
is
where
is
I
the
=
c.
nrm
&
firm k be such that c^
~
ci
minimum
cj
<
1> an<^
lower support). Using
we have that
*
ti k
=R
1-X>
^k
V
However, since
it
j,
has to be that
k € P;
=
c.j
n
•
The
the
distribution
first
Ffc
,
cjj.
[
the proof of
[as in
=
M x
,
c.
(
•
)
,
5]
then
ft/.
= Rj^.
For this to be true,
Therefore,
c]
=R
1
has an atom at the
part of the proof Ck
Lemma
t
—
c
and
- ]T Ci
=
i?
f
1
j
maximum upper
flj
=
26
R(l
+
c
—
+
c
- ]T
c.
.
1
1
support, for
j
some k
^ j. Then
by
Yli=i ci)- Moreover, j,k G Pi, which
implies that
J
3
We
conclude that
ck
{
jc k
{r[
N
n^i x(-),c]=R[l + c-J2 c <)i
>
Q.E.D.
Oligopoly Equilibria With
7.2
In this section,
we provide
Firms
TV
a characterization of Oligopoly Equilibria capacities with
Similar to the analysis for two firms,
we
N
>
2 firms.
will use this characterization to establish the efficiency
properties of Oligopoly Equilibria.
Proposition 13 Assume that
>
i-e., Cfc
c % for all k
£
{1,
.
A
N}.
,
.
< R
7j
.
some
for
maximum
Let k be a firm with the
i.
capacity vector
c is
an
OE capacity
if
and only
if
capacity,
J2i-i c i
=
1
and
R-li
for all
i
^
+
{(H
2/1-7*
..
<
c fc )
Ci
<
(24)
ck,
k.
Proof. (Sufficiency)
We
Note that since 53i=i
—
ci
first
li
show that X^i=i
—
ci
1
Proposition 8 implies that the profits of firm
n,[p(c),x,
C]
=
(i?- 70c.
1
profit for firm
are
i
is
from
by firm
C{
Cj
>
C{.
...,
N} and
is
Y^ J7a cj
<
q,
Clearly,
in the original vector to
{1,
if c\
1 in
=
0,
1-
m
Firm
i
has the
maximum
(i?
c,:,
-
Proposition 8
7j)£j
<
t
{1, ••AT}, are
(25)
Ilj[p(c)
in this case
still
,
x,
applies
c],
is
(R, ...,R).
and the resulting
establishing that there
X2i=i c i
—
1
an d
c,
=
1 (i.e.,
firm
the new), Proposition 11 applies and tli\p
not profitable.
<
=
<
e
C{.
ms
Therefore,
we must have
case, Proposition 10 applies
a mixed strategy continuation equilibrium
•
If c\
i.
EIj[p (cj, c_;) ,x, (q,c_,)]
so that the deviation
£
^
no profitable deviations with
Next consider
j
c\
?',i
capacity.
,
where p(c) denotes the continuation equilibrium price vector, which
Consider a deviation
OE
together with (24) define an
li.
We
capacity in the
27
i
changed
{c\,
Y2j=a cj
c_j)
+Q >
,
its
capacity
x, (c;, C-i)]
1,
and the deviation
Cj
will
>
—
for
induce
consider the following two cases:
new subgame,
i.e.,
c\
>
Cj,
for all j.
Then,
Proposition 12 implies that the deviation profits of firm
%{),
Ililfi,
Thus, in this case, there
Firm
i
Cfc
is
the
= R
1
- Y^Gj
I
~
liCi,
=
j& J
V
(R- 7O a - ji {di -
<
U
t
Ci)
,
\p{c) ,x,c).
no profitable deviation.
is
maximum
does not have the
such that
&, c-i)}
are
i
maximum
new subgame,
capacity in the
capacity and
Ui[ii,x(-),(ci,c-i)]
=
>
c\z
i.e.,
there exists
Then by Proposition
c{.
—
'
some k
12,
(26)
7iCj.
Let c™ ax denote the capacity that maximizes (26), given by
-max
<
_
~
all c;
<
c\
<
_
^j^k,i
2R
2
Front equation (24) we have that <>r2L
Therefore, for
c kli
1
'
(
ci
2
+
c k)
<
c%
<
(Necessity)
Any
all j,
< R
Y2i=\
profits of all firms are equal to 0.
capacity equilibrium with JZi=i
with Y] i=1
Ci
-
Cj
<
1.
together with (24)
1
capacity vector c such that
capacity, since the firm with 7;
vector c such that for
~
Cj
°i
is
Xw=i
an
ci
fc
[)u l
<
b\
OE
<
c"
iax
<
Ci.
1
c&.
capacity.
cannot be a pure strategy
can increase profits by raising
~~ c
j
—
1
Cj.
cannot be a pure strategy
Similarly,
OE
>
1, c»
>
for
i
£
{1, ..-,N}
s(0,c]
>
capacity, since the
and suppose that there
Cj for all j.
= ij(l-^Ci] - lk c k
28
.
OE
any capacity
Suppose, to obtain a contradiction, that there exists an
Consider the profits firm k for which Ck
n
which implies that
,
Ui\p(c),x,c],
establishing that there are no profitable deviations with
c%
,„-*
[()
we have
Cfe,
tli[lJ.,x(-),(ci,C-i))
This proves that any J2i=i
Cj
OE
exists j
Then, we have
(28)
Consider the deviation to
n
ck
fc
=
1
-
<
Y/i^k c *
b y nrm
ck
= R
[/j,,x(-),(c fc ,c_,c )]
-
1
.
fc
&
1
-
profits
-7fe4,
X/>
/
£-eH
#*
V
=
which yields
¥*=
\
>
'
-TfcCfc,
/
ni[//,a:(-),c],
establishing that such an equilibrium cannot exist.
—
Next, to obtain a contradiction, suppose that there exists an equilibrium with J2i=\ ci
but (24)
Without
violated.
is
assume that firm
loss of generality,
ci<^=-^-(ci +
Now
consider a deviation by firm
from Proposition
to c\
1
violates (24),
i.e.,
c fc ).
(29)
c™ ax as given by
=
1
1>
(27). In
view of
(29),
c™ ax >
c\
and
deviation profits are given by
12, the
,
,#
n^xQ.tcrv-r
-
(i
Ej#i,/c cjj
-
(
i
-
Ej-i,/c
<y
-
ci
j
2
71
4fl(l-£#i <*-ci)
lfc
>
i?ci-7ici,
=
Ili[/z,a;(-),c],
This establishes that there cannot be any equilibrium
not satisfy (24), completing the proof.
7.3
We
OE
capacity with
N
=
1
that does
Firms
next investigate the Price of Anarchy and Price of Stability
The
ci
Q.E.D.
Efficiency of Oligopoly Equilibria With
firms.
Ei=i
for oligopoly equilibria
with TV
following example shows that the efficiency loss in the worst oligopoly equilibrium
(Price of Anarchy) can again be arbitrarily high.
Example
<
e
<
2 Consider a price-capacity competition
=
min{l,i?}, 72
...
=
7jv
— R—
e
2
-
game with two
The unique
firms,
social capacity
and 71
is
= R—
(cf,^)
=
with social surplus
§{C
Using Proposition
.
oe _
13,
(r
it
OE
[l
S
=
)
€.
follows that the capacity vector
pE\ = ( _i
r OE
r
'* '"' w
\R +
'
e'
R
R
\
(R + t)(N-iy-'{R + e){N-l))>
29
e
for
some
(1,0, ...,0)
is
an
OE
capacity with social surplus
t '(i
OE _
Therefore, as
e
—
>
0,
+
fl)
R+e
v
the efficiency metric satisfies
^=
lim r({ 7i },c^)
The
R)
z; ~^+
_ lim
-L__i
=
preceding example implies the following efficiency result:
Theorem
4
Consider the price-competition game with
inf
inf
{0< 7i <iJ}cO^ C({ 7,})
i.e.,
0.
N firms, N >2.
r({ 7l },c
OE )=0,
the Price of Anarchy of the price-capacity competition
Next we provide a non-zero lower bound on the Price
Then
game
is 0.
of Stability of a price-capacity competition
game.
Theorem
i
5
Consider the price-competition game with
N firms, N >
2.
Then,
<
for all
7,
< R,
G {1, ...,N}, we have
sup
""*-
'
c° £ e c({ 7i })
i.e.,
OJ5w -VW-1
},0>2
—"
"
r({
"
V L 7i
/'J
/
>
jit
game
the Price of Stability of the price-capacity competition
bound
Proof.
is
1
-1
iV
is
2(yN — 1)/(N —
and
1)
this
tight.
We
assume without
then by definition jj
==
capacity vector (ef cf
,
,
loss of generality
R for
...,
all j,
cfj)
=
that 71
<
so that the equilibrium
(1, 0,
...,
0)
is
.m ji
min$e /2
and
and that
71
< R
[if
=
R,
Then, the
social surpluses coincide].
a social capacity, with social surplus S(c
71
s
)
= R — 71.
Using the definition of the efficiency metric r({ji},x OE ), we consider the following optimization
problem:
n
sup
c O£ €
Since for
all c
0E G
clearly attained at
maximum
C({7;}),
we have
^=1^'^
"-71
C({ 7l })
51
$2J=i cf
=
1,
the
(30)
,
supremum
some c OE G C({7i}) with the maximum value
E at an OE capacity is given by
in the
of cf
above expression
E By Proposition
.
is
13, the
value of cf
c
R
OE =
.
R + zZ'LiR-ir)
30
(31)
Substituting c?
B
---
i?/(i?
{2, ...,7V} in the objective function in (30),
R - Ry/jR +
n]
(R -
-
see that the optimal value
0<ii<R
is
bound on the preceding over
<
all
i
6
71
<
for
given by
^IW + ^(^ ~
Z?=2( R ~
- Rji/jR + Y;?= 2 (R - -ft) - Ejl 2 (-^ 5
R — Ji
fl
mf
+ YL 2 {R- 7i))
7i )/(i?
7,))
75
< R with
we consider the following optimization problem:
e ->
i-
lit
we
£f=2 (fl ~ 7j) -
-We-are-i-nterested in finding a lower
min^g/2
and c° E
+ EL(^-7i))
+ Ef=2 (i? - 7j))
707i/ (^
,„,
(32)
•
~ti<ii
This problem has a compact constraint set and a lower semicontinuous objective function [note
=
that for 7!
R, the efficiency metric
solution (71,72,
...,7jv)>
For 72
there should exist at least an
function
is
i
=
...
—
7/v
=
Therefore
it
has an optimal
R, the objective function value
is
1,
<
such that yz
R.
For
showing that 71
strictly increasing in 71,
OE ) =
satisfies r({-y1 },c
=
1].
Moreover
optimal solution to (32) will satisfy (71,72, ...,7jv)=(0, 7, •,7),
then that the optimal solution
value
—
given by (71,72, •,7/v)
5
>
bound
(0,
72
=
R,
,
j^j_i
...
=
7jv
iv-i
=
7-
It
follows
R) with optimal
In this case, as 5
optimum
An
tight,
=
-=7y= N _ 1
is
2
—
Vn-i
J^_-}
iV-l
.
>
OE
capacity of the
game with
the Price of
R-
the ratio of the surplus in the equilibrium and the surplus in the social
Q.E.D.
interesting implication of this result
Anarchy equal
coordination with a limited
there are
consider the best
is
and
72
is
=
i.e.,
not hard to see that the
it is
2^f^.
Finally, to see that this
71
is
[R,---,R), the objective
all (72, ...,7at) r"
0.
showing that
many competing
is
that as the
number
of players increases not only
is
to zero, but the Price of Stability also goes to zero. Therefore, while
number
of players can ensure that inefficiencies remain
firms even the best equilibrium has
interesting in part because
it
unbounded
bounded when
inefficiency.
goes against a naive conjecture that increasing the
This result
number
of
oligopolistic competitors should increase efficiency (or even ensure that the equilibrium limits to
a competitive allocation).
as the
number
The reason why
this naive intuition does not
of firms increases, investment incentives
31
become
apply in this case
potentially
more
distorted.
is
that
Stackelberg Leader Game
8
We
have so
far characterized the set of
pure strategy equilibria
in the baseline price-capacity
competition game, where a set of competing firms choose capacity simultaneously
compete
demands
in prices (and users allocate their
in the third stage).
The
first
and then
analysis has
shown
that different equilibria within this set have widely differing efficiency features. In particular, the
worst equilibrium from the set of pure strategy equilibria can have arbitrarily low
if
we
from the
select the best equilibrium
/
be 2(v ]V
—
—
l)/(iV"
1)
(in particular,
worst efficiency performance will
set of equilibria, the
2\/2
—
with two
2
This raises the question of
firms).
may be
a natural and simple multi-stage
we
of
used to affect equilibrium selection.
While an analysis on equilibrium selection
is
how
some type
the equilibrium will be selected from the set of pure strategy equilibria and whether
regulation
while
efficiency,
game
beyond the scope
is
of the current paper, there
that implements the best equilibrium. In this section,
discuss this multi-stage game, which involves the firms choosing their capacities sequentially,
acting in reverse order of their capacity costs. In the special case with two firms, this
to the lower-cost firm acting as the Stackelberg leader
To
simplify the exposition, in this section
and chooses
Cj.
Then
game works
firm —i, after observing
two firms simultaneously choose
their
demand.
=
If 7$
This game form
firm.
Alternatively,
7_j, the
may
if
prices,
and
it is
as follows:
is
first.
again use
if 7,
<
game where
let
may have
game
us suppose that 71
two firms choose their prices simultaneously.
its
A
i
to
moves
advantage to the low cost
an incumbent in the industry, we
firm 1 chooses
—i
and
7_j, firm
chosen
its
may
think that this
capacity before the
possible to imagine situations in which the lower cost firm
For the rest of this section,
i
same time.
their capacities at the
not the incumbent, in which case such a Stackelberg
to the multi-stage
—2 and
result as a focal point, giving the first-mover
the low cost firm
capacity
and prices are revealed, users allocate
after capacities
two firms choose
its
equivalent
chooses c_^. After the capacity choices, the
Cj,
equilibrium will arise naturally, since the incumbent
entrant. However,
N
we suppose that
denote the two firms. In this case, the Stackelberg
first
and choosing
is
<
will
72,
capacity
is
new
the entrant
not arise naturally.
and by a Stackelberg game, we
first,
followed
by firm
2,
refer
and then the
pure strategy Stackelberg equilibrium
is
defined
as follows.
Definition 5
[Stackelberg Equilibrium]
best response capacities for firm
BR.2{c\)
For a given
c\
>
0, let
BR2{c\) denote the
2, i.e.,
=
arg
max
cn>0
32
IT^i,
.t(-),
ci^co].
set of
A vector [c SE ,p(c SE ),x(c SE
€BR2 (c^E ),
PE(cSE ),4E
U
for all ci
>
0,
)] is
£(•)]
[//,
1
[p(c
SE ),x(c SE
),(cf
e MP-B(ci,c 2 ), and
<
71
72
<
c2
i?.
SE
E ,c! E
> U1
)}
5# 2 (ci)
G
Then
SE
),
x(c SE )} G
J,,x(-),c 1
[f
,c 2 },
(33)
.
there exists a unique Stackelberg equilibrium
—
#-
,
i
2i?
_
_
SjB
C2
p=
pf* = p
and xf E
j?
Proof. (Existence)
cf^
is
1, first
It
=
cf
c\
,
follows
a best response for firm
note that any
£
<
cf
E
xf
s =
+
c2
>
cf
72
-
ii-72
2i?- 72
72
'
B
.
from the sufficiency part of proof of Proposition 7 that given cf
2, i.e.,
S
cf G BR2
(cf
E
)-
To
see that there
Next consider
gives lower profits.
to that in the proof of Proposition 7
>.
>
-
(Uniqueness) From Proposition
other choice of
Denote the
we have the
i
=
E An argument
.
identical
c\ will satisfy
72
the analysis in the proof of Proposition 7 establishes that firm
R,
cf
for firm
^ - 72
1
1
2i?
<
>
no deviation
,
Since
1.
Theorem
c\
is
shows that the best response of firm 2 to such
C\
7i
\p{c
which
r
Ci
c\
if
&nd
Proposition 14 Suppose that
in
a (pure strategy) Stackelberg Equilibrium, (SE)
c\
7, this is
by
make lower
profits.
the equilibrium with the highest level of
can be improved upon by firm
set of Stackelberg equilibria
1 will
1
deviating to cf E
SE ({71})
•
.
Combining
Any
c\.
Q.E.D.
this result
with Theorem
2.
following result.
6
Consider the Stackelberg game described above with two firms. Then, for
1,2,
all
we have
se €
c SE
£L
SE({
1
/({7*}.^)=
7i })
SU P
r({7;},c
c se € SE{{ lt })
S£
)
= 2v/2-2,
i.e.,
both the Price of Anarchy and the Price of Stability of the Stackelberg game are 2y2 —
this
bound
is
<
tight.
33
2
and
we
In this section,
Game
Simultaneous Capacity-Price Selection
9
consider the alternative one-stage competition between the two firms: firms
simultaneously choose the capacity levels c t on their links and the price pi they will charge per
Given the price and the capacity
unit bandwidth.
firm
i
set
C-i, the profit of
firm, p_j,
is
= PiXi -
Ui[{pi,p-i),x, (a, c-i)]
where x £ W\p,c],
The
c.
by the other
£
i.e.,
7iCj,
a flow equilibrium given the price vector p and the capacity vector
is
objective of each firm
to
is
maximize
We
profits.
next define the one-stage Oligopoly
Equilibrium for this competition model.
A
Definition 6
vector [c*,p*,x*\
x* £ W\p*,c*} and for
all
n
>
for all pi
0, q,
>
0,
i
i
and
£
<
with
ji,
1
x £ W[(pi,pti),
for all
ji
N
<
if
{1, ...,N},
[(^,pi i ), a: *,(ct ci i )]>ni [(pi ,pi i ) ]a;i
Proposition 15 Consider
Given any
a (pure strategy) one-stage Oligopoly Equilibrium (OE)
is
(cj,
i
£
ci i )]
(34)
1
clj].
firms playing the one-stage
R,
( Cil
game described above with
N
>
2.
does not exist a one-stage Oligopoly Equilib-
{1, ...,N}, there
rium.
Proof. Suppose, to obtain a contradiction, that there
We
[c*,p*,x*].
J2i = i c*
<
first
show that
in this equilibrium,
the profit of firm
1 is
<
[c*,p*,x*]
R, by increasing
c\ slightly,
firm
c*
>
*•
Then
affecting its price
and flow
for
0,
firm
some
e
an<^ V\
p £ [0,-R]^ and
—
R-
If
^=i c t <
1>
=
(
Pl -7i)ci.
increases
1
(35)
its profits,
contradicting the claim that
1
>
firm j changes
can increase
0.
its
Then
=
(x*j,pj), since it
reduces
its
<
c*.
Clearly
it
capacity costs without
allocation.
Hence, we must have Xw=i c\
—
1
there exists j € {2,3,.. .,N} for which x*
profitable for firm j to deviate to {cj,pj)
x\
for all
—
a one-stage OE.
is
Consider next YliLi
is
c]
c*
given by
lW,z*,c*]
Since 71
we must have J2i=i
then since the flow allocation vector c £ W.\p,
1,
Equilibrium
exists a one-stage Oligopoly
its
—
1
and
^so
Pi
capacity level and
= R
make
by equation
positive profits.
the profit of any firm j £ {2, ...,N}
capacity and price to (cj,pj)
—
34
(1,
(35).
R—
is
5) for
at
5
0,
Assume next
most (R
some
=
If c\
>
—
7/)(l
and
8
—
then since
that c\
e).
< (R —
=
But
e
if
~fj)t, it
make
will
a profit of
R-5-
Oligopoly Equilibrium.
-yj
> (R -
showing that there does not
e),
Conclusions
efficiency of oligopoly equilibria in a
we studied the
over capacities and prices.
This problem
may
it
is
relevant
arise in the process of capacity
modern communication networks.
extension in
To
model where firms compete
not only of theoretical interest, but
is
understanding the extent of potential inefficiencies that
for
exist a one-stage
Q.E.D.
10
In this paper,
-
jj){l
main economic
isolate the
interactions,
we considered the following simple game form.
Second, after the capacity levels are observed,
firms independently choose their capacity levels.
they set prices.
Finally,
consumers allocate
First,
demands
their
across the firms.
This game has
an obvious similarity to Kreps and Scheinkman's model of quantity precommitment and price
competition,
rely
on
[9],
but
it is
simpler because
demand
is
inelastic
and because
results
do not have to
specific rationing rules.
Using similar ideas to the analysis
egy equilibria.
by mixed
A
in
[9]
and
in
[1],
we
characterized the entire set of pure strat-
pure strategy oligopoly equilibrium always exists in this game but
strategies off-the-equilibrium path.
The complete
is
supported
characterization of the equilibrium set
enables us to investigate the worst-case efficiency properties of oligopoly equilibria.
Our
first result
here
is
Price of Anarchy) of this
oligopoly equilibrium
Stability)
is
that efficiency in the worst oligopoly equilibria (also referred to as the
game can be
arbitrarily low.
if
the best
selected, the worst-case efficiency loss (also referred to as the Price of
can be bounded.
With two
firms, this
an arbitrary number of firms, N, the bound
Interestingly, this
However, we also show that
bound goes
to zero as the
is
bound
is
and equal to 2\/2 —
tight
again tight and equal to 2(\/~N
number
of firms,
N,
—
With
2.
l)/(-/V
—
1).
increases. This result contrasts
with a naive intuition that the efficiency of oligopoly equilibrium should improve as the number
of firms increases.
The reason why
this intuition does not
apply
in the
current context
is
that
with the greater number of competitors, ex ante investment incentives become potentially more
distorted.
We
also suggested a simple
way
of implementing the best oligopoly equilibrium,
which involves
the lower cost firms acting before higher cost firms as the "Stackelberg leaders" and choosing their
capacities.
loss
With two
firms, the Stackelberg
bounded by 2\/2 —
Finally,
game
gives a unique equilibrium, with the efficiency
2.
we studied an
alternative
game form where
35
capacities
and prices are chosen simulta-
neously and showed that
it
always
fails
to have a pure strategy equilibrium.
that the timing of capacity and price choices in oligopolistic environments
These
is
results suggest
important both
for
the existence of equilibrium and the extent of efficiency losses.
Many
features of the
model analyzed here were chosen
to simplify the exposition.
The
analysis
here can be easily generalized to arbitrary (convex)_costs functions for investmen t in ca pacities,
without changing the essence of the analysis or the
Another more important generalization
important feature of
libria
and
many communication
is
results.
to include potential congestion costs,
networks. Existence and efficiency of oligopoly equi-
with congestion costs (but without capacity investments) are analyzed
efficiency of oligopoly equilibria
which are an
in
[1],
and existence
with congestion costs and with capacity investments
case with symmetric firms are studied in
[15].
The problem
is
much more
challenging
in
when
the
there
are asymmetries, either in the costs of investing in capacity or in the extent of congestion costs
within a subnetwork.
We
leave the analysis of this general
model
to future work.
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37
352
Date Due
Lib-26-67
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