CHAPTER
4
CoLLUSIVE OuTcoMES IN A SINGLE SHoT GAME
AN ExAMPLE
INTRODUCTION
In a simple homogeneous product setting this chapter explores the
possibility of collusive outcomes in price competition in a single
The 1 i terature pertaining to this area says that
shot game.
possibility
of
implicit
collusion
clearly
requires
the
repeated
interactions because in a one shot game the free-rider problem is
too stark to sustain collusion.
However this chapter shows with
the help of a parametric example that such collusive outcomes are
possible even
in a single shot game of price competition;
thus
con.tradicting the relevant 1 i terature in some sense. Before coming
to the main thread of our argument we will provide a brief prelude
to our exercise.
The notion that oligopolists may collude has been around since the
virtual
inception of economics. -In the theoretical
oligopoly an
output
(or
price)
configuration
is
literature on
thought
of
as
comprising a "collusive" outcome if it maximises the joint profit
of all firms,
no
other
or, more generally,
output
feasible
if it is the case that
configuration
which
the1~e
gives
is
all
oligopolists as much profit as before and at least one oligopolist
a
larger
cartels
profit.
Most
with explicit
of
the
early
price or output
theoretical
agreements.
years explicit call us ion was not very uncommon.
passing of, the Sherman act
Case,
wa~3
1899,
to fall
work
In the
was
on
early
However after the
1890 and the Addyston Pipe and Steel
it was realised by the firms that to collude openly
foul
of the law.
Gradu3.lly the practice of explicit
64
co 11 us ion was driven underground
1
Now it was time for
collusion to come to the surface.
described
in
the courts as
For
"conscious
instance,
what
parallelism"
implicit
has been
(see
Bast•,
1993) consists of a set of firms behaving in union and setting a
high price. Casual inspection also leads us to believe that tacit
collusion is possible.
The number of industries that have been
accused of "conscious parallelism" is large and includes cement,
drugs,
dyes,
lumber,
theaters
and
tobacco
(see
Jacquemin
and
Slade , 1989 ) .
In
this
connection
suggested that
one
may
refer
to
Chamberlin
(1929)
who
in an oligopoly producing a homogeneous product,
firms would recognise their interdependence and, therefore, might
be able to sustain the monopoly price without explicit collusion.
The threat of vigorous price war would be sufficient to deter the
temptation to cut prices. Hence, the oligopolists might be able to
collude
in a
purely noncooperative
manner.
The
possibility of
collusive outcomes stemming out of noncooperative interactions is
also inherent in the story of the kinked demand curve (Hall and
Hitch, 1939 and Sweezy, 1939), in which each firm conjectures that
its rivals will match price cuts but will stay put if it raises
its price.
1
Here
it
susutainable
may
be
because
mentioned
of
the
that
explicit
Incentive
for
familiar story of the instability of cartels.
65
collusions
free
riding.
may
This
not
be
is
the
imp~icit
Regarding the possibility of
collusion in an infinitely
repeated play of an oligopoly one may refer to the simple trigger
- strategy argument which was developed by Friedman (1971) or to
to
the
"Folk
theorem"
which
holds
that
any
outcome
collusive outcome down to the competitive outcome,
from
the
including all
manner of assymetric outcomes, are outcomes of a Nash Equilibrium,
if the discount factor is close enough to one
It may be noted that
output
2
in the above models of collusion aggregate
and hence p:--ice also
is constant from period to period,
never changes. Price wars, which are so much a part 0f r'eality can
An ingenious model for explaining price wars within
never occur.
the
ambit
of
equilibrium analysis
Porter (1984).
was
developed
The central feature of their model
Cournot game with imperfect monitoring.
by Green
and
is a r'epeated
In their model,
price is
determined by the industry output and a random variable 8.
Each
firm observes this price and its output but cannot observe the
output of its rivals.
When an unusually low price is realised,
~s
there are two possibilities. Either demand
has
cheated
and
increased
production.
It
very low or a rival
is
therefore
always
possible for cheating to go undetected. Green and Porter show that
cheating can be deterred by threatening to produce at Cournot-Nash
levels for a period of fixed duration wheneveY'the market pr'ice
2
The
be
more
range
shown
to
of
be
campi i cated
behaviour
quite
than
theorem intensive venture
supportable
large
trigger
if
we
under
allow
strategies.
in this direction,
66
subgame
for
For
perfection
strategies
a
I uc i d,
see Abreau (1986).
which
if
can
are
somewhat
co ll us i ve
price,
trigger
below some
drops
output
l
q
=
q
•
'
The
P.
idea
is
to
a trigger price p and a
•
period T. An equilibrium trigger price strategy (q
exists because q
•
select
a
punishment
T) always
P,
can be taken to be equal to the Cournot level.
3
There will in general, however, be many equilibria
It is arguable that, even if the predictions of the above models
perform well,
infinite
they are not dependable,
repitition
repeatedly over
is
unrealistic.
because the assumption of
In reality firms
long stretches of time but
the
may meet
length of
the
interactions is nevertheless finite, and firms know this. Consider
a Cournot oligopoly with a unique Nash equilibrium. It is easy to
to see that if the Cournot oligopoly is played a finite number of
times then there
is a unique subgame perfect equilibrium which
consists of the Cournot outcome in each stage game.
Introspection, however, suggests that it is unrealistic to expect
firms
to
produce
the
Cournot
outcome
year
after
year
without
striking some implicitly collusive behaviour. How can we formally
explain collusion in a finite time framework ? One
interesting
possibility is to relax the assumption that rationality is common
knowledge.
This
has
been done
in
many different
literature. One of them is by Kreps,
3
In
the
downturns.
are
at
Green
and
Rotemberg
least
as
Porter
and
common
price
model,
Sa loner
during
Wilson,
(1986)
booms
this phenomenon.
67
and
build
in
the
Milgrom and Roberts
wars
claim,
ways
OCCUJ'
only
however,
that
a
to
model
during
wars
explain
( W82) whose basic insight is that a small uncertainty about the
preferences of the players can have a significant influence on the
players'
behaviour if the game is repeated long enough (but not
necessarily repeated an infinite number of times) and so it
possible
that
for
a
sufficiently
large
horizon,
each
is
player
cooperates at the begining of the game.
A third line of analysis involves giving more structure to the
Cournot stage game. Production takes time and it seems reasonable
to suppose that an oligopolist can observe
~hether
its competitors
are planning to produce a lot or little; and can respond to this
by
adjusting
finally
its
made
own
production
available
on
the
plans
market.
before
the
Basu
(1992)
product
offers
is
a
stylised view of this. This entails thinking of each stage game as
broken up into
two substages.
In the fir·st
substage each firm
produces some amount. They observe this and in the second substage
produce more or dispose off any amount of the output produced in
the first substage. Then their total production is offered on the
market and price and profits are determined in the usual Cournot
style.
If this modified Cournot game is played a finite number of
times, collusion becomes possible under subgame perfection as Basu
( 1!392) has shown.
The above discussion shows
collude
in
a
purely
that ol igopol ists might
noncooperative
manner
if
be able
they
to
interact
repeatedly. However now we will show the possibility of collusion
in price competition in a
one shot game.
68
A PARAMETRIC
Let
us
costs.
consider, a
homogeneous
The demand function
EXAMPLE
product
duopoly
with
is given by Q == A 2
functions are given by C (Q) == cQ,
I
I
I
i == 1, 2.
symmetric
P and the cost
We will consider a
game in prices.
Now define the following
rr ( P) = PF(P) - C (F(P))
I
I
1
.!pf(P) -C(-F(P))
rr (P)
I
p
I
2
s. t. rr (P
P .s.t. rr ( pI
I
)
2
0
= rr (P I
)
I
)
From Chapter 2 (Proposition 1) we know that that any P E [P., P
I
is a
pure strategy Nash equilibrium in price competition.
firm i
If a
quotes a price in this range then it is best for the jth
firm to quote the same price and not undercut it or charge more.
In our example P
example
any
P
I
=
cA/(2+c) and P
E
cA/(2+c),
I
== 3cA/(2+3c).
3cA/(2+3c)
is
That
a
pure
is
in our
strategy
Bertrand equilibrium.
It
may also be noted that
profit
maximisation
each firm
will
(i.e.
charge a
if both the firms goes
if there
price PJ
69
is an explicit
=
A(l+c)/(2+c)
in for
cartel)
and
joint
then
each will
•
QJ
proquce
1
::::
AI( 4+2c).
Now A(l+c)/(2+c) > cA/(2+c)
And A(l+c)/(2+c)
Hence for c
~
~ 2,
3cA/(2+3c) for any c
PJ
E
ca/(2+c),
~
2.
3cA/(2+3c)
].
That
is
the
collusive outcome price PJ can be sustained in a one shot game of
price competition when output is demand determined.
CoNCLUSION
This chapter shows the possibility of collusive outcome stemming
out of non-cooperative interaction in a single shot game of price
competiton in a homogeneous product duopoly.
demand that
comes up to their door and
If firms supply all
if costs are strictly
convex then such collusive outcomes are clearly sustainable (which
the example shows) because there is no incentive to deviate.
70