Market Structures Oligopoly and Repeated Games

About This Presentation

Title:

Market Structures Oligopoly and Repeated Games

Description:

We will use the discount factor, 0 1, to set all profits in terms of ... Each firm j attempts to maximize the discounted value of profits dt -1 jt, where ... – PowerPoint PPT presentation

Number of Views:119

Avg rating:3.0/5.0

Slides: 81

Provided by: debk6

Category:

more less

Transcript and Presenter's Notes

Title: Market Structures Oligopoly and Repeated Games

1
Topic 4 Part VI
Market Structures Oligopoly and Repeated Games
2
Infinitely Repeated Games and Cartel

Assume that a simultaneous-quantity-setting game
is played repeatedly
The strategic situation is similar to the
Prisoners Dilemma players will be better-off by
producing the level of output (cartel quota) that
maximizes the industrys profits, but they have
incentives to deviate and unilaterally produce a
level of output higher than their cartel quota

3
Infinitely Repeated Games and Cartel

A way of solving the instability of cartel can be
to address the problem under the perspective of
infinitely repeated games and apply a punishment
strategy

4
Infinitely Repeated Games and Cartel

A punishment strategy that supports the cartel
output as a SPNE is the grim-trigger strategy
(Friedman, J., Review of Economic Studies,1971)
Each firm produces the cartel level of output
as long as both produce the cartel output in the
past. But if either ever deviates (i.e., by
producing something other than the cartel
output), each firm revert to producing the
Cournot output forever (nice discussion in
Kreps, A Course in Microeconomic Theory)

5
Verifying SPNE in Infinitely Repeated Games

Note that if firm i believes that firm j ( i ? j)
will produce the Cournot level of output (as a
punishment for the deviation of firm i) in a
given period, then the optimal response of firm i
is to produce the Cournot level of output as well
(by definition of Cournot equilibrium)

6
Infinitely Repeated Games and Cartel

Note also that firm i will find it profitable to
stick to the cartel output level only if the
present value of its profits from cooperation are
greater than the present value of its profits
from deviation
Let ?di be the profits from deviating, ?ci be the
Cournot profits and ?i be the cartel profits,
for firm i

7
Infinitely Repeated Games and Cartel

We will use the discount factor, 0 lt ? lt 1, to
set all profits in terms of the present period
Remember that the present value of x dollars
received one period from today is ? x, two
periods from today is ? 2x, etc.

8
Infinitely Repeated Games and Cartel

Remember also that the sum of the discounted
payoff stream if an agent receives x dollars each
period for an infinite number of periods is
x ? x ? 2x x (1 ? ? 2 )
x/ (1 - ?)

9
Infinitely Repeated Games and Cartel

Then, firm i will stick to the Cartel output if
?i ? ?i ? 2 ?i ? ?di ? ?ci ? 2
?ci
? ?di ? (?ci ? ?ci )
? ?di ? ?ci (1 ? ? 2 )
i.e.,
?i /(1 ?) ? ?di ? ?ci /(1 ?)

10
Infinitely Repeated Games

Hence,
?i ? ?di (1 ?) ? ?ci
?di ? - ? ?ci ? ?di - ?i
Therefore, if
? ? ?di - ?i / ?di - ?ci

11
Infinitely Repeated Games

Hence, we find that if the parties are patiently
enough (i.e., if they have a discount factor
greater than some threshold) and value future
cooperation more than immediate benefits, then
cooperation (i.e., stick on the cartel output)
will be supported as a SPNE of the infinitely
repeated game

12
Verifying SPNE in Infinitely Repeated Games

For a detail analysis of why the threat
represented by the punishment strategy allows to
sustain the cartel solution as a SPNE in an
infinitely repeated game, we will consider a
simple 2X2 stage game and impose a strategy space
on firms
Strategy 1 Produce the cartel quota (cooperative
strategy)
Strategy 2 Produce the Cournot level of output
(defect strategy)

13
An lllustration A Simple 2X2 Stage-Game
Firm 2
F I R M 1
14
Verifying SPNE in Infinitely Repeated Games

This simple game resembles the Prisoner Dilemma
the cooperative solution (cartel solution)
maximizes the joint payoffs but each firm
receives a higher payoff playing the Cournot
solution (for any strategy chosen by the other
player)
In this simple example, the cooperative solution
of the stage game is (defect, defect)

15
Verifying SPNE in Infinitely Repeated Games

The grim-trigger strategy specify
that firms select (cooperate, cooperate) each
period as long as this profile was always played
in the past by both firms
otherwise, firms are to play (defect, defect)
forever

16
Verifying SPNE in Infinitely Repeated Games

We need to understand whether the firms have the
incentive to play (cooperate, cooperate) each
period under the threat that they will revert to
(defect, defect) forever if one or both of them
cheat

17
Verifying SPNE in Infinitely Repeated Games

Consider the incentives of firm i (i 1, 2) from
the perspective of period 1
Suppose the other firm, firm j, behaves according
to the grim-trigger strategy

18
Verifying SPNE in Infinitely Repeated Games

Player i has two options
It can follow the prescription of the
grim-trigger strategy, which means cooperating if
firm j does
In this case, firm i obtains a payoff of ? each
period, for a discounted total of ?i/(1 d)

19
Verifying SPNE in Infinitely Repeated Games

Firm i could defect in the first period, which
yields an immediate payoff of ?d (because firm j
cooperates in the first period). However, firm
is defection induces firm j to defect in each
period thereafter. So the best that firm I can do
is to keep defecting and get ?ic each period,
starting in the second period
Thus, by defecting in period 1, firm i obtains
the payoff ?id d ?ic/(1 d)

20
Verifying SPNE in Infinitely Repeated Games

If
?i/(1 d) ? ?id d ?ic/(1 d)
or
? ? ?di - ?i / ?di - ?ci ,
then player i earns a higher payoff by
perpetually cooperating against the grim-trigger
than by defecting in the first period

21
Verifying SPNE in Infinitely Repeated Games

So far, we see that the players have no incentive
to cheat in the first period as long as
? ? ?di - ?i / ?di - ?ci
The same analysis establishes that the players
have no incentive to deviate from the
grim-trigger in ANY period

22
Verifying SPNE in Infinitely Repeated Games

For example, suppose the players have cooperated
through period (t 1)
Then, because the game is infinitely repeated,
the continuation game from period t looks just
like the game from period 1

23
Verifying SPNE in Infinitely Repeated Games

Hence, the analysis starting in period t is
exactly the same as the analysis at period 1
Discounting the payoffs to period t, we see that
cooperating from period t yields each firm
?i/(1 d) and defecting against the
grim-trigger leads to the payoff ?id d ?ic/(1
d)

24
Verifying SPNE in Infinitely Repeated Games

Thus, neither player has an incentive to defect
in period t if ? ? ?di - ?i / ?di - ?ci
Then, the analysis performed is enough to
establish whether cooperation can be supported in
a SPNE

25
Verifying SPNE in Infinitely Repeated Games

For the simple simultaneous-quantity-setting game
proposed, the cartel equilibrium can be sustained
by the reputation mechanism if and only if the
discount factor
? ? ?di - ?i / ?di - ?ci

26
SPNE of the Infinitely Repeated 2 X 2 Game

Firm j will not have incentive to deviate from
cooperation iff
(a2/8b) (1/1 d) ? 5a2/36b d/(1 d) (a2/9b)
d ? 1/2

27
SPNE of the Infinitely Repeated Game

Consider now a general case, where firms are not
restricted to play the two specified strategies
Evaluate under which conditions cooperation can
be sustained as a SPNE

28
SPNE of the Infinitely Repeated Game

We will first need to obtain the optimal
deviation for firm j
The maximum gain from deviation for firm j can
be obtained by playing the best response to the
other firms output equal to the quota (i.e.,
evaluating the reaction function for firm j at
firm is output equal to the quota)
Then, we get ?di
Finally, applying the same line of reasoning, we
get that the condition on d can be expressed as
? ? ?di- ?i / ?di - ?ci

29
SPNE of the Infinitely Repeated Game

Given that
yj a/4b (cartel quota)
yi fi(yj) (a byj)/2b
(reaction function for firm i)
p(Y) a - bY (inverse market demand)

30
SPNE of the Infinitely Repeated Game

Then, the optimal deviation for firm i (when firm
j
remains producing the quota) is
yi d fi(yj) (a byj)/2b a b(a/4b)/2b
3a/8b
So, Y yi d yj 5a/8b and p(Y) 3a/8
Hence, ?di 9a2/64b

31
SPNE of the Infinitely Repeated Game

To sustain cooperation as a SPNE we need
(a2/8b) (1/1 d) ? 9a2/64b d/(1 d) (a2/9b)
d ? 9/17

32
Infinitely Repeated Games

As in the case of the Prisoners Dilemma, there
are a multiplicity of other equilibria in this
model (Folk Theorem)

33
Bertrand Model of Price Competition

Model Setup
Two firms, firm 1 and firm 2 (duopoly)
Demand function x(p)

34
Bertrand Model of Price Competition

Both firms have NO capacity constraint (i.e.,
that prevents to produce more than some maximal
amount). Then, a price announcement represents a
commitment to provide any quantity demanded
Both firms have same cost c gt 0 per unit produced

35
Bertrand Model of Price Competition

Competition takes place as follows the two firms
simultaneously name their prices p1 and p2
Sales for firm j are give by
x(pj) if pj lt pk
xj(pj, pk) ½ x(pj) if pj pk
0 if pj gtpk

36
Bertrand Model of Price Competition

The firms produce to order. So, they incur
production costs only form an output level equal
to their actual sales
Given prices pj and pk, firm js profits are
equal to (pj c) xj(pj, pk)

37
Bertrand Model of Price Competition

The Bertrand model constitutes a well-defined
one-shot simultaneous-move game
We can apply the Nash equilibrium concept. We
will restrict to pure-strategy N.E.
There is a unique Nash equilibrium (p1, p2) in
the Bertrand duopoly model. In this equilibrium,
both firms set their prices equal to cost p1
p2 c

38
Bertrand Model of Price Competition

Proof
(1) We will prove first that p1 p2 c is a
N.E.
At these prices, both firms earn zero profits.
Neither firm can gain by raising its price
because it will then make no sales (and still
earn zero profits)
By lowering its price below c a firm increases
its sales but incurs losses

39
Bertrand Model of Price Competition

(2) Now, we need to prove uniqueness (i.e.,
there are no other N.E.)
Suppose that the lower of the two prices named is
less than c. In this case, the firm naming this
price incurs losses. But by raising its price
above c, the worse it can do is earn zero. Thus,
these price choices could not constitute a N.E.

40
Bertrand Model of Price Competition

Now suppose that one firms price is equal to c
and that the others is strictly greater than c
pj c, pk gt c
In this case, firm j is selling to the entire
market
but making zero profits. By raising its price a
little, say to pj c (pk c)/2, firm j would
still
make all the sales in the market, but a strictly
positive profits. Thus, these price choices could
not constitute a N.E.

41
Bertrand Model of Price Competition

Finally, suppose that both prices are strictly
greater than c
pj gt c, pk gt c
Without loss of generality, assume pj ? pk
In this case, firm k can be earning at most ½(pj
c) x(pj)
But setting its price equal to pj - ?, for ? gt 0,
that is, by undercutting
firm js price, firm k will get the entire market
and earn
(pj - ? - c)x(pj - ?)
Since (pj - ? - c) x(pj - ?) gt ½(pj c) x(pj),
for small-enough ? gt 0,
firm k can strictly increase its profits by doing
so. Thus, these price
choices are not a N.E.

42
Bertrand Model of Price Competition

The three types of price configurations we just
ruled out constitute all the possible price
configurations other than p1 p2 c
So, we proved uniqueness

43
Bertrand Model and Repeated Interaction

One unrealistic assumption of this model is that
it is a one-shot game
In this model, a firm never had to consider the
reaction of its competitors to its price choice
In this model, a firm undercut its rival by a
penny and steal all the rivals customers

44
Bertrand Model and Repeated Interaction

In practice, however, a firm may well worry that
if it does undercut its rival, the rival will
respond by cutting its own price, ultimately
leading to only a short-run gain in sales but a
long-run reduction in the price level in the
market
Now we consider a Bertrand repeated game

45
Bertrand Model and Repeated Interaction

Model Setup
Two identical firms
They compete form sales repeatedly
Competition in each period t is described by the
Bertrand model

46
Bertrand Model and Repeated Interaction

The two firms know all the prices that have been
chosen (by both firms) previously
There is a discount factor 0 lt d lt 1
Each firm j attempts to maximize the discounted
value of profits ? dt -1 ?jt, where ?jt, is firm
js profits in period t
Dynamic game repeated play of the same static
simultaneous-move game

47
Bertrand Model and Repeated Interaction

In this repeated Bertrand game, firm js strategy
specifies what price pjt it will charge in each
period t as a function of the history of all past
price choices by the two firms,
t 1
Ht-1 p1?, p2 ?
? 1

48
Bertrand Model and Repeated Interaction

Strategies of this form allow for a range of
behavior
For example, a firms strategy could call for
retaliation if the firms rival ever lowers its
price below some threshold price
This retaliation could be brief, calling for the
firm to lower its price for only a few periods
after the rival crosses the line, or it could
be unrelenting

49
Bertrand Model and Repeated Interaction

The retaliation could be tailored to the amount
by which the firms rival undercut it, or it
could be severe no matter how minor the rivals
transgression
The firm could respond with increasingly
cooperative behavior in return for its rival
acting cooperatively in the past
The other option is that firms strategy could
also make the firms behavior in any period t
independent of past history

50
Bertrand Model and Cooperation Finite Repetitions

Consider first the case when firms compete only a
finite number of times T (finitely repeated game)
Can the set of behavior just described sustain
cooperation as a SPNE? NO
The unique SPNE of the finitely repeated Bertrand
game involves T repetitions of the static
Bertrand equilibrium in which prices equal cost

51
Bertrand Model and Cooperation Finite Repetitions

This is a consequence of backward induction
In the last period T, we must be at the Bertrand
solution, and therefore profits are zero in that
period regardless of what has happened earlier
But then in period T 1, we are, strategically
speaking, at the last period, and the Bertrand
solution must arise again
And so on, until we get to the first period

52
Bertrand Model and Cooperation Finite Repetitions

Hence, backward induction rules out the
possibility of more cooperative behavior in the
finitely repeated Bertrand game

53
Bertrand Model and Cooperation Infinite
Repetitions

Now we will extend the horizon to an infinite
number of periods (infinitely repeated game)
Consider the following strategies for firms
j 1,2
pm if all elements of Ht -1 equal
(pm, pm) or t 1
pjt(Ht 1)
c otherwise

54
Bertrand Model and Cooperation Infinite
Repetitions

This means that, firm js strategy calls for it
to initially play the monopoly price pm in period
1
Then, in each period t gt 1, firm j plays pm if in
every previous period both firms have charged
price pm, and otherwise, charges a price equal to
its cost

55
Bertrand Model and Cooperation Infinite
Repetitions

This type of grim-trigger strategy is also called
Nash reversion strategy firms cooperate until
someone deviates and, any deviation triggers a
permanent retaliation in which both firms
thereafter set their prices equal to cost, to
one-period Nash strategy

56
Bertrand Model and Cooperation Infinite
Repetitions

If both firms follow these strategies, then both
firms will end up charging the monopoly price in
every period they start by charging pm, and
therefore no deviation from pm will ever be
triggered

57
Bertrand Model and Cooperation Infinite
Repetitions

The strategies described constitute a SPNE of the
infinitely repeated Bertrand duopoly game if and
only if d ? ½
Proof
(1) First lets state that a set of strategies is
a SPNE of an infinite-horizon game iff it
specifies NE play in every subgame

58
Bertrand Model and Cooperation Infinite
Repetitions

(2) Note that although each subgame of this
repeated game has a distinct history of play
leading to it, all these subgames have an
identical structure each is an infinitely
repeated Bertrand duopoly game exactly like the
game as a whole
(3) Then, to establish that the specified
strategies constitute a SPNE, we need to show
that after any previous history of play, these
strategies for the remainder of the game
constitute a NE of an infinitely repeated
Bertrand game

59
Bertrand Model and Cooperation Infinite
Repetitions

(4) Given the specified strategies, we need to be
concerned about only two types of previous
histories
those in which there has been a previous
deviation (a price not equal to pm), and
those in which there has not been a previous
deviation

60
Bertrand Model and Cooperation Infinite
Repetitions

(5) Consider first a subgame arising after a
deviation has occurred
The strategies call for each firm to set its
price equal to c in every future period
regardless of its rivals behavior
The pair of strategies is a N.E. of an
infinitely repeated Bertrand game because each
firm j can earn at most zero when its opponent
always sets its price equal to c, and it earns
exactly this amount by itself setting its price
equal to c in every remaining period

61
Bertrand Model and Cooperation Infinite
Repetitions

(6) Now consider a subgame starting in, say,
period t after no previous deviation has occurred
Each firm j knows that its rivals strategy calls
for it to charge pm until it encounters a
deviation from pm and to charge c thereafter
Is it in firm js interest to use this strategy
itself given that its rival does? That is, do
these strategies constitute a N.E. of this
subgame? It depends on the PV of deviation
versus PV of keeping pm

62
Bertrand Model and Cooperation Infinite
Repetitions

Suppose that firm j contemplates deviating from
price pm in period ? ? t of the subgame if no
deviation has occurred prior to period ?
We know that once a deviation has occurred within
this subgame, firm j can do no better than to
play c in every period given that its rival will
do so. Hence, to check whether these strategies
form a N.E. in this subgame, we need only check
whether firm j will wish to deviate from pm if no
such deviation has yet occurred

63
Bertrand Model and Cooperation Infinite
Repetitions

From period t through period ? -1, firm j will
earn ½(pm c) x(pm) in each period, exactly as
it does if it never deviates
Starting in period ?, however, its payoffs will
differ from those that would arise if it does not
deviate
In periods after it deviates (periods ? 1, ?
2, ), firm js rival charges a price of c
regardless of the form of firm js deviation in
period ?, and so firm j can earn at most zero in
each of these periods

64
Bertrand Model and Cooperation Infinite
Repetitions

In period ?, firm j optimally deviates in a
manner that maximizes its payoff in that period
(note that the payoffs firm j receives in later
periods are the same for any deviation from pm
that it makes)
It will therefore charge pm - ? for some
arbitrary small ? gt 0, make all sales in the
market, and earn a one-period payoff of (pm c -
?) x(pm - ?)

65
Bertrand Model and Cooperation Infinite
Repetitions

Thus, its overall discounted payoff from period ?
onward, discounted to period ?, can be
arbitrarily close to (pm c) x(pm)

66
Bertrand Model and Cooperation Infinite
Repetitions

On the other hand, if firm j never deviates, it
earns a discounted payoff from period ? onward,
discounted to period ?, of ½(pm c) x(pm)/(1
d)
Hence, for any t and ? ? t, firm j will prefer
no deviation to deviation in period ? iff
½(pm c) x(pm)/(1 d) ? (pm c) x(pm),
d ? 1/2

67
Bertrand Model and Cooperation Infinite
Repetitions

Thus, the specified strategies constitute a SPNE
iff d ? ½
This means that the perfectly competitive outcome
of the static Bertrand game may be avoided if the
firms foresee infinitely repeated interactions

68
Bertrand Model and Cooperation Infinite
Repetitions

The reason is that, in contemplating deviation,
each firm takes into account not only the
one-period gain it earns from undercutting its
rival but also the profits forgone by triggering
retaliation
The size of the discount factor d is important
here because it affects the relative weights put
on the future losses versus the present gains
from a deviation

69
Bertrand Model and Cooperation Infinite
Repetitions

The monopoly price is sustainable iff the present
value of these future losses is large enough
relative to the possible current gain from
deviation to keep the firms from going for
short-run profits

70
Bertrand Model and Cooperation Infinite
Repetitions

Although the specified strategies constitute a
SPNE when d ? ½, they are not the only SPNE of
the infinitely repeated Bertrand model

71
Bertrand Model and Cooperation Infinite
Repetitions

(1) In the infinitely repeated Bertrand duopoly
game, when d ? ½ repeated choice of any price
p ?c, pm can be supported as a SPNE outcome
path using Nash reversion strategies

72
Bertrand Model and Cooperation Infinite
Repetitions

Proof of (1)
We have already proved that repeated choice of
price pm can be sustained as a SPNE outcome
when d ? ½. The proof for any price p ? c, pm)
follows exactly the same lines. We need only to
change price pm in the specified strategies to
p ? c, pm)

73
Bertrand Model and Cooperation Infinite
Repetitions

A general result in the theory of repeated
games, known as the Folk Theorem, tell us that
In an infinitely repeated game, any feasible
discounted payoffs that give each player, on a
per-period basis, more than the lowest payoff
that he could guarantee himself in a single play
of the simultaneous-move component game can be
sustained as the payoffs of a SPNE if players
discount the future to a sufficiently small
degree

74
Bertrand Model and Cooperation Infinite
Repetitions

Hence, although infinitely repeated games allow
for cooperative behavior, they also allow for an
extremely wide range of possible behavior
Historical focal points in the industry can solve
the multiplicity of equilibria
Self-enforcing agreements (secret collusion
because it is prohibited by law) can make the
cooperative equilibrium more likely to occur

75
Bertrand Model and Cooperation Infinite
Repetitions under Many Firms

We will now investigate how the number of firms
in a market affect its competitiveness
We will show that with J firms, repeated choice
of any price p ? (c, pm can be sustained as a
stationary SPNE outcome path of the infinitely
repeated Bertrand game using Nash reversion
strategies iff d ? (J - 1)/J

76
Bertrand Model and Cooperation Infinite
Repetitions under Many Firms

We will see that there is a relationship between
having more firms and the difficulty of
sustaining collusion

77
Bertrand Model and Cooperation Infinite
Repetitions under Many Firms

Proof
Let ? gt 0 be the firms equilibrium joint
profits, which in equilibrium are split among
firms
The best deviation for a firm is to undercut the
rivals by a small ?, in which case it can steal
all the demand and obtain almost as much as ? in
every period

78
Bertrand Model and Cooperation Infinite
Repetitions under Many Firms

(3) Given the Nash reversion strategies, in the
following punishing phase, the deviator will get
zero forever after
Therefore deviation will imply a payoff equal to
?
(4) If the firm does not deviate, its payoff is
(1/1 d) (?/J)

79
Bertrand Model and Cooperation Infinite
Repetitions under Many Firms