Student:
Matyukhin Anton,

2 group.

Teacher: Alla Friedman.

Международный Институт Экономики и Финансов, 3 курс.

Высшая Школа
Экономики

Essay in
Microeconomics.

Topic:

Is Collusion
Possible?

18.12.2000

Contents:

1.  
Introduction.

2.  
Two types of
behaviour (Collusive and non-collusive).

3.  
Game theory.

a.)     
Concept.

b.)     
The problem
of collusion.

c.)     
Predatory
pricing.

4.  
Repeated games
approach.

a.)     
Concept.

b.)     
Finite game
case.

c.)     
Infinite game
case.

i.)             
“Trigger”
strategy

ii.)          
Tit-for-Tat.

    d.)    Finite game case, Kreps approach.

5.  
The motives for
retaliation.

6.  
Conclusion.

7.  
Bibliography.

1.  
Introduction.

   In this essay I would
discuss the price and output determination under the one essential type of
imperfect competition markets- oligopoly. Inter-firm interactions in imperfect
markets take many forms. Oligopoly theory, those name refers to “competition
among the few”, lack unambiguous results of these interactions unlike monopoly
and perfect competition. There is a variety of results derived from many
different behavioural assumptions, with each specific model potentially
relevant to certain real-world situations, but not to others.

   Here we are interested
in the strategic nature of competition between firms. “Strategic” means the
dependence of each person’s proper choice of action on what he expects the
other to do. A strategic move of a person influences the other person’s choice,
the other person’s expectation of how would this particular person behave, in
order to produce the favourable outcome for him.

2.  
Two types of behaviour (Collusive and
non-collusive).

   Models of
enterprise decision making in oligopoly derive their special features from the
fact that firms in an oligopolistic industry are interdependent and this is
realised by these firms. When there are only a few producers, the reaction of
rivals should be taken into account. There are two broad approaches to this
problem.

   First,
oligopolists may be thought of as agreeing to co-operate in setting price and
quantity. This would be the Collusive model. According to this model, firms
agree to act together in their price and quantity decisions and this would to
exactly the same outcome as would have been under monopoly. Thus the explicit
or co-operative collusion or Cartel would take place.

   Second approach of the
oligopoly analysis is based on the assumption that firms do not co-operate, but
make their decisions on the basis of guesses, expectations, about the variables
to which their competitors are reaching and about the form and the nature of the
reactions in question. The Non-collusive behaviour deals with this model. Here,
though in equilibrium the expectations of each firm about the reactions of
rivals are realised, the parties never actually communicate directly with each
other about their likely reactions. The extreme case of this can even imply
competitive behaviour. Such a situation is much less profitable for firms than
the one in which they share the monopolistic profit. The purpose of this paper
is to analyse the case of the possibility of collusion between firms in order
to reach the monopolistic profits for the industry, assuming that they do not
co-operate with each other. This would be the most interesting and ambiguous
case to look at.

3.  
Game
theory.

a.) Concept.

   The notion
of game theory would a good starting point in the study of strategic
competition and would be very helpful in realising the model and the problems
facing oligopolistic firms associated with it.

   Game theory
provides a framework for analysing situations on which there is interdependence
between agents in the sense that the decisions of one agent affect the other
agents. This theory was developed by von Neumann and Morgenstern and describes
the situation, which is rather like that found in the children’s game “Scissors&Stones”.
Each firm is trying to second-guess the others, i.e. the behaviour of one firm
depends on what it expects the others to do, and the in turn are making their
decisions based upon their expectations of what the rivals (including the first
firm) will do. In our case, the players of the game are the firms in the
industry and each of them wants to maximise its pay-off. The pay-off that a
player receives measures how well he achieves his objective. Let’s assume in
our model the pay-off to be a profit. Their profits depend upon the decisions
they make  (the strategies chosen by the various players including themselves).
A strategy in this model is a plan of action, or a complete contingency plan,
which specifies what the player will do in any of the circumstances in which he
might find himself. The game also depends on the move order and the information
conditions.

   Games can be
categorised according to the degree of harmony or disharmony between the
players’ interests. The pure coordination game is the one extreme, in which
players have the same objectives. The other extreme is the pure conflict of the
opposite interests of players. And usually there is a mixture of coordination
and conflict of interests- mixed motive games.

   Although the
importance of the other players’ choices takes place, sometimes a player has a
strategy that is the best irrespective of what others do. This strategy is
called dominant, and the other inferior ones are called dominated. A situation
in which each player is choosing the best strategy available to him, given the
strategies chosen by others, is called a Nash equilibrium. This equilibrium
corresponds to the idea of self-fulfilled expectations, tacit, self-supporting
agreement. If the players have somehow reached Nash equilibrium, then none
would have an incentive to depart from this agreement. Any agreement that is
not a Nash equilibrium would require some enforcement.

b.) The problem of collusion.

   Now I would
like to use an example of a game in which the Cournot output deciding duopoly
is involved. This game is illustrated by the table below:    

   Here a firm
chooses between two alternatives: high and low output strategies. The
corresponding pay-offs (profits) are given in the boxes. In this game, the best
thing that can happen for a firm is to produce a high level of output while its
rival produces low. Low output of the rival provides that price is not driven
down too much, thus a firm could earn a good profit margin. The worst thing for
a firm is to change places with its rival assuming the same situation takes
place. If both firms produce high levels of output, then the price would be
low, allowing each of them to earn still positive but very small profits.
Nevertheless, (HIGH;HIGH) would be the dominant strategy of this game (we would
observe a Nash equilibrium in strictly dominant strategies here). It is the
best response of firm A whenever B produces a high or low output and this is
also true for firm B. The non-co-operative outcome for each firm would be to
get the pay-off of 1. But as we can see, it would be better for both to lower
their output and thereby to raise price, as their profits would increase to 2 for
each firm instead of 1 in NE. Strategy (LOW;LOW) would be the collusive
outcome. The problem of collusion is for the firms to achieve this superior
outcome notwithstanding the seemingly compelling argument that high output
levels will be chosen.

   This was an
example of a “one-shot” game and we saw that the collusive outcome was not
available for that case. But in reality these games are being played over and
over (on a long-term basis) and we will see later in this essay how the
collusion can be sustained by threats of retaliation against non-co-operative
behaviour.

c.) Predatory pricing. 

   The
potential entrant chooses between entering and staying out of the industry. In
the case of his entering, the incumbent firm can either fight this entry (which
as we see would be costly to both), or acquiesce and arrive at some peaceful
co-existence (which is obviously more profitable). The best thing for incumbent
is for entry not to take place at all. There are in fact two Nash equilibria:
(IN;ACQUIESCE) and (OUT;FIGHT). But the last mentioned (OUT;FIGHT) is
implausible, as if the incumbent were faced with the fact of entry, it would
more profitable for him to acquiesce rather than to fight the entry. Due to
this fact the potential entrant would choose to enter the industry and the only
equilibrium would be (IN;ACQUIESCE). Thus we can conclude, that in this case
the incumbent’s threat to fight was empty threat that wouldn’t be believed,
i.e. that threat was not a credible one. The concept of perfect equilibrium,
developed by Selten (1965;1975), requires that the “strategies chosen by the
players be a Nash equilibrium, not only in the game as a whole, but also in
every subgame of the game”. (In our model on the picture, the subgame starts
with the word “incumbent”). We have got the perfect equilibrium to rule out the
undesirable one.

4.   Repeated games approach.

a.)  Concept.

   As I have
already mentioned, in practice firms are likely to interact repeatedly. Such
factors as technological know-how, durable investments and entry barriers
promote long-run interactions among a relatively stable set of firms, and this
is especially true for the industries with only a few firms. With repeated
interaction every firm must take into account not only the possible increase in
current profits, but also the possibility of a price war and long-run losses
when deciding whether to undercut a given price directly or by increasing its
output level. Once the instability of collusion has been formulated in the
“one-shot” prisoners dilemma game, it raises the question of whether there is
any way to play the game in order to ensure a different, and perhaps more
realistic, outcome. Firms do in practice sometimes solve the co-ordination
problem either via formal or informal agreements. I would focus on the more
interesting and complicated case of how collusive outcomes can be sustained by
non-co-operative behaviour (informal), i.e. in the absence of explicit,
enforceable agreements between firms. We have seen that collusion is not
possible in the “one-shot” version of the game and we will now stress upon a
question of whether it is possible in a repeated version. The answer depends on
at least four factors:

1.  
Whether the game is repeated infinitely or there
is some finite number of times;

2.  
Whether there is a full information available to
each firm about the objectives of, and opportunities available to, other firms;

3.  
How much weight the firms attach to the future
in their calculations;

4.  
Whether the “cheating” can/can not be detected due to the knowledge/lack
of knowledge about the prior moves of the firm’s rivals. 

   The fact of
repetition broadens the strategies available to the players,

because they
can make their strategy in any currant round contingent on the others’ play in
previous rounds. This introduction of time dimension permits strategies, which
are damaging to be punished in future rounds of the game. This also permits
players to choose particular strategies with the explicit purpose of
establishing a reputation, e.g. by continuing to co- operate with the other
player even when short-term self-interest indicates that an agreement to do so
should be breached.

b.) Finite game case.

   But
repetition itself does not necessarily resolve the prisoner’s dilemma. Suppose
that the game is repeated a finite number of times, and that there is complete
and perfect information. Again, we assume firms to maximise the (possibly
discounted) sum of their profits in the game as a whole. The collusive low
output for the firms again, unfortunately for the firms, could not be
sustained. Suppose, they play a game for a total of five times. The repetition
for a predetermined finite number of plays does nothing to help them in
achieving a collusive outcome. This happens because, though each player
actually plays forward in sequence from the first to the last round of the
game, that player needs to consider the implications of each round up to and
including the last, before making its first move. While choosing its strategy
it’s sensible for every firm to start by taking the final round into
consideration and then work backwards. As we realise the backward induction, it
becomes evident that the fifth and the final round of the game would be
absolutely identical to a “one-shot” game and, thus, would lead to exactly the
same outcome. Both firms would cheat on the agreement at the final round. But
at the start of the fourth round, each firm would find it profitable to cheat
in this round as well. It would gain nothing from establishing a reputation for
not cheating if it knew that both it and its rival were bound to cheat next
time. And this crucial fact of inevitable cheating in the final round
undermines any alternative strategy, e.g. building a reputation for not
cheating as the basis for establishing the collusion. Thus cheating remains the
dominant strategy.

* NOTE: the is however one assumption about slightly
incomplete information, which allows collusive outcome to occur in the finitely repeated game, but I will left it for the discussion
some paragraphs later.

c.)_ Infinite game case.

   Now lets
consider the infinitely repeated version of the game. In this kind of game
there is always a next time in which a rival’s behaviour can be influenced by
what happens this time. In such a game, solutions to the problems represented
by the prisoners dilemma are feasible.

i.) “Trigger” strategy

   Suppose that
firms discount the future at some rate “w”, where “w” is a number between O and
1. That is, players attach weight “w” to what happens next period. Provided
that “w” is not too small, it is now possible for non-co-operative collusion to
occur. Suppose that firm B plays “trigger” strategy, which is to choose low
output in period 1 and in any subsequent period provided that firm A has never
produced high output, but to produce high output forever more once firm A ever produces
high output. That is B co-operates with A unless A “defects”, in which case B
is triggered into perpetual non-co-operation. If A were also to adopt the
“trigger” strategy, then there would always be collusion and each firm would
produce low output. Thus the discounted value of this profit flow is:

2+2w+2w^2+2w^3+…=2/(1-w)

   If fact A
gets this pay-off with any strategy in which he is not the first to defect. If
A chooses a strategy in which he defects at any stage, then he gets a pay-off
of 3 in the first period of defection (as B still produces low output), and a
pay-off of no more than 1 in every subsequent period, due to B being triggered
into perpetual non-co-operation. Thus, A’s pay-off is at most

3+w+w^2+w^3+…=3+w/(1-w)

   If we will
compare these two results, we will get that it is better not to defect so long
as

W > (or =)
½

   We can
conclude that is the firms give enough weight to the future, then
non-co-operative collusion can be sustained, for example, by “trigger”
strategies. The “trigger” strategies constitute a Nash equilibrium =
self-sufficient agreement. However it is not enough for a firm to announce a
punishment strategy in order to influence the behaviour of rivals. The strategy
that is announced must also be credible in the sense that it must be understood
to be in the firm’s self-interest to carry out its threat at the time when it
becomes necessary. It must also be severe in a sense that the gain from
defection should be less than the losses from punishment. But because it is possible
that mistakes will be made in detecting cheating (if, for example, the effects
of unexpected shifts in output demand are misinterpreted as the result of
cheating), the severity of punishment should be kept to the minimum required to
deter the act of cheating.

ii.) Tit-for-Tat.

   Trigger
strategies are not the only way to reach the non-co-operative collusion.
Another famous strategy is Tit-for-Tat, according to which a player chooses in
the current period what the other player chose in the previous period. Cheating
by either firm in the previous round is therefore immediately punished by
cheating, by the other, in this round. Cheating is never allowed to go
unpunished. Tit-for-Tat satisfies a number of criteria for successful
punishment strategies. It carries a clear threat to both parties, because it is
one of the simplest conceivable punishment strategies and is therefore easy to
understand. It also has the characteristics that the mode of punishment it
implies does not itself threaten to undermine the cartel agreement. This is
because firms only cheat in reaction to cheating be others; they never initiate
a cycle of cheating themselves. Although it is a tough strategy, it also offers
speedy forgiveness for cheating, because once punishment has been administered
the punishing firm is willing once again to restore co-operation. Its weakness
is in the fact that information is imperfect in reality, so it is hard to
detect whether a particular outcome is the consequence of unexpected external
events such as a lower demand than forecast, or cheating, Tit-for-Tat has a
capacity to set up a chain reaction in a response to an initial mistake.

d.)  Finite game case, Kreps
approach.

   Lets now
return to the question of how collusion might occur non-co-operatively even in
the finitely repeated game case. Intuition said that collusion could happen- at
least at the earlier rounds- but the game theory apparently said that it could
not. Kreps et al. (1982) offered the elegant solution to this paradox. They
relax the assumption of complete information and instead suppose that one
player has a small amount of doubt in his mind as to the motivation of the
other player. Suppose A attaches some tiny probability p to B referring- or
being committed- to playing the “trigger” strategy. In fact it turns out that
even if p is very small, the players will effectively collude until some point
towards the end of the game. This occurs because its not worth A detecting in
view of the risk that the no-collusive outcome will obtain for the rest of the
game, and because B wishes to maintain his reputation for possibly preferring,
or being committed to, the “trigger” strategy. Thus even the small degree of
doubt about the motivation of one of the players can yield much effective
collusion.

5. 
 The motives for
retaliation.

   The motives
for retaliation differ in three approaches. In the first approach, the price
war is a purely self-fulfilling phenomenon. A firm charges a lower price
because of its expectations about the similar action from the other one. The
signal that triggers such a non-co-operative phase is previous undercutting by
one of the firms. The second approach presumes short-run price rigidities; the
reaction by one firm to a price cut by another one is motivated by its desire to
regain a market share. The third approach (reputation) focuses on intertemporal
links that arise from the firm’s learning about each other. A firm reacts to a
price cut by charging a low price itself because the previous price cut has
conveyed the information that its opponent either has a low cost or cannot be
trusted to sustain collusion and is therefore likely to charge relatively low
prices in the future.

6.  Conclusion.

   So far I
have discussed the collusion using some simple example with a choice of output
levels made by the two firms. But there may be several firms in the industry,
and in fact firms have a much broader choice. It may be that their decision
variable is price, investment, R&D and advertising. Nevertheless the more
or less the same analysis could be applied in each of the case.

   I have
examined different assumptions and predictions, which allow or do not allow the
possibility of collusion. In reality such thing as collusion definitely takes
place, if it had not, there would not have been any strong an ambiguous
discussion of this topic. But I think it would be appropriate to end this essay
with an explicit reminder that once we leave the world of perfect competition,
we lose the identity of interests between consumers and producers. So, the
discussion of benefits to firms in oligopoly that arise from finding strategies
to enforce collusive behaviour might well have been the discussion of the
expenses of consumers.

7.   Bibliography.

1.  
J.Vickers, “Strategic competition among
the few- Some recent developments in the economics of industry”.

2.  
J.Tirole, “The theory of industrial
organisation”. Ch 6.

3.  
Estrin & Laidler. “Introduction to
microeconomics”. Ch 17.

4.  
W.Nicholson, “Microeconomic theory”. Ch 20.