Student: Matyukhin Anton,
Teacher: Alla Friedman.
Международный Институт Экономики и Финансов, 3 курс.
Высшая Школа Экономики
Essay in Microeconomics.
Is Collusion Possible?
Two types of behaviour (Collusive and non-collusive).
The problem of collusion.
Repeated games approach.
Finite game case.
Infinite game case.
d.) Finite game case, Kreps approach.
The motives for retaliation.
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.
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.
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
Firm B’s output level
Firm A’s output level HIGH (1;1) (3;0)
LOW (0;3) (2;2)
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.
Here we need to introduce the explicit order of moves in the model.
There are again two players-firms on the market- an incumbent firm and a
potential entrant in the market. The game is illustrated below:
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
Repeated games approach.
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
Whether the game is repeated infinitely or there is some finite number
Whether there is a full information available to each firm about the
objectives of, and opportunities available to, other firms;
How much weight the firms attach to the future in their calculations;
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
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
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
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:
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
If we will compare these two results, we will get that it is better
not to defect so long as
W > (or =) 1/2
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.
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.
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.
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.
J.Vickers, “Strategic competition among the few- Some recent
developments in the economics of industry”.
J.Tirole, “The theory of industrial organisation”. Ch 6.
Estrin & Laidler. “Introduction to microeconomics”. Ch 17.
W.Nicholson, “Microeconomic theory”. Ch 20.