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 =) Ѕ
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.
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