Topic: Is Collusion Possible?

                                                   Student: Matyukhin Anton,

                                                                    2 group.
                                                     Teacher: Alla Friedman.

            Международный Институт Экономики и Финансов, 3 курс.
                           Высшая Школа Экономики

                          Essay in Microeconomics.

                           Is Collusion Possible?



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:

|       |      |Firm B’s output level         |
|       |      |HIGH          |LOW           |
|Firm   |HIGH  |(1;1)         |(3;0)         |
|A’s    |      |              |              |
|output |      |              |              |
|level  |      |              |              |
|       |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 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
*  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:
   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 =) 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
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
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.