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

• (5) Therefore, deviation is not profitable iff
• (1/1 d) (?/J) ? ?
• d ? (J - 1)/J
• Note that (J 1)/J is increasing in J. Then, as
  
• increases, d has to increase in order for
  collusion
• to be still sustainable

80
Bertrand Model and Cooperation Infinite
Repetitions under Many Firms

• Hence, as the number of firms increases, it is
• harder to sustain a collusive outcome