More on Supergames

1
More on Supergames

• We have seen that in infinitely repeated games
  (supergames) collusive outcomes can be supported
  by the credible threat of future punishment
• The stronger the punishment, the more likely that
  firms will be able to sustain collusion
• The longer the delay before detection of
  cheating, the less likely a collusive agreement
  will be sustainable
• If there is some probability that a firms
  cheating will not be detected, then the expected
  payoff to cheating rises and the less likely a
  collusive outcome can be supported

2

• Stigler conjectured that if the collusive
  agreement (at industry profit maximizing output
  (aggregate monopoly))was not sustainable (say
  because cheating was not easily detected) firms
  would reduce the extent of their coordination by
  setting prices closer to the competitive level so
  that the incentive to cheat would be smaller

3

• Suppose the demand curve is given byP 130 - Q
• Constant per unit cost for both firms is 10
• Then the best response function for both firms
  is

4

• Above table shows corresponding output and
  profits for different strategies
• The collusive output referred to here is the
  industry monopoly output - i.e., the total output
  produced by all firms maximizes industry profit
• The defection output is the best-response to
  the other firms collusive output

5

• Recall our former condition that a firm will not
  find it profitable to deviate from the collusive
  output enforced by a grim Cournot strategy
• In our numerical example, we find, by
  substituting the values from Table 10.1 into the
  above condition, that the collusive agreement
  will be sustained if the discount factor for each
  firm exceeds .53 (or discount rate is less than
  89)
• (2025 - 1800) / (2025 - 1600) 225 / 425 .529

6

• If the discount factor of each firm is less than
  .53 then the collusive agreement cannot be
  sustained with the credible threat of reverting
  to a Cournot stage game
• The interesting question is Are there other
  credible punishments that are more severe,
  thereby increasing the sustainability off the
  collusion?

7
Optimal Penal Code

• An optimal penal code inflicts the most severe
  credible punishment on cheaters it minimizes the
  present value of the cheaters profits
• Abreu path - a possible future to the game - an
  action specified for every player for all periods
  in the future

8
Simple Penal Code

• A simple penal code defines a path or punishment
  for each firm that is to be followed if that firm
  deviates from either the collusive path or from a
  punishment path
• The punishments are simple in that they are
  independent of the history of the game or the
  nature of the deviation

9
Simple Optimal Penal Code

• A simple optimal penal code (SOPC) supports the
  maximal degree of collusion given the discount
  factor
• An SOPC in a game of symmetric quantities is as
  follows
• Phase 1 The Stick - Punishment, if necessary
  lasts only for a single period, as long as
  cheating firm(s) produces its punishment output
  in that period
• Phase 2 The Carrot - Reward all firms for
  cooperating with the punishment by returning to
  collusive output
• Failure by any firm to cooperate in the stick
  phase restarts the punishment path and delays a
  return to the collusive outcome by 1 period

10

• To find the maximal supportable level of
  collusion, first determine if the
  stick-and-carrot punishment paths can sustain the
  collusive outcome
• The harshest credible punishment qp satisfies

where qp is a firms punishment output, qm is the
firms collusive output, ?(qM) is a firms share
of monopoly profit, and ?r(qp) is the profit a
firm earns if it optimally cheats when all other
firms are playing their punishment outputs
11

• No firm will have an incentive to cheat from qM
  in the first place if the punishment output qP is
  such that

where ?r(qM) is the firms profit that results
when it optimally deviates from it industry
profit-maximizing output
12

• If the harshest credible punishment is not
  sufficient to inspire a firm to stay on the
  collusive path that maximizes industry profit,
  then the optimal stick and carrot punishment is
  defined by the largest qP and the smallest q
  that simultaneously satisfy the following 2
  conditions

13

• The optimal simple penal code paths are
  implemented through the following strategy for
  each firm
• At t play q if all firms played q at t-1
• If a firm deviated from q at t-1, play qP at t
• If all firms played qP at t-1, play q at t
• If a firm deviated from qP at t-1, play qP at t

14

• Go through numerical exercise

15

• Trigger price strategies - 342-343

16

• Facilitating practices