ECO290E: Game Theory
Lecture 11
Repeated Games
Bertrand Puzzle
• Firms receive 0 profit under the (oneshot) Bertrand competition.
• But the actual firms engaging a price
competition, e.g., gas stations locating
next to each other, seem to earn
positive profits.
 How come they can achieve positive
profits?
Long-term Relationship
• Firms need some devises to prevent them
from deviation, i.e., cutting its own price.
• Contracts (explicit cartels): If deviation
happens, a deviator must be punished by a
court or a third party.
 Illegal by antitrust law.
• Long-term relationship (implicit cartels):
Firms collude until someone deviates. After
deviation, firms engage in a price war.
 Long-term relationship helps achieve
cooperation.
Remarks
Long-term relationship has an advantage over
contracts when
• Deviation is difficult to be detected by a court.
• The definition of “cooperation” is vague.
• There is no court, e.g., medieval history
(economic history), developing countries
(development economics), global warming
(international relationship).
 The best way to study the interaction
between immediate gains and long-term
incentives is to examine a repeated game.
Repeated Games
• A repeated game is played over time,
t=1,2,…,T where T can be a finite number or
can be infinity.
• The same static game, called “a stage
game,” is played in each period.
• The players observe the history of play, i.e.,
the sequence of action profiles from the first
period through the previous period.
• The payoff of the entire game is defined as
the sum of the stage-game payoffs possibly
with discounting (especially in cases of
infinitely repeated games).
SPNE in Repeated Games
• After all history of play, each player
cannot become better off by changing
her strategy only.
which is equivalent to
• After all history of play and for every
player, immediate gains by deviation
must be smaller than future losses
triggered by deviation.
Repeated Bertrand Games
The following “trigger” strategies achieve
collusion if δ≥1/2.
• Each firm charges a monopoly price until
someone undercuts the price, and after such
deviation she will set a price equal to the
marginal cost c, i.e., get into a price war.
collusion
deviation
t
π
2π
t+1
π
0
t+2
π
0
…
…
…
Calculation
2       2  ...
      2  ...
 

1 
    1/ 2
“Formula”
r  (0,1)
S : a  ra  r 2 a  ...
rS : ra  r 2 a  r 3 a  ...
You can use the
following formula.
S  rS  (1  r ) S  a
a
S
1 r