ECO290E: Game Theory

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ECO290E: Game Theory. Lecture 11 Repeated Games. Bertrand Puzzle. Firms receive 0 profit under the (one-shot) Bertrand competition. But the actual firms engaging a price competition, e.g., gas stations locating next to each other, seem to earn positive profits.
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ECO290E: Game TheoryLecture 11Repeated GamesBertrand Puzzle
  • Firms receive 0 profit under the (one-shot) 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.
  • RemarksLong-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 GamesThe 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.
  • You can use the following formula.“Formula”Calculation
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