Repeated Games and Strategies in Oligopoly

Repeated Games and Strategic Behavior in Oligopoly

Introduction to Repeated Games

In microeconomics, especially in the study of oligopoly, firms often interact repeatedly over time. These interactions are modeled as repeated games, where the same strategic situation (game) is played multiple times, allowing players to adjust their strategies based on past outcomes.

• One-time game: A game that is played only once, with no future consequences for current actions.

• Repeated game: A game that is played multiple times, allowing for strategies that depend on the history of play.

Additional info: Repeated games are important in understanding how cooperation can be sustained among firms in an oligopoly, even when each firm has an incentive to cheat in a one-time game.

Strategies in Repeated Games

Players in repeated games can use strategies that respond to the actions of their opponents in previous rounds. Two common strategies are:

• Tit-for-tat strategy: "I cooperate this period. If you don't cooperate, I won't cooperate next period." The player's current choice depends on the opponent's previous choice.

• Trigger strategy: "I will cooperate until you don't cooperate. Then, I will never cooperate again." This strategy punishes defection by permanently switching to non-cooperation.

Additional info: These strategies can help enforce cooperation in repeated interactions, as the threat of future punishment can deter cheating.

Payoff Table: Jack and Jill's Oligopoly Game

The following table summarizes the profits (payoffs) for two firms, Jack and Jill, based on their production decisions. The numbers represent profits in dollars for each combination of choices.

Decision	Jack's Profit	Jill's Profit
Both Cooperate	$1,800	$1,800
Jack Cheats, Jill Cooperates	$2,000	$1,600
Jack Cooperates, Jill Cheats	$1,500	$2,000
Both Cheat	$1,600	$1,600

Additional info: The table is inferred from the context and typical oligopoly payoff structures.

Practice Questions and Applications

• Question 1: Jack employs a tit-for-tat strategy. If Jill cooperates this period, how many gallons will Jack produce?

  • Answer: Jack will match Jill's cooperative output, which is typically the lower, collusive quantity (e.g., 30 gallons). Additional info: The exact number is inferred based on standard collusive outcomes.

• Question 2: Jack employs a tit-for-tat strategy. If Jill cheated last period, what will Jack's total profit equal this period?

  • Answer: Jack will retaliate by cheating as well, resulting in the 'both cheat' outcome. Jack's profit will be $1,600.

Key Terms and Concepts

• Oligopoly: A market structure with a small number of firms whose decisions affect each other.

• Collusion: An agreement among firms to limit competition, often by fixing prices or output.

• Cheating: In the context of collusion, producing more than the agreed-upon quantity to increase individual profit at the expense of the group.

Mathematical Representation

In repeated games, the present value of cooperating can be compared to the short-term gain from cheating. The discount factor ($\delta$) represents how much future profits are valued relative to current profits.

The condition for cooperation to be sustained is:

$\text{Present Value of Cooperation} \geq \text{Short-term Gain from Cheating} + \text{Present Value of Future Punishment}$

Or, more formally:

$\frac{\text{Profit from Cooperation}}{1-\delta} \geq \text{Profit from Cheating} + \delta \cdot \frac{\text{Profit from Punishment}}{1-\delta}$

Additional info: If the discount factor is high (firms care a lot about the future), cooperation is more likely to be sustained.