Repeated Games and Strategies in Oligopoly: Tit-for-Tat and Trigger Strategies

Game Theory and Strategic Play

Repeated Games in Oligopoly

In microeconomics, repeated games are strategic interactions that occur multiple times between the same players, such as firms in an oligopoly. Unlike one-time games, repeated games allow players to condition their strategies on past behavior, which can foster cooperation or punishment.

• One-time game: Played only once; players choose strategies without concern for future consequences.

• Repeated game: The same game is played over and over, allowing for strategies that depend on previous actions.

Example: Two firms, Jack and Jill, repeatedly decide how many gallons to produce each period. Their choices and profits depend on whether they cooperate or compete.

Strategies in Repeated Games

Players in repeated games can use more complex strategies than in one-time games. Two important strategies are tit-for-tat and trigger strategies.

• 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.

Example: If Jack uses tit-for-tat, he will match Jill's previous action. If Jill cooperated last period, Jack will cooperate this period; if Jill cheated, Jack will cheat in response.

Payoff Table: Jack and Jill's Production Choices

The following table summarizes the profits for Jack and Jill based on their production 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: Table entries inferred from context and typical payoff structures in oligopoly games.

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 cooperate and produce the same amount as Jill did when cooperating (e.g., 30 gallons if that is the cooperative amount).

• 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 cheat in response, earning the profit associated with cheating (e.g., $1,600 if both cheat).

Key Terms and Definitions

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

• Repeated game: A strategic interaction played multiple times by the same players.

• Tit-for-tat strategy: A strategy in which a player mimics the opponent's previous action.

• Trigger strategy: A strategy that cooperates until the opponent defects, then switches to permanent defection.

Equations and Concepts

• Payoff in repeated games: The total profit is the sum of profits over all periods, often discounted by a factor $\delta$ (the discount rate): $\text{Total Payoff} = \sum_{t=0}^{\infty} \delta^t \cdot \text{Profit}_t$

• Discount factor ($\delta$): Represents how much future profits are valued compared to current profits, where $0 < \delta < 1$.

Summary Table: Strategies in Repeated Games

Strategy	Description	Response to Defection
Tit-for-tat	Copy opponent's previous move	Defect once, then return to cooperation if opponent does
Trigger	Cooperate until opponent defects	Defect forever after first defection

Additional info: Table inferred from standard game theory strategies.