If both players cooperate (C, C), each receives 3 points.

If both players defect (D, D), each receives 1 point.

If one player cooperates and the other defects (C, D or D, C), the defector receives 5 points, and the cooperator receives 0 points.

This game is played repeatedly for an indefinite number of rounds (indefinite horizon).

Players can adopt different strategies to maximize their long-term payoffs. Common strategies include:

**Tit-for-Tat (TFT):** Start by cooperating, then in each subsequent round, do what the opponent did in the previous round.

**Grim Trigger (GT):** Start by cooperating, but if the opponent defects even once, defect forever.

**Always Defect (AD):** Always choose to defect in every round.

**Random:** Cooperate or defect randomly.

**Pavlov (Win-Stay, Lose-Shift):** Cooperate if the previous round’s outcome was good (i.e., both cooperated or both defected), and defect otherwise.

Analyze the stability and effectiveness of each strategy in this context. Consider the following points:

How does each strategy perform against itself and other strategies?

What are the conditions under which each strategy can be considered a Nash equilibrium?

Which strategies are likely to evolve and dominate in a population of rational players?

**Analysis:**

**Tit-for-Tat (TFT):** This strategy is known for its simplicity and effectiveness in promoting mutual cooperation. TFT performs well against itself, as both players will continue to cooperate, resulting in a consistent payoff of 3 points per round. Against GT, TFT will also maintain cooperation as long as GT does. However, against AD, TFT will defect after the first defection, leading to a lower payoff of 1 point per round. TFT is effective in fostering cooperation but can be exploited initially by AD.

**Grim Trigger (GT):** GT is a harsh strategy that can enforce cooperation if players are wary of permanent punishment. When playing against itself or TFT, GT will result in a consistent payoff of 3 points per round. However, against AD, GT will defect after the first defection, resulting in a low payoff of 1 point per round. GT can enforce cooperation but may be too unforgiving in scenarios where mistakes or misunderstandings occur.

**Always Defect (AD):** This strategy exploits cooperators in the short term but results in a low long-term payoff when mutual defection becomes the norm. AD performs poorly against TFT and GT, as it leads to a consistent payoff of 1 point per round after the initial rounds. AD can dominate in a population with many naive cooperators but will not sustain high payoffs in a population of rational players.

**Random:** A random strategy is unpredictable but generally results in lower payoffs due to the lack of consistency. It does not foster cooperation and can be easily exploited by more systematic strategies. Random strategies are unlikely to be stable or dominate in a rational population.

**Pavlov:** This strategy adapts based on the previous outcome, promoting cooperation if it was beneficial and switching to defection if it was not. Pavlov can perform well against TFT and GT by maintaining cooperation and switching to defection only when necessary. However, it may be less effective against AD and random strategies. Pavlov can be stable in a mixed population and promote cooperation while being more forgiving than GT.

**Conclusion:**

In an indefinite horizon iterated game, strategies that promote cooperation (like TFT and Pavlov) tend to perform better in the long run compared to AD and random strategies. TFT and Pavlov can create a stable environment of mutual cooperation, resulting in higher payoffs. However, TFT’s vulnerability to initial exploitation and GT’s unforgiving nature must be considered. Pavlov offers a balanced approach, promoting cooperation while adapting to the opponent’s actions. Rational players are likely to adopt TFT or Pavlov, leading to a cooperative equilibrium in the population.

4o