Question

Consider the infinitely repeated prisoners' dilemma and recall the definition of the grim-trigger strategy. Here is the definition of another simple strategy called Tit-for-tat: Select \(\mathrm{C}\) in the first period; in each period thereafter, choose the action that the opponent selected in the previous period. \({ }^{8}\) Is the tit-for-tat strategy profile a Nash equilibrium of the repeated game for discount factors close to one? Is this strategy profile a subgame perfect Nash equilibrium of the repeated game for discount factors close to one? Explain.

Short answer

Yes, tit-for-tat is a Nash equilibrium and subgame perfect Nash equilibrium for high discount factors.

Step-by-step solution

Understanding Tit-for-tat Strategy

The tit-for-tat strategy involves initially cooperating (choosing C) in the first period of the infinitely repeated game. In each subsequent period, the player selects the action that their opponent chose in the previous period. This encourages mutual cooperation, as deviations are met with direct reprocity.

Definition of Nash Equilibrium

A Nash Equilibrium occurs when no player can benefit by unilaterally changing their strategy, given the strategies of the other players. In this context, we need to determine if any player has an incentive to deviate from the tit-for-tat strategy when the discount factor is close to one.

Checking for Nash Equilibrium

Consider a situation where both players follow tit-for-tat. If one player deviates, the other will retaliate by mimicking the deviation in the next period. When the discount factor is close to one, the present value of future losses from retaliation outweighs the short-term gains from deviating, disincentivizing deviations. Thus, it is a Nash Equilibrium.

Definition and Importance of Subgame Perfect Nash Equilibrium

A Subgame Perfect Nash Equilibrium (SPNE) is a refinement of Nash Equilibrium, where the strategy profiles provide Nash Equilibria in every subgame of the original game. SPNE accounts for credibility of threats and responses throughout the game.

Evaluating Subgame Perfect Nash Equilibrium

For tit-for-tat to be a SPNE, it must be a Nash Equilibrium in every subgame. Since tit-for-tat punishes deviations consistently (with the opponent mirroring the move) across all subgames, and given high enough discount factors, this results in credible deterrence and consistent outcomes, confirming that tit-for-tat is a SPNE.

Key concepts

Prisoner's Dilemma

The Prisoner's Dilemma is a fundamental concept in game theory that illustrates why individuals might not cooperate, even when collaboration would lead to the best overall outcome. Imagine two partners in crime who are caught and interrogated separately. Each has the option to either betray the other (defect) or remain silent (cooperate). If both betray each other, they receive a moderate penalty. If one betrays while the other stays silent, the betrayer walks away free while the silent partner receives a harsh penalty. If both stay silent, they each get a minor penalty.

The dilemma arises because mutual cooperation is collectively optimal, yet from an individual perspective, betrayal seems tempting as it could potentially lead to a better personal outcome. The key takeaway is that rational individuals, acting independently, can end up in worse situations when they don't collaborate. This scenario describes many real-life situations, making the Prisoner's Dilemma a compelling study in strategic interaction.

Nash Equilibrium

Nash Equilibrium refers to a stable state of a system where no player can gain an advantage by solely changing their strategy, assuming other players' strategies remain unchanged. It is named after mathematician John Nash, who introduced the concept in 1950.

In the context of the infinitively repeated Prisoner's Dilemma, when players adopt the tit-for-tat strategy, cooperation appears to be maintained. When both stick to tit-for-tat, neither has a unilateral incentive to deviate, particularly when the discount factor is near one. This implies the future payoff values are significant enough to ensure cooperation.

Thus, when each player considers the long-term consequences of their actions, continuing with tit-for-tat becomes a Nash Equilibrium. They avoid deviating as the potential negative outcomes of future retaliations outweigh short-term individual gains.

Subgame Perfect Nash Equilibrium

A Subgame Perfect Nash Equilibrium (SPNE) extends the Nash Equilibrium by ensuring players' strategies represent a Nash Equilibrium in every subgame of the original game. This concept helps assess whether a player's threat or course of action is credible and rational at every stage of the game.

In the repeated Prisoner's Dilemma, a tit-for-tat strategy qualifies as a SPNE because it consistently maintains Nash Equilibrium in every subgame. This means after any history or series of moves, the strategy chosen by each player still follows a Nash Equilibrium path.

• Players respond to a deviation with a similar deviation in future rounds.
• High discount factors imply future losses will outweigh any immediate gain from defecting, ensuring compliance.

Thus, tit-for-tat functions effectively across the infinitely repeated game, providing durable peace and cooperation.

Discount Factor

The concept of a discount factor is crucial when determining how future rewards are valued compared to immediate rewards in repeated games. It essentially measures a player's preference for current rewards versus future gains. A discount factor ranges between 0 and 1. Values closer to 1 indicate players significantly value future payoffs, while values near 0 imply preference for immediate rewards.

In the context of the repeated Prisoner's Dilemma and the tit-for-tat strategy, a discount factor close to one is key to maintaining cooperation. It ensures that future retaliation losses due to defection are significant enough to deter players from rogue behavior.

• It emphasizes the importance of long-term outcomes in strategic decision-making.
• Encourages stable cooperation by aligning immediate actions with future benefits.

Hence, high discount factors harmonize the players' strategies, holding them accountable to choices that ensure mutual long-term cooperation.