Question

Consider an infinite-period repeated prisoners' dilemma game in which a long- run player 1 faces a sequence of short-run opponents. (You dealt with games like this in Exercise 8 of Chapter 22.) Formally, there is an infinite number of players - denoted \(2^{1}, 2^{2}, 2^{3}, \ldots\)-who play as player 2 in the stage game. In period \(t\), player 1 plays the following prisoners' dilemma with player \(2^{t}\). Assume that all of the players observe the history of play. Let \(\delta\) denote the discount factor of player \(1 .\) (a) Show that the only subgame perfect equilibrium involves play of (D, D) each period. (b) Next suppose that the players \(2^{t}\) have the opportunity to buy and sell the right to play the game each period. Formally, suppose that in order for player \(2^{t}\) to play the stage game in period \(t\), this player must purchase the right from player \(2^{t-1}\). Model the negotiation between successive players \(2^{t-1}\) and \(2^{t}\) as a joint decision. Further, assume the outcome of negotiation is consistent with the standard bargaining solution. Under what conditions on bargaining weights and the discount factor can \((\mathrm{C}, \mathrm{C})\) be supported over time?

Short answer

(a) (D, D) is subgame perfect; (b) (C, C) if \(\delta > \frac{1}{2}\) and bargaining weights favor cooperation.

Step-by-step solution

Analyze the Stage Game Payoffs

In the basic prisoner's dilemma, each player has two strategies: Cooperate (C) or Defect (D). The payoff structure is typically such that mutual defection (D, D) leads to a payoff of 1 for each player, mutual cooperation (C, C) results in a payoff of 3 for each player, and a player who defects while the other cooperates gets 5 (the cooperator gets 0).

Determine the Nash Equilibrium in Stage Game

In a one-shot interaction, no player has an incentive to deviate from mutual defection because if Player 1 cooperates, Player 2 benefits by defecting. Hence, (D, D) becomes the Nash Equilibrium in the stage game.

Analyze Subgame Perfect Equilibrium for Infinite Game

For player 1 facing a sequence of short-lived Player 2s, history observed does not incentivize Player 2 to deviate from defecting, as there are no future gains from sustained cooperation by short-lived players. Thus, (D, D) in each period remains the subgame perfect equilibrium in the infinite game.

Introduce the Market for Rights to Play

In the modified game, each Player 2 must buy the right to play the prisoner's dilemma from the previous Player 2. This introduces an additional layer of negotiation.

Apply Bargaining Solution Analysis

The negotiation result between successive players is modeled with a standard bargaining solution. The key is to identify conditions where the bargaining weights \((\theta)\) and player 1's discount factor \(\delta\) make cooperation between players more beneficial than defection.

Derive Conditions for Cooperation (C, C)

For (C, C) to prevail, future players must assign high value to cooperation outcomes. This depends on \(\delta > \frac{1}{2}\) ensuring the benefit from future cooperative agreements outweighs the immediate gain from defecting, and the bargaining weights must skew to favor cooperative outcomes.

Key concepts

Prisoner's Dilemma

The Prisoner's Dilemma is a fundamental concept in game theory. It involves two players each having two options: Cooperate (C) and Defect (D).
The twist is that mutual cooperation yields a higher reward than mutual defection, but the selfish choice of defection offers an immediate gain if the other cooperates.
In our example, the payoffs are structured such that if both players cooperate, they each receive 3 units of reward. If only one defects while the other cooperates, the defector gets 5 and the cooperator gets nothing. If both defect, they receive 1 each. This setup creates a tension between individual rationality (defecting) and collective benefit (cooperating).

• Mutual Cooperation (C, C): 3 for each player
• Mutual Defection (D, D): 1 for each player
• One Defects, One Cooperates: Defector gets 5, Cooperator gets 0

Understanding this dilemma helps explain why rational individuals might not cooperate, even if it seems in their best interest.

Repeated Games

Repeated games address scenarios where players encounter the same game multiple times. This repetition allows for strategic evolution and potential cooperation, as players can adjust their strategies based on previous interactions.
For example, in an infinitely repeated prisoners’ dilemma, each round’s outcome can influence future decisions. This creates a complex strategic environment where players weigh immediate gains against long-term rewards.
In the context of the exercise, player 1 interacts with a perpetually changing sequence of short-lived opponents across infinite periods. As these opponents observe historical actions, their strategies are influenced by previous periods.

• Repetition allows for potential cooperation
• Strategic choices depend on past interactions
• Complex dynamics emerge with infinite repetition

Repeated games illustrate how the possibility of future interactions can reshape current strategic decisions.

Subgame Perfect Equilibrium

The concept of Subgame Perfect Equilibrium (SPE) refines Nash Equilibrium to consider the strategies in every possible subgame of the original game.
In an infinite-period prisoner's dilemma, SPE requires that players’ strategies be optimal in every period, given the game's history.
In this exercise, the only SPE involves consistent defection (D, D), because each new player faces no future repercussions from Player 1 and sees immediate gain in defecting since they only play once.

• Considers all possible history scenarios
• Ensures optimal strategy at every point
• Infinite games often result in (D, D) due to lack of future incentive to deviate

Subgame Perfect Equilibrium helps analyze strategic consistency across multiple rounds in repeated games.

Nash Equilibrium

Nash Equilibrium is a key concept in game theory where no player has an incentive to unilaterally change their strategy. It signifies stability in strategic interactions.
In the one-shot version of the prisoner's dilemma, both players choosing to defect achieves Nash Equilibrium. This is because neither player can increase their payoff by changing their individual strategy, given the strategy of the other.
However, when extended to infinitely repeated games, as in our exercise, Nash Equilibrium helps identify stable strategy profiles across time. Here, (D, D) persists as the Nash Equilibrium in every stage, as it ensures neither player can improve their outcome by unilaterally changing their strategy in a single period.

• Stability in strategy choice
• No unilateral incentive to change strategy
• In repeated games, equilibrium can extend over time

Nash Equilibrium provides a foundation for understanding strategic stability in both one-shot and repeated interactions.