Question

Consider an infinite-period repeated game in which a "long-run player" faces a sequence of "short-run" opponents. Formally, player 1 plays the stage game with a different player 2 in each successive period. Denote by \(2^{t}\) the player who plays the role of player 2 in the stage game in period \(t\). Assume that all players observe the history of play. Let \(\delta\) denote the discount factor of player 1 . Note that such a game has an infinite number of players. (a) In any subgame perfect equilibrium, what must be true about the behavior of player \(2^{t}\) with respect to the action selected by player 1 in period \(t\) ? (b) Give an example of a stage-game and subgame perfect equilibrium where the players select an action profile in the stage game that is not a stage Nash equilibrium. (c) Show by example that a greater range of behavior can be supported when both players are long-run players than when only player 1 is a long-run player.

Short answer

(a) Player \(2^t\)'s action is a best response to player 1's strategy. (b) Use contingent punishment in repeated games to support \(C, C\) despite it not being a Nash equilibrium. (c) Two long-run players can sustain more cooperative outcomes than one.

Step-by-step solution

Understanding Subgame Perfect Equilibrium

In a subgame perfect equilibrium (SPE), each player's strategy must constitute a Nash equilibrium in every subgame of the original game. For each stage game, the new player 2 (i.e., \(2^t\)) reacts optimally to player 1's strategy based on the history of play. Therefore, the behavior of \(2^t\) must correspond to a best response to player 1's actions, taking the history into account.

Example Stage Game Without Stage Nash Equilibrium

Consider a standard prisoners' dilemma as a stage game: both players choose \(C\) (cooperate) or \(D\) (defect). Normally, (\(D, D\)) is the Nash equilibrium. For a SPE where \(C, C\) is played, player 1 can 'punish' player \(2^{t}\) in future periods if \(D\) is chosen by \(2^{t}\). This discourages defections, even though \(C, C\) is not a Nash equilibrium for the stage game.

Comparison of Long-run Player Dynamics

For an infinite horizon game with two long-run players, consider a repeated prisoners' dilemma. Both can support cooperative behavior (playing \(C\)) using trigger strategies, where deviation leads to \(D\) forever. However, when only player 1 is persistent, player 2 can deviate without future repercussions for them, limiting cooperation. Therefore, the scope for supporting cooperative behavior widens with two long-run players, as future consequences are more substantial for both.

Key concepts

Infinite-period Repeated Game

An infinite-period repeated game is a type of game in game theory that involves players interacting repeatedly over an infinite number of time periods. Unlike finite games, there is no predetermined end to the interaction.
In this type of game, the "long-run player" remains a participant throughout all periods.
In contrast, the "short-run" opponents change in each subsequent period. This dynamic gives the long-run player unique leverage due to the ability to impact future outcomes.
These games allow for strategies extending over time since the future consequences of current actions must be considered. Players' decisions can depend on the entire history of play, not just the current stage.

• Key Characteristics:
  • Limitless time periods.
  • One persistent player and multiple short-term opponents.
  • Strategies dependent on past actions.

Understanding the infinite-period repeated game is essential for analyzing how cooperation and punishment strategies evolve over time.

Subgame Perfect Equilibrium

Subgame perfect equilibrium (SPE) is a refinement of Nash equilibrium used in dynamic games of complete information. In the context of repeated games, every player’s strategy must constitute a Nash equilibrium in every subgame of the original game.
This means that whatever the history of play may be, at any point in the game, the players' strategies in the remaining part of the game must be optimal responses to each other.
For the infinite-period repeated game, each player 2 (denoted as \(2^t\)) responds optimally to player 1's strategy based on this history, ensuring their actions are the best response given player 1's past actions.

• Key Points:
  • Strategies must be Nash equilibria in all subgames.
  • Decisions at each stage are influenced by past play.
  • Ensures rational behavior throughout the game.

This concept ensures consistency and rationality across the entire span of the repeated game.

Prisoners' Dilemma

The prisoners' dilemma is a standard example in game theory, illustrating why individuals might not cooperate even if it appears that it's in their collective best interest.
In a typical prisoners' dilemma, each player has the option to either cooperate or defect.
The dilemma arises because the temptation to defect and the fear of being exploited when the other player defects leads both players to defect, despite mutual cooperation yielding a better outcome.
This can be modeled in the context of an infinite-period repeated game, where cooperation can be sustained if players are aware that defection will lead to future negative consequences.
Trigger strategies, like "punishing" the defecting player with defections in future rounds, can stabilize cooperative behavior.

• Key Aspects:
  • Players must choose between cooperation and defection.
  • Punishment strategies can enforce cooperation over time.
  • Illustrates the challenge of achieving mutually beneficial outcomes.

The prisoners' dilemma demonstrates the complexities of decision-making in the face of potential short-term gains versus long-term benefits.

Discount Factor

The discount factor, denoted as \(\delta\), plays a critical role in repeated games. It represents the weight or value that players assign to future payoffs relative to immediate ones. In other words, it's a measure of patience in regards to potential future gains.
If the discount factor is high, players value future rewards almost as much as present ones, promoting cooperative behavior, as the repercussions for defection are more severe.
Conversely, a low discount factor means players are more focused on immediate gains and may be less cooperative.
This is especially relevant for the "long-run player," who can influence the game by affecting future outcomes over the long term.

• Highlights of the Discount Factor:
  • Indicates the degree of players' patience.
  • A higher factor encourages cooperation.
  • Crucial for sustaining long-term strategies in infinite games.

Discount factors are vital for determining how the promise of future interactions can enforce cooperative behavior.

Long-run Player

In an infinite-period repeated game, the long-run player is a participant who continues to engage with different opponents over each period. This player uniquely impacts the game's dynamics, both immediately and over time.
The advantage of a long-run player is their ability to establish strategies and reputations influencing the behavior of others. They can reward or "punish" actions by affecting the strategy in future rounds, leveraging the continuity of the game.

• Advantages include:
  • Ability to influence the entire course of play.
  • Strategic utilization of history to guide future interactions.
  • Capability to drive long-term cooperation through continuous involvement.

The long-run player plays a significant role in sustaining cooperation and determining outcomes throughout the infinite horizon.