🎱Game Theory Unit 7 – Repeated Games and Cooperation

What Are Repeated Games?

• Repeated games involve players interacting with each other over multiple rounds or periods
• In each round, players choose strategies and receive payoffs based on the combination of strategies chosen
• The game is played for a finite or infinite number of rounds, and players have the opportunity to observe and respond to each other's actions
• Repeated games allow for the possibility of cooperation and punishment strategies that are not possible in one-shot games
• The repeated nature of the game can lead to different equilibrium outcomes compared to the corresponding one-shot game
  • Players may be able to sustain cooperation in a repeated Prisoner's Dilemma, even though defection is the dominant strategy in the one-shot version
• Repeated games can be modeled with a discount factor that represents the weight players place on future payoffs relative to current payoffs
• The concept of repeated games is applicable to various real-world situations (business relationships, international relations, and social interactions)

Key Concepts in Repeated Games

• Discount factor ($\delta$) represents the weight players place on future payoffs
  • A higher discount factor means players are more patient and value future payoffs more
  • A lower discount factor means players are more impatient and focus more on immediate payoffs
• Subgame perfect equilibrium is a refinement of Nash equilibrium that requires players' strategies to be optimal at every decision point (subgame) of the repeated game
• Trigger strategies involve players cooperating until someone defects, then punishing the defector by switching to a less cooperative strategy for the remainder of the game
• Tit-for-tat is a common trigger strategy where a player starts by cooperating and then mimics the opponent's previous action in subsequent rounds
• Grim trigger strategy involves players cooperating until someone defects, then permanently switching to the Nash equilibrium strategy of the one-shot game
• Folk theorem states that any feasible and individually rational payoff can be sustained as a subgame perfect equilibrium in an infinitely repeated game with sufficiently patient players
• Reputation effects can arise in repeated games, where players' actions in early rounds influence how others perceive and interact with them in later rounds

Strategies in Repeated Games

• Players can employ various strategies in repeated games to maximize their long-term payoffs
• Cooperative strategies involve players coordinating their actions to achieve mutually beneficial outcomes
  • Players may agree to play a Pareto-optimal strategy profile that maximizes joint payoffs
• Punishment strategies are used to deter players from deviating from a cooperative agreement
  • Players may threaten to switch to a less cooperative strategy if someone defects, reducing the defector's future payoffs
• Forgiving strategies allow players to return to cooperation after a defection occurs
  • Tit-for-tat is an example of a forgiving strategy, as it will cooperate if the opponent cooperates in the previous round, even if there was an earlier defection
• Non-forgiving strategies, such as grim trigger, permanently punish defections and do not allow for a return to cooperation
• Players may also use mixed strategies, randomizing between different actions to keep opponents guessing and prevent exploitation
• Signaling strategies involve players using their actions to convey information about their intentions or types to influence the behavior of other players

The Folk Theorem

• The Folk Theorem is a fundamental result in repeated game theory that characterizes the set of payoffs that can be sustained as equilibria in infinitely repeated games
• It states that any feasible and individually rational payoff can be sustained as a subgame perfect equilibrium if players are sufficiently patient (i.e., have a high enough discount factor)
  • Feasible payoffs are those that can be achieved by some combination of players' strategies in the one-shot game
  • Individually rational payoffs are those that give each player at least their minimax payoff (the lowest payoff they can guarantee themselves, regardless of the other players' actions)
• The Folk Theorem implies that a wide range of outcomes, including cooperative ones, can be sustained in repeated games through the use of appropriate strategies and punishments
• The theorem relies on the idea that players can use the threat of future punishments to deter deviations from a desired outcome
• Different versions of the Folk Theorem exist for games with finite and infinite time horizons, as well as for games with imperfect monitoring or incomplete information
• The Folk Theorem highlights the importance of patience and the ability to punish in achieving cooperation in repeated interactions

Cooperation and Defection

• Cooperation in repeated games refers to players coordinating their actions to achieve mutually beneficial outcomes
  • Players may agree to play strategies that maximize joint payoffs, even if these strategies are not individually optimal in the one-shot game
• Defection occurs when a player deviates from a cooperative agreement in order to gain a short-term advantage
  • In the Prisoner's Dilemma, defection refers to a player choosing the "betray" strategy when the cooperative agreement was to "remain silent"
• The possibility of future interactions and punishments can make cooperation sustainable in repeated games, even when it is not an equilibrium in the one-shot game
• Players must weigh the short-term gains from defection against the long-term losses from punishment when deciding whether to cooperate or defect
• The effectiveness of cooperation in repeated games depends on factors such as the players' patience (discount factor), the ability to monitor and punish defections, and the clarity of the cooperative agreement
• Cooperation can break down if players have different time horizons, face uncertainty about the game's duration, or have imperfect information about each other's actions

Equilibrium in Repeated Games

• Equilibrium in repeated games refers to a situation where players' strategies are optimal given the strategies of the other players, and no player has an incentive to unilaterally deviate
• Subgame perfect equilibrium (SPE) is a common equilibrium concept used in repeated games
  • SPE requires that players' strategies are optimal at every decision point (subgame) of the repeated game, not just at the beginning
• The Folk Theorem characterizes the set of payoffs that can be sustained as SPE in infinitely repeated games with sufficiently patient players
• In finitely repeated games, the unique SPE is often the repeated play of the Nash equilibrium of the one-shot game (by backward induction)
  • However, cooperation can still be sustained in finitely repeated games if there is uncertainty about the game's duration or if players have incomplete information
• Equilibrium strategies in repeated games often involve the use of trigger strategies, such as tit-for-tat or grim trigger, to punish deviations and sustain cooperation
• Multiple equilibria may exist in repeated games, and the selection among them can depend on factors such as focal points, communication, and cultural norms
• Equilibrium analysis in repeated games helps predict the outcomes of long-term strategic interactions and understand the conditions under which cooperation can be sustained

Real-World Applications

• Repeated games have numerous applications in economics, political science, and other social sciences
• In business, repeated interactions between firms can lead to tacit collusion and price-fixing, as the threat of future price wars can deter deviations from the collusive agreement
• In international relations, countries engage in repeated interactions over trade, environmental policies, and security issues, and the prospect of future cooperation or retaliation shapes their behavior
• In social interactions, repeated games can explain the emergence and maintenance of social norms, as individuals who violate norms may face punishment or exclusion from future interactions
• In labor markets, repeated interactions between employers and employees can lead to implicit contracts and efficiency wages, as firms may pay above-market wages to deter shirking and maintain a good reputation
• In online marketplaces and platforms, reputation systems and feedback mechanisms can sustain cooperation by allowing users to punish or reward each other based on past behavior
• In environmental conservation, repeated games can model the dynamics of common-pool resource management, where individuals must balance their short-term incentives to overexploit with the long-term benefits of sustainable use

Common Mistakes and Pitfalls

• Ignoring the difference between one-shot and repeated games
  • Strategies that are optimal in one-shot games may not be optimal in repeated games, and vice versa
• Assuming that repetition always leads to cooperation
  • While repeated interactions can facilitate cooperation, it is not guaranteed, and the specific conditions of the game (payoffs, discount factors, information structure) matter
• Failing to consider the credibility of threats and punishments
  • For a punishment strategy to be effective, it must be in the punisher's interest to carry out the punishment if a deviation occurs
• Neglecting the role of communication and coordination
  • In many repeated games, players can improve their outcomes by communicating and coordinating their strategies, but this is not always captured in the basic models
• Overestimating the impact of reputation
  • While reputation can be important in repeated games, its effectiveness depends on factors such as the observability of actions, the presence of noise, and the possibility of forgiveness
• Assuming perfect rationality and foresight
  • In reality, players may have bounded rationality, face cognitive limitations, or be influenced by emotions and biases, which can lead to deviations from the predictions of standard repeated game models
• Ignoring the potential for renegotiation
  • If players can renegotiate their strategies during the course of the game, some punishment strategies may not be credible, and the set of sustainable outcomes may be different
• Failing to account for heterogeneity among players
  • Players may have different time preferences, risk attitudes, or beliefs, which can affect their behavior in repeated games and the prospects for cooperation