🎲Game Theory and Business Decisions Unit 4 – Repeated Games: Fostering Cooperation

What Are Repeated Games?

• Repeated games involve players interacting with each other multiple times over a period of time
• Players have the opportunity to observe and respond to each other's actions in previous rounds
• Repeated interactions allow for the possibility of cooperation, even in situations where defection is the dominant strategy in a one-shot game
• Players can establish reputations and build trust through their actions in repeated games
• The number of repetitions can be finite (known endpoint) or infinite (unknown or indefinite endpoint)
• Repeated games are often used to model long-term relationships, such as business partnerships, international relations, and social interactions
• The outcome of a repeated game depends on the players' strategies, which can be influenced by factors such as trust, reciprocity, and the shadow of the future

Key Concepts in Repeated Games

• Discount factor ($\delta$) represents the weight players assign to future payoffs relative to present payoffs
  • A higher discount factor indicates a greater value placed on future outcomes
  • Discount factors close to 1 suggest a long-term orientation, while discount factors close to 0 indicate a focus on short-term gains
• Trigger strategies involve players cooperating until one player defects, at which point all players switch to punishment mode
  • Grim trigger strategy: players cooperate until a defection occurs, then permanently switch to defection
  • Tit-for-tat strategy: players start with cooperation and then mirror the opponent's previous action
• Nash equilibrium in repeated games can differ from the equilibrium in one-shot games due to the possibility of cooperation
• Subgame perfect equilibrium requires that players' strategies constitute a Nash equilibrium in every subgame of the repeated game
• Reputation effects can influence players' behavior, as they may seek to establish a cooperative reputation to encourage future cooperation
• Signaling involves players using their actions to convey information about their intentions or type

Strategies for Cooperation

• Tit-for-tat (TFT) is a simple yet effective strategy that promotes cooperation
  • Players start by cooperating and then mirror the opponent's previous action
  • TFT is forgiving (returns to cooperation after a defection is reciprocated) and retaliatory (punishes defections)
• Grim trigger strategy involves players cooperating until a defection occurs, then permanently switching to defection
  • This strategy can deter defections but may lead to suboptimal outcomes if a defection occurs accidentally or due to noise
• Pavlov (win-stay, lose-shift) strategy involves players repeating their previous action if it led to a favorable outcome and switching actions otherwise
• Generous tit-for-tat (GTFT) is a variant of TFT that forgives defections with a certain probability, allowing for the restoration of cooperation
• Gradual strategies, such as tit-for-two-tats (TF2T), are more forgiving and can help maintain cooperation in the presence of noise or mistakes
• Reward and punishment strategies involve players offering incentives for cooperation and imposing costs for defection

The Folk Theorem

• The Folk Theorem states that any feasible and individually rational payoff can be sustained as a Nash equilibrium in an infinitely repeated game with sufficiently patient players
• Feasible payoffs are those that can be obtained through some combination of players' actions
• Individually rational payoffs are those that exceed the minimax payoff (the lowest payoff a player can guarantee themselves)
• The Folk Theorem implies that cooperation can be sustained in repeated games if players are patient enough (i.e., have a high discount factor)
  • Intuitively, players are more likely to cooperate if they value future payoffs and the long-term benefits of cooperation
• The Folk Theorem does not specify which equilibrium will be played, as there may be multiple equilibria that satisfy the conditions
• The Folk Theorem highlights the importance of the shadow of the future in promoting cooperation

Tit-for-Tat and Other Famous Strategies

• Tit-for-tat (TFT) has been successful in various tournaments and simulations, demonstrating its effectiveness in promoting cooperation
  • TFT won Axelrod's famous prisoner's dilemma tournament, which involved competing against other submitted strategies
  • TFT performs well against a wide range of strategies and is robust to noise and mistakes
• Grim trigger strategy is a simple and effective deterrent against defections but can lead to suboptimal outcomes if a defection occurs
  • The threat of permanent defection can be credible if players have a high discount factor and value future payoffs
• Pavlov (win-stay, lose-shift) strategy has been shown to outperform TFT in certain environments, particularly when there is noise or uncertainty
• Generous tit-for-tat (GTFT) can help restore cooperation after a defection and is more resilient to noise than standard TFT
• Tit-for-two-tats (TF2T) and other gradual strategies are more forgiving and can maintain cooperation in the presence of occasional defections
• Zero-determinant strategies allow players to unilaterally set the opponent's payoff, but their effectiveness depends on the specific game and conditions

Real-World Applications

• Repeated interactions are common in business relationships, such as supplier-buyer partnerships and joint ventures
  • Companies can foster cooperation by establishing long-term contracts, investing in relationship-specific assets, and building trust
  • Reputation effects can incentivize firms to cooperate, as a reputation for trustworthiness can attract future business partners
• International relations often involve repeated interactions between countries, such as trade agreements, environmental treaties, and military alliances
  • Countries can use trigger strategies and sanctions to enforce cooperation and deter defections
  • International institutions, such as the United Nations and World Trade Organization, can facilitate cooperation by providing a framework for repeated interactions
• Social norms and conventions can emerge through repeated interactions in social settings
  • Individuals may cooperate to maintain a positive reputation within their social network
  • Repeated interactions can lead to the development of trust and reciprocity, which can support cooperative behavior

Challenges and Limitations

• Bounded rationality and cognitive limitations can prevent players from accurately assessing the long-term consequences of their actions
  • Players may focus on short-term gains and fail to fully consider the impact of their decisions on future interactions
• Uncertainty and incomplete information can hinder cooperation, as players may be unsure about the other players' intentions or payoffs
  • Noisy environments, where actions or payoffs are subject to random disturbances, can disrupt cooperation and lead to unintended defections
• Coordination problems can arise when there are multiple equilibria, and players may have difficulty agreeing on which equilibrium to play
• The Folk Theorem assumes an infinite or indefinite time horizon, which may not always be realistic in practical situations
  • In finitely repeated games, cooperation can unravel through backward induction, as players anticipate the end of the game
• Power asymmetries and unequal bargaining positions can limit the ability of some players to enforce cooperation or punish defections

Advanced Topics in Repeated Games

• Stochastic games involve players interacting in an environment where the game structure or payoffs can change over time based on a probability distribution
  • Players must adapt their strategies to the changing conditions and consider the long-term implications of their actions
• Repeated games with imperfect monitoring involve situations where players cannot perfectly observe each other's actions or payoffs
  • Players must rely on noisy signals or public correlating devices to coordinate their actions and sustain cooperation
• Evolutionary game theory explores how strategies evolve over time in a population of players through processes of selection and mutation
  • Strategies that perform well against the current population are more likely to be adopted, leading to dynamic changes in the prevalence of different strategies
• Repeated games with incomplete information involve players having private information about their own payoffs or types
  • Players may engage in signaling or reputation-building to convey information and facilitate cooperation
• Higher-order beliefs and recursive reasoning can influence players' strategies in repeated games
  • Players may consider not only their own beliefs but also their beliefs about the other players' beliefs, leading to more complex strategic considerations