🆚Game Theory and Economic Behavior Unit 6 – Repeated Games: Fostering Cooperation

What Are Repeated Games?

• Repeated games involve players interacting with each other over multiple rounds or periods
• Players can observe and react to each other's past actions, allowing for more complex strategies and behaviors
• Repeated interactions allow players to establish reputations, build trust, and potentially foster cooperation
• The number of repetitions can be finite (known end point) or infinite (unknown or indefinite end point)
• The stage game, or the game played in each round, can be any type of game (Prisoner's Dilemma, Coordination Game, etc.)
• Players' actions in one round can influence the outcomes and payoffs in future rounds
• Repeated games can be used to model various real-world situations (business relationships, international diplomacy, etc.)

Key Concepts in Repeated Games

• Discount factor ($\delta$) represents the weight players assign to future payoffs relative to current payoffs
  • A higher discount factor indicates more patience and a greater value placed on future outcomes
• Subgame perfect equilibrium (SPE) is a refinement of Nash equilibrium for repeated games
  • In an SPE, players' strategies must be optimal at every decision point (subgame) of the repeated game
• Trigger strategies involve players cooperating until someone defects, then punishing the defector in future rounds
• Grim trigger strategy is an extreme form of punishment where players permanently revert to non-cooperation after a single defection
• Reputation effects can emerge in repeated games, as players' past actions influence how others perceive and interact with them
• Bounded rationality acknowledges that players may have cognitive limitations and may not always make perfectly optimal decisions

Strategies for Cooperation

• Tit-for-Tat (TFT) starts by cooperating and then mimics the opponent's previous action in subsequent rounds
• Generous Tit-for-Tat (GTFT) is similar to TFT but occasionally forgives defection to avoid prolonged punishment cycles
• Win-Stay, Lose-Shift (WSLS) continues with the same action if it yields a favorable outcome and switches otherwise
• Pavlov strategy cooperates if both players chose the same action in the previous round and defects otherwise
• Grim Trigger cooperates until the first defection and then permanently defects thereafter
• Forgiving strategies allow for occasional defections without triggering permanent punishment
• Signaling strategies involve players communicating their intentions or commitments through their actions

The Folk Theorem

• The Folk Theorem states that any feasible and individually rational payoff can be sustained as an equilibrium outcome in an infinitely repeated game with sufficiently patient players
  • Feasible payoffs are those that can be achieved through some combination of players' actions
  • Individually rational payoffs are those that exceed each player's minimax payoff (the lowest payoff a player can guarantee themselves)
• The theorem implies that cooperation can be sustained in repeated games, even if it is not an equilibrium in the stage game
• Multiple equilibria may exist, and the specific equilibrium reached depends on factors such as players' expectations and coordination
• The Folk Theorem relies on the assumption of perfect information and the ability to implement complex strategies
• The theorem highlights the importance of patience (high discount factor) in fostering cooperation

Tit-for-Tat and Other Famous Strategies

• Tit-for-Tat (TFT) has been successful in various computer tournaments and simulations
  • TFT is easy to understand, nice (never defects first), provocable (responds to defection), and forgiving (returns to cooperation after punishment)
• Generous Tit-for-Tat (GTFT) can prevent lock-in to mutual defection by occasionally forgiving defections
• Win-Stay, Lose-Shift (WSLS) is a simple strategy that adapts based on the outcome of the previous round
• Grim Trigger is an unforgiving strategy that permanently punishes any defection
• Pavlov (Win-Stay, Lose-Shift) encourages cooperation by rewarding mutual cooperation and mutual defection
• Zero-Determinant (ZD) strategies allow players to unilaterally set the opponent's payoff, but they rely on specific assumptions and have limitations

Real-World Applications

• Repeated interactions in business relationships (suppliers, manufacturers, retailers) can foster trust and cooperation
• International relations and diplomacy often involve repeated interactions and the potential for cooperation or conflict
• Environmental agreements (climate change, resource management) require ongoing cooperation among nations
• Repeated games can help explain the emergence and maintenance of social norms and conventions
• Reputation systems (online marketplaces, credit ratings) rely on the principles of repeated interactions to encourage good behavior
• Iterative Prisoner's Dilemma has been used to study the evolution of cooperation in biological systems

Challenges and Limitations

• Repeated games assume that players have perfect information about past actions, which may not always be realistic
• The strategies and equilibria predicted by repeated game models may be too complex for humans to implement in practice
• The assumption of a fixed discount factor may not capture the variability in how people value future outcomes
• Repeated games often assume that players are rational and aim to maximize their payoffs, which may not always be true
• In real-world situations, the number of repetitions may be unknown or change over time, complicating the analysis
• Repeated game models may not fully capture the richness and complexity of real-world interactions and institutions

Advanced Topics in Repeated Games

• Stochastic games involve transition probabilities between different states, adding complexity to the repeated interaction
• Incomplete information in repeated games means that players may not know certain aspects of the game or their opponents
  • Reputation building and signaling become important in games with incomplete information
• Evolutionary game theory studies the dynamics of strategy adoption and adaptation in populations over time
• Repeated games with imperfect monitoring involve situations where players cannot perfectly observe each other's actions
• Repeated games with communication allow players to send messages and coordinate their strategies
• Repeated games with renegotiation consider the possibility of players agreeing to change their strategies during the game
• Finitely repeated games with a known end point can have different equilibria and dynamics compared to infinitely repeated games