11.3 Repeated games and the Folk Theorem

Repeated games add depth to strategic interactions, allowing players to consider past behavior and future consequences. This complexity enables cooperation in scenarios where one-time interactions might lead to selfish choices.

The Folk Theorem is a key concept, stating that patient players can achieve any reasonable outcome in infinitely repeated games. This broadens our understanding of possible equilibria and highlights the importance of long-term thinking in strategic situations.

Impact of Repeated Interactions

Repeated Games and Strategic Complexity

• Repeated games involve multiple iterations of the same game allowing players to condition their strategies on past behavior
• Shadow of the future influences current decision-making in repeated games by considering potential future interactions
• Strategies in repeated games incorporate elements such as punishment, forgiveness, and reciprocity
• Discounting future payoffs affects the relative importance of short-term gains versus long-term cooperation
  • Higher discount factors increase the value of future payoffs
  • Lower discount factors prioritize immediate gains
• Repeated interactions can lead to cooperative behavior even in games where defection dominates in one-shot scenarios
  • Example: Infinitely repeated prisoner's dilemma demonstrates how cooperation can emerge over time

Complexity and Examples in Repeated Games

• Players can use more complex strategies in repeated games compared to one-shot games
  • Tit-for-tat strategy (cooperate initially, then mirror opponent's previous move)
  • Grim trigger strategy (cooperate until opponent defects, then defect forever)
• Infinitely repeated prisoner's dilemma serves as a canonical example for studying repeated interactions
  • Players: Two suspects
  • Actions: Confess (defect) or remain silent (cooperate)
  • Payoffs: Determined by combination of actions chosen by both players
• Other examples of repeated games in real-world scenarios
  • Business partnerships with ongoing transactions
  • International trade agreements with multiple rounds of negotiations
  • Repeated auctions in online marketplaces

Cooperation in Repeated Games

Factors Influencing Cooperation

• Discount factor represents players' patience and probability of future interactions
  • Higher discount factors increase likelihood of sustained cooperation
  • Lower discount factors may lead to short-term thinking and defection
• Trigger strategies enforce cooperative behavior through punishment threats for deviation
  • Grim trigger (switch to permanent defection after any deviation)
  • Tit-for-tat (mimic opponent's previous action)
• Folk theorem states any feasible and individually rational payoff can be sustained as equilibrium in infinitely repeated games if players are sufficiently patient
• Indefinite repetition increases likelihood of cooperation compared to known finite repetitions
  • Backward induction in finite games can lead to unraveling of cooperation
  • Uncertainty about game end maintains cooperative incentives

Monitoring and Credibility in Repeated Games

• Subgame perfection analyzes credible threats and promises in repeated games
  • Ensures strategies are optimal in every subgame, not just the overall game
  • Eliminates non-credible threats that players would not carry out if tested
• Monitoring and information structures affect ability to detect and punish deviations
  • Perfect monitoring (players observe all past actions)
  • Imperfect monitoring (players receive noisy signals about past actions)
• Renegotiation possibility impacts credibility of punishment threats and cooperation sustainability
  • Players may be tempted to forgive deviations and restart cooperation
  • Renegotiation-proof equilibria must be resistant to this temptation

Equilibrium Outcomes in Repeated Games

Folk Theorem and Payoff Sets

• Folk Theorem characterizes Nash equilibrium payoffs in infinitely repeated games with patient players
• Feasible payoff set represents all possible average payoffs achievable through different stage-game action combinations
  • Convex hull of all possible stage-game payoff vectors
• Minmax payoff for each player represents lowest payoff other players can force upon them in stage game
  • $v_i = \min_{a_{-i}} \max_{a_i} u_i(a_i, a_{-i})$
  • Where $v_i$ is player i's minmax payoff, $a_i$ is player i's action, and $a_{-i}$ are other players' actions
• Individually rational payoff set consists of feasible payoffs giving each player at least their minmax payoff
  • $V^* = \{v \in V | v_i \geq \underline{v}_i \text{ for all } i\}$
  • Where $V^*$ is the individually rational payoff set, $V$ is the feasible payoff set, and $\underline{v}_i$ is player i's minmax payoff

Applying the Folk Theorem

• Folk Theorem states any payoff in individually rational payoff set can be sustained as subgame perfect equilibrium if discount factor is sufficiently close to 1
• Constructing appropriate equilibrium strategies often involves combination of cooperative play and credible punishment threats
  • Example: Using grim trigger strategies to support cooperation in prisoner's dilemma
  • Example: Employing optimal penal codes to minimize punishment phase length
• Folk Theorem implications for understanding repeated games
  • Multiplicity of equilibria in repeated games
  • Potential for cooperation in long-term relationships
  • Importance of patience and long-term thinking in achieving efficient outcomes
• Applications of Folk Theorem in various fields
  • Industrial organization (analyzing collusion in oligopolies)
  • International relations (explaining cooperation among nations)
  • Labor economics (modeling employer-employee relationships)