Glossary

11.3 Repeated games and the Folk Theorem

4 min read•august 16, 2024

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 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

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  • 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)
    • (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

  • 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

  • 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 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
    • vi=min⁡a−imax⁡aiui(ai,a−i)v_i = \min_{a_{-i}} \max_{a_i} u_i(a_i, a_{-i})
    • Where viv_i is player i's minmax payoff, aia_i is player i's action, and a−ia_{-i} are other players' actions
  • Individually rational payoff set consists of feasible payoffs giving each player at least their minmax payoff
    • V∗={v∈V∣vi≥v‾i for all i}V^* = \{v \in V | v_i \geq \underline{v}_i \text{ for all } i\}
    • Where V∗V^* is the individually rational payoff set, VV is the feasible payoff set, and v‾i\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 in oligopolies)
    • International relations (explaining cooperation among nations)
    • Labor economics (modeling employer-employee relationships)

Key Terms to Review (16)

Collusion: Collusion is a strategic agreement between competing firms to coordinate their actions, typically to increase profits by reducing competition. This can involve setting prices, limiting production, or dividing markets among themselves. In repeated games, collusion becomes more feasible as firms can use past interactions to enforce cooperative behavior and sustain agreements over time.
Cooperative equilibrium: Cooperative equilibrium is a situation in repeated games where players work together to achieve mutually beneficial outcomes, rather than competing against each other. This concept relies heavily on the idea of sustaining cooperation over time through strategies that encourage players to stick to cooperative behavior, ensuring that everyone benefits in the long run. In such equilibria, players can coordinate their actions to achieve better payoffs than they would receive in a non-cooperative setting.
David Gale: David Gale was an influential mathematician and economist known for his pioneering work in game theory and cooperative games. His contributions laid the groundwork for understanding repeated games and the Folk Theorem, which explains how cooperation can emerge in long-term interactions between rational players.
Discount Factor: The discount factor is a numerical value used to determine the present value of future payoffs or benefits in economic decision-making. It reflects how much a future payoff is valued today and is crucial in repeated games, as it influences players' strategies and their willingness to cooperate over time. A higher discount factor indicates that future payoffs are valued more highly, which can lead to increased cooperation in repeated interactions.
Folk Theorem: The Folk Theorem refers to a concept in game theory that suggests, under certain conditions, players in repeated games can sustain cooperation as an equilibrium outcome. It shows that if players interact multiple times, they can establish and maintain cooperative strategies, even in situations where a single interaction would lead to non-cooperation, highlighting the importance of future consequences in decision-making.
Grim trigger strategy: The grim trigger strategy is a specific type of punishment strategy used in repeated games where a player cooperates until the other player defects, after which they defect for all subsequent rounds. This strategy is characterized by its harshness, as it permanently retaliates against any deviation from cooperation, making it a powerful deterrent against non-cooperative behavior. By establishing a high cost for defection, it encourages players to maintain cooperation throughout the game.
Imperfect information: Imperfect information occurs when one or more participants in a game do not have complete knowledge about the other players' actions, payoffs, or types. This uncertainty can significantly impact decision-making processes, as players may strategize based on assumptions rather than definitive knowledge. In the context of games, this concept plays a critical role in influencing strategies and outcomes, especially in how players react to the behavior of others under varying conditions of information availability.
Infinite horizon: An infinite horizon refers to a time frame in economic models where the decision-making process extends indefinitely into the future. This concept is crucial for analyzing situations where future consequences significantly affect current choices, particularly in repeated interactions like games. In this context, strategies can evolve based on long-term payoffs rather than just immediate rewards, making it essential for understanding cooperation and competition over time.
Nash Equilibrium: Nash Equilibrium is a concept in game theory where players in a strategic interaction choose their optimal strategy, given the strategies of others, resulting in no player having an incentive to deviate from their chosen strategy. This concept is crucial in understanding how firms operate in competitive markets, particularly where their decisions are interdependent.
Payoff Matrix: A payoff matrix is a table that shows the potential outcomes or payoffs for each player in a game, based on the combination of strategies chosen by all players. It visually represents how players' choices affect their payoffs, helping to analyze strategic interactions. The matrix can be used in both static and dynamic settings to understand the incentives and possible equilibria in various game scenarios.
Repeated game: A repeated game is a strategic scenario in which players engage in the same game multiple times, allowing their previous actions to influence future decisions. This setup enables players to develop strategies based on past interactions, creating opportunities for cooperation or retaliation depending on the behavior of others. The essence of repeated games lies in their ability to sustain cooperative behavior through the threat of future punishment or the promise of future rewards.
Robert J. Aumann: Robert J. Aumann is a renowned mathematician and game theorist best known for his contributions to the understanding of repeated games and the Folk Theorem. His work emphasizes how players can achieve cooperative outcomes in infinitely repeated games, where the future payoffs can influence present behavior, allowing for strategies that promote long-term collaboration over short-term gains.
Strategy profile: A strategy profile is a combination of strategies chosen by all players in a game, determining how each player will behave based on their individual strategies. This term is crucial in understanding the outcome of games, especially in scenarios where players interact repeatedly, as it influences the possible equilibria and payoffs they can achieve. In the context of repeated interactions, a strategy profile helps to analyze how cooperation can emerge and persist among rational players.
Subgame perfection: Subgame perfection is a refinement of Nash equilibrium used in dynamic games, ensuring that players' strategies constitute a Nash equilibrium in every subgame of the original game. This concept emphasizes that players will not just play optimally at the start, but will also respond optimally to all possible future decisions, leading to credible threats and promises. It highlights the importance of strategies that remain optimal even when considering different points in the game.
Sustainable cooperation: Sustainable cooperation refers to a situation where individuals or entities consistently work together over the long term in a way that is mutually beneficial and stable. This concept is crucial in repeated games, as it allows players to achieve outcomes that are more favorable than those obtainable through one-time interactions, fostering a collaborative environment that can withstand challenges and encourage ongoing partnerships.
Trigger Strategy: A trigger strategy is a plan in repeated games where a player threatens to revert to a less favorable action if another player deviates from a cooperative agreement. This strategy helps maintain cooperation by imposing consequences for non-compliance, ensuring that players are incentivized to stick to mutually beneficial outcomes. Trigger strategies are essential in analyzing how players can sustain cooperation over multiple rounds of play, thus connecting deeply with concepts of reputation and long-term benefits in game theory.
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