Repeated Games

Repeated Games are strategic scenarios in which the same game (known as the stage game) is played multiple times by the same players. Unlike one-shot games, where players interact only once, repeated games allow players to condition their actions on the history of play, enabling strategies that reward cooperation or punish defection over time. This temporal structure introduces the possibility of sustaining cooperation and more complex strategic behavior that would not be feasible in a single interaction.

Definition and Basic Structure

Repeated games consist of:

• A stage game, which is a finite strategic form game played at each period.
• A sequence of time periods indexed by t = 1, 2, 3, ..., where the same stage game is played.
• Players who observe the history of past actions (perfect or imperfect monitoring) before choosing their current actions.
• A method to aggregate payoffs over time, often through discounted or average payoffs.

Formally, the repeated game can be finite or infinite in horizon. Infinite repetition with discounting or indefinitely uncertain ending is common in theoretical models to analyze sustainability of cooperation.

Payoff Aggregation and Discounting

Players evaluate their overall payoff as an aggregate of the stage-game payoffs across all periods. Two common methods are:

• Discounted sum of payoffs: Each period’s payoff is weighted by a discount factor δ, where 0 < δ < 1, representing the player’s patience or valuation of future rewards.

Here, $U\_i$ is the total payoff of player i, and $u\_i^t$ is the payoff in period t.

• Average payoff: The limit of the average stage payoffs as the number of periods grows large.

Discounting is more common in economic settings reflecting time preferences.

Strategies in Repeated Games

A strategy in a repeated game is a complete contingent plan that specifies a player's action at each stage as a function of the entire history of play up to that point. This allows players to:

• Condition current actions on past behavior of other players.
• Implement punishment and reward mechanisms, such as cooperation if others cooperated previously, or punishment if others defected.
• Use strategies like trigger strategies, tit-for-tat, or grim trigger that enforce cooperation by threatening future punishment.

Equilibrium Concepts

The main equilibrium concept in repeated games is the Subgame Perfect Equilibrium (SPE), which refines the Nash equilibrium by requiring strategies to form a Nash equilibrium in every subgame, including after any history of play.

• In repeated games, SPE often support cooperative outcomes that are not equilibria in the one-shot stage game.
• The Folk Theorem describes how a wide range of payoff profiles can be sustained as SPE outcomes when players are sufficiently patient (high δ).

Role of Cooperation and Punishment

Repeated interactions enable the possibility of cooperation even in games where the one-shot equilibrium is non-cooperative, such as the Prisoner’s Dilemma. This is because:

• Players can reward mutual cooperation by continuing cooperation.
• Players can punish deviations by reverting to non-cooperative strategies indefinitely or for a specified duration.

The threat of future punishment deters deviation, making cooperation an equilibrium outcome.

Types of Repeated Games

• Finitely repeated games: The stage game is repeated a known finite number of times. Backward induction often leads to unraveling of cooperation in the last period and consequently in all previous periods.
• Infinitely repeated games: The horizon is infinite or uncertain, allowing sustained cooperation under appropriate conditions.
• Discounted repeated games: Future payoffs are discounted, balancing present and future rewards.
• Imperfect monitoring repeated games: Players do not observe actions perfectly but receive noisy signals, complicating enforcement of cooperation.

Applications and Importance

Repeated games model strategic interactions in economics, politics, and business where agents interact repeatedly over time. Examples include:

• Pricing strategies among competing firms.
• Collusion and cartel enforcement.
• Bargaining and contract enforcement.
• Regulation and compliance monitoring.
• Social norms and reputation building.

The framework explains how long-term incentives can promote cooperation and deter opportunistic behavior.

Mathematical Example: Grim Trigger Strategy in Prisoner’s Dilemma

Consider a Prisoner’s Dilemma repeated infinitely with discount factor δ.

• The Grim Trigger strategy cooperates initially and continues cooperating as long as the other player cooperates.
• If the opponent defects once, the player defects forever as punishment.

The condition for cooperation to be sustained is:

Where:

• R = reward for mutual cooperation
• T = temptation payoff for defecting while the other cooperates
• P = punishment payoff for mutual defection

If the discount factor δ is high enough, future losses from punishment outweigh the short-term gain from defection, making cooperation stable.

Summary

Repeated games enrich the analysis of strategic interaction by incorporating time and history, enabling strategies that reward or punish past actions. This temporal dimension allows cooperation to emerge in environments where one-shot incentives favor defection, highlighting the importance of future consequences in strategic decision-making.