Repeated Game

A stage game played repeatedly, allowing histories to influence future play.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

Given stage game $G$ , the $T$ -stage repeated game has payoff $\sum_{t=1}^T \delta^{t-1}u_i(a^t)$ with discount factor $\delta\in[0,1]$ ; $T=\infty$ gives the infinitely repeated game.

Strategies are history-dependent maps $\sigma_i:\bigcup_t H_t\to\Delta(A_i)$ .

Intuition

Players can condition today's play on past behavior, supporting reputations and threats.

Repetition transforms static dilemmas into opportunities for cooperation via credible punishments.

Worked example

Tit-for-Tat sustains cooperation in the infinitely repeated Prisoner's Dilemma for $\delta$ large enough.

Grim-trigger supports any individually rational feasible payoff in repeated oligopoly.

The math

Folk Theorem: any individually rational feasible payoff can be sustained as a subgame-perfect equilibrium of the infinitely repeated game for $\delta$ close enough to $1$ .

Finitely repeated games with unique stage-game equilibria collapse by backward induction to stage-game play.

Where it is used

Models long-run business relationships, cartel stability, and reputation in platforms.

Connects to reinforcement-learning convergence in multi-agent systems.

More in Game theory

Assembled from the ReLU.chat curated knowledge base. These explanations are concise on purpose; check the sources for anything important.