Folk Theorem

Any individually rational feasible payoff can be sustained in an infinitely repeated game with patient enough players.

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

Definition

Let $V$ be the convex hull of stage-game payoffs and $\underline v_i=\min_{\sigma_{-i}}\max_{\sigma_i}u_i(\sigma_i,\sigma_{-i})$ (min-max). Any $v\in V$ with $v_i>\underline v_i$ is a subgame-perfect equilibrium payoff for $\delta$ close enough to $1$ .

Perfect Folk Theorem (Fudenberg-Maskin) handles the full-dimension and imperfect-monitoring cases.

Intuition

Patience lets players threaten long-run punishments that sustain cooperation today.

Almost anything is an equilibrium — this is both a strength (explains cooperation) and weakness (weak predictions) of repeated games.

Worked example

In repeated Prisoner's Dilemma, grim-trigger sustains $(R,R)$ for $\delta\ge(T-R)/(T-P)$ .

Aumann-Shapley (1994) provided the first characterization with observable mixed actions.

The math

Classical proof constructs strategies with reward/punishment phases; requires individually rational payoffs and sufficient discount factors.

Versions exist for imperfect public monitoring (Abreu-Pearce-Stacchetti) and private monitoring (Kandori).

Where it is used

Rationalizes cartel stability, relational contracts, and community enforcement.

Explains why reputation and long relationships matter.

More in Game theory

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