Folk theorem
In a long-enough relationship, almost any outcome — including cooperation — can be sustained.
What it means
A family of results in repeated game theory showing that when players interact indefinitely and value the future sufficiently, a vast range of outcomes can be supported as equilibria, including mutually cooperative ones impossible in the one-shot game. The mechanism is the credible threat of future punishment: deviating today triggers retaliation tomorrow that outweighs the short-term gain. The theorem is double-edged — it explains how repetition rescues cooperation but also implies a multiplicity of equilibria, so theory alone cannot predict which one occurs. It matters because it formalizes 'the shadow of the future' and underpins why ongoing relationships, unlike one-off encounters, can police good behavior.
How it works
Repeated (or "super-") games embed a one-shot stage game that the same players face again and again. Each player discounts future payoffs by a factor between zero and one that captures both impatience and the chance the relationship ends; a factor close to one means the future looms large. The engine of the result is that a stream of future losses can be made to dwarf any single-period gain. A common construction is the "grim trigger": everyone cooperates until someone deviates, after which all revert forever to the uncooperative stage-game equilibrium. If players are patient enough, the once-and-for-all gain from cheating is smaller than the discounted value of the cooperation forgone, so no one cheats. The reachable payoffs are bounded by each player's minmax value — the worst payoff the others can jointly force on them — and by feasibility. Any profile that gives every player strictly more than their minmax, and that lies inside the achievable payoff set, can be sustained as an equilibrium once the discount factor is near one. The Nash-reversion version only requires the threatened punishment to be a Nash equilibrium of the stage game; the sharper "perfect" version insists the punishment itself be something players are willing to carry out, which calls for more elaborate strategies that also punish those who fail to punish.
The original demonstration
The result carries the name "folk theorem" because its simplest form — that any individually rational payoff can be sustained by the threat of reverting to the stage-game equilibrium — circulated informally among game theorists through the late 1950s and 1960s before anyone set it down in print; it belonged to the profession's oral tradition, its authorship untraceable. James Friedman published the first widely cited formal version in 1971, showing that in a discounted supergame, cooperation above the Nash payoff could be supported by permanent Nash reversion. For the case of no discounting, where players weigh average long-run payoffs, versions were established by Robert Aumann and Lloyd Shapley and, independently, by Ariel Rubinstein. The decisive modern statement came from Drew Fudenberg and Eric Maskin in 1986, who proved a folk theorem for subgame-perfect equilibrium under discounting: with a mild "full-dimensionality" condition on the payoff set, essentially any individually rational, feasible payoff is sustainable as patience rises, using punishments that are themselves credible. Dilip Abreu's work on optimal penal codes showed that punishments could be organized economically, and Fudenberg, Levine and Maskin later extended the theorem to settings where players observe only a noisy public signal of each other's actions rather than the actions themselves, a step toward realism.
What the evidence shows
Because the folk theorem is a mathematical result, it cannot be "confirmed" empirically; what experiments test is whether real people behave as the theory says they can. The clearest evidence comes from laboratory studies of the indefinitely repeated prisoner's dilemma, where a random stopping rule lets the continuation probability play the role of the discount factor. Pedro Dal Bó found that cooperation was substantially higher when the future mattered more — under a higher continuation probability, and in indefinitely repeated games compared with finitely repeated ones of the same expected length — direct evidence that the "shadow of the future" changes behavior. But later work sharpened the picture in a way that matters. Dal Bó and Fréchette showed that a discount factor high enough to make cooperation an equilibrium is necessary but not sufficient for people to actually cooperate. What predicts cooperation is whether the cooperative strategy is also "risk dominant" — roughly, whether it pays off well enough given uncertainty about what the other side will do. Their 2018 survey, pooling many experiments, reached the same conclusion: cooperation rises with patience and with the size of cooperation's "basin of attraction," and subjects converge toward cooperation with experience only when conditions favor it, and away from it when they do not. The lesson is that the folk theorem describes what cooperation is possible, not how much occurs; a second condition, absent from the theorem itself, does much of the predictive work.
Where it shows up
Its most studied application is tacit collusion. Firms in a market they expect to share for years may keep prices high without any explicit pact, because the extra profit from undercutting is wiped out by the price war that follows. The same logic underpins "relational contracts" inside and between firms — informal understandings about effort, quality or payment that no court could enforce but that survive because the relationship is worth more continued than exploited once. It explains how reputation can substitute for legal enforcement in trade among parties who lack recourse to courts, how informal lending and rotating savings groups hold together, and how self-enforcing international agreements on trade or the environment can persist when no authority stands above the signatories. In each case the shared ingredient is the folk theorem's: a valued, open-ended future whose withdrawal is a credible penalty. The theorem does not claim any of these arrangements are inevitable — only that they sit within the range of outcomes that patient, repeated interaction can support, which is why it is treated as a possibility result rather than a prediction.
Limits and caveats
The theorem's power is also its embarrassment. By showing that almost any reasonable outcome can be an equilibrium, it strips the equilibrium concept of predictive bite: told only that players are patient, the theory cannot say whether they will cooperate fully, collude partially, or grind through cycles of punishment. This indeterminacy is why the empirical finding above — that a second condition governs actual behavior — matters so much. Several further conditions constrain the result. Players must be patient enough; when the future is discounted heavily, or the relationship is expected to end soon, the cooperative equilibria vanish. When the horizon is finite and its end is common knowledge, backward induction can unravel cooperation entirely, because the last period has no future to protect and the logic rolls back to the first. The perfect folk theorem needs its full-dimensionality condition, which can fail. And the tidy results assume players can at least observe a common signal of one another's conduct; under purely private, noisy monitoring, "anti-folk-theorems" show that cooperation can be far harder or even impossible to sustain, because players cannot coordinate on when to punish. Finally, the theorem assumes payoff-maximizing agents; real behavior is also shaped by fairness, limited foresight and habit, which is why observed cooperation departs, in both directions, from what pure patience would predict.
Examples
Two firms can sustain high prices indefinitely if each will punish a price cut with a lasting price war that erases the gain from cheating.
Two firms selling near-identical products in a market they both expect to serve for years may hold prices above the competitive level with no explicit agreement, because each knows that undercutting to grab share this quarter would touch off a price war that erases the gains for many quarters afterward.
A supplier and a manufacturer with a long-running relationship often honor informal quality and payment commitments that no court could enforce, because either side's cheating would end a stream of future orders worth far more than the one-time gain from cutting corners.
A rotating savings group, where members pay into a common pot each cycle and take turns receiving it, holds together without legal contracts because a member who collected a payout and then stopped contributing would be shut out of every future cycle.
In a finitely repeated version of the same situation, with a known and publicly announced last round, cooperation can unravel: with no future to protect in the final period, defection there is anticipated, and the reasoning rolls back to the first round — showing how the theorem's leverage depends on the horizon never visibly closing.
First described in Repeated-games result; Friedman (1971); Fudenberg & Maskin (1986).
Key references
- Friedman, J. W. (1971). A non-cooperative equilibrium for supergames. The Review of Economic Studies, 38(1), 1-12. doi.org/10.2307/2296617
- Rubinstein, A. (1979). Equilibrium in supergames with the overtaking criterion. Journal of Economic Theory, 21(1), 1-9. doi.org/10.1016/0022-0531(79)90002-4
- Fudenberg, D., & Maskin, E. (1986). The folk theorem in repeated games with discounting or with incomplete information. Econometrica, 54(3), 533-554. doi.org/10.2307/1911307
- Abreu, D. (1988). On the theory of infinitely repeated games with discounting. Econometrica, 56(2), 383-396. doi.org/10.2307/1911077
- Fudenberg, D., Levine, D., & Maskin, E. (1994). The folk theorem with imperfect public information. Econometrica, 62(5), 997-1039. doi.org/10.2307/2951505
- Dal Bó, P. (2005). Cooperation under the shadow of the future: Experimental evidence from infinitely repeated games. American Economic Review, 95(5), 1591-1604. doi.org/10.1257/000282805775014434
- Dal Bó, P., & Fréchette, G. R. (2011). The evolution of cooperation in infinitely repeated games: Experimental evidence. American Economic Review, 101(1), 411-429. doi.org/10.1257/aer.101.1.411
- Dal Bó, P., & Fréchette, G. R. (2018). On the determinants of cooperation in infinitely repeated games: A survey. Journal of Economic Literature, 56(1), 60-114. doi.org/10.1257/jel.20160980