Folk Theorem
What is Folk Theorem?
The folk theorem states that in an infinitely repeated game with patient players, almost any reasonable (individually rational) outcome can be sustained as an equilibrium.
When a stage game is repeated indefinitely and players discount the future little, credible threats of future punishment let them support cooperative outcomes that are impossible in a one-shot game. So the set of equilibrium payoffs balloons to include nearly all feasible, individually rational payoffs. This formalizes why repeated interaction enables collusion, relational contracts, and reputation effects.
Folk Theorem: a worked example
Take a stage game where mutual restraint pays each firm 8, a mutual price war pays each 3, and undercutting alone pays 10 against 1. Played once, undercutting is dominant and the only equilibrium is the price war at 3 apiece. Repeated forever, each player can still guarantee itself 3 whatever the rival does, so 3 is the floor, and the folk theorem says any feasible average pair above (3, 3) can be an equilibrium. Alternate the outcomes: firm A undercuts in odd rounds and both restrain in even rounds. A averages (10 + 8) / 2 = 9 and B averages (1 + 8) / 2 = 4.5, both above the floor, with a permanent price war standing by to police the deal.
The mistake students make with folk theorem
The folk theorem gets read as proof that repetition produces cooperation. It proves something less comforting: repetition makes almost everything possible, including lopsided splits and the bad one-shot outcome itself, so the theory stops picking a single prediction. Patience and credible punishment are conditions, not decoration, and players who discount the future steeply lose the extra equilibria entirely. The singular name misleads as well, since this is a family of theorems with different assumptions about monitoring and punishment.
Folk Theorem questions
Why is it called the folk theorem?
The folk theorem earned its name because game theorists knew and used the result long before anyone wrote a formal proof, so it circulated without an author, like folklore. The label stuck even after rigorous versions appeared. The name is also misleadingly singular, since a family of folk theorems exists with different assumptions about punishment, how well players observe each other, and how heavily they discount.
What does individually rational mean in the folk theorem?
Individually rational means a payoff at least as large as the minmax value, which is the most a player can guarantee even when everyone else is trying to hold that player down. No equilibrium can pay less than this floor, because the player could just switch to the strategy that guarantees it. The folk theorem covers payoffs that clear the floor and are feasible in the stage game.
Does the folk theorem apply to finitely repeated games?
The folk theorem is stated for infinitely repeated games, and its logic usually breaks once a final round is known, because backward induction strips away the future punishment that supported cooperation. Finite versions do exist when the stage game has more than one equilibrium, since players can then threaten to switch to the worse equilibrium for the remaining rounds instead of needing an endless horizon.
Related terms
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