Repeated Games & the Folk Theorem

How patience and the threat of punishment sustain cooperation.

The idea

Theorem (Folk theorem).

Consider a stage game played infinitely often, with payoffs discounted by a factor $\delta \in (0,1)$, and for each player $i$ let $m_{i}$ be that player's minmax value, the lowest payoff the other players can force on them. Then every feasible payoff profile in which each player $i$ receives strictly more than $m_{i}$ is an equilibrium payoff of the repeated game for all $\delta$ close enough to $1$.

A repeated game plays the same stage game period after period with the same players. A strategy is then no longer a single action but a plan naming an action after every possible history of play, so what a player does today, the others can answer tomorrow.

Payoffs are discounted: fix a discount factor $\delta \in (0,1)$; a stream $u_{0}, u_{1}, u_{2}, \ldots$ is then worth

$U = (1-\delta)\sum_{t=0}^{\infty} \delta^{t} u_{t},$

the average discounted payoff. The leading $(1-\delta)$ is a normalization: it makes a constant stream of $c$ per period worth exactly $c$, so repeated-game values sit on the same scale as stage payoffs. A $\delta$ near $1$ describes patient players; a $\delta$ near $0$, players who barely see past today.

A plan is an equilibrium when no player, after any history, can raise their average discounted payoff by deviating from it, given that the others follow the plan — including its punishments — from then on. Cooperation is therefore sustainable when the one-period gain from deviating is outweighed by the discounted loss the punishment inflicts afterwards, which holds once $\delta$ is large enough. That is the mechanism behind the theorem. In a two-player game, writing $m_{1}$ and $m_{2}$ for the players' minmax values, the sustainable region is the part of the feasible set lying strictly above both, as the figure shows.

Ways to work on it

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