deviated, play the strategy profile m which minmaxes i forever after. 3. Ignore multilateral deviations. If player i gets ε more than his minmax payoff each stage by following 1, then the potential loss from punishment is If δ is close to 1, this outweighs any finite one-stage gain, making the strategy a Nash equilibrium. An alternative statement of this folk theorem allows the equilibrium payoff profile u to be any individually rational feasible payoff profile; it only requires there exist an individually rational feasible payoff profile that strictly dominates the minmax payoff profile. Then, the folk theorem guarantees that it is possible to approach u in equilibrium to any desired precision (for every ε there exists a Nash equilibrium where the payoff profile is a distance ε away from u). Subgame perfection Attaining a subgame perfect equilibrium in discounted games is more difficult than in undiscounted games. The cost of punishment does not vanish (as with the limit-of-means criterion). It is not always possible to punish the non-punishers endlessly (as with the overtaking criterion) since the discount factor makes punishments far away in the future irrelevant for the present. Hence, a different approach is needed: the punishers should be rewarded. This requires an additional assumption, that the set of feasible payoff profiles is full dimensional and the min-max profile lies in its interior. The strategy is as follows. 1. All players start by playing a and continue to play a if no deviation occurs. 2. If any one player, say player i, deviated, play the strategy profile m which minmaxes i for N periods. (Choose N and δ large enough so that no player has incentive to deviate from phase 1.) 3. If no players deviated from phase 2, all player j ≠ i gets rewarded ε above j's min-max forever after, while player i continues receiving his min-max. (Full-dimensionality and the interior assumption is needed here.) 4. If player j deviated from phase 2, all players restart phase 2 with j as target. 5. Ignore multilateral deviations. Player j ≠ i now has no incentive to deviate from the punishment phase 2. This proves the subgame perfect folk theorem. Finitely-repeated games without discount Assume that the payoff of player i in a game that is repeated T times is given by a simple arithmetic mean: A folk theorem for this case has the following additional requirement: In the