grim trigger strategy. All players start by playing the prescribed action and continue to do so until someone deviates. If player i deviates, all other players switch to picking the action which minmaxes player i forever after. The one-stage gain from deviation contributes 0 to the total utility of player i. The utility of a deviating player cannot be higher than his minmax payoff. Hence all players stay on the intended path and this is indeed a Nash equilibrium. Subgame perfection The above Nash equilibrium is not always subgame perfect. If punishment is costly for the punishers, the threat of punishment is not credible. A subgame perfect equilibrium requires a slightly more complicated strategy. The punishment should not last forever; it should last only a finite time which is sufficient to wipe out the gains from deviation. After that, the other players should return to the equilibrium path. The limit-of-means criterion ensures that any finite-time punishment has no effect on the final outcome. Hence, limited-time punishment is a subgame-perfect equilibrium. Coalition subgame-perfect equilibria: An equilibrium is called a coalition Nash equilibrium if no coalition can gain from deviating. It is called a coalition subgame-perfect equilibrium if no coalition can gain from deviating after any history. With the limit-of-means criterion, a payoff profile is attainable in coalition-Nash-equilibrium or in coalition-subgame-perfect-equilibrium, if-and-only-if it is Pareto efficient and weakly-coalition-individually-rational. Overtaking Some authors claim that the limit-of-means criterion is unrealistic, because it implies that utilities in any finite time-span contribute 0 to the total utility. However, if the utilities in any finite time-span contribute a positive value, and the value is undiscounted, then it is impossible to attribute a finite numeric utility to an infinite outcome sequence. A possible solution to this problem is that, instead of defining a numeric utility for each infinite outcome sequence, we just define the preference relation between two infinite sequences. We say that agent (strictly) prefers the sequence of outcomes over the sequence , if: For example, consider the sequences and . According to the limit-of-means criterion, they provide the same utility to player i, but according to the overtaking criterion, is better than for player i. See overtaking criterion for more information. The folk theorems with the overtaking criterion are slightly weaker than with the limit-of-means criterion. Only outcomes that are strictly individually rational, can be attained in Nash equilibrium. This is because, if an agent deviates,