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In evolutionary game theory, evolutionarily stable states are characterisedby the folk theorem because exact solutions to the replicator equation aredifficult to obtain. It is generally assumed that the folk theorem, which isthe fundamental theory for non-cooperative games, defines all Nash equilibriain infinitely repeated games. Here, we prove that Nash equilibria that are notcharacterised by the folk theorem do exist. By adopting specific reactivestrategies, a group of players can be better off by coordinating their actionsin repeated games. We call it a type-k equilibrium when a group of k playerscoordinate their actions and they have no incentive to deviate from theirstrategies simultaneously. The existence and stability of the type-kequilibrium in general games is discussed. This study shows that the sets ofNash equilibria and evolutionarily stable states have greater cardinality thanclassic game theory has predicted in many repeated games.

On Nash Equilibrium and Evolutionarily Stable States that Are Not Characterised by the Folk Theorem