  In evolutionary game theory, evolutionarily stable states are characterised
by the folk theorem because exact solutions to the replicator equation are
difficult to obtain. It is generally assumed that the folk theorem, which is
the fundamental theory for non-cooperative games, defines all Nash equilibria
in infinitely repeated games. Here, we prove that Nash equilibria that are not
characterised by the folk theorem do exist. By adopting specific reactive
strategies, a group of players can be better off by coordinating their actions
in repeated games. We call it a type-k equilibrium when a group of k players
coordinate their actions and they have no incentive to deviate from their
strategies simultaneously. The existence and stability of the type-k
equilibrium in general games is discussed. This study shows that the sets of
Nash equilibria and evolutionarily stable states have greater cardinality than
classic game theory has predicted in many repeated games.
