Group-separations based on the repeated prisoners' dilemma games

We model group-separations in an n-player set. In the n-player set, every two players play an infinitely repeated two-player prisoners' dilemma game. Each player takes a mixed strategy to play the game and trigger strategy is used to punish the deviator. Let all players share a common discount factor δ. We find that with the variation of δ, the n-player set is separated into several subsets such that (1) for any two players in any two different subsets, their strategy profile is not a subgame perfect equilibrium and (2) each subset cannot be separated into several subsets that satisfy (1). Such subsets are called groups and the separation is called group-separation. We aim to specify the intervals (of δ) such that group-separations emerge. Particularly, we focus on the relationship between the interval and the form of each group-separation.