Sub-game Perfect Nash Equilibrium of an Infinitely Repeated Game (Selton 1965):
A Nash equilibrium is sub-game perfect if the players' strategies constitute a Nash equilibrium in every sub-game of the infinitely repeated game. Sub-game perfect Nash equilibrium is refinement of the concept of Nash equilibrium, which means for a strategy profile to be sub- game perfect, it must be a Nash equilibrium first and then it must satisfy an additional test, that is, it must be a Nash equilibrium in every sub-game of the game.
Our objective of describing all the definition above was to show that the trigger strategy that we have already defined in the game of infinitely repeated prisoners' dilemma is sub-game perfect. Therefore, we need to show that the trigger strategy constitute a Nash equilibrium on every sub-game of that infinitely repeated game. Recall that every sub-game of an infinitely repeated game is identical to the game as a whole. In the trigger strategy, Nash equilibrium of the infinitely repeated prisoners' dilemma sub-games could be grouped into two classes:
i) sub-games in which all the outcomes of the earlier stages have been (RI, R2)
ii) sub-games in which the outcome of at least one stage differs from (Rl, R2).
If the players adopt the trigger strategy for the game as a whole, then (i) the players' strategies in a sub-game in the first case are again the trigger strategy, which we have already shown to be the Nash equilibrium of the game as a whole.
The players' strategies in a sub-game in the second case are simply to repeat the stage game equilibrium (L1, L2) forever, which is also Nash equilibrium of the game as a whole. Therefore, the trigger strategy Nash equilibrium of the infinitely repeated prisoners' dilemma is sub-game-perfect.