I am doing an exercise in Steve Tadelis An Introduction to Game Theory:
(10.4)Trust Off the Equilibrium Path: Recall the trust game depicted in Figure 10.1. We argued that for $δ ≥ 1/2$ the following pair of strategies is a subgameperfect equilibrium. For player 1: “In period 1 I will trust player 2, and as long as there were no deviations from the pair (T , C) in any period, then I will continue to trust him. Once such a deviation occurs then I will not trust him forever after.” For player 2: “In period 1 I will cooperate, and as long as there were no deviations from the pair (T , C) in any period, then I will continue to do so. Once such a deviation occurs then I will deviate forever after.” Show that if instead player 2 uses the strategy “as long as player 1 trusts me I will cooperate” then the path (T , C) played forever is a Nash equilibrium for $δ ≥ 1/2$ but is not a subgame-perfect equilibrium for any value of $δ$.
My attempt: this is a Nash equilibrium: the equilibrium path is followed because neither player benefits from deviating as they both believe that a deviation will call for the continuation of grim trigger.
But i'm not exactly sure why it is not a subgame perfect equilibrium. Could anyone please explain.