# Why is the path (T,C) not a subgame-perfect equilibrium for any value of $\delta$

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.

• Find the conditions on $\delta$ for subgames following every possible type of history (e.g. histories where neither player has deviated before, histories where 1 player has deviated before etc etc). Then you can comment on SPNE. This is the key difference between checking for NE and SPNE. For NE, you just need to check for deviations along the proposed equilibrium path. For SPNE you need to check at every possible history. Check our Osborne-Rubinstein for a simple and elegant exposition. – user28372 Jun 13 '20 at 19:54