This is an exercise in the Steve Tadelis An Introduction to Game Theory book:
(10.12) Folk Theorem Revisited: Consider the infinitely repeated trust game described in Figure 10.1
(a) Draw the convex hull of average payoffs.
So, this is pretty easy:
The vector of payoffs is $V=\{(0,0),(0,0),(-1,2),(1,1)\}$
So, here is my sketch in paint:
(b) Are the average payoffs $(\overline{v_1}, \overline{v_2}) = (−0.4, 1.1)$ in the convex hull of average payoffs? Can they be supported by a pair of strategies that form a subgame-perfect equilibrium for a large enough discount factor $δ$?
I don't have an idea how to do (b). If someone could explain that would be great.
Yes, to get $(-.4,1.1)$, play $(N,D)$ 1/2 of the time, $(T,C)$ 1/10 of the time, and $(T,D)$ 4/10 of the time. That averages the payoffs to $(-.4,1.1)$.
No, $(-.4,1.1)$ cannot be supported. Player 1 can always deviate by playing $N$ in every period, thereby guaranteeing an average payoff of 0. The proposed payoffs give him a strictly negative payoff.