# Are the average payoffs in the convex hull?

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.

• Your "vector of possible payoffs" in (a) is actually a set and needs only one $(0,0)$ entry. The first part of (b) shouldn't be a problem. Just draw the given point into your diagram. Is it contained in the red triangle? For the second part, consider the following question: What minimum average payoff can player 1 guarantee for himself? – VARulle Jun 14 '20 at 0:28