Simple hawk-dove type super-game
Finally, I believe it is easy to answer it. Well, in case where some player will not cooperate and the triger strategy is enabled, we have the following soloutions:
Soppuse that p2 does not cooperate in the first round so, he will gain a payoof of 1, but in the second round p1 plays $a_2$ and he won't cooperate with p2 in perpetual. Thus, we have that p2 will play $a_1$, because by playing $a_2$ he increases his looses and the value function is:
$$V_2^{NC}=1-1\cdot\delta-1\cdot\delta^2-1\cdot\delta^3+...=1-\delta\sum_{j=1}^{+\infty}\delta^{j}=1-\dfrac{\delta}{1-\delta}$$
Hence, they will cooperate if:
$$0&gt;1-\dfrac{\delta}{1-\delta}\Rightarrow \delta&gt;\dfrac{1}{2}$$
Differently, if both players choose trigger strategies, so p2 also chooses $a_2$, then both will make looses of -2 and the value function is:
$$V_2^{NC}=1-2\cdot\delta-2\cdot\delta^2-2\cdot\delta^3+...=1-2\delta\sum_{j=1}^{+\infty}\delta^{j}=1-\dfrac{2\delta}{1-\delta}$$
and hence they will cooperate if:
$$0&gt;1-\dfrac{2\delta}{1-\delta}\Rightarrow \delta&gt;\dfrac{1}{3}$$