Suppose there are two possible states $H$ and $L$, with prior probability $p$ and $1-p$ respectively. There are infinite rounds with a discount factor $ d$. In round 1, you could choose a value $t_1$ between 0 and constant $a$ ($a<1$), If the true state is $H$, you get $H(t_1)$, otherwise you get $L(t_1)$. $H$ is increasing in t, while $L$ is decreasing in t. However, if you choose $t_1$, there is probability of $t_1$ that you are informed the true state is $L$, if the true state is L. Next in round 2, if you are informed, you know the state is L and choose $t_i=0$ forever. If you are not informed, you bayesian update the prior prob and choose $t_2$ again, and so on.
How to maximize your expectation?
The original question is a distance problem ($H$ and $L$ make an interval, and you choose a point in between). I just generalize it a bit. Any kind of input (like the tool needed) is appreciated!