Solution
= Solution
Continue at time zero. At time one stop on the lower branch and continue on the upper branch:
$$
\boxed{\tau_0=\begin{cases}1,&\xi_1=7,\\2,&\xi_1=11.\end{cases}}
$$
This is a <stopping time> since the branch decision is <measurable> at time one. Its expected reward is
$$
\boxed{\mathbb E\xi_{\tau_0}=\frac35\,7+\frac25\left(\frac23\,13+\frac13\,10\right)=9.}
$$
It therefore attains the upper bound in part (iii).