 Solution

Under the uniform prior, the posterior density after $X=x$ is proportional to ${\theta }^x(1-\theta {)}^{n-x}$. For $0<x<n$, differentiating posterior risk gives the unique stationary pointStrict convexity makes it the unique minimum. For $x=0$, every $a\ne 0$ gives infinite posterior risk at zero and $a=0$ is optimal; symmetrically $a=1$ is optimal for $x=n$. Hence the unique Bayes rule is $\delta (x)=x/n$.