 Constant-risk binomial proportion estimator

For $X\sim \mathrm{Bin}(n,\theta )$, the posterior mean under $\mathrm{Beta}(\sqrt{n}/2,\sqrt{n}/2)$ has constant squared-error risk $1/[4(\sqrt{n}+1{)}^2]$. Its unique Bayes property and constant risk make it the unique minimax estimator.