= Solution
The posterior expected loss at decision $\delta$ is
$$
R(\delta)
=2\int_{\gamma\leq\delta}(\delta-\gamma)\,dF(\gamma)
+\int_{\gamma>\delta}(\gamma-\delta)\,dF(\gamma).
$$
At continuity points its derivative is
$$
R'(\delta)=2F(\delta)-[1-F(\delta)]
=3F(\delta)-1.
$$
Thus the risk is minimized at a posterior one-third <quantile>:
$$
\boxed{
\widehat\gamma_B=F^{-1}\left(\frac13\right)}.
$$
More generally, any $\delta$ satisfying
$$
F(\delta-)\leq\frac13\leq F(\delta)
$$
is optimal. The quantile lies below the posterior median, reflecting the smaller penalty assigned to underestimation.
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