Consider a decision problem in which the model parameter, $\theta$, is any integer, the distribution for the integer observation, y, given $\theta$ is $P(y|\theta) = 1/3$ if $y \in [\theta - 1, \theta + 1]$ and 0 otherwise. The action space is the integers, and the loss function is 0-1 loss function. The decision rule is $\delta(y) = y$. Find decision rules $\delta'$ and $\delta''$ such that $\delta'$ dominates $\delta$ and $\delta''$ dominates $\delta'$ and is admissible.
I think that here we use the improper prior $f(\theta) = 1$ and so the posterior is the same as the prior (i.e. discrete uniform on $[\theta - 1, \theta + 1]$). Then with the 0-1 loss the associated Bayes estimator is the posterior mode, which in this case could be one of three values. Neither, however, seem to dominate $\delta(y) = y$. Then I tried randomized rules (for example with 1/2 prob. take $y+1$ and with 1/2 prob take $y-1$), but that also doesn't seem to work....