Estimating a binomial parameter: Is robust bayes real bayes?

In robust Bayesian analysis, a prior is assumed to belong to a family instead of being specified exactly. The multiplicity of priors naturally leads to a collection of Bayes actions (estimates), and these often form a convex set (an interval in the case of a real parameter). It is clearly essential to be able to recommend one action from this set to the user. We address the following problem: if we systematically choose one action for each X thereby constructing a decision rule, is it going to be Bayes? Is it Bayes with respect to a prior in the original prior family? Even if it is not genuine Bayes, is it admissible? This problem is addressed in the context of estimating an unknown Binomial parameter. Several prior families are considered. We look at the midpoint of the interval of Bayes estimates; this has a minimax interpretation, apart from its obvious simplistic appeal. We establish that unless the prior family includes unreasonable priors, use of this estimate guarantees good behavior and indeed it is usually admissible or even genuine Bayes.