On the natural selection rule in general linear models

We consider a problem of selecting the best treatment in a general linear model. We look at the properties of the natural selection rule. It is shown that the natural selection rule is minimax under to ''0-1'' loss function and it is a Bayes rule under a monotone permutation invariant loss function with respect to a permutation invariant prior for every variance balanced design. Some other condition on the design matrix is given so that a Bayes rule with respect to a normal prior will be of simple structure.