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.