The framework for multistage comparison procedures in the present paper is roughly that introduced by Duncan and treated more fully by Tukey. In the present paper we consider the problem of finding the optimum allocation of nominal significance levels for successive stages. The optimum procedure we obtain when the number $s$ of treatments is odd, and the compromise procedure we propose for even $s$, essentially agree with a procedure suggested by Tukey. The agreement is exact when $s$ is even and close when $s$ is odd. The results of the present paper apply among others to the problem of distinguishing normal distributions with known variances, multinomial distributions, Poisson distributions, and distributions in certain nonparametric settings. However, they do not apply exactly to the comparison of normal distributions with a common unknown variance. When the variances are completely unknown, the method applies in principle but faces the difficulty that no exact test is then available for testing the homogeneity of a set of means.