Multistage decission problems

K treatments that yield dichotomous responses are considered in a multistage clinical trial. The total number of patients to be involved in the trial is N, which is random. The optimal length for each treatment at each stage is to be decided. The objective is to maximize the expected total number of successes of the trial. Two cases are considered: (1) Two stages. It is shown that the rate of the optimal length of the first stage is no greater than √E(N) as E(N) goes to infinity, when the distribution of N, Q, yields a regular discount sequence. The rate may be smaller than √E(N) which depends on the distribution of N and the prior distributions on the probabilities of success of the treatments. Under certain conditions, the rate is exactly √E(N). Nevertheless, the rate may be greater than E(N) without the regularity of Q. (2) r stages. It is shown that M∗/E(N) converges to zero in probability as E(N) goes to infinity when Q has the regularity, where M∗ is the sum of the optimal lengths for the first r - 1 stages.