Multistage bandit problems

Consider two or more treatments with dichotomous responses. The total number N of experimental units are to be allocated in a fixed number r of stages. The problem is to decide how many units to assign to each treatment in each stage. Responses from selections in previous stages are available and can be considered but responses in the current stage are not available until the next group of selections is made. Information is updated via the Bayes theorem after each stage. The goal is to maximize the overall expected number of successes in the N units.Two forms of prior information are considered: (i) All arms have beta priors, and (ii) prior distributions have continuous densities. Various characteristics of optimal decisions are presented. For example, in most cases of (i) and (ii), the rate of the optimal size of the first stage cannot be greater than √N when r = 2.