Two treatments that yield Bernoulli outcomes are available in a clinical trial. One success probability is known. A probability distribution reflects opinion about the other success rate. $N$ patients are to be treated, with $N$ possibly unknown, in a multistage trial. The goal is to maximize the total number of successes on the $N$ patients. Optimal lengths for each stage and optimal treatment allocations are found for two-stage trials with $N$ known. When $N$ is unknown the problem is shown to be equivalent to that of discounting future observations. Optimal stage lengths and treatment allocations are characterized for distributions on $N$ that yield regular discount sequences. This class of distributions includes the geometric family, which is given special consideration. It is shown that if the number of stages in the trial is fixed and if the distribution on $N$ yields a regular discount sequence, then it is optimal to use the known treatment in the last stage only. This extends the work of Berry and Fristedt (1979).
"Bayesian Multistage Decision Problems." Ann. Statist. 14 (1) 283 - 297, March, 1986. https://doi.org/10.1214/aos/1176349856