Bounded deficiency in the comparison of the means of two normal populations

Consider the problem of sequentially testing which of two normally distributed treatments has a larger mean response. A Bayesian formulation of this problem, where the treatment means have conjugate priors, is used. If we restrict our attention to the class of procedures using the Kiefer and Sacks shopping time then we will not exceed the minimum Bayes risk by more than the cost of a fixed number of observations, uniformly over conjugate priors. Using this we then develop procedures with a bounded deficiency. If the sampling costs were increasing than these procedures will reduce the expected number of patients assigned to the inferior (lower mean) treatment.