  Medical research has evolved conventions for choosing sample size in
randomized clinical trials that rest on the theory of hypothesis testing.
Bayesians have argued that trials should be designed to maximize subjective
expected utility in settings of clinical interest. This perspective is
compelling given a credible prior distribution on treatment response, but
Bayesians have struggled to provide guidance on specification of priors. We use
the frequentist statistical decision theory of Wald (1950) to study design of
trials under ambiguity. We show that {\epsilon}-optimal rules exist when trials
have large enough sample size. An {\epsilon}-optimal rule has expected welfare
within {\epsilon} of the welfare of the best treatment in every state of
nature. Equivalently, it has maximum regret no larger than {\epsilon}. We
consider trials that draw predetermined numbers of subjects at random within
groups stratified by covariates and treatments. The principal analytical
findings are simple sufficient conditions on sample sizes that ensure existence
of {\epsilon}-optimal treatment rules when outcomes are bounded. These
conditions are obtained by application of Hoeffding (1963) large deviations
inequalities to evaluate the performance of empirical success rules.
