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 E-optimal rules exist when trials have large enough sample size. An E-optimal rule has expected welfare within E of the welfare of the best treatment in every state of nature. Equivalently, it has maximum regret no larger than E. 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 E-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.