As electronic medical records enable increasingly ambitious studies of treatment outcomes, ethical issues previously important only to limited clinical trials become relevant to unlimited whole populations. For randomized clinical trials, adaptive assignment strategies are known to expose substantially fewer patients to avoidable treatment failures than strategies with fixed assignments (e.g., equal sample sizes). An idealized adaptive case--the two-armed Bernoulli bandit problem--can be exactly optimized for a variety of ethically motivated cost functions that embody principles of duty-to-patient, but the solutions have been thought computationally infeasible when the numbers of patients in the study (the "horizon") is large. We report numerical experiments that yield a heuristic approximation that applies even to very large horizons, and we propose a near-optimal strategy that remains valid even when the horizon is unknown or unbounded, thus applicable to comparative effectiveness studies on large populations or to standard-of-care recommendations. For the case in which the economic cost of treatment is a parameter, we give a heuristic, near-optimal strategy for determining the superior treatment (whether more or less costly) while minimizing resources wasted on any inferior, more expensive, treatment. Key features of our heuristics can be generalized to more complicated protocols.