Neyman allocation is minimax optimal for best arm identification with two arms

This note describes the optimal policy rule, according to the local asymptotic minimax regret criterion, for best arm identification when there are only two treatments. It is shown that the optimal sampling rule is the Neyman allocation, which allocates a constant fraction of units to each treatment in a manner that is proportional to the standard deviation of the treatment outcomes. When the variances are equal, the optimal ratio is one-half. This policy is independent of the data, so there is no adaptation to previous outcomes. At the end of the experiment, the policy maker adopts the treatment with higher average outcomes.