  This study investigates the experimental design problem for identifying the
arm with the highest expected outcome, referred to as best arm identification
(BAI). In our experiments, the number of treatment-allocation rounds is fixed.
During each round, a decision-maker allocates an arm and observes a
corresponding outcome, which follows a Gaussian distribution with variances
that can differ among the arms. At the end of the experiment, the
decision-maker recommends one of the arms as an estimate of the best arm. To
design an experiment, we first discuss lower bounds for the probability of
misidentification. Our analysis highlights that the available information on
the outcome distribution, such as means (expected outcomes), variances, and the
choice of the best arm, significantly influences the lower bounds. Because
available information is limited in actual experiments, we develop a lower
bound that is valid under the unknown means and the unknown choice of the best
arm, which are referred to as the worst-case lower bound. We demonstrate that
the worst-case lower bound depends solely on the variances of the outcomes.
Then, under the assumption that the variances are known, we propose the
Generalized-Neyman-Allocation (GNA)-empirical-best-arm (EBA) strategy, an
extension of the Neyman allocation proposed by Neyman (1934). We show that the
GNA-EBA strategy is asymptotically optimal in the sense that its probability of
misidentification aligns with the lower bounds as the sample size increases
infinitely and the differences between the expected outcomes of the best and
other suboptimal arms converge to the same values across arms. We refer to such
strategies as asymptotically worst-case optimal.
