  We study the problem of best-arm identification with fixed budget in
stochastic multi-armed bandits with Bernoulli rewards. For the problem with two
arms, also known as the A/B testing problem, we prove that there is no
algorithm that (i) performs as well as the algorithm sampling each arm equally
(referred to as the {\it uniform sampling} algorithm) in all instances, and
that (ii) strictly outperforms uniform sampling on at least one instance. In
short, there is no algorithm better than the uniform sampling algorithm. To
establish this result, we first introduce the natural class of {\it consistent}
and {\it stable} algorithms, and show that any algorithm that performs as well
as the uniform sampling algorithm in all instances belongs to this class. The
proof then proceeds by deriving a lower bound on the error rate satisfied by
any consistent and stable algorithm, and by showing that the uniform sampling
algorithm matches this lower bound. Our results provide a solution to the two
open problems presented in \citep{qin2022open}. For the general problem with
more than two arms, we provide a first set of results. We characterize the
asymptotic error rate of the celebrated Successive Rejects (SR) algorithm
\citep{audibert2010best} and show that, surprisingly, the uniform sampling
algorithm outperforms the SR algorithm in some instances.
