  I analyse the frequentist regret of the famous Gittins index strategy for
multi-armed bandits with Gaussian noise and a finite horizon. Remarkably it
turns out that this approach leads to finite-time regret guarantees comparable
to those available for the popular UCB algorithm. Along the way I derive
finite-time bounds on the Gittins index that are asymptotically exact and may
be of independent interest. I also discuss some computational issues and
present experimental results suggesting that a particular version of the
Gittins index strategy is a modest improvement on existing algorithms with
finite-time regret guarantees such as UCB and Thompson sampling.
