  We extend the notion of regret with a welfarist perspective. Focussing on the
classic multi-armed bandit (MAB) framework, the current work quantifies the
performance of bandit algorithms by applying a fundamental welfare function,
namely the Nash social welfare (NSW) function. This corresponds to equating
algorithm's performance to the geometric mean of its expected rewards and leads
us to the study of Nash regret, defined as the difference between the -- a
priori unknown -- optimal mean (among the arms) and the algorithm's
performance. Since NSW is known to satisfy fairness axioms, our approach
complements the utilitarian considerations of average (cumulative) regret,
wherein the algorithm is evaluated via the arithmetic mean of its expected
rewards.
  This work develops an algorithm that, given the horizon of play $T$, achieves
a Nash regret of $O \left( \sqrt{\frac{{k \log T}}{T}} \right)$, here $k$
denotes the number of arms in the MAB instance. Since, for any algorithm, the
Nash regret is at least as much as its average regret (the AM-GM inequality),
the known lower bound on average regret holds for Nash regret as well.
Therefore, our Nash regret guarantee is essentially tight. In addition, we
develop an anytime algorithm with a Nash regret guarantee of $O \left(
\sqrt{\frac{{k\log T}}{T}} \log T \right)$.
