  We study the stochastic multi-player multi-armed bandit problem. In this
problem, $m$ players cooperate to maximize their total reward from $K > m$
arms. However the players cannot communicate and are penalized (e.g. receive no
reward) if they pull the same arm at the same time. We ask whether it is
possible to obtain optimal instance-dependent regret $\tilde{O}(1/\Delta)$
where $\Delta$ is the gap between the $m$-th and $m+1$-st best arms. Such
guarantees were recently achieved in a model allowing the players to implicitly
communicate through intentional collisions.
  Surprisingly, we show that with no communication at all, such guarantees are
not achievable. In fact, obtaining the optimal $\tilde{O}(1/\Delta)$ regret for
some values of $\Delta$ necessarily implies strictly sub-optimal regret in
other regimes. Our main result is a complete characterization of the Pareto
optimal instance-dependent trade-offs that are possible with no communication.
Our algorithm generalizes that of Bubeck, Budzinski, and the second author. As
there, our algorithm succeeds even when feedback upon collision can be
corrupted by an adaptive adversary, thanks to a strong no-collision property.
Our lower bound is based on topological obstructions at multiple scales and is
completely new.
