  Motivated by cognitive radios, stochastic multi-player multi-armed bandits
gained a lot of interest recently. In this class of problems, several players
simultaneously pull arms and encounter a collision - with 0 reward - if some of
them pull the same arm at the same time. While the cooperative case where
players maximize the collective reward (obediently following some fixed
protocol) has been mostly considered, robustness to malicious players is a
crucial and challenging concern. Existing approaches consider only the case of
adversarial jammers whose objective is to blindly minimize the collective
reward. We shall consider instead the more natural class of selfish players
whose incentives are to maximize their individual rewards, potentially at the
expense of the social welfare. We provide the first algorithm robust to selfish
players (a.k.a. Nash equilibrium) with a logarithmic regret, when the arm
performance is observed. When collisions are also observed, Grim Trigger type
of strategies enable some implicit communication-based algorithms and we
construct robust algorithms in two different settings: the homogeneous (with a
regret comparable to the centralized optimal one) and heterogeneous cases (for
an adapted and relevant notion of regret). We also provide impossibility
results when only the reward is observed or when arm means vary arbitrarily
among players.
