Coordinating the Unknown Lipschitz Constant in Multiplayer Bandits

Motivated by decentralized applications, we study cooperative multi-agent bandits in continuous (Lipschitz) action spaces when the Lipschitz constant is unknown. We consider three information structures: (A)~unobserved actions with common rewards, (B)~observed actions with independent rewards, and (C)~unobserved actions with independent rewards. In each case we design and analyze an algorithm that estimates the Lipschitz constant, chooses a discretization of the joint action space, and applies a cooperative bandit method to the induced discrete problem. Players never communicate once learning starts, so the central difficulty is that they must reach the \emph{same} discretization from their own data. We prove regret guarantees showing that common rewards and observable actions each supply this agreement for free, and that in their absence agreement can still be bought, through a dithered quantization of the estimate, at no cost in the leading order of the regret.