  We propose and study the known-compensation multi-arm bandit (KCMAB) problem,
where a system controller offers a set of arms to many short-term players for
$T$ steps. In each step, one short-term player arrives to the system. Upon
arrival, the player aims to select an arm with the current best average reward
and receives a stochastic reward associated with the arm. In order to
incentivize players to explore other arms, the controller provides a proper
payment compensation to players. The objective of the controller is to maximize
the total reward collected by players while minimizing the compensation. We
first provide a compensation lower bound $\Theta(\sum_i {\Delta_i\log T\over
KL_i})$, where $\Delta_i$ and $KL_i$ are the expected reward gap and
Kullback-Leibler (KL) divergence between distributions of arm $i$ and the best
arm, respectively. We then analyze three algorithms to solve the KCMAB problem,
and obtain their regrets and compensations. We show that the algorithms all
achieve $O(\log T)$ regret and $O(\log T)$ compensation that match the
theoretical lower bound. Finally, we present experimental results to
demonstrate the performance of the algorithms.
