  We study the problem of combinatorial multi-armed bandits (CMAB) under
strategic manipulations of rewards, where each arm can modify the emitted
reward signals for its own interest. Our setting elaborates a more realistic
model of adaptive arms that imposes relaxed assumptions compared to adversarial
corruptions and adversarial attacks. Algorithms designed under strategic arms
gain robustness in real applications while avoiding being overcautious and
hampering the performance. We bridge the gap between strategic manipulations
and adversarial attacks by investigating the optimal colluding strategy among
arms under the MAB problem. We then propose a strategic variant of the
combinatorial UCB algorithm, which has a regret of at most $O(m\log T + m
B_{max})$ under strategic manipulations, where $T$ is the time horizon, $m$ is
the number of arms, and $B_{max}$ is the maximum budget. We further provide
lower bounds on the strategic budgets for attackers to incur certain regret of
the bandit algorithm. Extensive experiments corroborate our theoretical
findings on robustness and regret bounds, in a variety of regimes of
manipulation budgets.
