Motivated by economic applications such as recommender systems, we study the behavior of stochastic bandits algorithms under strategic behavior conducted by rational actors, i.e., the arms. Each arm is a self-interested strategic player who can modify its own reward whenever pulled, subject to a cross-period budget constraint, in order to maximize its own expected number of times of being pulled. We analyze the robustness of three popular bandit algorithms: UCB, ε-Greedy, and Thompson Sampling. We prove that all three algorithms achieve a regret upper bound 𝒪(max{ B, Kln T}) where B is the total budget across arms, K is the total number of arms and T is the running time of the algorithms. This regret guarantee holds for arbitrary adaptive manipulation strategy of arms. Our second set of main results shows that this regret bound is tight— in fact, for UCB, it is tight even when we restrict the arms' manipulation strategies to form a Nash equilibrium. We do so by characterizing the Nash equilibrium of the game induced by arms' strategic manipulations and show a regret lower bound of Ω(max{ B, Kln T}) at the equilibrium.