#### The Intrinsic Robustness of Stochastic Bandits to Strategic Manipulation

##### Zhe Feng, David C. Parkes, Haifeng Xu

We study the behavior of stochastic bandits algorithms under \emph{strategic behavior} conducted by rational actors, i.e., the arms. Each arm is a strategic player who can modify its own reward whenever pulled, subject to a cross-period budget constraint. Each arm is \emph{self-interested} and seeks to maximize its own expected number of times of being pulled over a decision horizon. Strategic manipulations naturally arise in various economic applications, e.g., recommendation systems such as Yelp and Amazon. We analyze the robustness of three popular bandit algorithms: UCB, $\varepsilon$-Greedy, and Thompson Sampling. We prove that all three algorithms achieve a regret upper bound $\mathcal{O}(\max \{ B, \ln T\})$ under \emph{any} (possibly adaptive) strategy of the strategic arms, where $B$ is the total budget across arms. Moreover, we prove that our regret upper bound is \emph{tight}. Our results illustrate the intrinsic robustness of bandits algorithms against strategic manipulation so long as $B=o(T)$. This is in sharp contrast to the more pessimistic model of adversarial attacks where an attack budget of $\mathcal{O}(\ln T)$ can trick UCB and $\varepsilon$-Greedy to pull the optimal arm only $o(T)$ number of times. Our results hold for both bounded and unbounded rewards.

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