  We study a strategic version of the multi-armed bandit problem, where each
arm is an individual strategic agent and we, the principal, pull one arm each
round. When pulled, the arm receives some private reward $v_a$ and can choose
an amount $x_a$ to pass on to the principal (keeping $v_a-x_a$ for itself). All
non-pulled arms get reward $0$. Each strategic arm tries to maximize its own
utility over the course of $T$ rounds. Our goal is to design an algorithm for
the principal incentivizing these arms to pass on as much of their private
rewards as possible.
  When private rewards are stochastically drawn each round ($v_a^t \leftarrow
D_a$), we show that:
  - Algorithms that perform well in the classic adversarial multi-armed bandit
setting necessarily perform poorly: For all algorithms that guarantee low
regret in an adversarial setting, there exist distributions $D_1,\ldots,D_k$
and an approximate Nash equilibrium for the arms where the principal receives
reward $o(T)$.
  - Still, there exists an algorithm for the principal that induces a game
among the arms where each arm has a dominant strategy. When each arm plays its
dominant strategy, the principal sees expected reward $\mu'T - o(T)$, where
$\mu'$ is the second-largest of the means $\mathbb{E}[D_{a}]$. This algorithm
maintains its guarantee if the arms are non-strategic ($x_a = v_a$), and also
if there is a mix of strategic and non-strategic arms.
