  Incentivized exploration in multi-armed bandits (MAB) has witnessed
increasing interests and many progresses in recent years, where a principal
offers bonuses to agents to do explorations on her behalf. However, almost all
existing studies are confined to temporary myopic agents. In this work, we
break this barrier and study incentivized exploration with multiple and
long-term strategic agents, who have more complicated behaviors that often
appear in real-world applications. An important observation of this work is
that strategic agents' intrinsic needs of learning benefit (instead of harming)
the principal's explorations by providing "free pulls". Moreover, it turns out
that increasing the population of agents significantly lowers the principal's
burden of incentivizing. The key and somewhat surprising insight revealed from
our results is that when there are sufficiently many learning agents involved,
the exploration process of the principal can be (almost) free. Our main results
are built upon three novel components which may be of independent interest: (1)
a simple yet provably effective incentive-provision strategy; (2) a carefully
crafted best arm identification algorithm for rewards aggregated under unequal
confidences; (3) a high-probability finite-time lower bound of UCB algorithms.
Experimental results are provided to complement the theoretical analysis.
