  We study the problem of incentivizing exploration for myopic users in linear
bandits, where the users tend to exploit arm with the highest predicted reward
instead of exploring. In order to maximize the long-term reward, the system
offers compensation to incentivize the users to pull the exploratory arms, with
the goal of balancing the trade-off among exploitation, exploration and
compensation. We consider a new and practically motivated setting where the
context features observed by the user are more informative than those used by
the system, e.g., features based on users' private information are not
accessible by the system. We propose a new method to incentivize exploration
under such information gap, and prove that the method achieves both sublinear
regret and sublinear compensation. We theoretical and empirically analyze the
added compensation due to the information gap, compared with the case that the
system has access to the same context features as the user, i.e., without
information gap. We also provide a compensation lower bound of our problem.
