  We propose and study a novel mechanism design setup where each bidder holds
two kinds of private information: (1) type variable, which can be misreported;
(2) information variable, which the bidder may want to conceal or partially
reveal, but importantly, not to misreport. We refer to bidders with such
behaviors as strategically reticent bidders. Among others, one direct
motivation of our model is the ad auction in which many ad platforms today
elicit from each bidder not only their private value per conversion but also
their private information about Internet users (e.g., user activities on the
advertiser's websites) in order to improve the platform's estimation of
conversion rates.
  We show that in this new setup, it is still possible to design mechanisms
that are both Incentive and Information Compatible (IIC). We develop two
different black-box transformations, which convert any mechanism $\mathcal{M}$
for classic bidders to a mechanism $\bar{\mathcal{M}}$ for strategically
reticent bidders, based on either outcome of expectation or expectation of
outcome, respectively. We identify properties of the original mechanism
$\mathcal{M}$ under which the transformation leads to IIC mechanisms
$\bar{\mathcal{M}}$. Interestingly, as corollaries of these results, we show
that running VCG with bidders' expected values maximizes welfare, whereas the
mechanism using expected outcome of Myerson's auction maximizes revenue.
Finally, we study how regulation on the auctioneer's usage of information can
lead to more robust mechanisms.
