  We study information design in click-through auctions, in which the
bidders/advertisers bid for winning an opportunity to show their ads but only
pay for realized clicks. The payment may or may not happen, and its probability
is called the click-through rate (CTR). This auction format is widely used in
the industry of online advertising. Bidders have private values, whereas the
seller has private information about each bidder's CTRs. We are interested in
the seller's problem of partially revealing CTR information to maximize
revenue. Information design in click-through auctions turns out to be
intriguingly different from almost all previous studies in this space since any
revealed information about CTRs will never affect bidders' bidding behaviors --
they will always bid their true value per click -- but only affect the
auction's allocation and payment rule. In some sense, this makes information
design effectively a constrained mechanism design problem.
  Our first result is an FPTAS to compute an approximately optimal mechanism
under a constant number of bidders. The design of this algorithm leverages
Bayesian bidder values which help to ``smooth'' the seller's revenue function
and lead to better tractability. The design of this FPTAS is complex and
primarily algorithmic. Our second main result pursues the design of ``simple''
mechanisms that are approximately optimal yet more practical. We primarily
focus on the two-bidder situation, which is already notoriously challenging as
demonstrated in recent works. When bidders' CTR distribution is symmetric, we
develop a simple prior-free signaling scheme, whose construction relies on a
parameter termed optimal signal ratio. The constructed scheme provably obtains
a good approximation as long as the maximum and minimum of bidders' value
density functions do not differ much.
