  We analyze the optimal information design in a click-through auction with
fixed valuations per click, but stochastic click-through rates. While the
auctioneer takes as given the auction rule of the click-through auction, namely
the generalized second-price auction, the auctioneer can design the information
flow regarding the click-through rates among the bidders. A natural requirement
in this context is to ask for the information structure to be calibrated in the
learning sense. With this constraint, the auction needs to rank the ads by a
product of the bid and an unbiased estimator of the click-through rates, and
the task of designing an optimal information structure is thus reduced to the
task of designing an optimal unbiased estimator.
  We show that in a symmetric setting with uncertainty about the click-through
rates, the optimal information structure attains both social efficiency and
surplus extraction. The optimal information structure requires private (rather
than public) signals to the bidders. It also requires correlated (rather than
independent) signals, even when the underlying uncertainty regarding the
click-through rates is independent. Beyond symmetric settings, we show that the
optimal information structure requires partial information disclosure.
