  A decision maker looks to take an active action (e.g., purchase some goods or
make an investment). The payoff of this active action depends on his own
private type as well as a random and unknown state of nature. To decide between
this active action and another passive action, which always leads to a safe
constant utility, the decision maker may purchase information from an
information seller. The seller can access the realized state of nature, and
this information is useful for the decision maker (i.e., the information buyer)
to better estimate his payoff from the active action.
  We study the seller's problem of designing a revenue-optimal pricing scheme
to sell her information to the buyer. Suppose the buyer's private type and the
state of nature are drawn from two independent distributions, we fully
characterize the optimal pricing mechanism for the seller in closed form.
Specifically, under a natural linearity assumption of the buyer payoff
function, we show that an optimal pricing mechanism is the threshold mechanism
which charges each buyer type some upfront payment and then reveals whether the
realized state is above some threshold or below it. The payment and the
threshold are generally different for different buyer types, and are carefully
tailored to accommodate the different amount of risks each buyer type can take.
The proof of our results relies on novel techniques and concepts, such as
upper/lower virtual values and their mixtures, which may be of independent
interest.
