Optimal Pricing of Information

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.