Selling Data to an Agent with Endogenous Information

We consider a model of a data broker selling information to a single agent to maximize his revenue. The agent has a private valuation of the additional information, and upon receiving the signal from the data broker, the agent can conduct her own experiment to refine her posterior belief on the states with additional costs. To maximize expected revenue, only offering full information in general is suboptimal, and the optimal mechanism may contain a continuum of menu options with partial information to prevent the agent from having incentives to acquire additional information from other sources. However, our main result shows that the additional benefit from price discrimination is limited, i.e., posting a deterministic price for revealing full information obtains at least half of the optimal revenue for arbitrary prior and cost functions.