Selling Data to an Agent with Endogenous Information

We consider the model of the data broker selling information to a single agent to maximize his revenue. The agent has private valuation for 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. In this paper, we show that in the optimal mechanism, there is no distortion at the top and the agent has no incentive to acquire any additional costly information under equilibrium. Still, the ability to acquire additional information distorts the incentives of the agent, and reduces the optimal revenue of the data broker. Finally, we show that posting a deterministic price on fully revealing the states is optimal when the prior distribution is sufficiently informative.