When to persuade and when to pay: Designing incentives for private learning

This paper studies how a principal can influence an agent’s costly private learning about the state of the world through two instruments: (i) a public signal, and (ii) signal-realization-contingent monetary transfers. The agent gains from trade in the good state but loses in the bad state, while the principal benefits from trade in all states. Both parties share a common prior. Although the principal can, at no cost, disclose full information by providing a perfect signal, this never happens in equilibrium. Instead, the agent acquires costly information under some conditions. Specifically, when the prior is intermediate, the principal optimally designs a partially informative public signal that induces the agent’s private learning. Transfers are used only when the agent is sufficiently pessimistic to retain those at the participation threshold. Combining a public signal with monetary incentives yields strictly better outcomes for the principal than either instrument alone. Finally, when transfers may be negative—or in procurement environments where the agent must always be compensated—the principal optimally deters the agent’s private learning.