Active Identification of Private Information

Actions can reveal what an agent privately knows. We study a problem in which a Bayesian decision-maker receives a private signal from an unknown experiment before choosing actions, and an analyst infers the decision-maker's private experiment from the chosen actions. Importantly, the analyst actively designs probing experiments that provide additional signals to shape the decision-maker's posterior beliefs. We characterize the necessary and sufficient conditions for identifiability. For a quantal-response decision-maker, by using sufficiently rich probing experiments, the analyst can identify all private experiments of the decision-maker if and only if no two states are payoff-equivalent. Interestingly, this identifiability fails for a best-response decision-maker. We also characterize the condition for any specific private experiment to be identifiable. In the case where identifiability holds, we explicitly construct a probing experiment to achieve identification. Finally, we provide a sample complexity result for approximate identification when the analyst only observes finitely many probe-action samples.