Robust scoring rules

We study elicitation of latent (prior) beliefs when the agent can acquire information via a costly attention strategy. We introduce a mechanism that simultaneously makes it strictly dominant to (a) not acquire any information, and (b) report truthfully. We call such a mechanism a robust scoring rule. Robust scoring rules are important for different reasons. Theoretically, they are crucial both for establishing that decision-theoretic models under uncertainty are testable. From an applied point of view, they are needed for eliciting unbiased estimates of population beliefs. We prove that a robust scoring rule exists under mild axioms on the attention costs. These axioms are shown to characterize the class of posterior-separable cost functions. Our existence proof is constructive, thus identifying an entire class of robust scoring rules. Subsequently, we show that we can arbitrarily approximate the agent's prior beliefs with a quadratic scoring rule. The same holds true for a discrete scoring rule. Finally, we show that the prior beliefs can be approximated, even when we are uncertain about the exact specification of the agent's attention costs.