Incentive-compatible surveys via posterior probabilities

We consider the problem of eliciting truthful responses to a survey question when the respondents share a common prior that the survey planner is agnostic about. The planner would therefore like to have a “universal” mechanism, which would induce honest answers for all possible priors. If the planner also requires a locality condition that ensures that the mechanism payoffs are determined by the respondents' posterior probabilities of the true state of nature, we prove that, under additional smoothness and sensitivity conditions, the payoff in the truth-telling equilibrium must be a logarithmic function of those posterior probabilities. Moreover, the respondents are necessarily ranked according to those probabilities. Finally, we discuss implementation issues.