Voting on tricky questions

We study a common-value voting model, in which private signal is typically informative but may be unreliable. Reliability determines the precision and the meaning of voters' private signals. These private signals are negatively correlated between different reliabilities. Each voter also receives noisy signals about reliability itself. When the population is sufficiently large, a bad equilibrium exists, in which all voters ignore reliability signals. It is thus possible that, at equilibrium, the majority rule makes an incorrect decision with a probability close to one.