Collective Mistake in a Tricky Question Under Strategic Voting

We consider a common-value voting in which the state of the nature is a pair of payoff-relevant state and a variable that determines the precision and the meaning of voters' private signals. Each voter receives noisy signals about both of the states. When the number of voters is sufficiently large, we show the existence of a bad equilibrium in which all voters ignore the signal about the variable, and vote as if one of the variables is true. In the equilibrium, majority voting makes an incorrect decision with a probability that can be sufficiently close to 1.