Condorcet Jury Theorem: An example in which informative voting is rational but leads to inefficient information aggregation

Recent research on the Condorcet Jury Theorem has proven that informative voting (that is, voting according to one's signal) is not necessarily rational. With two alternatives, rational voting typically leads to the election of the correct alternative, in spite of the fact that not all voters vote informatively. We prove that with three alternatives, there are cases in which informative voting is rational and yet leads to the election of a wrong alternative.