There is a theorem that a group composed of rational individuals which have collective beliefs decided by majority rule cannot be rational:

Consider a group in which everyone agrees with "C if and only if (A and B)". 1/3 of the group believes A is true and B is false and C is false, 1/3 believes A is false and B is true and C is false, and 1/3 believes A and B and C are all true. By majority vote, the group as a whole would assert that A and B are true and C is false.

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