A Democracy Principle and Strategy-Proofness

In some social choice applications we want more than one alternative to be selected in some situations. This allows the construction of strategy-proof social choice rules that are not dictatorial. But if we also require x alone to be selected if it is at the top of some ordering that is submitted by more than half of the individuals then the rule cannot be strategy-proof. We prove this for rules that sometimes select one alternative, and sometimes two, but never more than two.