In the presence of Pareto, non-dictatorship, full domain, and transitivity, an extremely weak independence condition disallows both anonymity and neutrality.

There are two feasible alternatives. We impose a weak version of anonymity on an arbitrary social choice function, which selects a single alternative at each profile. It implies majority rule in the presence of neutrality and monotonicity.

Anonymity and neutrality conflict if the number of individuals is even and a single alternative is selected. Limited neutrality, anonymity, and monotonicity imply majority rule when the agenda is a two-element set.