Necessary gradient restrictions at the core of a voting rule

This paper generalizes known gradient restrictions for the core of a voting rule parameterized by an arbitrary quota. For the special case of majority rule with an even number of voters, the result implies that given any pointed, finitely generated, convex cone C, the difference between the number of voters with gradients in C and the number with gradients in −C cannot exceed the number of voters with zero gradient, plus a dimensional adjustment. When the cone has dimensionality less than three, the adjustment is zero. A difficulty in the proof of a result of Schofield (1983), which neglects the dimensional adjustment term, is identified, and a counterexample (in three dimensions) presented.