Order Independence in Sequential, Issue-by-Issue Voting

We study when the voting outcome is independent of the order of issues put up for vote in a spatial multidimensional voting model. Agents equipped with norm-based preferences that use a norm to measure the distance from their ideal policy vote sequentially and issue by issue via simple majority. If the underlying norm is generated by an inner product—such as the Euclidean norm—then the voting outcome is order independent if and only if the issues are orthogonal. If the underlying norm is a general one, then the outcome is order independent if the basis defining the issues to be voted upon satisfies the following property; for any vector in the basis, any linear combination of the other vectors is Birkhoff–James orthogonal to it. We prove a partial converse in the case of two dimensions; if the underlying basis fails this property, then the voting order matters. Finally, despite existence results for the two-dimensional case and for the general l_p case, we show that nonexistence of bases with this property is generic.