The Mediancentre-Borda rule and one-way monotonicity

The Mediancentre-Borda rule Ω is a voting rule which associates each vote with a vertex on a convex polytope and finds the point which minimizes the sums of the distances to these weighted vertices in order to return a winning set of candidates. In this paper, we motivate the study of such an unusual rule in terms of the Borda count, manipulability, and decisiveness. We will particularly focus on monotonicity properties – properties that insist the outcome of an election respond appropriately to changes in the electorate. We invoke the assistance of computer algorithms both to help us visualize the underlying geometry and to reveal truths to us about the relationship between Ω and the monotonicity properties that would otherwise not be available, and we rigorously justify our methods. Specifically, our algorithm enumerates all possible voting situations for a set number of voters and attempts to generate monotonicity violations from each situation. We find that Ω does not violate resolute one-way monotonicity for up to 12 voters, but does violate irresolute one-way monotonicity for 7, 8, 9, 10, 11, 12, and 13 voters. We define an equivalence relation called similarity on violations which are “structurally identical” and note that for 7, 8, 9, 10, and 11 voters, Ω only has one violation up to similarity, but that for 12 and 13 voters it has more. We study the violations we have detected for Ω and observe that we have found a violation of irresolute half-way monotonicity, a particularly nasty violation of irresolute one-way monotonicity, and we discuss the interpretations of this violation in terms of sensible virtues and participation